Research Article The Interval-Valued Intuitionistic Fuzzy …downloads.hindawi.com/journals/mpe/2015/560690.pdf · 2019. 7. 31. · Multiple criteria decision making methods have

Research ArticleThe Interval-Valued Intuitionistic Fuzzy MULTIMOORAMethod for Group Decision Making in Engineering

Edmundas Kazimieras Zavadskas1 Jurgita Antucheviciene1

Seyed Hossein Razavi Hajiagha2 and Shide Sadat Hashemi3

1Department of Construction Technology and Management Vilnius Gediminas Technical University Sauletekio Aleja 11LT-10223 Vilnius Lithuania2Department of Management Khatam Institute of Higher Education No 30 Hakim Arsquozam Alley Mollasadra StreetNorth Shiraz Avenue Tehran 19941633356 Iran3Department of Management Islamic Azad University Kashan Branch Kashan 8715998151 Iran

Correspondence should be addressed to Jurgita Antucheviciene jurgitaantuchevicienevgtult

Received 12 March 2015 Revised 8 May 2015 Accepted 11 May 2015

Academic Editor David Bigaud

Copyright copy 2015 Edmundas Kazimieras Zavadskas et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Multiple criteria decision making methods have received different extensions under the uncertain environment in recent yearsThe aim of the current research is to extend the application of the MULTIMOORA method (Multiobjective Optimization byRatio Analysis plus Full Multiplicative Form) for group decision making in the uncertain environment Taking into account theadvantages of IVIFS (interval-valued intuitionistic fuzzy sets) in handling the problem of uncertainty the development of theinterval-valued intuitionistic fuzzy MULTIMOORA (IVIF-MULTIMOORA) method for group decision making is considered inthe paper Two numerical examples of real-world civil engineering problems are presented and ranking of the alternatives basedon the suggested method is described The results are then compared to the rankings yielded by some other methods of decisionmaking with IVIF information The comparison has shown the conformity of the proposed IVIF-MULTIMOORA method withother approaches The proposed algorithm is favorable because of the abilities of IVIFS to be used for imagination of uncertaintyand the MULTIMOORA method to consider three different viewpoints in analyzing engineering decision alternatives

1 Introduction

Multicriteria decision making (MCDM) is a growing fieldof operations research both from theoretical and imple-mentation perspectives The importance of MCDM can bedrawn from Zeleny [1] ldquoit has become more and moredifficult to see the world around us in a uni-dimensionalway and to use only a single criterion when judging whatwe see We always compare rank and order the objects ofour choice with respect to multiple criteria of choicerdquo TheMCDM field is further divided into two classes includingmultiobjective decision making (MODM) and multiattributedecision making (MADM) These classes are respectivelyassociatedwith planning and selection classification of Simon

[2] for decision making problems The main topic of thispaper covers MADM

A MADM problem can be formally characterized as thetask to evaluate compare and rank a set of finite alternativesoptions or choices with regard to a set of finite attributesAccording to Yu [3] a MADM problem is composed ofthe set of substitutive alternatives (1) the set of evaluationcriteria (2) the outcome (or decision) matrix with regard tothe alternatives scored based on the evaluation criteria (3)and the preference structure of decision making about thecriterion significances or weights (4) The information aboutthe third and fourth parts of a decision making problemis determined exactly However a decision maker alwaysdeals with approximate or partial information [4] Therefore

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 560690 13 pageshttpdxdoiorg1011552015560690

2 Mathematical Problems in Engineering

exactness is an unrealistic assumption Different frameworkshave been proposed to handle uncertainty in practice Liuand Lin [5] classified these frameworks into probability andstatistics (1) and grey system theory and fuzzy set theory(2) Fuzzy set theory introduced by Zadeh [6] is widelyapplied to decision making problems In other words thefuzzy set theory is a generalization of ordinal sets wherea membership degree is assigned to each element of a setEach type of uncertainty has its own characteristics and isappropriate for special cases While probability is concernedwith the occurrence of well-defined events fuzzy sets dealwith gradual concepts and describe their boundaries [7] Infact the fuzzy set theory is appropriate for recognition-baseduncertainty that is common in decision making

