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Research ArticleInterval-Valued Hesitant Fuzzy HamacherSynergetic Weighted Aggregation Operators and TheirApplication to Shale Gas Areas Selection
Liang-Guo Li1 and Ding-Hong Peng12
1 Kunming University of Science and Technology Kunming 650093 China2 School of Management Harbin University of Science and Technology Harbin 150040 China
Correspondence should be addressed to Ding-Hong Peng pengdinghong2009163com
Received 26 October 2013 Revised 21 January 2014 Accepted 23 January 2014 Published 27 April 2014
Academic Editor Wudhichai Assawinchaichote
Copyright copy 2014 L-G Li and D-H Peng This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We investigate the multiple criteria decision making (MCDM) problem concerns on the selection of shale gas areas with interval-valued hesitant fuzzy information First some Hamacher operations of interval-valued hesitant fuzzy information are introducedwhich generalize and extend the existing onesThen some interval-valued hesitant fuzzyHamacherweighted aggregation operatorsespecially the interval-valued hesitant fuzzyHamacher synergetic weighted averaging (IVHFHSWA) operators and their geometricversion (IVHFHSWG) operators that weight simultaneously the argument variables themselves and their position orders andthus generalize the ideas of the weighted averaging and the ordered weighted averaging are proposed The distinct advantagesof these operators are that they can provide more choices for the decision makers and considerably enhance or deteriorate theperformance of aggregation The essential properties of these operators are studied and their specific cases are discussed Basedon the IVHFHSWA operator we propose a practical approach to shale gas areas selection with interval-valued hesitant fuzzyinformation Finally an illustrative example for selecting the shale gas areas is used to demonstrate the practicality and effectivenessof the proposed approach and a comparative analysis is performed with other approaches to highlight the distinctive advantages ofthe proposed operators
1 Instruction
As a novel generalization of fuzzy sets hesitant fuzzy sets(HFSs) [1 2] introduced by Torra and Narukawa have beensuccessfully used in the decision making field as a powerfultool for processing with uncertain and vague informationUnlike the other generalizations of fuzzy sets HFSs whichpermit the membership degree of an element to a set tobe represented as several possible values between 0 and 1are quite suited for describing the situation where we havea set of possible values rather than a margin of error orsome possibility distribution on the possible values and thusHFSs are very useful in dealing with the practical decisionmaking situations where people hesitate among several val-ues to express their opinions [3ndash5] or their opinions withincongruity [6ndash8] especially the group decision making
with anonymity [9ndash12] Moreover HFSs could also avoidperforming information aggregation and can directly reflectthe differences of the opinions of different experts [1 13 14]In addition it is proven that the envelope of hesitant fuzzy setis an intuitionistic fuzzy set (IFS) all HFSs are type-2 fuzzy setand hesitant fuzzy set and fuzzy multiset have the same formbut their operations are different [2] Thus HFSs open newperfectives for further research on decision making underhesitant environments and have received much attentionfrom many authors Torra and Narukawa [1 2] proposedsome set theoretic operations such as union intersection andcomplement on HFSs Subsequently Xia and Xu [6] definedsome new operations on HFSs based on the interconnectionbetween HFSs and the IFSs and then made an intensivestudy of hesitant fuzzy information aggregation techniquesand their applications in decision making Xu and Xia [7]
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 181050 25 pageshttpdxdoiorg1011552014181050
2 Mathematical Problems in Engineering
investigated some distancemeasures forHFSs drawing on thewell-known Hamming distance the Euclidean distance theHausdorff metric and their generalizations Following thesepioneering studies many subsequent studies on the aggre-gation operators [8 9 12ndash15] the discrimination measures[16] (including distance measures [3ndash5 13ndash15] similaritymeasures [3 7] correlation measures [3 17] entropy andcross-entropy [18]) for hesitant fuzzy sets (HFSs) and thefurther extensions of the HFSs such as the interval-valuesHFSs (IVHFSs) [11 17] the dual (or generalized) HFSs(DHFSs) [10 19] and the hesitant fuzzy linguistic term sets(HFLTSs) [20 21] have been conducted
In some practical decision making problems howeverthe precise membership degrees of an element to a set aresometimes hard to be specified To overcome the barrierChen et al [11 17] proposed the concept of interval-valuedhesitant fuzzy sets (IVHFSs) that represent the membershipdegrees of an element to a set with several possible intervalvalues and then presented some interval-valued hesitantfuzzy aggregation operators Wei [22] developed some hesi-tant interval-valued fuzzy aggregation operators (which areessential interval-valued hesitant fuzzy aggregation opera-tors) such as the hesitant interval-valued fuzzy weightedaggregation operators (HIVFWAandHIVFWG) the hesitantinterval-valued fuzzy ordered weighted aggregation oper-ators (HIVFOWA and HIVFOWG) the hesitant interval-valued fuzzy choquet ordered aggregation operators (HIVF-COA and HIVFCOG) the hesitant interval-valued fuzzyprioritized aggregation operator and the hesitant interval-valued fuzzy power aggregation operator
It is well known that the aggregation operators introducedabove are based on the basic algebraic 119905-norms (119905-norm and119905-conorm) ofHFSs (or IVHFSs) for carrying the combinationprocess which are not the unique 119905-norms that can be chosento model the intersection and union of HFSs (or IVHFSs)For instance Wei and Zhao [23] presented the Einstein oper-ations of interval-valued hesitant fuzzy sets based the Einstein119905-norms and then developed some interval-valued hesitantfuzzy Einstein aggregation operators and induced interval-valued hesitant fuzzy Einstein aggregation operators Besidesthere are a lot of 119905-norms that can be used to construct theoperations ofHFSs (or IVHFSs) one of them is theHamacher119905-norms [24ndash27] which have proven that the basic algebraic119905-norms and the Einstein 119905-norms are the special cases ofthe Hamacher 119905-norms and can supply a wide class of 119905-norm operators ranging from the probabilistic product tothe weakest 119905-norm by the choice of a parameter Thus theHamacher 119905-norms can considerably enhance or deterioratethe performance of aggregation Given the advantages of theHamacher 119905-norms in this paper we will investigate theinterval-valued hesitant fuzzy aggregation operators basedon the Hamacher 119905-norms and apply them to the multiplecriteria decision making
To do so the remainder of this paper is organized as fol-lows Section 2 introduces some preliminary concepts relatedto the interval-valued hesitant fuzzy sets and their operationsbased on the Hamacher 119905-norms In Section 3 based onthe defined operations we first develop the interval-valuedhesitant fuzzy Hamacher weighted averaging operators
and the interval-valued hesitant fuzzy Hamacher orderedweighted averaging operators then based on which wefurther propose the interval-valued hesitant fuzzy Hamachersynergetic weighted aggregation operators that simultane-ously consider the weights of argument variables them-selves and their position orders Moreover some essentialproperties and special cases of these operators are studiedIn Section 4 we develop a practical approach based onthe IVHFHSWA operators to multicriteria decision makingunder interval-valued hesitant fuzzy environments Section 5an illustrative example for selecting the shale gas areas isused to demonstrate the practicality and effectiveness of theproposed approach and Section 6 a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators Finally we summarizethe main conclusions of the paper in Section 7
2 Preliminaries
To overcome the barrier that the precise membership degreesof an element to a set are sometimes hard to be specifiedChen et al [11 17] introduce interval-valued hesitant fuzzyset (IVHFS) that represents the membership degrees of anelement to a set with several possible interval values
Definition 1 (see [11]) Let119883 be a reference set and let119863[0 1]
be the set of all closed subintervals of [0 1] Then an IVHFSon119883 is defined as
119864 = ⟨119909 ℎ
119864
(119909)⟩ | 119909 isin 119883 (1)
where ℎ
119864
(119909) 119909 rarr 119863[0 1] denotes all possible interval-valued membership degrees of the element 119909 isin 119883 to the set119864 For convenience we call ℎ
119864
(119909) an interval-valued hesitantfuzzy element (IVHFE) which reads
ℎ
119864
(119909) = ⋃
120574isin
ℎ
119864
(119909)
120574 = [120574119871
120574119880
] | 0 le 120574119871
le 120574119880
le 1 (2)
The operational laws of IVHFSs can be constructed by119905-norms (119905-norm and 119905-conorm) which satisfy the require-ments of the conjunction and disjunction operators respec-tively The existing interval-valued hesitant fuzzy operationallaws include the ones based on the algebraic 119905-norms [611 13ndash15 22] and the ones based on the Einstein 119905-norms[23] It is well known that the Hamacher 119905-norms are moregeneralized and flexible than the algebraic 119905-norms and theEinstein 119905-norms and they are defined as follows
Definition 2 (see [24 25]) The Hamacher 119905-norm 120601120578
and itsconorm 120593
120578
are defined as
120601120578
(119909 119910) =119909119910
120578 + (1 minus 120578) (119909 + 119910 minus 119909119910) 120578 gt 0
120593120578
(119909 119910) =119909 + 119910 minus 119909119910 minus (1 minus 120578) 119909119910
1 minus (1 minus 120578) 119909119910 120578 gt 0
(3)
Similar to the existing operations of IVHFEs based onthe Hamacher 119905-norms we can establish some fundamentalHamacher operations of IVHFEs
Mathematical Problems in Engineering 3
Definition 3 Let ℎ ℎ1
and ℎ2
be three IVHFEs then theHamacher operations of IVHFSs are defined as follows
(1)
120582ℎ = ⋃
120574isin
ℎ
[
[
(1 + (120578 minus 1) 120574119871
)120582
minus (1 minus 120574119871
)120582
(1 + (120578 minus 1) 120574119871
)120582
+ (120578 minus 1) (1 minus 120574119871
)120582
(1 + (120578 minus 1) 120574119880
)120582
minus (1 minus 120574119880
)120582
(1 + (120578 minus 1) 120574119880
)120582
+ (120578 minus 1) (1 minus 120574119880
)120582
]
]
(120582 gt 0)
(4)
(2)
ℎ120582
= ⋃
120574isin
ℎ
[
[
120578(120574119871
)120582
(1 + (120578 minus 1) (1 minus 120574119871
))120582
+ (120578 minus 1) (120574119871
)120582
120578(120574119880
)120582
(1 + (120578 minus 1) (1 minus 120574119880
))120582
+ (120578 minus 1) (120574119880
)120582
]
]
(120582 gt 0)
(5)
(3)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
minus (1 minus 120578) 120574119871
1
120574119871
2
1 minus (1 minus 120578) 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
minus (1 minus 120578) 120574119880
1
120574119880
2
1 minus (1 minus 120578) 120574119880
1
120574119880
2
]
(6)
(4)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
120578 minus (1 minus 120578) (120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
)
120574119880
1
120574119880
2
120578 minus (1 minus 120578) (120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
)]
(7)
TheHamacher operations of IVHFEs contain a wide classof special cases Especially if 120578 = 1 then we have (1)ndash(4)reduced to the following forms which are presented by Chenet al [11]
(11015840)
120582ℎ = ⋃
120574isin
ℎ
[1 minus (1 minus 120574119871
)120582
1 minus (1 minus 120574119880
)120582
] (8)
(21015840)
ℎ120582
= ⋃
120574isin
ℎ
[(120574119871
)120582
(120574119880
)120582
] (9)
(31015840)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
]
(10)
(41015840)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
120574119880
1
120574119880
2
] (11)
If 120578 = 2 then we have (1)ndash(4) reduced to the followingforms which are presented by Wei and Zhao [23]
(110158401015840)
120582ℎ = ⋃
120574isin
ℎ
[
[
(1 + 120574119871
)120582
minus (1 minus 120574119871
)120582
(1 + 120574119871
)120582
+ (1 minus 120574119871
)120582
(1 + 120574119880
)120582
minus (1 minus 120574119880
)120582
(1 + 120574119880
)120582
+ (1 minus 120574119880
)120582
]
]
(12)
(210158401015840)
ℎ120582
= ⋃
120574isin
ℎ
[
[
2(120574119871
)120582
(2 minus 120574119871
)120582
+ (120574119871
)120582
2(120574119880
)120582
(2 minus 120574119880
)120582
+ (120574119880
)120582
]
]
(13)
(310158401015840)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
1 + 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
1 + 120574119880
1
120574119880
2
] (14)
(410158401015840)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
1 + (1 minus 120574119871
1
) (1 minus 120574119871
2
)
120574119880
1
120574119880
2
1 + (1 minus 120574119880
1
) (1 minus 120574119880
2
)]
(15)
4 Mathematical Problems in Engineering
Based on Definition 3 the following Theorem 4 can beeasily proven
Theorem 4 Let ℎ1
and ℎ2
be two IVHFEs then
(1) ℎ1
oplus ℎ2
= ℎ2
oplus ℎ1
(2) ℎ1
otimes ℎ2
= ℎ2
otimes ℎ1
(3) 1205821
ℎ1
oplus 1205822
ℎ1
= (1205821
+ 1205822
)ℎ1
1205821
1205822
gt 0
(4) 120582(ℎ1
oplus ℎ2
) = 120582ℎ2
oplus 120582ℎ1
120582 gt 0
(5) ℎ12058211
otimes ℎ120582
2
1
= ℎ(120582
1
+120582
2
)
1
1205821
1205822
gt 0
(6) ℎ1205822
otimes ℎ120582
1
= (ℎ1
otimes ℎ2
)120582 120582 gt 0
The proofs of Theorem 4 are straightforward and omittedhere for saving space
Chen et al [11] defined the score function of IVHFE andgave a comparison approach of the score values of two IVHFEswith the possibility degree
Definition 5 For an IVHFE ℎ 119904(ℎ) = (1ℎ) sum120574isin
ℎ
120574 =
[(1ℎ) sum120574isin
ℎ
120574119871
(1ℎ) sum120574isin
ℎ
120574119880
] is called the score functionof ℎ Moreover for two IVHFEs ℎ
1
and ℎ2
if
119875 (ℎ1
ge ℎ2
)
= max
1 minusmax((1
ℎ2
sum
120574
2
isin
ℎ
2
120574119880
2
minus1
ℎ1
sum
120574
1
isin
ℎ
1
120574119871
1
)
times (1
ℎ1
sum
120574
1
isin
ℎ
1
(120574119880
1
minus 120574119871
1
)
+1
ℎ2
sum
120574
2
isin
ℎ
2
(120574119880
2
minus 120574119871
2
))
minus1
0) 0
gt 05
(16)
then ℎ1
gt ℎ2
if 119875(ℎ1
ge ℎ2
) = 05 then ℎ1
= ℎ2
Based on the possibility degree we further give the
definition of the relative possibility degree to rank or comparemultiple IVHFEs ℎ
119895
(119895 = 1 2 119899)
Definition 6 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs then the relative possibility degreeof ℎ
119895
that dominates all ℎ119896
(119896 = 1 2 119899) is defined as
119875 (ℎ119895
) =1
119899 (119899 minus 1)
times (
119899
sum
119896=1
max
1 minusmax ((1
ℎ119896
sum
120574
119896
isin
ℎ
119896
120574119880
119896
minus1
ℎ119895
sum
120574
119895
isin
ℎ
119895
120574119871
119895
)
times (1
ℎ119895
sum
120574
119895
isin
ℎ
119895
(120574119880
119895
minus 120574119871
119895
)
+1
ℎ119896
times
120574
119896
isin
ℎ
119896
sum(120574119880
119896
minus120574119871
119896
))
minus1
)0
+119899
2minus 1)
(17)
Definition 7 (see [28]) An ordered weighted averaging(OWA) operator of dimensions 119899 is a mapping OWA 119877
119899
rarr
119877 that has an associated weight vector 119908 = (1199081
1199082
119908119899
)119879
with the properties 0 le 119908119895
le 1(119895 = 1 2 119899) and sum119899
119895=1
119908119895
=
1 such that
OWA (1198861
1198862
119886119899
) = 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
(18)
where120590 defines a permutation of 1 2 119899 such that 119886120590(119895)
ge
119886120590(119895+1)
for all 119895It is well known that the weights in OWA operator are
assigned by the positions of argument variables that is thereis a one-to-one relative relation for each associatedweight andits corresponding value of argument variable [4 29 30]Thuswe can find a permutation 120588 1 2 119899 rarr 1 2 119899which is the inverse permutation of 120590 that is 120588 = 120590
minus1 andthe OWA operator can be alternatively defined as
OWA1015840
(1198861
1198862
119886119899
) = 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
(19)
where 120588 = 120590minus1
1 2 119899 rarr 1 2 119899 is the inversepermutation of 120590 119886
119895
is the 120588(119894)th largest element of thecollection of 119886
119895
(119894 = 1 2 119899) and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weight vector with119908119894
isin [0 1] andsum119899
119894=1
119908119894
= 1
Mathematical Problems in Engineering 5
Theorem 8 119908 = (1199081
1199082
119908119899
)119879 is the associated weight
vector with 119908119894
isin [0 1] sum119899
119894=1
119908119894
= 1 and 120588(sdot) and 120590(sdot) are twopermutations of 1 2 119899 if 120588(sdot) = 120590(sdot)
minus1 then
119874119882119860(1198861
1198862
119886119899
) = 1198741198821198601015840
(1198861
1198862
119886119899
) (20)
Proof Suppose
120588 997891997888rarr OWA (1198861
1198862
119886119899
)
= 119908120588(1)
119886120588(120590(1))
+ 119908120588(2)
119886120588(120590(2))
+ sdot sdot sdot + 119908120588(119899)
119886120588(120590(119899))
= 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
= OWA1015840
(1198861
1198862
119886119899
)
120590 997891997888rarr OWA1015840
(1198861
1198862
119886119899
)
= 119908120590(120588(1))
119886120590(1)
+ 119908120590(120588(2))
119886120590(2)
+ sdot sdot sdot + 119908120590(120588(119899))
119886120590(119899)
= 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
= OWA (1198861
1198862
119886119899
)
(21)
Hence OWA(1198861
1198862
119886119899
) = OWA1015840
(1198861
1198862
119886119899
)
3 Interval-Valued Hesitant Fuzzy HamacherSynergetic Weighted Aggregation Operators
The weighted averaging operator and the ordered weightedaveraging operator are the most common and basic aggre-gation operators In the section based on the aboveHamacher operations of IVHFSs we first develop theinterval-valued hesitant fuzzy Hamacher weighted averag-ing (IVHFHWA) operator and the interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator then based on which we further propose theinterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator to unify the IVHFHWAand IVHFHOWA operators Furthermore based on thegeometric mean we propose the interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operatorsThe essential properties of the operators arestudied and special cases are discussed
31 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Averaging Operator
Definition 9 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is the
relative weighting vector of ℎ119895
(119894 = 1 2 119899) with 120596119894
isin
[0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitant
fuzzy Hamacher weighted averaging (IVHFHWA) operatoris a mapping IVHFWAHamacher
119899
rarr such that
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578minus1)
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
)
minus1
]
]
(22)
Because the algebraic 119905-norms and Einstein 119905-norms arethe special cases of the Hamacher 119905-norms the followingtheorems hold
Theorem 10 The IVHFWA operator proposed by Chen et al[11] is a special case of the IVHFHWAoperator that is if 120578 = 1then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(23)
6 Mathematical Problems in Engineering
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
= IVHFWA (ℎ1
ℎ2
ℎ119899
)
(24)
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=1
997888997888997888rarr IVHFWA(ℎ
1
ℎ2
ℎ119899
)
Theorem 11 The IVHFEWA operator proposed by Wei andZhao [23] is a special case of the IVHFHWA operator that isif 120578 = 2 then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(25)
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFEWA (ℎ1
ℎ2
ℎ119899
)
(26)
Mathematical Problems in Engineering 7
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=2
997888997888997888rarr IVHFEWA(ℎ
1
ℎ2
ℎ119899
)From the above analysis we know that the IVHFWA
operator [11] and the IVHFEWA operator [23] are thespecial cases of the IVHFHWA operator and the IVHFHWAoperator can provide more special cases by selecting differentvalues of parameter 120578 which can providemore choices for thedecision makers and considerably enhance or deteriorate theperformance of aggregation Thus the IVHFHWA operatoris more general and more flexible Similar to the IVHFWAoperator and the IVHFEWA operator the IVHFHWA oper-ator is also monotonic bounded and idempotent
Example 12 Given the collection of IVHFEs ℎ1
= [02 04][05 07] [06 08] and ℎ
2
= [04 05][07 08] the relativeweights are 120596
1
= 03 1205962
= 07 and suppose that the 120578 = 1then
IVHFWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
times [1 minus (1 minus 02)03
(1 minus 04)07
1 minus (1 minus 04)03
(1 minus 05)07
]
[1 minus (1 minus 02)03
(1 minus 07)07
1 minus (1 minus 04)03
(1 minus 08)07
]
[1 minus (1 minus 05)03
(1 minus 04)07
1 minus (1 minus 07)03
(1 minus 05)07
]
[1 minus (1 minus 05)03
(1 minus 07)07
1 minus (1 minus 07)03
(1 minus 08)07
]
[1 minus (1 minus 06)03
(1 minus 04)07
1 minus (1 minus 08)03
(1 minus 05)07
]
[1 minus (1 minus 06)03
(1 minus 07)07
1 minus (1 minus 08)03
(1 minus 08)07
]
= [03459 04719] [05974 07219]
[04319 05710] [06503 07741]
[04687 06202] [0673 08]
(27)
Definition 13 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weighting vector of ℎ119895
(119894 = 1 2 119899) with119908119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 Then an interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator is a mapping IVHFOWAHamacher
119899
rarr suchthat
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
minus1
]
]
(28)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
120590(119895)
is the 120590(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)On the other hand
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119901(119895)
ℎ119895
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
![Page 2: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/2.jpg)
2 Mathematical Problems in Engineering
investigated some distancemeasures forHFSs drawing on thewell-known Hamming distance the Euclidean distance theHausdorff metric and their generalizations Following thesepioneering studies many subsequent studies on the aggre-gation operators [8 9 12ndash15] the discrimination measures[16] (including distance measures [3ndash5 13ndash15] similaritymeasures [3 7] correlation measures [3 17] entropy andcross-entropy [18]) for hesitant fuzzy sets (HFSs) and thefurther extensions of the HFSs such as the interval-valuesHFSs (IVHFSs) [11 17] the dual (or generalized) HFSs(DHFSs) [10 19] and the hesitant fuzzy linguistic term sets(HFLTSs) [20 21] have been conducted
In some practical decision making problems howeverthe precise membership degrees of an element to a set aresometimes hard to be specified To overcome the barrierChen et al [11 17] proposed the concept of interval-valuedhesitant fuzzy sets (IVHFSs) that represent the membershipdegrees of an element to a set with several possible intervalvalues and then presented some interval-valued hesitantfuzzy aggregation operators Wei [22] developed some hesi-tant interval-valued fuzzy aggregation operators (which areessential interval-valued hesitant fuzzy aggregation opera-tors) such as the hesitant interval-valued fuzzy weightedaggregation operators (HIVFWAandHIVFWG) the hesitantinterval-valued fuzzy ordered weighted aggregation oper-ators (HIVFOWA and HIVFOWG) the hesitant interval-valued fuzzy choquet ordered aggregation operators (HIVF-COA and HIVFCOG) the hesitant interval-valued fuzzyprioritized aggregation operator and the hesitant interval-valued fuzzy power aggregation operator
It is well known that the aggregation operators introducedabove are based on the basic algebraic 119905-norms (119905-norm and119905-conorm) ofHFSs (or IVHFSs) for carrying the combinationprocess which are not the unique 119905-norms that can be chosento model the intersection and union of HFSs (or IVHFSs)For instance Wei and Zhao [23] presented the Einstein oper-ations of interval-valued hesitant fuzzy sets based the Einstein119905-norms and then developed some interval-valued hesitantfuzzy Einstein aggregation operators and induced interval-valued hesitant fuzzy Einstein aggregation operators Besidesthere are a lot of 119905-norms that can be used to construct theoperations ofHFSs (or IVHFSs) one of them is theHamacher119905-norms [24ndash27] which have proven that the basic algebraic119905-norms and the Einstein 119905-norms are the special cases ofthe Hamacher 119905-norms and can supply a wide class of 119905-norm operators ranging from the probabilistic product tothe weakest 119905-norm by the choice of a parameter Thus theHamacher 119905-norms can considerably enhance or deterioratethe performance of aggregation Given the advantages of theHamacher 119905-norms in this paper we will investigate theinterval-valued hesitant fuzzy aggregation operators basedon the Hamacher 119905-norms and apply them to the multiplecriteria decision making
To do so the remainder of this paper is organized as fol-lows Section 2 introduces some preliminary concepts relatedto the interval-valued hesitant fuzzy sets and their operationsbased on the Hamacher 119905-norms In Section 3 based onthe defined operations we first develop the interval-valuedhesitant fuzzy Hamacher weighted averaging operators
and the interval-valued hesitant fuzzy Hamacher orderedweighted averaging operators then based on which wefurther propose the interval-valued hesitant fuzzy Hamachersynergetic weighted aggregation operators that simultane-ously consider the weights of argument variables them-selves and their position orders Moreover some essentialproperties and special cases of these operators are studiedIn Section 4 we develop a practical approach based onthe IVHFHSWA operators to multicriteria decision makingunder interval-valued hesitant fuzzy environments Section 5an illustrative example for selecting the shale gas areas isused to demonstrate the practicality and effectiveness of theproposed approach and Section 6 a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators Finally we summarizethe main conclusions of the paper in Section 7
2 Preliminaries
To overcome the barrier that the precise membership degreesof an element to a set are sometimes hard to be specifiedChen et al [11 17] introduce interval-valued hesitant fuzzyset (IVHFS) that represents the membership degrees of anelement to a set with several possible interval values
Definition 1 (see [11]) Let119883 be a reference set and let119863[0 1]
be the set of all closed subintervals of [0 1] Then an IVHFSon119883 is defined as
119864 = ⟨119909 ℎ
119864
(119909)⟩ | 119909 isin 119883 (1)
where ℎ
119864
(119909) 119909 rarr 119863[0 1] denotes all possible interval-valued membership degrees of the element 119909 isin 119883 to the set119864 For convenience we call ℎ
119864
(119909) an interval-valued hesitantfuzzy element (IVHFE) which reads
ℎ
119864
(119909) = ⋃
120574isin
ℎ
119864
(119909)
120574 = [120574119871
120574119880
] | 0 le 120574119871
le 120574119880
le 1 (2)
The operational laws of IVHFSs can be constructed by119905-norms (119905-norm and 119905-conorm) which satisfy the require-ments of the conjunction and disjunction operators respec-tively The existing interval-valued hesitant fuzzy operationallaws include the ones based on the algebraic 119905-norms [611 13ndash15 22] and the ones based on the Einstein 119905-norms[23] It is well known that the Hamacher 119905-norms are moregeneralized and flexible than the algebraic 119905-norms and theEinstein 119905-norms and they are defined as follows
Definition 2 (see [24 25]) The Hamacher 119905-norm 120601120578
and itsconorm 120593
120578
are defined as
120601120578
(119909 119910) =119909119910
120578 + (1 minus 120578) (119909 + 119910 minus 119909119910) 120578 gt 0
120593120578
(119909 119910) =119909 + 119910 minus 119909119910 minus (1 minus 120578) 119909119910
1 minus (1 minus 120578) 119909119910 120578 gt 0
(3)
Similar to the existing operations of IVHFEs based onthe Hamacher 119905-norms we can establish some fundamentalHamacher operations of IVHFEs
Mathematical Problems in Engineering 3
Definition 3 Let ℎ ℎ1
and ℎ2
be three IVHFEs then theHamacher operations of IVHFSs are defined as follows
(1)
120582ℎ = ⋃
120574isin
ℎ
[
[
(1 + (120578 minus 1) 120574119871
)120582
minus (1 minus 120574119871
)120582
(1 + (120578 minus 1) 120574119871
)120582
+ (120578 minus 1) (1 minus 120574119871
)120582
(1 + (120578 minus 1) 120574119880
)120582
minus (1 minus 120574119880
)120582
(1 + (120578 minus 1) 120574119880
)120582
+ (120578 minus 1) (1 minus 120574119880
)120582
]
]
(120582 gt 0)
(4)
(2)
ℎ120582
= ⋃
120574isin
ℎ
[
[
120578(120574119871
)120582
(1 + (120578 minus 1) (1 minus 120574119871
))120582
+ (120578 minus 1) (120574119871
)120582
120578(120574119880
)120582
(1 + (120578 minus 1) (1 minus 120574119880
))120582
+ (120578 minus 1) (120574119880
)120582
]
]
(120582 gt 0)
(5)
(3)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
minus (1 minus 120578) 120574119871
1
120574119871
2
1 minus (1 minus 120578) 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
minus (1 minus 120578) 120574119880
1
120574119880
2
1 minus (1 minus 120578) 120574119880
1
120574119880
2
]
(6)
(4)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
120578 minus (1 minus 120578) (120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
)
120574119880
1
120574119880
2
120578 minus (1 minus 120578) (120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
)]
(7)
TheHamacher operations of IVHFEs contain a wide classof special cases Especially if 120578 = 1 then we have (1)ndash(4)reduced to the following forms which are presented by Chenet al [11]
(11015840)
120582ℎ = ⋃
120574isin
ℎ
[1 minus (1 minus 120574119871
)120582
1 minus (1 minus 120574119880
)120582
] (8)
(21015840)
ℎ120582
= ⋃
120574isin
ℎ
[(120574119871
)120582
(120574119880
)120582
] (9)
(31015840)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
]
(10)
(41015840)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
120574119880
1
120574119880
2
] (11)
If 120578 = 2 then we have (1)ndash(4) reduced to the followingforms which are presented by Wei and Zhao [23]
(110158401015840)
120582ℎ = ⋃
120574isin
ℎ
[
[
(1 + 120574119871
)120582
minus (1 minus 120574119871
)120582
(1 + 120574119871
)120582
+ (1 minus 120574119871
)120582
(1 + 120574119880
)120582
minus (1 minus 120574119880
)120582
(1 + 120574119880
)120582
+ (1 minus 120574119880
)120582
]
]
(12)
(210158401015840)
ℎ120582
= ⋃
120574isin
ℎ
[
[
2(120574119871
)120582
(2 minus 120574119871
)120582
+ (120574119871
)120582
2(120574119880
)120582
(2 minus 120574119880
)120582
+ (120574119880
)120582
]
]
(13)
(310158401015840)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
1 + 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
1 + 120574119880
1
120574119880
2
] (14)
(410158401015840)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
1 + (1 minus 120574119871
1
) (1 minus 120574119871
2
)
120574119880
1
120574119880
2
1 + (1 minus 120574119880
1
) (1 minus 120574119880
2
)]
(15)
4 Mathematical Problems in Engineering
Based on Definition 3 the following Theorem 4 can beeasily proven
Theorem 4 Let ℎ1
and ℎ2
be two IVHFEs then
(1) ℎ1
oplus ℎ2
= ℎ2
oplus ℎ1
(2) ℎ1
otimes ℎ2