Determining a single membership degree is a difficulttask for decision makers and therefore Grattan-Guinness[8] believes that presentation of a linguistic expression inthe form of a fuzzy set is not sufficient In 1986 Atanassov[9] introduced the notion of intuitionistic fuzzy sets (IFSs)as an extension of ordinal fuzzy sets In addition to amembership degree of each element IFS assigns a degree ofnonmembership to each element Later Atanassov and Gar-gov [10] extended the interval-valued intuitionistic fuzzy sets(IVIFSs) where membership and nonmembership degreesare stated as closed intervals

When it became obvious that the type 1 fuzzy sets werenot always sufficient for MADM under uncertain environ-ment type 2 fuzzy sets involving interval-valued as wellas intuitionistic fuzzy sets were proposed As a result theMULTIMOORA method was updated by using generalizedinterval-valued trapezoidal fuzzy numbers [11] or intuition-istic fuzzy numbers [12] The method has been successfullyapplied to making economic technological or managementdecisions The applications of various approaches in 2006ndash2013 including crisp and extended methods were summa-rized by T Balezentis and A Balezentis [13] The most recentapplications cover the extended versions of the method inparticular the fuzzy MULTIMOORA [14 15] or MULTI-MOORA based on the interval 2-tuple linguistic variables[16] Also Li [17] extended MULTIMOORA with hesitantfuzzy numbers where membership degree of elements isdefined over a set of different values

The current research aims to extend the MULTIMOORAmethod using the interval-valued intuitionistic fuzzy setsTaking into account the advantages of IVIFS the interval-valued intuitionistic fuzzy MULTIMOORA (IVIF-MULTI-MOORA) method for group decision making is presentedin the paper For this reason three parts of MULTIMOORAwere extended under IVIF information and the appropriatealgorithm was proposed

The paper is organized as follows In Section 2 theliterature review is presented and then the crisp version ofMULTIMOORA is briefly overviewed in Section 3 Section 4includes a brief description of the interval-valued intuitionis-tic fuzzy setsThe proposed IVIF-MULTIMOORA algorithmis presented in Section 5 The solution of two numericalexamples of civil engineering problems is given in Section 6while the results obtained are compared with the data yieldedby some other methods of IVIF decision making Finally the

discussion and the conclusions are presented in Sections 7and 8

2 Literature Review

Taking into account the flexibility and general characterof IVIFS researchers developed decision making methodsunder the IVIFS environment Some researchers developedthe well-known MADM methods in the IVIF form Someauthors [18ndash20] proposed different frameworks of TOPSISmethod under the IVIF environment Li [21 22] developedsomemathematical programming-basedmethod to solve theMADM problems with both ratings of the alternatives onattributes and weights of attributes expressed with the helpof IVIFS Park et al [23] presented an IVIF version of theVIKOR method Li [24] proposed a closeness coefficientbased on the nonlinear programming method for solvingIVIF MADM problems Chen et al [25] solved the MADMproblems based on the interval-valued intuitionistic fuzzyweighted average operator and newly defined fuzzy rank-ing method for intuitionistic fuzzy values Yu et al [26]introduced some IVIF aggregation operators and appliedthem to solving various decision making problems Zhangand Yu [27] presented an optimization model to determinethe attribute weights in an IVIF decision making problemand then used these weights in an extended TOPSIS torank the alternatives Ye [28] introduced a cosine similaritymeasure and a weighted cosine similarity measure toMADMproblemsMeng et al [29] also proposed twonew aggregationoperators that is the arithmetical interval-valued intuition-istic fuzzy generalized 120582-Shapley Choquet operator and thegeometric interval-valued intuitionistic fuzzy generalized 120582-Shapley Choquet operator and investigated their applicationto solving MADM problems Meng et al [30] used Shapleyfunction in extending a generalized IVIF hybrid Shapleyaveraging operator and proposed its application to solvingMADM problems Razavi Hajiagha et al [31] developedComplex Proportional Assessment (COPRAS) method withIVIF data Chai et al [32] proposed a rule-based decisionmodelwhendecision information in a groupdecisionmakingproblem is provided as IVIF values Wan and Dong [33]defined the possibility degree of comparison between twoIVIF numbers and introduced two ordered averaging opera-tors based on the Karnik-Mendel algorithm to solve MADMproblems with IVIF information Chen [34] developed amethod based on the traditional linear assignment methodfor solving decision making problems in the IVIF contextXu and Shen [35] presented the IVIF outranking choicemethod to solve MADM problems while Zavadskas et al[36] extended the IVIF weighted aggregated sum productassessment (WASPAS) method Geetha et al [37] proposeda new ranking method of IVIF numbers and extended itsapplication in decision making problems Zhang et al [38]proposed a new definition of IVIF entropy an entropy-based MADMmethod Later Wei and Zhang [39] proposedapplying an entropy measure for IFSs and IVIFSs to assessthe expertsrsquo weights for multicriteria fuzzy group decisionmaking Tong and Yu [40] introduced a novel approach for