= ℎ2
otimes ℎ1
(3) 1205821
ℎ1
oplus 1205822
ℎ1
= (1205821
+ 1205822
)ℎ1
1205821
1205822
gt 0
(4) 120582(ℎ1
oplus ℎ2
) = 120582ℎ2
oplus 120582ℎ1
120582 gt 0
(5) ℎ12058211
otimes ℎ120582
2
1
= ℎ(120582
1
+120582
2
)
1
1205821
1205822
gt 0
(6) ℎ1205822
otimes ℎ120582
1
= (ℎ1
otimes ℎ2
)120582 120582 gt 0
The proofs of Theorem 4 are straightforward and omittedhere for saving space
Chen et al [11] defined the score function of IVHFE andgave a comparison approach of the score values of two IVHFEswith the possibility degree
Definition 5 For an IVHFE ℎ 119904(ℎ) = (1ℎ) sum120574isin
ℎ
120574 =
[(1ℎ) sum120574isin
ℎ
120574119871
(1ℎ) sum120574isin
ℎ
120574119880
] is called the score functionof ℎ Moreover for two IVHFEs ℎ
1
and ℎ2
if
119875 (ℎ1
ge ℎ2
)
= max
1 minusmax((1
ℎ2
sum
120574
2
isin
ℎ
2
120574119880
2
minus1
ℎ1
sum
120574
1
isin
ℎ
1
120574119871
1
)
times (1
ℎ1
sum
120574
1
isin
ℎ
1
(120574119880
1
minus 120574119871
1
)
+1
ℎ2
sum
120574
2
isin
ℎ
2
(120574119880
2
minus 120574119871
2
))
minus1
0) 0
gt 05
(16)
then ℎ1
gt ℎ2
if 119875(ℎ1
ge ℎ2
) = 05 then ℎ1
= ℎ2
Based on the possibility degree we further give the
definition of the relative possibility degree to rank or comparemultiple IVHFEs ℎ
119895
(119895 = 1 2 119899)
Definition 6 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs then the relative possibility degreeof ℎ
119895
that dominates all ℎ119896
(119896 = 1 2 119899) is defined as
119875 (ℎ119895
) =1
119899 (119899 minus 1)
times (
119899
sum
119896=1
max
1 minusmax ((1
ℎ119896
sum
120574
119896
isin
ℎ
119896
120574119880
119896
minus1
ℎ119895
sum
120574
119895
isin
ℎ
119895
120574119871
119895
)
times (1
ℎ119895
sum
120574
119895
isin
ℎ
119895
(120574119880
119895
minus 120574119871
119895
)
+1
ℎ119896
times
120574
119896
isin
ℎ
119896
sum(120574119880
119896
minus120574119871
119896
))
minus1
)0
+119899
2minus 1)
(17)
Definition 7 (see [28]) An ordered weighted averaging(OWA) operator of dimensions 119899 is a mapping OWA 119877
119899
rarr
119877 that has an associated weight vector 119908 = (1199081
1199082
119908119899
)119879
with the properties 0 le 119908119895
le 1(119895 = 1 2 119899) and sum119899
119895=1
119908119895
=
1 such that
OWA (1198861
1198862
119886119899
) = 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
(18)
where120590 defines a permutation of 1 2 119899 such that 119886120590(119895)
ge
119886120590(119895+1)
for all 119895It is well known that the weights in OWA operator are
assigned by the positions of argument variables that is thereis a one-to-one relative relation for each associatedweight andits corresponding value of argument variable [4 29 30]Thuswe can find a permutation 120588 1 2 119899 rarr 1 2 119899which is the inverse permutation of 120590 that is 120588 = 120590
minus1 andthe OWA operator can be alternatively defined as
OWA1015840
(1198861
1198862
119886119899
) = 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
(19)
where 120588 = 120590minus1
1 2 119899 rarr 1 2 119899 is the inversepermutation of 120590 119886
119895
is the 120588(119894)th largest element of thecollection of 119886
119895
(119894 = 1 2 119899) and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weight vector with119908119894
isin [0 1] andsum119899
119894=1
119908119894
= 1
Mathematical Problems in Engineering 5
Theorem 8 119908 = (1199081
1199082
119908119899
)119879 is the associated weight
vector with 119908119894
isin [0 1] sum119899
119894=1
119908119894
= 1 and 120588(sdot) and 120590(sdot) are twopermutations of 1 2 119899 if 120588(sdot) = 120590(sdot)
minus1 then
119874119882119860(1198861
1198862
119886119899
) = 1198741198821198601015840
(1198861
1198862
119886119899
) (20)
Proof Suppose
120588 997891997888rarr OWA (1198861
1198862
119886119899
)
= 119908120588(1)
119886120588(120590(1))
+ 119908120588(2)
119886120588(120590(2))
+ sdot sdot sdot + 119908120588(119899)
119886120588(120590(119899))
= 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
= OWA1015840
(1198861
1198862
119886119899
)
120590 997891997888rarr OWA1015840
(1198861
1198862
119886119899
)
= 119908120590(120588(1))
119886120590(1)
+ 119908120590(120588(2))
119886120590(2)
+ sdot sdot sdot + 119908120590(120588(119899))
119886120590(119899)
= 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
= OWA (1198861
1198862
119886119899
)
(21)
Hence OWA(1198861
1198862
119886119899
) = OWA1015840
(1198861
1198862
119886119899
)
3 Interval-Valued Hesitant Fuzzy HamacherSynergetic Weighted Aggregation Operators
The weighted averaging operator and the ordered weightedaveraging operator are the most common and basic aggre-gation operators In the section based on the aboveHamacher operations of IVHFSs we first develop theinterval-valued hesitant fuzzy Hamacher weighted averag-ing (IVHFHWA) operator and the interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator then based on which we further propose theinterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator to unify the IVHFHWAand IVHFHOWA operators Furthermore based on thegeometric mean we propose the interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operatorsThe essential properties of the operators arestudied and special cases are discussed
31 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Averaging Operator
Definition 9 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is the
relative weighting vector of ℎ119895
(119894 = 1 2 119899) with 120596119894
isin
[0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitant
fuzzy Hamacher weighted averaging (IVHFHWA) operatoris a mapping IVHFWAHamacher
119899
rarr such that
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578minus1)
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
)
minus1
]
]
(22)
Because the algebraic 119905-norms and Einstein 119905-norms arethe special cases of the Hamacher 119905-norms the followingtheorems hold
Theorem 10 The IVHFWA operator proposed by Chen et al[11] is a special case of the IVHFHWAoperator that is if 120578 = 1then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(23)
6 Mathematical Problems in Engineering
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
= IVHFWA (ℎ1
ℎ2
ℎ119899
)
(24)
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=1
997888997888997888rarr IVHFWA(ℎ
1
ℎ2
ℎ119899
)
Theorem 11 The IVHFEWA operator proposed by Wei andZhao [23] is a special case of the IVHFHWA operator that isif 120578 = 2 then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(25)
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFEWA (ℎ1
ℎ2
ℎ119899
)
(26)
Mathematical Problems in Engineering 7
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=2
997888997888997888rarr IVHFEWA(ℎ
1
ℎ2
ℎ119899
)From the above analysis we know that the IVHFWA
operator [11] and the IVHFEWA operator [23] are thespecial cases of the IVHFHWA operator and the IVHFHWAoperator can provide more special cases by selecting differentvalues of parameter 120578 which can providemore choices for thedecision makers and considerably enhance or deteriorate theperformance of aggregation Thus the IVHFHWA operatoris more general and more flexible Similar to the IVHFWAoperator and the IVHFEWA operator the IVHFHWA oper-ator is also monotonic bounded and idempotent
Example 12 Given the collection of IVHFEs ℎ1
= [02 04][05 07] [06 08] and ℎ
2
= [04 05][07 08] the relativeweights are 120596
1
= 03 1205962
= 07 and suppose that the 120578 = 1then
IVHFWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
times [1 minus (1 minus 02)03
(1 minus 04)07
1 minus (1 minus 04)03
(1 minus 05)07
]
[1 minus (1 minus 02)03
(1 minus 07)07
1 minus (1 minus 04)03
(1 minus 08)07
]
[1 minus (1 minus 05)03
(1 minus 04)07
1 minus (1 minus 07)03
(1 minus 05)07
]
[1 minus (1 minus 05)03
(1 minus 07)07
1 minus (1 minus 07)03
(1 minus 08)07
]
[1 minus (1 minus 06)03
(1 minus 04)07
1 minus (1 minus 08)03
(1 minus 05)07
]
[1 minus (1 minus 06)03
(1 minus 07)07
1 minus (1 minus 08)03
(1 minus 08)07
]
= [03459 04719] [05974 07219]
[04319 05710] [06503 07741]
[04687 06202] [0673 08]
(27)
Definition 13 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weighting vector of ℎ119895
(119894 = 1 2 119899) with119908119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 Then an interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator is a mapping IVHFOWAHamacher
119899
rarr suchthat
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
minus1
]
]
(28)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
120590(119895)
is the 120590(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)On the other hand
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119901(119895)
ℎ119895
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
![Page 3: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/3.jpg)
Mathematical Problems in Engineering 3
Definition 3 Let ℎ ℎ1
and ℎ2
be three IVHFEs then theHamacher operations of IVHFSs are defined as follows
(1)
120582ℎ = ⋃
120574isin
ℎ
[
[
(1 + (120578 minus 1) 120574119871
)120582
minus (1 minus 120574119871
)120582
(1 + (120578 minus 1) 120574119871
)120582
+ (120578 minus 1) (1 minus 120574119871
)120582
(1 + (120578 minus 1) 120574119880
)120582
minus (1 minus 120574119880
)120582
(1 + (120578 minus 1) 120574119880
)120582
+ (120578 minus 1) (1 minus 120574119880
)120582
]
]
(120582 gt 0)
(4)
(2)
ℎ120582
= ⋃
120574isin
ℎ
[
[
120578(120574119871
)120582
(1 + (120578 minus 1) (1 minus 120574119871
))120582
+ (120578 minus 1) (120574119871
)120582
120578(120574119880
)120582
(1 + (120578 minus 1) (1 minus 120574119880
))120582
+ (120578 minus 1) (120574119880
)120582
]
]
(120582 gt 0)
(5)
(3)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
minus (1 minus 120578) 120574119871
1
120574119871
2
1 minus (1 minus 120578) 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
minus (1 minus 120578) 120574119880
1
120574119880
2
1 minus (1 minus 120578) 120574119880
1
120574119880
2
]
(6)
(4)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
120578 minus (1 minus 120578) (120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
)
120574119880
1
120574119880
2
120578 minus (1 minus 120578) (120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
)]
(7)
TheHamacher operations of IVHFEs contain a wide classof special cases Especially if 120578 = 1 then we have (1)ndash(4)reduced to the following forms which are presented by Chenet al [11]
(11015840)
120582ℎ = ⋃
120574isin
ℎ
[1 minus (1 minus 120574119871
)120582
1 minus (1 minus 120574119880
)120582
] (8)
(21015840)
ℎ120582
= ⋃
120574isin
ℎ
[(120574119871
)120582
(120574119880
)120582
] (9)
(31015840)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
minus 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
minus 120574119880
1
120574119880
2
]
(10)
(41015840)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
120574119880
1
120574119880
2
] (11)
If 120578 = 2 then we have (1)ndash(4) reduced to the followingforms which are presented by Wei and Zhao [23]
(110158401015840)
120582ℎ = ⋃
120574isin
ℎ
[
[
(1 + 120574119871
)120582
minus (1 minus 120574119871
)120582
(1 + 120574119871
)120582
+ (1 minus 120574119871
)120582
(1 + 120574119880
)120582
minus (1 minus 120574119880
)120582
(1 + 120574119880
)120582
+ (1 minus 120574119880
)120582
]
]
(12)
(210158401015840)
ℎ120582
= ⋃
120574isin
ℎ
[
[
2(120574119871
)120582
(2 minus 120574119871
)120582
+ (120574119871
)120582
2(120574119880
)120582
(2 minus 120574119880
)120582
+ (120574119880
)120582
]
]
(13)
(310158401015840)
ℎ1
oplus ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
+ 120574119871
2
1 + 120574119871
1
120574119871
2
120574119880
1
+ 120574119880
2
1 + 120574119880
1
120574119880
2
] (14)
(410158401015840)
ℎ1
otimes ℎ2
= ⋃
120574
1
isin
ℎ
1
120574
2
isin
ℎ
2
[120574119871
1
120574119871
2
1 + (1 minus 120574119871
1
) (1 minus 120574119871
2
)
120574119880
1
120574119880
2
1 + (1 minus 120574119880
1
) (1 minus 120574119880
2
)]
(15)
4 Mathematical Problems in Engineering
Based on Definition 3 the following Theorem 4 can beeasily proven
Theorem 4 Let ℎ1
and ℎ2
be two IVHFEs then
(1) ℎ1
oplus ℎ2
= ℎ2
oplus ℎ1
(2) ℎ1
otimes ℎ2
= ℎ2
otimes ℎ1
(3) 1205821
ℎ1
oplus 1205822
ℎ1
= (1205821
+ 1205822
)ℎ1
1205821
1205822
gt 0
(4) 120582(ℎ1
oplus ℎ2
) = 120582ℎ2
oplus 120582ℎ1
120582 gt 0
(5) ℎ12058211
otimes ℎ120582
2
1
= ℎ(120582
1
+120582
2
)
1
1205821
1205822
gt 0
(6) ℎ1205822
otimes ℎ120582
1
= (ℎ1
otimes ℎ2
)120582 120582 gt 0
The proofs of Theorem 4 are straightforward and omittedhere for saving space
Chen et al [11] defined the score function of IVHFE andgave a comparison approach of the score values of two IVHFEswith the possibility degree
Definition 5 For an IVHFE ℎ 119904(ℎ) = (1ℎ) sum120574isin
ℎ
120574 =
[(1ℎ) sum120574isin
ℎ
120574119871
(1ℎ) sum120574isin
ℎ
120574119880
] is called the score functionof ℎ Moreover for two IVHFEs ℎ
1
and ℎ2
if
119875 (ℎ1
ge ℎ2
)
= max
1 minusmax((1
ℎ2
sum
120574
2
isin
ℎ
2
120574119880
2
minus1
ℎ1
sum
120574
1
isin
ℎ
1
120574119871
1
)
times (1
ℎ1
sum
120574
1
isin
ℎ
1
(120574119880
1
minus 120574119871
1
)
+1
ℎ2
sum
120574
2
isin
ℎ
2
(120574119880
2
minus 120574119871
2
))
minus1
0) 0
gt 05
(16)
then ℎ1
gt ℎ2
if 119875(ℎ1
ge ℎ2
) = 05 then ℎ1
= ℎ2
Based on the possibility degree we further give the
definition of the relative possibility degree to rank or comparemultiple IVHFEs ℎ
119895
(119895 = 1 2 119899)
Definition 6 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs then the relative possibility degreeof ℎ
119895
that dominates all ℎ119896
(119896 = 1 2 119899) is defined as
119875 (ℎ119895
) =1
119899 (119899 minus 1)
times (
119899
sum
119896=1
max
1 minusmax ((1
ℎ119896
sum
120574
119896
isin
ℎ
119896
120574119880
119896
minus1
ℎ119895
sum
120574
119895
isin
ℎ
119895
120574119871
119895
)
times (1
ℎ119895
sum
120574
119895
isin
ℎ
119895
(120574119880
119895
minus 120574119871
119895
)
+1
ℎ119896
times
120574
119896
isin
ℎ
119896
sum(120574119880
119896
minus120574119871
119896
))
minus1
)0
+119899
2minus 1)
(17)
Definition 7 (see [28]) An ordered weighted averaging(OWA) operator of dimensions 119899 is a mapping OWA 119877
119899
rarr
119877 that has an associated weight vector 119908 = (1199081
1199082
119908119899
)119879
with the properties 0 le 119908119895
le 1(119895 = 1 2 119899) and sum119899
119895=1
119908119895
=
1 such that
OWA (1198861
1198862
119886119899
) = 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
(18)
where120590 defines a permutation of 1 2 119899 such that 119886120590(119895)
ge
119886120590(119895+1)
for all 119895It is well known that the weights in OWA operator are
assigned by the positions of argument variables that is thereis a one-to-one relative relation for each associatedweight andits corresponding value of argument variable [4 29 30]Thuswe can find a permutation 120588 1 2 119899 rarr 1 2 119899which is the inverse permutation of 120590 that is 120588 = 120590
minus1 andthe OWA operator can be alternatively defined as
OWA1015840
(1198861
1198862
119886119899
) = 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
(19)
where 120588 = 120590minus1
1 2 119899 rarr 1 2 119899 is the inversepermutation of 120590 119886
119895
is the 120588(119894)th largest element of thecollection of 119886
119895
(119894 = 1 2 119899) and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weight vector with119908119894
isin [0 1] andsum119899
119894=1
119908119894
= 1
Mathematical Problems in Engineering 5
Theorem 8 119908 = (1199081
1199082
119908119899
)119879 is the associated weight
vector with 119908119894
isin [0 1] sum119899
119894=1
119908119894
= 1 and 120588(sdot) and 120590(sdot) are twopermutations of 1 2 119899 if 120588(sdot) = 120590(sdot)
minus1 then
119874119882119860(1198861
1198862
119886119899