Mathematical Problems in Engineering 3

ranking the alternatives based on the IVIF cross entropyand TOPSIS in the IVIF environment Wu and Chiclana[41 42] introduced a risk attitudinal ranking method forIVIF numbers and applied this ranking method in a MADMproblem They imposed a risk attitude parameter over IVIFnumbers ordinal score and accuracy functions [43] and usedthis method for solving MADM problems

As mentioned before the current paper has focused onthe extension of the MULTIMOORA (Multiobjective Opti-mization by Ratio Analysis plus Full Multiplicative Form)method This method belongs to the group of completeaggregationmethods based on the reference point techniqueThe crisp MOORA method (Multiobjective Optimization byRatio Analysis) was presented by Brauers and Zavadskas in2006 [44]This method was further supplemented by the fullmultiplicative form The MULTIMOORA was suggested byBrauers and Zavadskas in 2010 [45] The method is based onthe theory of dominance [46] and enables us to summarizeMOORA consisting of the ratio system and the referencepoint approaches as well as the full multiplicative formThe robustness of the method has been proved [47] and ithas been successfully applied to evaluate decisions by usinga single approach or its combinations with other MADMmethods such as TOPSIS [48] and WASPAS [49 50]

Several developments of MULTIMOORA for the uncer-tain environment have been presented The fuzzy MUL-TIMOORA was suggested by Brauers et al [52] To over-come some drawbacks of the fuzzy set theory the two-tuple linguistic representation method for computing withwords can be applied [53] Accordingly the consideredmethod has been extended for group decision making basedon two tuples (MULTIMOORA-2T-G) [54 55] The fuzzyMULTIMOORA based on triangular fuzzy numbers anddesigned for group decision making (MULTIMOORA-FG)was introduced [56 57] As a further modification of themethodMOORAwith grey numbers was developed [58 59]

3 MULTIMOORA

Themethod summarizes three approaches that is MOORAconsisting of the ratio system and the reference point and thefull multiplicative form

Regardless of the approach used initial decision criteriaare transformed by applying vector normalization

119909119894119895=

119909119894119895

radicsum119898

119894=1 1199092

119894119895

(1)

where 119909119894119895is the initial criterion value that is the response of

the alternative 119894 to objective 119895 119894 = 1 2 119898119898 is the numberof alternatives 119895 = 1 2 119899 119899 is the number of decisioncriteria (objectives) 119909

119894119895is a dimensionless value of a decision

criterionThe first part of the approach is based on the ratio

system [44] To calculate the relative significance 119910119894 of each

alternative 119894 with respect to all objectives 119895 the weightednormalized criteria values should be added in the caseof maximization while in the case of minimization the

weighted normalized criteria values should be subtracted asfollows

119910119894=

119895=119892

sum

119895=1119909119894119895119908119895minus

119895=119899

sum

119895=119892+1119909119894119895119908119895 (2)

where 119895 = 1 2 119892 are maximized decision criteria 119895 =

119892 + 1 119892 + 2 119899 are minimized decision criteria 119908119895is the

weight (or relative significance) of a criterionThe values of variables 119910

119894show the preference of the

alternatives according to the ratio system approachThe second part of the approach is based on the maximal

objective reference point [44] First the desirable ideal alter-native should to be established The virtual ideal alternativeconsists of the best values of the considered criteria 119903

119895 It

is formed by selecting the best criteria values from everydecision alternative based on their optimization directionthat is the maximal values from the criteria set 119895 = 1 2 119892and the minimal values from the criteria set 119895 = 119892 + 1 119892 +

2 119899All criteria values were transformed by applying vector

normalization (1) Having the dimensionless values of thecriteria 119909

119894119895and relative significances (weights) of the criteria

119908119895 decision alternatives are ranked based on the Min-Max

metric of Tchebycheff [60]

Min(119894)

max(119895)

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (3)