) = 1198741198821198601015840
(1198861
1198862
119886119899
) (20)
Proof Suppose
120588 997891997888rarr OWA (1198861
1198862
119886119899
)
= 119908120588(1)
119886120588(120590(1))
+ 119908120588(2)
119886120588(120590(2))
+ sdot sdot sdot + 119908120588(119899)
119886120588(120590(119899))
= 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
= OWA1015840
(1198861
1198862
119886119899
)
120590 997891997888rarr OWA1015840
(1198861
1198862
119886119899
)
= 119908120590(120588(1))
119886120590(1)
+ 119908120590(120588(2))
119886120590(2)
+ sdot sdot sdot + 119908120590(120588(119899))
119886120590(119899)
= 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
= OWA (1198861
1198862
119886119899
)
(21)
Hence OWA(1198861
1198862
119886119899
) = OWA1015840
(1198861
1198862
119886119899
)
3 Interval-Valued Hesitant Fuzzy HamacherSynergetic Weighted Aggregation Operators
The weighted averaging operator and the ordered weightedaveraging operator are the most common and basic aggre-gation operators In the section based on the aboveHamacher operations of IVHFSs we first develop theinterval-valued hesitant fuzzy Hamacher weighted averag-ing (IVHFHWA) operator and the interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator then based on which we further propose theinterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator to unify the IVHFHWAand IVHFHOWA operators Furthermore based on thegeometric mean we propose the interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operatorsThe essential properties of the operators arestudied and special cases are discussed
31 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Averaging Operator
Definition 9 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is the
relative weighting vector of ℎ119895
(119894 = 1 2 119899) with 120596119894
isin
[0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitant
fuzzy Hamacher weighted averaging (IVHFHWA) operatoris a mapping IVHFWAHamacher
119899
rarr such that
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578minus1)
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
)
minus1
]
]
(22)
Because the algebraic 119905-norms and Einstein 119905-norms arethe special cases of the Hamacher 119905-norms the followingtheorems hold
Theorem 10 The IVHFWA operator proposed by Chen et al[11] is a special case of the IVHFHWAoperator that is if 120578 = 1then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(23)
6 Mathematical Problems in Engineering
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
= IVHFWA (ℎ1
ℎ2
ℎ119899
)
(24)
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=1
997888997888997888rarr IVHFWA(ℎ
1
ℎ2
ℎ119899
)
Theorem 11 The IVHFEWA operator proposed by Wei andZhao [23] is a special case of the IVHFHWA operator that isif 120578 = 2 then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(25)
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFEWA (ℎ1
ℎ2
ℎ119899
)
(26)
Mathematical Problems in Engineering 7
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=2
997888997888997888rarr IVHFEWA(ℎ
1
ℎ2
ℎ119899
)From the above analysis we know that the IVHFWA
operator [11] and the IVHFEWA operator [23] are thespecial cases of the IVHFHWA operator and the IVHFHWAoperator can provide more special cases by selecting differentvalues of parameter 120578 which can providemore choices for thedecision makers and considerably enhance or deteriorate theperformance of aggregation Thus the IVHFHWA operatoris more general and more flexible Similar to the IVHFWAoperator and the IVHFEWA operator the IVHFHWA oper-ator is also monotonic bounded and idempotent
Example 12 Given the collection of IVHFEs ℎ1
= [02 04][05 07] [06 08] and ℎ
2
= [04 05][07 08] the relativeweights are 120596
1
= 03 1205962
= 07 and suppose that the 120578 = 1then
IVHFWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
times [1 minus (1 minus 02)03
(1 minus 04)07
1 minus (1 minus 04)03
(1 minus 05)07
]
[1 minus (1 minus 02)03
(1 minus 07)07
1 minus (1 minus 04)03
(1 minus 08)07
]
[1 minus (1 minus 05)03
(1 minus 04)07
1 minus (1 minus 07)03
(1 minus 05)07
]
[1 minus (1 minus 05)03
(1 minus 07)07
1 minus (1 minus 07)03
(1 minus 08)07
]
[1 minus (1 minus 06)03
(1 minus 04)07
1 minus (1 minus 08)03
(1 minus 05)07
]
[1 minus (1 minus 06)03
(1 minus 07)07
1 minus (1 minus 08)03
(1 minus 08)07
]
= [03459 04719] [05974 07219]
[04319 05710] [06503 07741]
[04687 06202] [0673 08]
(27)
Definition 13 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weighting vector of ℎ119895
(119894 = 1 2 119899) with119908119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 Then an interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator is a mapping IVHFOWAHamacher
119899
rarr suchthat
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
minus1
]
]
(28)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
120590(119895)
is the 120590(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)On the other hand
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119901(119895)
ℎ119895
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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Stochastic AnalysisInternational Journal of
![Page 4: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/4.jpg)
4 Mathematical Problems in Engineering
Based on Definition 3 the following Theorem 4 can beeasily proven
Theorem 4 Let ℎ1
and ℎ2
be two IVHFEs then
(1) ℎ1
oplus ℎ2
= ℎ2
oplus ℎ1
(2) ℎ1
otimes ℎ2
= ℎ2
otimes ℎ1
(3) 1205821
ℎ1
oplus 1205822
ℎ1
= (1205821
+ 1205822
)ℎ1
1205821
1205822
gt 0
(4) 120582(ℎ1
oplus ℎ2
) = 120582ℎ2
oplus 120582ℎ1
120582 gt 0
(5) ℎ12058211
otimes ℎ120582
2
1
= ℎ(120582
1
+120582
2
)
1
1205821
1205822
gt 0
(6) ℎ1205822
otimes ℎ120582
1
= (ℎ1
otimes ℎ2
)120582 120582 gt 0
The proofs of Theorem 4 are straightforward and omittedhere for saving space
Chen et al [11] defined the score function of IVHFE andgave a comparison approach of the score values of two IVHFEswith the possibility degree
Definition 5 For an IVHFE ℎ 119904(ℎ) = (1ℎ) sum120574isin
ℎ
120574 =
[(1ℎ) sum120574isin
ℎ
120574119871
(1ℎ) sum120574isin
ℎ
120574119880
] is called the score functionof ℎ Moreover for two IVHFEs ℎ
1
and ℎ2
if
119875 (ℎ1
ge ℎ2
)
= max
1 minusmax((1
ℎ2
sum
120574
2
isin
ℎ
2
120574119880
2
minus1
ℎ1
sum
120574
1
isin
ℎ
1
120574119871
1
)
times (1
ℎ1
sum
120574
1
isin
ℎ
1
(120574119880
1
minus 120574119871
1
)
+1
ℎ2
sum
120574
2
isin
ℎ
2
(120574119880
2
minus 120574119871
2
))
minus1
0) 0
gt 05
(16)
then ℎ1
gt ℎ2
if 119875(ℎ1
ge ℎ2
) = 05 then ℎ1
= ℎ2
Based on the possibility degree we further give the
definition of the relative possibility degree to rank or comparemultiple IVHFEs ℎ
119895
(119895 = 1 2 119899)
Definition 6 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs then the relative possibility degreeof ℎ
119895
that dominates all ℎ119896
(119896 = 1 2 119899) is defined as
119875 (ℎ119895
) =1
119899 (119899 minus 1)
times (
119899
sum
119896=1
max
1 minusmax ((1
ℎ119896
sum
120574
119896
isin
ℎ
119896
120574119880
119896
minus1
ℎ119895
sum
120574
119895
isin
ℎ
119895
120574119871
119895
)
times (1
ℎ119895
sum
120574
119895
isin
ℎ
119895
(120574119880
119895
minus 120574119871
119895
)
+1
ℎ119896
times
120574
119896
isin
ℎ
119896
sum(120574119880
119896
minus120574119871
119896
))
minus1
)0
+119899
2minus 1)
(17)
Definition 7 (see [28]) An ordered weighted averaging(OWA) operator of dimensions 119899 is a mapping OWA 119877
119899
rarr
119877 that has an associated weight vector 119908 = (1199081
1199082
119908119899
)119879
with the properties 0 le 119908119895
le 1(119895 = 1 2 119899) and sum119899
119895=1
119908119895
=
1 such that
OWA (1198861
1198862
119886119899
) = 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
(18)
where120590 defines a permutation of 1 2 119899 such that 119886120590(119895)
ge
119886120590(119895+1)
for all 119895It is well known that the weights in OWA operator are
assigned by the positions of argument variables that is thereis a one-to-one relative relation for each associatedweight andits corresponding value of argument variable [4 29 30]Thuswe can find a permutation 120588 1 2 119899 rarr 1 2 119899which is the inverse permutation of 120590 that is 120588 = 120590
minus1 andthe OWA operator can be alternatively defined as
OWA1015840
(1198861
1198862
119886119899
) = 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
(19)
where 120588 = 120590minus1
1 2 119899 rarr 1 2 119899 is the inversepermutation of 120590 119886
119895
is the 120588(119894)th largest element of thecollection of 119886
119895
(119894 = 1 2 119899) and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weight vector with119908119894
isin [0 1] andsum119899
119894=1
119908119894
= 1
Mathematical Problems in Engineering 5
Theorem 8 119908 = (1199081
1199082
119908119899
)119879 is the associated weight
vector with 119908119894
isin [0 1] sum119899
119894=1
119908119894
= 1 and 120588(sdot) and 120590(sdot) are twopermutations of 1 2 119899 if 120588(sdot) = 120590(sdot)
minus1 then
119874119882119860(1198861
1198862
119886119899
) = 1198741198821198601015840
(1198861
1198862
119886119899
) (20)
Proof Suppose
120588 997891997888rarr OWA (1198861
1198862
119886119899
)
= 119908120588(1)
119886120588(120590(1))
+ 119908120588(2)
119886120588(120590(2))
+ sdot sdot sdot + 119908120588(119899)
119886120588(120590(119899))
= 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
= OWA1015840
(1198861
1198862
119886119899
)
120590 997891997888rarr OWA1015840
(1198861
1198862
119886119899
)
= 119908120590(120588(1))
119886120590(1)
+ 119908120590(120588(2))
119886120590(2)
+ sdot sdot sdot + 119908120590(120588(119899))
119886120590(119899)
= 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
= OWA (1198861
1198862
119886119899
)
(21)
Hence OWA(1198861
1198862
119886119899
) = OWA1015840
(1198861
1198862
119886119899
)
3 Interval-Valued Hesitant Fuzzy HamacherSynergetic Weighted Aggregation Operators
The weighted averaging operator and the ordered weightedaveraging operator are the most common and basic aggre-gation operators In the section based on the aboveHamacher operations of IVHFSs we first develop theinterval-valued hesitant fuzzy Hamacher weighted averag-ing (IVHFHWA) operator and the interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator then based on which we further propose theinterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator to unify the IVHFHWAand IVHFHOWA operators Furthermore based on thegeometric mean we propose the interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operatorsThe essential properties of the operators arestudied and special cases are discussed
31 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Averaging Operator
Definition 9 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is the
relative weighting vector of ℎ119895
(119894 = 1 2 119899) with 120596119894
isin
[0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitant
fuzzy Hamacher weighted averaging (IVHFHWA) operatoris a mapping IVHFWAHamacher
119899
rarr such that
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578minus1)
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
)
minus1
]
]
(22)
Because the algebraic 119905-norms and Einstein 119905-norms arethe special cases of the Hamacher 119905-norms the followingtheorems hold
Theorem 10 The IVHFWA operator proposed by Chen et al[11] is a special case of the IVHFHWAoperator that is if 120578 = 1then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(23)
6 Mathematical Problems in Engineering
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
= IVHFWA (ℎ1
ℎ2
ℎ119899
)
(24)
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=1
997888997888997888rarr IVHFWA(ℎ
1
ℎ2
ℎ119899
)
Theorem 11 The IVHFEWA operator proposed by Wei andZhao [23] is a special case of the IVHFHWA operator that isif 120578 = 2 then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(25)
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFEWA (ℎ1
ℎ2
ℎ119899
)
(26)
Mathematical Problems in Engineering 7
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=2
997888997888997888rarr IVHFEWA(ℎ
1
ℎ2
ℎ119899
)From the above analysis we know that the IVHFWA
operator [11] and the IVHFEWA operator [23] are thespecial cases of the IVHFHWA operator and the IVHFHWAoperator can provide more special cases by selecting differentvalues of parameter 120578 which can providemore choices for thedecision makers and considerably enhance or deteriorate theperformance of aggregation Thus the IVHFHWA operatoris more general and more flexible Similar to the IVHFWAoperator and the IVHFEWA operator the IVHFHWA oper-ator is also monotonic bounded and idempotent
Example 12 Given the collection of IVHFEs ℎ1
= [02 04][05 07] [06 08] and ℎ
2
= [04 05][07 08] the relativeweights are 120596
1
= 03 1205962
= 07 and suppose that the 120578 = 1then
IVHFWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
times [1 minus (1 minus 02)03
(1 minus 04)07
1 minus (1 minus 04)03
(1 minus 05)07
]
[1 minus (1 minus 02)03
(1 minus 07)07
1 minus (1 minus 04)03
(1 minus 08)07
]
[1 minus (1 minus 05)03
(1 minus 04)07
1 minus (1 minus 07)03
(1 minus 05)07
]
[1 minus (1 minus 05)03
(1 minus 07)07
1 minus (1 minus 07)03
(1 minus 08)07
]
[1 minus (1 minus 06)03
(1 minus 04)07
1 minus (1 minus 08)03
(1 minus 05)07
]
[1 minus (1 minus 06)03
(1 minus 07)07
1 minus (1 minus 08)03
(1 minus 08)07
]
= [03459 04719] [05974 07219]
[04319 05710] [06503 07741]
[04687 06202] [0673 08]
(27)
Definition 13 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weighting vector of ℎ119895
(119894 = 1 2 119899) with119908119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 Then an interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator is a mapping IVHFOWAHamacher
119899
rarr suchthat
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
minus1
]
]
(28)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
120590(119895)
is the 120590(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)On the other hand
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119901(119895)
ℎ119895
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 5: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/5.jpg)
Mathematical Problems in Engineering 5
Theorem 8 119908 = (1199081
1199082
119908119899
)119879 is the associated weight
vector with 119908119894
isin [0 1] sum119899
119894=1
119908119894
= 1 and 120588(sdot) and 120590(sdot) are twopermutations of 1 2 119899 if 120588(sdot) = 120590(sdot)
minus1 then
119874119882119860(1198861
1198862
119886119899
) = 1198741198821198601015840
(1198861
1198862
119886119899
) (20)
Proof Suppose
120588 997891997888rarr OWA (1198861
1198862
119886119899
)
= 119908120588(1)
119886120588(120590(1))
+ 119908120588(2)
119886120588(120590(2))
+ sdot sdot sdot + 119908120588(119899)
119886120588(120590(119899))
= 119908120588(1)
1198861
+ 119908120588(2)
1198862
+ sdot sdot sdot + 119908120588(119899)
119886119899
= OWA1015840
(1198861
1198862
119886119899
)
120590 997891997888rarr OWA1015840
(1198861
1198862
119886119899
)
= 119908120590(120588(1))
119886120590(1)
+ 119908120590(120588(2))
119886120590(2)
+ sdot sdot sdot + 119908120590(120588(119899))
119886120590(119899)
= 1199081
119886120590(1)
+ 1199082
119886120590(2)
+ sdot sdot sdot + 119908119899
119886120590(1)
= OWA (1198861
1198862
119886119899
)
(21)
Hence OWA(1198861
1198862
119886119899
) = OWA1015840
(1198861
1198862
119886119899
)
3 Interval-Valued Hesitant Fuzzy HamacherSynergetic Weighted Aggregation Operators