The second part of the approach is based on the fullmultiplicative form as presented by Brauers and Zavadskas[44] The full multiplicative form for calculating the utility ofthe alternatives 119880

119894is applied as follows

119880119894=119860119894

119861119894

(4)

where 119860119894and 119861

119894are calculated separately for maximized

decision criteria 119895 = 1 2 119892 and minimized decisioncriteria 119895 = 119892 + 1 119892 + 2 119899 respectively 119860

119894and 119861

119894are

calculated as follows

119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(5)

TheMULTIMOORA is based on the theory of dominance[46] and summarizes the MOORA involving the abovedescribed ratio system and the reference point approachesas well as the full multiplicative form [45]

Robustness of the method has been verified proving thataccuracy of aggregated approaches is larger comparing toaccuracy of single ones [47] The method has been widelyapplied to evaluate alternatives and to select the best decisionsin engineering technology ormanagement problems by usinga single approach or its combinations with other MADMmethods [13 48ndash50] However the farther the more complex


decisions should be made and decision makers usually facea larger amount of approximate or partial informationAccordingly extensions of conventional MADM methodsunder uncertain environment for group decision makingare essential The further developed IVIF-MULTIMOORAmethod could be applied in innovation themes in engineer-ing such as sustainable building life-cycle modelling andevaluating and selecting new business models to financebuild and manage public buildings and infrastructures

4 Interval-Valued Intuitionistic Fuzzy Sets

Zadeh [6] generalized the characteristic function of classicsets into membership function and introduced fuzzy setswhere a membership degree was assigned to each element ofa fuzzy set Later Atanassov [9] proposed the idea of intu-itionistic fuzzy sets (IFSs) when a nonmembership degree isassigned to each element of the set alongside its membershipThe interval-valued intuitionistic fuzzy sets (IVIFSs) presentan extended form of IFSs

Let 119863[0 1] be the set of all closed subintervals of theinterval [0 1] and let 119883( = Φ) be a given set An IVIFS in119883 was defined as 119860 = ⟨119909 120583

119860(119909) V119860(119909)⟩ | 119909 isin 119883 where

120583119860 119883 rarr 119863[0 1] and V

119860 119883 rarr 119863[0 1] with condition

0 ≺ sup119909120583119860(119909) + sup

119909V119860(119909) le 1 The intervals 120583

119860(119909) and

V119860(119909) denote the degree of membership and nonmembership

of the element 119909 of the set119860 Thus for each 119909 isin 119883 120583119860(119909) and

V119860(119909) are closed intervals whose lower and upper end points

are denoted by 120583119860119871(119909) 120583119860119880(119909) V119860119871(119909) and V

119860119880(119909)

The IVIFS was denoted by

119860 = ⟨119909 [120583119860119871 (119909) 120583119860119880 (119909)] [V119860119871 (119909) V119860119880 (119909)]⟩ | 119909

isin119883

(6)

where 0 ≺ 120583119860119880(119909) + V

119860119880(119909) le 1 120583

119860119871(119909) V119860119871(119909) ge 0 For con-

venience the IVIFS value was denoted by 119860 = ([119886 119887] [119888 119889])

and referred to as an interval-valued intuitionistic fuzzynumber (IVIFN)

The algebraic operations were extended over IVIFSs Let1198601 = ([1198861 1198871] [1198881 1198891]) and1198602 = ([1198862 1198872] [1198882 1198892]) be any twoIVIFNs then [43]

1198601 +1198602

= ([1198861 + 1198862 minus 11988611198862 1198871 + 1198872 minus 11988711198872] [11988811198882 11988911198892]) (7)

1198601 sdot 1198602

= ([11988611198862 11988711198872] [1198881 + 1198882 minus 11988811198882 1198891 +1198892 minus11988911198892]) (8)

1205821198601

= ([1minus (1minus 1198861)120582 1minus (1minus 1198871)

120582] [119888120582

1 119889120582

1])

120582 ge 0

(9)

The division and subtraction operations were defined asfollows using the extension principle [61]

1198601 divide1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (10)

1198601 minus1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (11)

Since the final rating of the alternatives is usually deter-mined as IVIFNs it requires their comparisonThis compar-ison was made using the score and accuracy functions [43]Let 119860 = ([119886 119887] [119888 119889]) be an IVIFN Then

119904 (119860) =12(119886 minus 119888 + 119887 minus 119889) (12)

was called the score function of119860 where 119904(119860) isin [minus1 1] while

ℎ (119860) =12(119886 + 119888 + 119887 + 119889) (13)

was referred to as the accuracy function of 119860 where ℎ(119860) isin[0 1]

For 1198601 and 1198602 as two IVIFNs it follows that [43]

(1) if 119904(1198601) ≺ 119904(1198602) then 1198601 is smaller than 1198602 1198601 ≺

1198602

(2) if 119904(1198601) = 119904(1198602) then

(a) if ℎ(1198601) = ℎ(1198602) then 1198601 = 1198602

(b) if ℎ(1198601) ≺ ℎ(1198602) then 1198601 is smaller than 11986021198601 ≺ 1198602

Another requirement of group decision making is asso-ciated with the aggregation of different judgments of deci-sion makers into a single estimate In this case aggre-gation operators on IVIFNs can be used Let 119860

119895=

([119886119895 119887119895] [119888119895 119889119895]) 119895 = 1 2 119899 be a collection of IVIFNs

Then the generalized interval intuitionistic fuzzy weightedaverage GIIFWA

119908(1198601 1198602 119860119899) was defined as

GIIFWA119908(1198601 1198602 119860119899)

= (1199081119860120582

1 +1199082119860120582

2 + sdot sdot sdot +119908119899119860120582

119899)1120582

(14)

where 120582 ≻ 0 and 119908 = (1199081 1199082 119908119899)119879 is the weight vector

with119908119895ge 0 119895 = 1 2 119899 andsum119899

119895=1 119908119895 = 1 It can be shown


thatGIIFWA is also an IVIFNand can be calculated as follows[62]

GIIFWA119908(1198601 1198602 119860119899)

= ([

[

(1minus119899

prod

119895=1(1 minus 119886120582

119895)119908119895)

1120582

(1minus119899

prod

119895=1(1 minus 119887120582

119895)119908119895)

1120582

]

]

[

[

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119888

119895)120582

)

119908119895

)

1120582

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119889

119895)120582

)

119908119895

)

1120582

]

]

)

(15)

If 120582 = 1 then GIIFWA is turned into the intervalintuitionistic fuzzy weighted average (IIFWA)

5 IVIF-MULTIMOORA forGroup Decision Making

51 IVIF-MULTIMOORA Assume that a group of 119870 deci-sionmakers (experts) wants to appraise a set of119898 alternatives1198601 1198602 119860119898 based on a set of 119899 criteria 1198621 1198622 119862119899Also the weight vector 1199081 1199082 119908119899 with 119908

119895ge 0 119895 =

1 2 119899 and sum119899

119895=1 119908119895 = 1 is determined that shows therelative importance of different criteria on final decision Dueto incomplete and ill-defined data the required informationin the considered problem is expressed by IVIFNs

At the first step each decision maker expresses hisherdecision matrix119883119896 119896 = 1 2 119870 as follows

119883119896=

[[[[[[[

[

119909119896

11 119909119896

12 sdot sdot sdot 119909119896

1119899

119909119896

21 119909119896


2119899

sdot sdot sdot

119909119896

1198981 119909119896

1198982 sdot sdot sdot 119909119896

119898119899

]]]]]]]

]

(16)

where 119909119896119894119895represents the IVIF performance of the alternative

119860119894over the criterion 119862

119895from the viewpoint of 119896th decision

maker The aim of a decision making group is to rank thealternatives

To solve this MAGDM problem an IVIF version ofthe MULTIMOORA method called IVIF-MULTIMOORAwas extended and described in this section Initially theaggregation was required to transform the set of 119870 decisionmatrices to an aggregated decision matrix This aggregateddecision matrix had the form of119883 = [119909

119894119895] If an equal weight

was assumed for decision makers then

119909119894119895= ([120583

119897

119894119895 120583119906

119894119895] [V119897119894119895 V119906119894119895]) = IIFWA (1199091

119894119895 119909

2119894119895 119909

119870

119894119895) (17)

The aggregation operation was performed using theGIIFWA operator and considering the weight vector 119881 =

(V1 V2 V119870) for different decision makersSince the aggregated decision matrix 119883 = [119909

119894119895] was

available the IVIF-MULTIMOORA method was proposedto solve the MAGDM problem As mentioned in Section 3the MULTIMOORA method involves three parts the ratiosystem (1) the reference point approach (2) and the fullmultiplicative form (3)