The weighted averaging operator and the ordered weightedaveraging operator are the most common and basic aggre-gation operators In the section based on the aboveHamacher operations of IVHFSs we first develop theinterval-valued hesitant fuzzy Hamacher weighted averag-ing (IVHFHWA) operator and the interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator then based on which we further propose theinterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator to unify the IVHFHWAand IVHFHOWA operators Furthermore based on thegeometric mean we propose the interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operatorsThe essential properties of the operators arestudied and special cases are discussed
31 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Averaging Operator
Definition 9 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is the
relative weighting vector of ℎ119895
(119894 = 1 2 119899) with 120596119894
isin
[0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitant
fuzzy Hamacher weighted averaging (IVHFHWA) operatoris a mapping IVHFWAHamacher
119899
rarr such that
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578minus1)
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
)
minus1
]
]
(22)
Because the algebraic 119905-norms and Einstein 119905-norms arethe special cases of the Hamacher 119905-norms the followingtheorems hold
Theorem 10 The IVHFWA operator proposed by Chen et al[11] is a special case of the IVHFHWAoperator that is if 120578 = 1then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(23)
6 Mathematical Problems in Engineering
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
= IVHFWA (ℎ1
ℎ2
ℎ119899
)
(24)
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=1
997888997888997888rarr IVHFWA(ℎ
1
ℎ2
ℎ119899
)
Theorem 11 The IVHFEWA operator proposed by Wei andZhao [23] is a special case of the IVHFHWA operator that isif 120578 = 2 then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(25)
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFEWA (ℎ1
ℎ2
ℎ119899
)
(26)
Mathematical Problems in Engineering 7
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=2
997888997888997888rarr IVHFEWA(ℎ
1
ℎ2
ℎ119899
)From the above analysis we know that the IVHFWA
operator [11] and the IVHFEWA operator [23] are thespecial cases of the IVHFHWA operator and the IVHFHWAoperator can provide more special cases by selecting differentvalues of parameter 120578 which can providemore choices for thedecision makers and considerably enhance or deteriorate theperformance of aggregation Thus the IVHFHWA operatoris more general and more flexible Similar to the IVHFWAoperator and the IVHFEWA operator the IVHFHWA oper-ator is also monotonic bounded and idempotent
Example 12 Given the collection of IVHFEs ℎ1
= [02 04][05 07] [06 08] and ℎ
2
= [04 05][07 08] the relativeweights are 120596
1
= 03 1205962
= 07 and suppose that the 120578 = 1then
IVHFWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
times [1 minus (1 minus 02)03
(1 minus 04)07
1 minus (1 minus 04)03
(1 minus 05)07
]
[1 minus (1 minus 02)03
(1 minus 07)07
1 minus (1 minus 04)03
(1 minus 08)07
]
[1 minus (1 minus 05)03
(1 minus 04)07
1 minus (1 minus 07)03
(1 minus 05)07
]
[1 minus (1 minus 05)03
(1 minus 07)07
1 minus (1 minus 07)03
(1 minus 08)07
]
[1 minus (1 minus 06)03
(1 minus 04)07
1 minus (1 minus 08)03
(1 minus 05)07
]
[1 minus (1 minus 06)03
(1 minus 07)07
1 minus (1 minus 08)03
(1 minus 08)07
]
= [03459 04719] [05974 07219]
[04319 05710] [06503 07741]
[04687 06202] [0673 08]
(27)
Definition 13 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weighting vector of ℎ119895
(119894 = 1 2 119899) with119908119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 Then an interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator is a mapping IVHFOWAHamacher
119899
rarr suchthat
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
minus1
]
]
(28)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
120590(119895)
is the 120590(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)On the other hand
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119901(119895)
ℎ119895
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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Stochastic AnalysisInternational Journal of
![Page 6: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/6.jpg)
6 Mathematical Problems in Engineering
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 1) 120574119880
119895
)120596
119895
+ (1 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
= IVHFWA (ℎ1
ℎ2
ℎ119899
)
(24)
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=1
997888997888997888rarr IVHFWA(ℎ
1
ℎ2
ℎ119899
)
Theorem 11 The IVHFEWA operator proposed by Wei andZhao [23] is a special case of the IVHFHWA operator that isif 120578 = 2 then
119868119881119867119865119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119871
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119871
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
minus prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
prod119899
119895=1
(1 + 120574119880
119895
)120596
119895
+ prod119899
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
(25)
Proof Suppose
IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119871
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (2 minus 1) 120574119880
119895
)120596
119895
+ (2 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
+
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFEWA (ℎ1
ℎ2
ℎ119899
)
(26)
Mathematical Problems in Engineering 7
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=2
997888997888997888rarr IVHFEWA(ℎ
1
ℎ2
ℎ119899
)From the above analysis we know that the IVHFWA
operator [11] and the IVHFEWA operator [23] are thespecial cases of the IVHFHWA operator and the IVHFHWAoperator can provide more special cases by selecting differentvalues of parameter 120578 which can providemore choices for thedecision makers and considerably enhance or deteriorate theperformance of aggregation Thus the IVHFHWA operatoris more general and more flexible Similar to the IVHFWAoperator and the IVHFEWA operator the IVHFHWA oper-ator is also monotonic bounded and idempotent
Example 12 Given the collection of IVHFEs ℎ1
= [02 04][05 07] [06 08] and ℎ
2
= [04 05][07 08] the relativeweights are 120596
1
= 03 1205962
= 07 and suppose that the 120578 = 1then
IVHFWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
times [1 minus (1 minus 02)03
(1 minus 04)07
1 minus (1 minus 04)03
(1 minus 05)07
]
[1 minus (1 minus 02)03
(1 minus 07)07
1 minus (1 minus 04)03
(1 minus 08)07
]
[1 minus (1 minus 05)03
(1 minus 04)07
1 minus (1 minus 07)03
(1 minus 05)07
]
[1 minus (1 minus 05)03
(1 minus 07)07
1 minus (1 minus 07)03
(1 minus 08)07
]
[1 minus (1 minus 06)03
(1 minus 04)07
1 minus (1 minus 08)03
(1 minus 05)07
]
[1 minus (1 minus 06)03
(1 minus 07)07
1 minus (1 minus 08)03
(1 minus 08)07
]
= [03459 04719] [05974 07219]
[04319 05710] [06503 07741]
[04687 06202] [0673 08]
(27)
Definition 13 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weighting vector of ℎ119895
(119894 = 1 2 119899) with119908119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 Then an interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator is a mapping IVHFOWAHamacher
119899
rarr suchthat
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
minus1
]
]
(28)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
120590(119895)
is the 120590(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)On the other hand
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119901(119895)
ℎ119895
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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Stochastic AnalysisInternational Journal of
![Page 7: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/7.jpg)
Mathematical Problems in Engineering 7
Thus IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)120578=2
997888997888997888rarr IVHFEWA(ℎ
1
ℎ2
ℎ119899
)From the above analysis we know that the IVHFWA
operator [11] and the IVHFEWA operator [23] are thespecial cases of the IVHFHWA operator and the IVHFHWAoperator can provide more special cases by selecting differentvalues of parameter 120578 which can providemore choices for thedecision makers and considerably enhance or deteriorate theperformance of aggregation Thus the IVHFHWA operatoris more general and more flexible Similar to the IVHFWAoperator and the IVHFEWA operator the IVHFHWA oper-ator is also monotonic bounded and idempotent
Example 12 Given the collection of IVHFEs ℎ1
= [02 04][05 07] [06 08] and ℎ
2
= [04 05][07 08] the relativeweights are 120596
1
= 03 1205962
= 07 and suppose that the 120578 = 1then
IVHFWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
]
]
times [1 minus (1 minus 02)03
(1 minus 04)07
1 minus (1 minus 04)03
(1 minus 05)07
]
[1 minus (1 minus 02)03
(1 minus 07)07
1 minus (1 minus 04)03
(1 minus 08)07
]
[1 minus (1 minus 05)03
(1 minus 04)07
1 minus (1 minus 07)03
(1 minus 05)07
]
[1 minus (1 minus 05)03
(1 minus 07)07
1 minus (1 minus 07)03
(1 minus 08)07
]
[1 minus (1 minus 06)03
(1 minus 04)07
1 minus (1 minus 08)03
(1 minus 05)07
]
[1 minus (1 minus 06)03
(1 minus 07)07
1 minus (1 minus 08)03
(1 minus 08)07
]
= [03459 04719] [05974 07219]
[04319 05710] [06503 07741]
[04687 06202] [0673 08]
(27)
Definition 13 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 119908 = (1199081
1199082
119908119899
)119879 is
the associated weighting vector of ℎ119895
(119894 = 1 2 119899) with119908119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 Then an interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator is a mapping IVHFOWAHamacher
119899
rarr suchthat
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
120590(119895))
119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
)
minus1
]
]
(28)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
120590(119895)
is the 120590(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)On the other hand
IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
120596119901(119895)
ℎ119895
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 8: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/8.jpg)
8 Mathematical Problems in Engineering
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(29)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899)Analogously because the algebraic 119905-norms and Einstein
119905-norms are the special cases of the Hamacher 119905-norms thefollowing theorems hold
Theorem 14 The IVHFOWA operator proposed by Chen et al[11] is a special case of the IVHFHOWA operator that is if 120578 =
1 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(30)
Theorem 15 The IVHFEOWA operator proposed by Wei andZhao [23] is a special case of the IVHFHOWA operator that isif 120578 = 2 then
119868119881119867119865119874119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119860(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119871
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119871
120590(119895)
)119908
119895
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + 120574119880
120590(119895)
)119908
119895
+
119899
prod
119895=1
(1 minus 120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(31)Analogously compared with IVHFOWA operator [11] and
the IVHFEOWA operator [23] the IVHFHOWA operatorcan provide more special cases by selecting different values ofparameter 120578 which can provide more choices for the decisionmakers and considerably enhance or deteriorate the perfor-mance of aggregationThus the IVHFHOWA operator is moregeneral and more flexible Similar to the IVHFOWA operatorand the IVHFEOWA operator the IVHFHOWA operator isalso commutative monotonic bounded and idempotent
Example 16 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05][07 08]the associatedweights are119908
1
= 06 and1199082
= 04 and supposethat the 120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max1 minusmax((12) (05+08)minus(13) (02+05+06)
(13) (02+05+06)+(12) (01+01)
0) 0 = 0267
(32)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 9: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/9.jpg)
Mathematical Problems in Engineering 9
we have ℎ1
lt ℎ2
and the assigned associated weights of ℎ1
andℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Then
IVHFOWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
119908120588(119895)
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
]
]
= [1 minus (1 minus 02)04
(1 minus 04)06
1 minus (1 minus 04)04
(1 minus 05)06
]
[1 minus (1 minus 02)04
(1 minus 07)06
1 minus (1 minus 04)04
(1 minus 08)06
]
[1 minus (1 minus 05)04
(1 minus 04)06
1 minus (1 minus 07)04
(1 minus 05)06
]
[1 minus (1 minus 05)04
(1 minus 07)06
1 minus (1 minus 07)04
(1 minus 08)06
]
[1 minus (1 minus 06)04
(1 minus 04)06
1 minus (1 minus 08)04
(1 minus 05)06
]
[1 minus (1 minus 06)04
(1 minus 07)06
1 minus (1 minus 08)04
(1 minus 08)06
]
= [03268 04622] [05559 06896]
[04422 05924] [06320 07648]
[04898 06534] [06634 08]
(33)
According to the above analysis we know that the IVHFHWAoperator weights only the interval-valued hesitant fuzzyargument variables its weights are the relative weights andrepresent the differential importance (salience significance)of argument variables themselves while the IVHFHOWAoperator weights only the ordered positions of the interval-valued hesitant fuzzy argument variables (or the magnitudesof the interval-valued hesitant fuzzy argument values) itsweights are the associated weights and depend on thecorresponding satisfaction values of argument variablesAccording to the associated weights are derived in view of thesatisfaction values of argument variables the IVHFHOWA
operator can relieve (or intensify) the influence of undulylarge or unduly small deviations on the aggregation results[31] or decide the portion of the criteria they feel is necessaryfor a good solution [28 32 33] Therefore weights representdifferent aspects in both the IVHFHWA and IVHFHOWAoperators However in general we need to consider thetwo weights because they represent different aspects ofdecision making problems Obviously both the IVHFHWAand IVHFHOWA operators have drawbacks In order tosolve these drawbacks according to Theorem 8 and inspiredby the idea of twofold weighting [4 34] we propose aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging (IVHFHSWA) operator in what follows
Definition 17 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted averaging (IVHFH-SWA) operator is a mapping IVHFSWAHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(34)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 10: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/10.jpg)
10 Mathematical Problems in Engineering
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
the IVHFEs ℎ
119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899) arereduced to HFEs ℎ
119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) then theIVHFHSWA operator becomes a hesitant fuzzy Hamachersynergetic weighted averaging (HFHSWA) operatorHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(35)
Example 18 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights are 120596
1
= 03 1205962
= 07 and the associatedweights are 119908
1
= 06 and 1199082
= 04 and suppose that the120578 = 1 then since
119875 (ℎ1
ge ℎ2
)
= max 1minusmax((12) (05+08) minus (13) (02+05+06)
(13) (02+02+02) + (12) (01+01)
0) 0 = 0267
(36)
we have ℎ1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are119908120588(1)
= 04 and 119908120588(2)
= 06 respectively Thus
IVHFSWAHamacher
(ℎ1
ℎ2
)
=
2
⨁
119895=1
120596119895
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
1 minus
2
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
1 minus
2
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
2
119895=1
120596
119895
119908
120588(119895)
]
]
= [1 minus (1 minus 02)022
(1 minus 04)078
1 minus (1 minus 04)022
(1 minus 05)078
]
[1 minus (1 minus 02)022