52 The Part of MOORA Based on the IVIF-Ratio SystemThe first step in the ratio system is the normalization of thedecision matrix However since IVIFNs are commensurablenumbers normalization is not required The set of criteriawas decomposed into two subsets of the benefit criteria(where more is better) and cost criteria (where less is better)Then the 119910

119894 119894 = 1 2 119898 score of alternatives was com-

puted by adding its weighted benefit criteria and subtractingthe weighted cost criteria If the benefit criteria were orderedas 1198621 1198622 119862119892 and cost criteria as 119862

119892+1 119862119892+2 119862119899 then

119910119894=

119892

sum

119895=1119908119895119909119894119895minus

119899

sum

119895=119892+1119908119895119909119894119895 (18)

This score could be computed by applying the IVIFNsalgebraic operators and the extension principle given inSection 4 However a slight modification could simplify thecomputations Let119882+ = sum119892

119895=1 119908119895 and119882minus= sum119899

119895=119892+1 119908119895Then1199081015840

119895= 119908119895119882+ for each 119895 isin 1 2 119892 and 11990810158401015840

119895= 119908119895119882minus

for each 119895 isin 119892 + 1 119892 + 2 119899 It was clear that 1199081015840119895ge

0 119895 = 1 2 119892 and sum119892119895=1 1199081015840

119895= 1 Similarly 11990810158401015840

119895ge 0 119895 =

1 2 119892 and sum119892119895=1 11990810158401015840

119895= 1 Then

1199101015840

119894=

119910119894

119882+119882minus (19)

Multiplying both sides of (18) by positive value of1(119882+119882minus) (18) was transformed as follows

1199101015840

119894=

1119882minus

119892

sum

119895=11199081015840

119895119909119894119895minus

1

119882+

119899

sum

119895=119892+111990810158401015840

119895119909119894119895 (20)

Based on the IIFWA operator definition (18) was repre-sented as

1199101015840

119894=

1119882minus

IIFWA119882+ (1199091198941 1199091198942 119909119894119892)

minus1119882+

IIFWA119882minus (119909119894119892+1 119909119894119892+2 119909119894119899)

(21)

Themultiplication of 1(119882minus) and 1(119882+)was performedusing (9) In addition the subtraction of two IVIFNs in(21) was performed by applying (16) 1199101015840

119894 119894 = 1 2 119898 is

relative significance of the alternative 119894 The comparison of1199101015840

119894 119894 = 1 2 119898 values based on the score and the accuracy

function resulted in the ranking of the alternatives based onthe ratio system


53 The IVIF-Reference Point Portion of the MOORAMethodAccording to the second part of the MOORA the maximalobjective reference point approach was used The desirableideal alternative in the IVIF environment was the IVIF vectorwith the coordinates 119903

119895 119895 = 1 2 119899 which was formed by

selecting the data from every considered decision alternativeand taking into account the optimization direction of everyparticular criterion To find the reference point based on theidea of the TOPSIS method the weighted decision matrix = [119899

119894119895] = 119882 sdot 119883 was constructed first where

119899119894119895= ([120578119897

119894119895 120578119906

119894119895] [120575119897

119894119895 120575119906

119894119895]) = 119908

119895119909119894119895 (22)

The scalar multiplication was performed by applying (9)Then for benefit criteria 1198621 1198622 119862119892 the reference pointwas constructed as follows

119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([max1le119894le119898

(120578119897

119894119895) max

1le119894le119898(120578119906

119894119895)]

[min1le119894le119898

(120575119897

119894119895) min

1le119894le119898(120575119906

119894119895)])

(23)

and for cost criteria 119862119892+1 119862119892+2 119862119899

119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([min1le119894le119898

(120583119897

119894119895) min

1le119894le119898(120583119906

119894119895)]

[max1le119894le119898

(V119897119894119895) max

1le119894le119898(V119906119894119895)])

(24)

Then the Min-Max metric of Tchebycheff [26] was usedfor ranking the alternatives

Min1le119894le119898

max1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (25)

Based on (22)ndash(24) (25) could be simplified The fol-lowing steps were taken to determine the rankings of thealternatives using the reference point approach