(1 minus 07)078
1 minus (1 minus 04)022
(1 minus 08)078
]
[1 minus (1 minus 05)022
(1 minus 04)078
1 minus (1 minus 07)022
(1 minus 05)078
]
[1 minus (1 minus 05)022
(1 minus 07)078
1 minus (1 minus 07)022
(1 minus 08)078
]
[1 minus (1 minus 06)022
(1 minus 04)078
1 minus (1 minus 08)022
(1 minus 05)078
]
[1 minus (1 minus 06)022
(1 minus 07)078
1 minus (1 minus 08)022
(1 minus 08)078
]
= [03608 04795] [06278 07453]
[04236 05531] [06643 07813]
[04512 05913] [06804 08]
(37)
By comparing Examples 12 16 and 18 we know thatthe results derived from the IVHFHWA IVHFHOWA andIVHFHSWA operators are different the reason is intu-itive the IVHFHWA operator focuses solely on the relativeweights and ignores the associated weights in the process ofaggregation while the IVHFHOWA operator focuses onlyon the associated weights and ignores the relative weightsin the process of aggregation the IVHFHSWA operatorcomprehensively and simultaneously considers the relativeweights and the associated weights and then its aggregatedresults are more feasible and effective
Inevitably we can see that the aggregation procedure ofthe interval-valued hesitant fuzzy information is complexespecially with the increases of the number of argumentvariables but it can surely better describe the situationswherepeople have hesitancy in providing their preferences in theprocess of decision making and the proposed operators canbe easily solved using Microsoft Excel Solver or integrated ina decision support system [4 10]
IVHFHSWA operator not only integrates the relativeweights and the associated weights into the weighted averag-ing operation and then generalizes the IVHFHWA operatorand IVHFHOWA operator but also satisfies the properties ofidempotency boundary and monotonicity and so on
Theorem 19 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs if ℎ119895
=
ℎ = ⋃120574isin
ℎ
120574 = [120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (38)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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Stochastic AnalysisInternational Journal of
![Page 11: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/11.jpg)
Mathematical Problems in Engineering 11
Proof Consider
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
⨁119899
119895=1
120596119895
ℎ119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578minus1)
119899
prod
119895=1
(1minus120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus120574119880
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
(1 + (120578 minus 1) 120574119871
) minus (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119871
) + (120578 minus 1) (1 minus 120574119871
)
(1 + (120578 minus 1) 120574119880
) minus (1 minus 120574119880
)
(1 + (120578 minus 1) 120574119880
) + (120578 minus 1) (1 minus 120574119880
)]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[120574119871
120574119880
] = ℎ
(39)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) = ℎ
Theorem 20 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(40)
Proof Suppose ⋂119899
119895=1
ℎ119895
= ⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min
1le119895le119899
120574119880
119895
] and
119899
⋃
119895=1
ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] (41)
For any a combination we always have
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 12: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/12.jpg)
12 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le [max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(42)
Then considering all possible combinations we have
⋃
120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1
+ (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
⋃
120574
119895
isin
ℎ
119895
119895=12119899
(
119899
prod
119895=1
(1
+(120578minus1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
le ⋃
120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
]
(43)
Hence we have ⋂119899
119895=1
ℎ119895
le IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
) le ⋃119899
119895=1
ℎ119895
Theorem 21 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
]
and ℎ1015840
119895
= ⋃120574
1015840
119895
isin
ℎ
1015840
119895
1205741015840
119895
= [1205741015840119871
119895
1205741015840119880
119895
](119895 = 1 2 119899) be two
collections of IVHFEs if only if 120574119895
ge 1205741015840
119895
120574119895
isin ℎ119895
and 1205741015840
119895
isin 1205741015840
119895
forall 119895 = 1 2 119899 then
119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) ge 119868119881119867119865119878119882119860119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(44)
Proof We have known that 120593120578
(119909 119910) = (119909 + 119910 minus 119909119910 minus (1 minus
120578)119909119910)(1 minus (1 minus 120578)119909119910) 120578 gt 0 is a strictly increasing functionSince 120574
119895
ge 1205741015840
119895
120574119895
isin ℎ119895
1205741015840119895
isin ℎ1015840
119895
for all 119895 = 1 2 119899 then
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 13: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/13.jpg)
Mathematical Problems in Engineering 13
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
ge [
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 1205741015840119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(45)
By considering all possible combinations we canget easily that IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899) ge
IVHFSWAHamacher(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)In the following we discuss some special cases of the
IVHFHSWA operator(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy synergetic weighted averaging(IVHFSWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr IVHFSWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
1 minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(46)
Case 2 If 120578 = 2 then the IVHFHSWA operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedaveraging (IVHFESWA) operator
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr IVHFESWA (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
119899
prod
119895=1
(1
minus120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(47)(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWA operator is reducedto the IVHFHOWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1minus120574119871
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 14: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/14.jpg)
14 Mathematical Problems in Engineering
(
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
times (
119899
prod
119895=1
(1+(120578minus1) 120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(1
minus120574119880
119895
)(1119899)119908
120588(119895)
sum
119899
119895=1
(1119899)119908
120588(119895)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)119908
120588(119895)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)119908
120588(119895)
)
minus1
]
]
= IVHFOWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(48)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFOWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 2 If the associated weighting vector 119908 =
(1199081
1199082
119908119899
) = (1119899 1119899 1119899) then the IVHFHSWAoperator is reduced to the IVHFHWA operator
Proof Suppose
IVHFSWAHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
+ (120578 minus 1)
times
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
(1119899)sum
119899
119895=1
120596
119895
(1119899)
)
minus1
]
]
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)120596
119895
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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![Page 15: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/15.jpg)
Mathematical Problems in Engineering 15
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)120596
119895
)
minus1
]
]
= IVHFWAHamacher
(ℎ1
ℎ2
ℎ119899
)
(49)
Thus IVHFSWAHamacher(ℎ1 ℎ2 ℎ119899)119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr
IVHFWAHamacher(ℎ1 ℎ2 ℎ119899)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) thenthe IVHFHSWA operator results in an interval-valued hes-itant fuzzy Hamacher averaging (IVHFHA) operator
IVHFAHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
1
119899ℎ119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119871
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119871
119895
)1119899
)
minus1
(
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
minus
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) 120574119880
119895
)1119899
+ (120578 minus 1)
119899
prod
119895=1
(1 minus 120574119880
119895
)1119899
)
minus1
]
]
(50)
From Examples 12 16 and 18 as well as Cases 1 and 2it follows that the IVHFHSWA operator generalizes both
the IVHFHWA operator and the IVHFHOWA operator andthus it can reflect the importance of both the consideredargument and its ordered position
32 Interval-Valued Hesitant Fuzzy Hamacher SynergeticWeighted Geometric Operator
Definition 22 Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
be a collection of IVHFEs and 120596 = (1205961
1205962
120596119899
)119879 is
the relative weighting vector of the ℎ119895
(119894 = 1 2 119899) with120596119894
isin [0 1] and sum119899
119894=1
120596119894
= 1 Then an interval-valued hesitantfuzzy Hamacher synergetic weighted geometric (IVHFH-SWG) operator is a mapping IVHFSWGHamacher
119899
rarr associated with a weighting vector119908 = (119908
1
1199082
119908119899
) suchthat 119908
119894
isin [0 1] and sum119899
119894=1
119908119894
= 1 according to the followingexpression
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119871
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119880
119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
(51)
where 120588 1 2 119899 rarr 1 2 119899 is a permutationfunction such that ℎ
119895
is the 120588(119895)th largest element of thecollection of ℎ
119895
(119894 = 1 2 119899) In particular if all 120574119871119895
= 120574119880
119895
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 16: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/16.jpg)
16 Mathematical Problems in Engineering
the IVHFEs ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
= [120574119871
119895
120574119880
119895
](119895 = 1 2 119899)
are reduced to HFEs ℎ119895
= ⋃120574
119895
isinℎ
119895
120574119895
(119895 = 1 2 119899) thenthe IVHFHSWG operator becomes a hesitant fuzzyHamacher synergetic weighted geometric (HFHSWG)operator
HFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119895
= ⋃
120574
119895
isinℎ
119895
119895=1119899
(120578
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1)
times (1 minus 120574119895
))120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+ (120578 minus 1)
times
119899
prod
119895=1
(120574119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(52)
Example 23 Given the collection of IVHFEs ℎ1
=
[02 04] [05 07] [06 08] and ℎ2
= [04 05] [07 08]the relative weights 120596
1
= 03 1205962
= 07 and the associatedweight vectors are 119908
1
= 06 1199082
= 04 and suppose that the120578 = 2 then since ℎ
1
lt ℎ2
and the associated weights of ℎ1
and ℎ2
are 119908120588(1)
= 04 and 119908120588(2)
= 06 respectively then
IVHFSWGHamacher
(ℎ1
ℎ2
)
=
2
⨂
119895=1
ℎ119908
120588(119895)
120596
119895
sum
2
119895=1
119908
120588(119895)
120596
119895
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=12
[
[
(2
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
(2
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
times (
2
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
+
2
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
)
minus1
]
]
= [0346 0477] [0551 0699] [0421 0541]
[0653 0778] [0439 0559] [0677 08]
(53)
Furthermore we can obtain the aggregated results corre-sponding to some special cases of the parameter 120578 which areshown in Table 1
FromTable 1 we know that the aggregated results derivedfrom the IVHFHSWG steadily increases as the parameter 120578increases which implies that the IVHFHSWG operator withparameters can provide the decision makers more choicesand thus the aggregated results are more flexible than theexisting ones because we can choose different values of theparameter according to the different situations
The IVHFHSWG operator has some essential propertiessuch as idempotency boundary and monotonicity
Theorem 24 (Idempotency) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
= ℎ = ⋃120574isin
ℎ
120574 =
[120574119871
120574119880
] for all 119895 = 1 2 119899 then
119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) = ℎ (54)
Theorem 25 (Boundedness) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
=
[120574119871
119895
120574119880
119895
](119895 = 1 2 119899) be a collection of IVHFEs ⋂119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[min1le119895le119899
120574119871
119895
min1le119895le119899
120574119880
119895
] and ⋃119899
119895=1
ℎ119895
=
⋃120574
119895
isin
ℎ
119895
119895=12119899
[max1le119895le119899
120574119871
119895
max1le119895le119899
120574119880
119895
] then
119899
⋂
119895=1
ℎ119895
le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
) le
119899
⋃
119895=1
ℎ119895
(55)
Theorem 26 (Monotonicity) Let ℎ119895
= ⋃120574
119895
isin
ℎ
119895
120574119895
(119895 =
1 2 119899) be a collection of IVHFEs If ℎ119895
le ℎ1015840
119895
for all 119895 =
1 2 119899 then
119868119881119867119865119878119882119866(ℎ1
ℎ2
ℎ119899
) le 119868119881119867119865119878119882119866119867119886119898119886119888ℎ119890119903
(ℎ1015840
1
ℎ1015840
2
ℎ1015840
119899
)
(56)
In the following we discuss the special cases of theIVHFHSWG operator
(1) From the perspective of parameter 120578
Case 1 If 120578 = 1 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy synergetic weighted geometric(IVHFSWG) operator
IVHFSWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
]
]
(57)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Stochastic AnalysisInternational Journal of
![Page 17: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/17.jpg)
Mathematical Problems in Engineering 17
Table 1 Aggregated results for the different 120578
ID 120578 Aggregated results Score values1 01 [0332 0474] [0470 0661] [0419 0534] [0644 0776] [0433 0546] [0675 08] [0496 0632]2 05 [0340 0475] [0509 0676] [0420 0537] [0648 0776] [0435 0551] [0676 08] [0505 0636]3 1 [0343 0476] [0531 0687] [0420 0538] [0650 0777] [0437 0554] [0677 08] [0510 0639]4 15 [0345 0476] [0543 0694] [0420 0540] [0652 0777] [0438 0557] [0677 08] [0513 0641]5 2 [0346 0477] [0551 0699] [0421 0541] [0653 0778] [0439 0559] [0677 08] [0515 0642]6 5 [0348 0477] [0570 0713] [0421 0543] [0656 0779] [0441 0566] [0678 08] [0519 0646]7 100 [0349 0478] [0587 0728] [0422 0546] [0659 0780] [0443 0575] [0679 08] [0523 0651]
Case 2 If 120578 = 2 then the IVHFHSWG operator results in aninterval-valued hesitant fuzzy Einstein synergetic weightedgeometric (IVHFESWG) operator
IVHFESWG (ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119871
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
times (
119899
prod
119895=1
(2 minus 120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
+
119899
prod
119895=1
(120574119880
119895
)120596
119895
119908
120588120582(119895)
sum
119899
119895=1
120596
119895
119908
120588120582(119895)
)
minus1
]
]
(58)
(2) From the perspective of the types of weights
Case 1 If the relative weighting vector 120596 = (1205961
1205962
120596119899
) =
(1119899 1119899 1119899) then the IVHFHSWG operator becomesan interval-valued hesitant fuzzy Hamacher orderedweighted geometric (IVHFHOWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
120588(119895)
119895
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))119908
120588(119895)