Step 1 Compute for each alternative 119860119894 119894 = 1 2 119898 the

score function in the weighted decision matrix

119878 (119899119894119895) = 120578119897

119894119895minus 120575119897

119894119895+ 120578119906

119894119895minus 120575119906

119894119895 (26)

Step 2 Compute the score function of the reference points

119878 (119903119895) = 120578lowast119897

119894119895minus 120575lowast119897

119894119895+ 120578lowast119906

119894119895minus 120575lowast119906

119894119895 (27)

Step 3 Find the maximum deviation from the referencepoints

119889119894= max

1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816= max

1le119895le119899

10038161003816100381610038161003816119878 (119903119895) minus 119878 (119899

119894119895)10038161003816100381610038161003816 (28)

Step 4 Rank the alternatives in the ascending order of 119889119894 119894 =

1 2 119898

54 The IVIF-Full Multiplicative Form Finally the full mul-tiplicative form was applied as follows

119894=119860119894

119861119894

(29)

where 119894denotes the overall utility of the alternative 119894 and

119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(30)

Multiplication of a set of IVIFNs was performed bysequentially applying (8) To give a compact form of thismultiplication it was supposed that 119909

119894119895 119895 = 1 2 119892 was

determined as an IVIFN as indicated in (17) First 119899119894119895

=

119908119895119909119894119895 119895 = 1 2 119892 was formed by applying (9) in the same

manner as previously performed in (22) Then

119860119894=

119892

prod

119895=1119899119894119895= ([120578

119897

119894119895 120578119897

119894119895] [120575119897

119894119895 120575119906

119894119895])

= ([

[

119892

prod

119895=1120578119897

119894119895

119892

prod

119895=1120578119906

119894119895]

]

[[[

[

119892

sum

119895=1120575119897

119894119895minus sum

1198951lt11989521198951 1198952isin12119892

120575119897

1198941198951sdot 120575119897

1198941198952

+ sdot sdot sdot + (minus1)119896+1 sum

1198951lt1198952ltsdotsdotsdotlt1198951198961198951 1198952 119895119896isin12119892

(120575119897

1198941198951sdot 120575119897

1198941198952sdot sdot sdot 120575119897

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751198971198941 sdot 120575119897

1198942 sdot sdot sdot 120575119897

119894119892)

119892

sum

119895=1120575119906

119894119895

minus sum

1198951lt11989521198951 1198952isin12119892

120575119906

1198941198951sdot 120575119906

1198941198952+ sdot sdot sdot

+ (minus1)119896+1 sum


(120575119906

1198941198951sdot 120575119906

1198941198952sdot sdot sdot 120575119906

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751199061198941 sdot 120575119906

1198942 sdot sdot sdot 120575119906

119894119892)]]]

]

)

(31)

Then 119861119894 119894 = 1 2 119898 was also calculated in a similar

way Then 119894 119894 = 1 2 119898 was determined by (10) for

division The ranking of the alternatives was specified bycalculating the score functions 119878(

119894) 119894 = 1 2 119898 and

ranking them in a descending orderApplying three distinct parts of MULTIMOORA three

sets of rankings of the alternatives were determined Theobtained rankings formed a ranking pool based onwhich thefinal ranking of the alternatives could be determined To find


Full

mul

tiplic

ativ

e for

m

Ratio

syste

m

Refe

renc

e poi

nt ap

proa

ch

Initi

aliz

atio

n

Form the ranking pool of the alternatives

Start

Determine the of the alternatives

Form a decision making group

Determine thedecision criteria

Construct individual decision matrices

Compute the aggregated decision matrix

Eq (17)

Compute the weighted decision matrix

Eq (22)

Determine the reference point

Eq (23) Eq (24)

Find the maximum deviation from the

reference point

Eq (28)

Rank the alternatives based on

Compute relative significance of the

alternatives

Eq (21)


Eq (11)

Compute the overallutility of the alternatives

Eq (29)


Eq (11)

Agg

rega

tion

Find the final ranking of the alternatives

(i) Theory of dominance(ii) Borda count(iii) Copeland pairwise aggregation

criteria weights WDefine the set A

Find y998400

i i = 1 2 m

S(y998400

i) i = 1 2 m

Find Ui i = 1 2 m

di i = 1 2 m

S(Ui) i = 1 2 m

Figure 1 The IVIF-MULTIMOORA algorithm

the final ranking of the alternatives the theory of dominance[46] could be used Also the techniques such as the Bordacountmethod [63] orCopeland pairwise aggregationmethod[64] could be applied to determine the final ranking of thealternatives