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(59)
On the other hand
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFOWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ119908
119895
120590(119895)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 18: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/18.jpg)
18 Mathematical Problems in Engineering
(120578
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
120590(119895)
))119908
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(60)
Furthermore similar toTheorems 10 and 11 the followingtheorems hold
Theorem 27 The IVHFOWG operator proposed by Chen etal [11] is the special case of the IVHFHOWG operator that isif 120578 = 1then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
]
]
(61)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
]
]
(62)
Theorem 28 The IVHFEOWG operator proposed byWei andZhao [23] is the special case of the IVHFHOWG operator thatis if 120578 = 2 then
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119871
120590(119895)
)119908
119895
)
minus1
(2
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
120590(119895)
))119908
119895
+
119899
prod
119895=1
(120574119880
120590(119895)
)119908
119895
)
minus1
]
]
(63)
On the other hand
119868119881119867119865119874119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119874119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(2
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119871
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119871
119895
)119908
120588(119895)
)
minus1
(2
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
times (
119899
prod
119895=1
(1 + (1 minus 120574119880
119895
))119908
120588(119895)
+
119899
prod
119895=1
(120574119880
119895
)119908
120588(119895)
)
minus1
]
]
(64)
Case 2 If the associated weighting vector 119908 = (1199081
1199082
119908
119899
) = (1119899 1119899 1119899) then the IVHFHSWG oper-ator becomes an interval-valued hesitant fuzzy Hamacherweighted geometric (IVHFHWG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
119908
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr IVHFWGHamacher
(ℎ1
ℎ2
ℎ119899
)
=
119899
⨂
119895=1
ℎ120596
119895
119895
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 19: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/19.jpg)
Mathematical Problems in Engineering 19
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)120596
119895
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))120596
119895
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)120596
119895
)
minus1
]
]
(65)
Furthermore the following theorems hold
Theorem 29 The IVHFWG operator proposed by Chen et al[11] is the special case of the IVHFHWG operator that is if120578 = 1 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=1
997888997888997888rarr 119868119881119867119865119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
119899
prod
119895=1
(120574119871
119895
)120596
119895
119899
prod
119895=1
(120574119880
119895
)120596
119895
]
]
(66)
Theorem 30 The IVHFEWG operator proposed by Wei andZhao [23] is the special case of the IVHFHWG operator thatis if 120578 = 2 then
119868119881119867119865119882119866119867119886119898119886119888ℎ119890119903
(ℎ1
ℎ2
ℎ119899
)
120578=2
997888997888997888rarr 119868119881119867119865119864119882119866(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
2prod119899
119895=1
(120574119871
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119871
119895
))120596
119895
+ prod119899
119895=1
(120574119871
119895
)120596
119895
2prod119899
119895=1
(120574119880
119895
)120596
119895
prod119899
119895=1
(1 + (1 minus 120574119880
119895
))120596
119895
+ prod119899
119895=1
(120574119880
119895
)120596
119895
]
]
(67)
Case 3 If 119908 = (1199081
1199082
119908119899
) = (1119899 1119899 1119899)
and 120596 = (1205961
1205962
120596119899
) = (1119899 1119899 1119899) then
the IVHFHSWG operator becomes an interval-valued hesi-tant fuzzy Hamacher geometric (IVHFHG) operator
IVHFSWGHamacher
(ℎ1
ℎ2
ℎ119899
)
120596
119895
=1119899119895=1119899
997888997888997888997888997888997888997888997888997888997888997888rarr119908
119895
=1119899119895=1119899
IVHFGHamacher
(ℎ1
ℎ2
ℎ119899
)
= ⋃
120574
119895
isin
ℎ
119895
119895=1119899
[
[
(120578
119899
prod
119895=1
(120574119871
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119871
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119871
119895
)1119899
)
minus1
(120578
119899
prod
119895=1
(120574119880
119895
)1119899
)
times (
119899
prod
119895=1
(1 + (120578 minus 1) (1 minus 120574119880
119895
))1119899
+ (120578 minus 1)
119899
prod
119895=1
(120574119880
119895
)1119899
)
minus1
]
]
(68)
When applying the IVHFHSW (IVHFHSWA andIVHFHSWG) operators for decision making the key issueis to determine the associated weights Different approacheshave been suggested for obtaining the associated weightvector [31 32] The most common of them is the onebased on the use of a linguistic quantifier [4 32 33] 119876which is a regular increasing monotonic (RIM) function119876 [0 1] rarr [0 1] that satisfies 119876(0) = 0 119876(1) = 1 and119876(119909) ge 119876(119910) if 119909 gt 119910 where119876 is substituted with a linguisticquantifier In the quantifier-guided aggregation processthe decision makers provide a decision strategy with alinguistic quantifier that indicates the portion of the criteriathey feel is necessary for a good solution such as ldquomanyrdquoor ldquomostrdquo The formal decision strategy is that ldquo119876 criteriamust be satisfied by an acceptable alternativerdquo Yager [32]recommended obtaining the descending orders associatedweights based on a linguistic quantifier as follows
119908119894
= 119876(119894
119899) minus 119876(
119894 minus 1
119899) 119894 = 1 119899 (69)
where 119876 is a linguistic quantifier and the simplest and mostcommon one is defined as
119876 (119903) = 119903120572
120572 ge 0 119903 isin [0 1] (70)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
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Stochastic AnalysisInternational Journal of
![Page 20: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/20.jpg)
20 Mathematical Problems in Engineering
Table 2 Guideline to select linguistic quantifiers and equivalent 120572values
ID Linguistic quantifier Parameter1 All (max) 120572 rarr infin (1000)2 Most 120572 = 5
3 Many 120572 = 2
4 Half (weighted averaging) 120572 = 1
5 Some 120572 = 05
6 Few 120572 = 02
7 At least one (min) 120572 rarr 0 (0001)
in which 120572 is a parameter indicating the decision strategies(rules) its changes represent a continuum of different deci-sion strategies between the two extreme cases of requiringldquoallrdquo and ldquoat least onerdquo of the criteria to be satisfiedThe com-mon decision strategies and their corresponding parameter 120572are listed in Table 2 [4 33]
Based on linguistic quantifier 119876 the decision makersprovide a decision strategy with a linguistic quantifier thatindicates the portion of the criteria they feel is necessary fora good solution
4 Approach for Selecting Shale Gas AreasBased on the IVHFHSWA Operator
The following assumptions or notations are used to rep-resent the MCDM problems with interval-valued hesitant
fuzzy information Let 1198601
1198602
119860119898
be a discrete setof alternatives and let 119862
1
1198622
119862119899
be the set of criteriaand its relative weight vector is 120596 = (120596
1
1205962
120596119899
)Decision makers provide interval values for the alternative119860
119894
under the criteria119862119895
with anonymity To avoid performinginformation aggregation and to directly reflect the differencesof the opinions of different experts these interval values areconsidered as an interval-valued hesitant fuzzy element ℎ
119894119895
and formed the decision matrix (ℎ119894119895
)119898times119899
In the following we apply the IVHFHSWA (IVHFH-
SWG) operator to multiple criteria decision making basedon interval-valued hesitant fuzzy information The methodinvolves the following steps
Step 1 Determine the associatedweights by utilizing (69) and(70) as
119908119895
= (119895
119899)
120572
minus (119895 minus 1
119899)
120572
119895 = 1 119899 (71)
Step 2 Rank the criteria values against alternatives by the rel-ative possibility degrees (17) and then assign the associatedweights as
119875 (ℎ119894119895
) =
sum119899
119896=1
max
1 minusmax((1ℎ
119894119896
)sum120574
119894119896
isin
ℎ
119894119896
120574119880
119894119896
minus (1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
120574119871
119894119895
(1ℎ119894119895
)sum120574
119894119895
isin
ℎ
119894119895
(120574119880
119894119895
minus 120574119871
119894119895
) + (1ℎ119894119896
)sum120574
119894119896
isin
ℎ
119894119896
(120574119880
119894119896
minus 120574119871
119894119896
)
0) 0
+ (1198992) minus 1
119899 (119899 minus 1)
119894 = 1 2 119898 119895 = 1 2 119899
(72)
Step 3 Utilize the IVHFHSWA operator (34) as
ℎ119894
= IVHFSWAHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) = (
⨁119899
119895=1
120596119895
ℎ119894119895
119908120588(119895)
sum119899
119895=1
120596119895
119908120588(119895)
)
119894 = 1 2 119898
(73)
Or the IVHFHSWG operator (51) as
ℎ119894
= IVHFSWGHamacher
(ℎ1198941
ℎ1198942
ℎ119894119899
) =
119899
⨂
119895=1
ℎ120596
119895
119908
120588(119895)
sum
119899
119895=1
120596
119895
119908
120588(119895)
119894119895
119894 = 1 2 119898
(74)
to aggregate decision information of ℎ119894119895
(119894 = 1 2 119898 119895 =
1 2 119899) and obtain the IVHFEs ℎ119894
(119894 = 1 2 119898) for thealternatives 119860
119894
(119894 = 1 2 119898)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 119898) of the IVHFEs ℎ119894
(119894 = 1 2 119898) by utilizing (17)as
119875 (ℎ119894
) =
sum119898
119896=1
max
1 minusmax((1ℎ
119896
)sum120574
119896
isin
ℎ
119896
120574119880
119896
minus (1ℎ119894
)sum120574
119894
isin
ℎ
119894
120574119871
119894
(1ℎ119894
)sum120574
119894
isin
ℎ
119894
(120574119880
119894
minus 120574119871
119894
) + (1ℎ119896
)sum120574
119896
isin
ℎ
119896
(120574119880
119896
minus 120574119871
119896
)
0) 0
+ (1198982) minus 1
119898 (119898 minus 1)
119894 = 1 2 119898
(75)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 21: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/21.jpg)
Mathematical Problems in Engineering 21
Table 3 IVHF decision matrix
1198621
1198622
1198623
1198624
1198601
[04 06] [08 09] [05 07] [06 08] [03 04] [04 05] [05 06]
1198602
[07 08] [04 06] [05 07] [08 09] [03 06] [07 09]
1198603
[03 04] [06 08] [03 04] [06 07] [06 08] [02 03] [05 07]
1198604
[04 05] [02 03] [04 05] [05 06] [07 08] [03 05]
1198605
[06 07] [08 09] [03 05] [07 08] [03 04] [05 07]
Step 5 Rank all the alternatives 119860119894
(119894 = 1 2 119898) andselect the best one(s) according to 119875(ℎ
119894
)(119894 = 1 2 119898) indescending order
Step 6 End
5 Numerical Example
With the increase of energy consumption conventionalnatural gas resources are gradually reducing and gettingincreasingly difficult to develop Due to the shale gas withthe advantages of great resource potential and low carbonemissions shale gas has recently been the focus of explorationin many countries and the development of shale gas becomesthe strategic choice of China [35 36] It is well known thatthe evaluation and selection of the areas are fundamental inthe process of shale gas development In order to preliminaryselect a best area from the five potential shale gas areas inSouth China the Hunan area the Sichuan area the Yunnanarea the Guizhou area and the Guangxi area denoted as119909119894
(119894 = 1 2 3 4 5) respectively The policy makers fromgovernment agencies environmental organizations localcommunities and so forth serve as the decision makersThe evaluation criteria is based on the SWOT perspectives[37] and they are summarized as follows (adopted from[36]) strengths (119862
1
) which consists of abundant resourcereserves great development potential high environmentalbenefits and long lifetime for exploitation weaknesses (119862
2
)which can be divided into lack of funds lack of key tech-nologies prominent water treatment problems and seriousenvironmental risks opportunities (119862
3
) which involve hugepotential market policy support increased investment andfinancing channels plentiful foreign development experi-ence and deepened international cooperation threats (119862
4
)which can include unconfirmed resource potential imper-fect policy system unsound management system deficientinvestment mechanism poor infrastructure and the relativeweight vector of the criteria which is 120596 = (120596
1
1205962
1205963
1205964
) =
(01 03 02 04) Given the experts who make such an eval-uation have different backgrounds and levels of knowledgeskills experience personality and so forth this could lead toa difference in the evaluation information To clearly reflectthe differences of the opinions of different experts the dataof evaluation information are represented by the IVHFEs andlisted in Table 3
Below we apply the proposed approach to the selection ofshale gas areas
Step 1 Determine the associated weights by (71) where theterm ldquomanyrdquo is selected as a linguistic quantifier The resultsare listed as follows
1199081
= (1
4)
2
minus (0
4)
2
= 006
1199082
= (2
4)
2
minus (1
4)
2
= 019
1199083
= (3
4)
2
minus (2
4)
2
= 031
1199084
= (4
4)
2
minus (3
4)
2
= 044
(76)
Step 2 Rank the criteria values against each alternative by(72) and then assign the associated weights according to therankings The results are listed in Table 4
Step 3 Utilize (73) and suppose that 120578 = 1 to aggregatedecision information of ℎ
119894119895
(119894 = 1 2 3 4 5 119895 = 1 2 3 4) andobtain the IVHFEs ℎ
119894
(119894 = 1 2 3 4 5) for the alternatives119860
119894
(119894 = 1 2 3 4 5)Consider
ℎ1
= IVHFSWAHamacher
(ℎ11
ℎ12
ℎ13
ℎ14
)
= [0385 0540] [0441 0591]
[0499 0643] [0419 0572]
[0473 0620] [0528 0670]
ℎ2
= IVHFSWAHamacher
(ℎ21
ℎ22
ℎ23
ℎ24
)
= [0382 0619] [0559 0779]
[0441 0665] [0606 0808]
ℎ3
= IVHFSWAHamacher
(ℎ31
ℎ32
ℎ33
ℎ34
)
= [0256 0366] [0435 0612]
[0363 0478] [0526 0691]
[0277 0400] [0454 0637]
[0383 0509] [0543 0712]
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Stochastic AnalysisInternational Journal of
![Page 22: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/22.jpg)
22 Mathematical Problems in Engineering
Table 4 Rankings and the assigned associated weights of criteria values against alternatives
Rankings of criteria values against alternatives 1198621
1198622
1198623
1198624
1198601
ℎ13
gt ℎ11
gt ℎ12
gt ℎ14
019 031 006 044119860
2
ℎ21
gt ℎ23
gt ℎ24
gt ℎ22
006 044 019 031119860
3
ℎ33
gt ℎ31
gt ℎ32
gt ℎ34
019 031 006 044119860
4
ℎ43
gt ℎ41
gt ℎ44
gt ℎ42
019 044 006 031119860
5
ℎ51
= ℎ53
gt ℎ54
gt ℎ52
0125 044 0125 031
Table 5 Score values obtained by the IVHFHSWA operator with different values of 120578
1198601
1198602
1198603
1198604
1198605
1 [04611 06115] [05042 0722] [04108 05582] [03254 04703] [04156 05836]2 [04575 06060] [04968 07181] [04048 05506] [03224 04673] [04074 05776]3 [04558 06035] [04932 07164] [04014 05469] [03207 04657] [04033 05749]4 [04548 06021] [04911 07155] [03993 05447] [03197 04648] [04009 05734]5 [04541 06012] [04897 07149] [03978 05432] [03190 04641] [03992 05725]6 [04536 06005] [04887 07145] [03966 05421] [03184 04637] [03981 05718]7 [04532 06000] [04879 07142] [03958 05413] [03180 04633] [03972 05713]8 [04529 05997] [04874 07140] [03951 05407] [03177 04630] [03965 05709]9 [04526 05994] [04869 07139] [03945 05402] [03174 04628] [03959 05706]10 [04525 05991] [04865 07137] [03941 05398] [03172 04626] [03955 05704]
ℎ4
= IVHFSWAHamacher
(ℎ41
ℎ42
ℎ43
ℎ44
)
= [0271 0418] [0283 0431]
[0362 0504] [0373 0516]
ℎ5
= IVHFSWAHamacher
(ℎ51
ℎ52
ℎ53
ℎ54
)
= [0358 0507] [0443 0632]
[0372 0525] [0456 0646]
(77)
Step 4 Compute the relative possibility degrees 119875(ℎ119894
)(119894 =
1 2 5) of the IVHFEs ℎ119894
(119894 = 1 2 5)
119875 (ℎ1
) = 0229 119875 (ℎ2
) = 0267 119875 (ℎ3
) = 0185
119875 (ℎ4
) = 0122 119875 (ℎ5
) = 0197
(78)
Step 5 Rank all the alternatives119860119894
(119894 = 1 2 119898) accordingto 119875(ℎ
119894
)(119894 = 1 2 119898) in descending order The rankingresults are 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
and the mostprospective shale gas area is 119860
2
(Sichuan)
Step 6 End
Furthermore we use the IVHFHSWG operator to thesame decision problem we can obtain the ranking results thatconsist with the ones above which are validate each other
6 Comparative Analysis
61 Performance Analysis of the IVHFHSWAOperator Fromthe definition of IVHFHSWA operator we know that theIVHFHSWAoperator provides awide class of interval-valuedhesitant fuzzy aggregation operators via the parameters 120578To understand the performance of aggregation in depth weadopt the parameter 120578 = 1 to 10 for the numerical exampleaboveWhen the 120578 takes the different values the scores valuesare obtained based on IVHFHSWA operator as shown inTable 5 and represented graphically in Figure 1
From Table 5 it is obvious that the score values derivedby using the IVHFHSWA operator are nonincreasing withrespect to 120578 which implies that decision makers can utilizetheir preferences to give the preferred values of 120578 accordingto practical decision situations On the basis of the scorevalues we can obtain the precisely ranking results of thealternatives by computing their relative possibility degreesMoreover from Figure 1 we find that the ranking resultsof the alternatives are the same when the values of 120578 aredifferent in the example and the consistent ranking resultsdemonstrate the stability of the proposed operators
62 Comparison With Other Operators Similar to theIVHFHSWA operator in order to integrate simultaneously
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 23: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/23.jpg)
Mathematical Problems in Engineering 23
Table 6 Score values obtained by the IVHFHSWA IVHFHWA and IVHFHOWA operator
1198601
1198602
1198603
1198604
1198605
IVHFSWA [04611 06115] [05042 07222] [04108 05582] [03254 04703] [04156 05836]IVHFWA [05045 06744] [05496 07500] [04582 06232] [03882 05290] [04940 06455]IVHFOWA [04999 06588] [05254 07284] [04312 05847] [03455 04781] [04651 06258]
075
0705
066
0615
057
0525
048
0435
039
0345
03
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
120578=1
120578=2
120578=3
120578=4
120578=5
120578=6
120578=7
120578=8
120578=9
120578=10
A1 A2 A3 A4 A5
Figure 1 Score values obtained by the IVHFHSWA operator with different values of 120578
the relative weights and the associated weights into averagingoperator Chen et al [11] have developed an interval-valuedhesitant fuzzy hybrid averaging (IVHFHA) operator basedon the hybrid weighted aggregation [37] operator which isdefined as follows
IVHFHA (ℎ1
ℎ2
ℎ119899
)
=
119899
⨁
119895=1
119908119895
ℎ120590(119895)
= ⋃
120590(119895)
isin
ℎ
120590(119895)
119895=1119899
[
[
1 minus
119899
prod
119895=1
(1 minus 120574119871
120590(119895))
119908
119895
1 minus
119899
prod
119895=1
(1 minus 120574119880
120590(119895))
119908
119895
]
]
(79)
where ℎ120590(119895)
is the 119895th largest of ℎ119895
= 119899120596119895
ℎ119895
(119895 = 1 2 119899)However the HWA operator has proven that it does notsatisfy boundary idempotent and so forth [4 38] Moreoverthe computation of IVHFHWA operator is simpler thanthe ones of IVHFHA operator When using the IVHFHAoperator we have to first calculate
ℎ119895
= 119899120596119895
ℎ119895
and com-
pare them and then calculate 119908119895
ℎ120590(119895)
after which we will
compute the aggregation values with oplus119899
119895=1
119908119895
ℎ120590(119895)
Since thecomputation with interval-valued hesitant fuzzy sets is verycomplex the results derived via the IVHFHA operator arehard to be obtained As for the IVHFHSWA operator theoperation of the relative weights and the ones of associatedweights in IVHFHWA operator are synchronized whichis in the mathematical form as 120596
119895
119908120588(119895)
Since both 120596119895
and 119908120588(119895)
are crisp numbers we only need to calculate(oplus
119899
119895=1
120596119895
ℎ119895
119908120588(119895)
)(sum119899
119895=1
120596119895
119908120588(119895)
) which makes the IVHFH-SWA operator easier to calculate than the IVHFHA operatorFurthermore the IVHFHWA operator can provide morespecial cases by selecting different values of parameter 120578which can provide more choices for the decision makersand considerably enhance or deteriorate the performance ofaggregation and thus is more general and more flexible
To show the advantage of IVHFHSWA operator againstthe VHFHWA and IVHFHOWA operators we use again theIVHFHSWA IVHFHWA and IVHFHOWA operators to thenumerical example above here we take 120578 = 1 as exampleand their final scores are listed in Table 6 and representedgraphically in Figure 2
From Table 6 and Figure 2 it is clear that despite thescore values obtained by the IVHFHSWA IVHFHWA andIVHFHOWA operators are different it is clear that despitethe score values of the alternatives obtained by the IVHFH-SWA IVHFHWA and IVHFHOWA operators are different
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 24: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/24.jpg)
24 Mathematical Problems in Engineering
030350404505055060650707508
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
IVHFS
WA
IVHFW
A
IVHFO
WA
A1 A2 A3 A4 A5
Figure 2 Score values obtained by the IVHFHSWA IVHFHWAand IVHFHOWA operator
the ranking results of the alternatives derived from themare the same that is 119860
2
≻ 1198601
≻ 1198605
≻ 1198603
≻ 1198604
The reasons about the difference of score values is intuitivethat as discussed above the IVHFHWA operator focusessolely on the relative weights and ignores the associatedweights while the IVHFHOWA operator focuses only onthe associated weights and ignores the relative weights TheIVHFHSWA operator comprehensively considers both theassociated weights and the relative weights Hence the resultsderived by IVHFHSWA operator are more feasible andeffective On the other hand the identical ranking resultsimply that the IVHFHSWA IVHFHWA and IVHFHOWAoperators all are effective
Finally our interval-valued hesitant fuzzy aggrega-tion (IVHFHWA IVHFHOWA IVHFHSWA IVHFHWGIVHFHOWG and IVHFHSWG) operators can also beapplied to deal with the situations when the interval-valuedhesitant fuzzy sets are reduced to hesitant fuzzy sets Incontrast the existing hesitant fuzzy aggregation operators(HFWA HFOWA HFHA HFWG HFOWG and HFHG)cannot be applied to deal with the interval-valued hesitantfuzzy situation In other words our interval-valued hesitantfuzzy aggregation operators have much wider applicationsthan the existing hesitant fuzzy aggregation operators
7 Conclusion
In this paper we first introduce the Hamacher operationsof interval-valued hesitant fuzzy sets and developed someinterval-valued hesitant fuzzy Hamacher operators includ-ing the interval-valued hesitant fuzzy Hamacher weightedaveraging (IVHFHWA)operator and interval-valued hesitantfuzzy Hamacher ordered weighted averaging (IVHFHOWA)operator The prominent advantages of the developed opera-tors are that they provide a family of interval-valued hesitantfuzzy aggregation operators that include the existing interval-valued hesitant fuzzy operators and interval-valued hesitantfuzzy Einstein operators as special cases and then providemore choices for the decision makers Then we proposed aninterval-valued hesitant fuzzyHamacher synergetic weightedaveraging operator to generalize further the IVHFHWA
and IVHFHOWA operator Some essential properties ofthe proposed operators are studied and their special cases arediscussed Based on the IVHFHSWA operator we developa practical approach to multiple criteria decision makingwith interval-valued hesitant fuzzy information Finally anillustrative example for selecting the shale gas areas is used toillustrate the proposed approach and a comparative analysis isperformed with other approaches to highlight the distinctiveadvantages of the proposed operators
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are very grateful to the editor WudhichaiAssawinchaichote and the anonymous referees for theirinsightful and constructive comments and suggestions whichhave helped to improve the paper This work was sup-ported in part by the National Natural Science Funds ofChina (nos 61364016 71272191 and 71072085) the scien-tific Research Fund Project of Educational Commission ofYunnan Province China (no 2013Y336) the Science andTechnology Planning Project of Yunnan Province China(no 2013SY12) the Natural Science Funds of KUST (noKKSY201358032) and the Graduate Innovation Funds ofHeilongjiang Province of China (no YJSCX2011-003HLJ)
References
[1] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 18th IEEE International Conference onFuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009
[2] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010
[3] Z S Xu and M Xia ldquoOn distance and correlation measures ofhesitant fuzzy informationrdquo International Journal of IntelligentSystems vol 26 no 5 pp 410ndash425 2011
[4] D-H Peng C-Y Gao and Z-F Gao ldquoGeneralized hesitantfuzzy synergetic weighted distance measures and their applica-tion to multiple criteria decision-makingrdquo Applied Mathemati-cal Modelling vol 37 no 8 pp 5837ndash5850 2013
[5] X Zhang and Z S Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience 2013
[6] M Xia and Z S Xu ldquoHesitant fuzzy information aggregationin decision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011
[7] Z S Xu and M Xia ldquoDistance and similarity measures forhesitant fuzzy setsrdquo Information Sciences vol 181 no 11 pp2128ndash2138 2011
[8] D Yu W Zhang and Y Xu ldquoGroup decision making underhesitant fuzzy environment with application to personnel eval-uationrdquo Knowledge-Based Systems vol 52 pp 1ndash10 2013
[9] B Farhadinia ldquoA novel method of ranking hesitant fuzzy valuesfor multiple attribute decision-making problemsrdquo InternationalJournal of Intelligent Systems vol 28 no 8 pp 752ndash767 2013
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 25: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/25.jpg)
Mathematical Problems in Engineering 25
[10] G Qian HWang and X Feng ldquoGeneralized hesitant fuzzy setsand their application in decision support systemrdquo Knowledge-Based Systems vol 37 pp 357ndash365 2013
[11] N Chen Z S Xu and M Xia ldquoInterval-valued hesitantpreference relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash540 2013
[12] M Xia Z S Xu andN Chen ldquoSome hesitant fuzzy aggregationoperators with their application in group decision makingrdquoGroupDecision andNegotiation vol 22 no 2 pp 259ndash279 2013
[13] G Wei ldquoHesitant fuzzy prioritized operators and their appli-cation to multiple attribute decision makingrdquo Knowledge-BasedSystems vol 31 pp 176ndash182 2012
[14] B Zhu Z S Xu and M Xia ldquoHesitant fuzzy geometricBonferroni meansrdquo Information Sciences vol 205 pp 72ndash852012
[15] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision makingrdquoInformation Sciences vol 234 pp 150ndash181 2013
[16] B Farhadinia ldquoInformationmeasures for hesitant fuzzy sets andinterval-valued hesitant fuzzy setsrdquo Information Sciences vol240 pp 129ndash144 2013
[17] N Chen Z S Xu and M Xia ldquoCorrelation coefficients ofhesitant fuzzy sets and their applications to clustering analysisrdquoApplied Mathematical Modelling vol 37 no 4 pp 2197ndash22112013
[18] Z S Xu and M Xia ldquoHesitant fuzzy entropy and cross-entropyand their use in multiattribute decision-makingrdquo InternationalJournal of Intelligent Systems vol 27 no 9 pp 799ndash822 2012
[19] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012
[20] R M Rodriguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012
[21] RM Rodrıguez L Martınez and F Herrera ldquoA group decisionmaking model dealing with comparative linguistic expressionsbased on hesitant fuzzy linguistic term setsrdquo Information Sci-ences vol 241 pp 28ndash42 2013
[22] G W Wei ldquoSome hesitant interval-valued fuzzy aggregationoperators and their applications to multiple attribute decisionmakingrdquo Knowledge-Based Systems vol 46 pp 43ndash53 2013
[23] G W Wei and X Zhao ldquoInduced hesitant interval-valuedfuzzy Einstein aggregation operators and their application tomultiple attribute decision makingrdquo Journal of Intelligent ampFuzzy Systems vol 24 no 4 pp 789ndash803 2013
[24] H Hamachar ldquoUber logische verknunpfungenn unssharferaussagen und deren zugenhorige bewertungsfunktionerdquo inProgress in Cybernatics and Systems Research R Trappl G JKlir and L Riccardi Eds vol 3 pp 276ndash288 HemisphereWashington DC USA 1978
[25] J Dombi ldquoTowards a general class of operators for fuzzysystemsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2 pp477ndash484 2008
[26] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Dodrecht The Netherlands 2000
[27] T Calvo G Mayor and R Mesiar Aggregation Operators NewTrends and Applications Physica Heidelberg Germany 2002
[28] R R Yager ldquoOn ordered weighted averaging aggregationoperators in multicriteria decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics vol 18 no 1 pp 183ndash1901988
[29] R R Yager ldquoTime series smoothing and OWA aggregationrdquoIEEE Transactions on Fuzzy Systems vol 16 no 4 pp 994ndash10072008
[30] R A Ribeiro and R A Marques Pereira ldquoGeneralized mixtureoperators using weighting functions a comparative study withWA and OWArdquo European Journal of Operational Research vol145 no 2 pp 329ndash342 2003
[31] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005
[32] R R Yager ldquoQuantifier guided aggregation using OWA oper-atorsrdquo International Journal of Intelligent Systems vol 11 no 1pp 49ndash73 1996
[33] D-H Peng C-Y Gao and L-L Zhai ldquoMulti-criteria groupdecision making with heterogeneous information based onideal points conceptrdquo International Journal of ComputationalIntelligence Systems vol 6 no 4 pp 616ndash625 2013
[34] D-H Peng Z-F Gao C-Y Gao and H Wang ldquoA directapproach based on C2-IULOWA operator for group decisionmaking with uncertain additive linguistic preference relationsrdquoJournal of Applied Mathematics vol 2013 Article ID 420326 14pages 2013
[35] A Kalantari-Dahaghi and S D Mohaghegh ldquoA new practicalapproach in modelling and simulation of shale gas reservoirsapplication to New Albany Shalerdquo International Journal of OilGas and Coal Technology vol 4 no 2 pp 104ndash133 2011
[36] X Zhao J Kang and B Lan ldquoFocus on the development ofshale gas in Chinamdashbased on SWOT analysisrdquo Renewable andSustainable Energy Reviews vol 21 pp 603ndash613 2013
[37] C-Y Gao and D-H Peng ldquoConsolidating SWOT analy-sis with nonhomogeneous uncertain preference informationrdquoKnowledge-Based Systems vol 24 no 6 pp 796ndash808 2011
[38] J Lin and Y Jiang ldquoSome hybrid weighted averaging operatorsand their application to decision makingrdquo Information Fusionvol 16 pp 18ndash28 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 26: Research Article Interval-Valued Hesitant Fuzzy Hamacher … · 2019. 7. 31. · 1. Instruction As a novel generalization of fuzzy sets, hesitant fuzzy sets (HFSs) [ , ] introduced](https://reader035.pdfslide.net/reader035/viewer/2022071610/61490aa19241b00fbd674d67/html5/thumbnails/26.jpg)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