55 The IVIF-MULTIMOORA Algorithm MULTIMOORAis the extension of the MOORA method and the full mul-tiplicative form of multiple-objective The distinct parts ofthe MULTIMOORA method are presented in Sections 51 to53 The algorithmic scheme of MULTIMOORA is presented

in the current section The IVIF-MULTIMOORA algorithmincludes five different stages (1) initialization (2) IVIF-ratio system (3) IVIF-reference point approach (4) IVIF-fullmultiplicative form and (5) final ranking These stages arepresented in Figure 1

6 A Numerical Example

In this section two applications of the proposed methodare presented and the results are compared to the rankingsyielded by other multiple criteria decision making methods


Table 1 Decision matrix of rational revitalization of derelict and mismanaged buildings

Criteria Optimization direction Weight A1 A2 A3

1199091 Max 00600 ([070 080] [010 015]) ([045 060] [020 030]) ([030 040] [045 050])1199092 Min 00727 ([060 070] [015 020]) ([045 055] [020 030]) ([025 040] [045 055])1199093 Max 00747 ([075 080] [010 015]) ([045 055] [025 035]) ([030 040] [040 050])1199094 Max 00627 ([080 090] [0 005]) ([050 065] [015 025]) ([035 050] [030 040])1199095 Max 00673 ([070 085] [005 010]) ([055 065] [020 025]) ([035 045] [030 040])1199096 Max 00627 ([075 085] [010 015]) ([055 065] [020 025]) ([035 045] [030 040])1199097 Min 00667 ([080 085] [010 015]) ([045 055] [025 035]) ([015 035] [040 050])1199098 Max 00667 ([085 090] [0 005]) ([020 035] [035 050]) ([010 015] [075 080])1199099 Max 00667 ([080 090] [0 005]) ([010 025] [055 065]) ([005 010] [080 085])11990910 Max 00667 ([045 060] [025 035]) ([010 025] [055 065]) ([005 010] [080 085])11990911 Max 00667 ([075 080] [010 015]) ([015 030] [030 050]) ([005 010] [075 085])11990912 Max 00667 ([075 085] [005 010]) ([025 040] [020 040]) ([070 080] [010 015])11990913 Min 00667 ([070 085] [005 010]) ([010 015] [075 080]) ([050 055] [035 040])11990914 Min 00667 ([070 080] [010 015]) ([050 055] [030 040]) ([010 015] [070 080])11990915 Max 00667 ([060 075] [015 020]) ([045 060] [020 030]) ([025 045] [025 040])

61The First Example Zavadskas et al [36] solved a problemof rational revitalization of derelict and mismanaged build-ings in Lithuanian rural areas Three alternatives includedthe reconstruction of rural buildings and their adaptation toproduction (or commercial) activities (119860

1) as well as their

improvement and use for farming (1198602) or dismantling and

recycling the demolition waste materials (1198603) The following

criteriawere taken into consideration the average soil fertilityin the area119909

1(points) quality of life of the local population119909

2

(points) population activity index 1199093() GDP proportion

with respect to the average GDP of the country 1199094()

material investment in the area 1199095(LTL per resident) foreign

investments in the area 1199096(LTL times 103 per resident) building

redevelopment costs 1199097(LTL times 106) increase in the local

populationrsquos income 1199098(LTL times 106 per year) increase in sales

in the area1199099() increase in employment in the area119909

10()

state income from business and property taxes 11990911(LTL times 106

per year) business outlook 11990912 difficulties in changing the

original purpose of the site 11990913 the degree of contamination

11990914 and the attractiveness of the countryside (ie image and

landscape) 11990915

A decision matrix is presented in Table 1This problem was solved with the proposed IVIF-

MULTIMOORA method as follows

611 Ratio System In the ratio system ranking the relativeimportance (significance) of the alternatives was determinedby (21) For this purpose initially the benefit criteriaincluding 119909

1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15

were separated from the cost criteria including 1199097 11990913 and

11990914Then the aggregations of the benefit criteriaweights119882+

and cost criteria weights119882minus were computed These weightswere then used in (21) and the values of 1199101015840

119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])

(32)Ranking these values according to their score functions

led to the ranking of the alternatives as 1198601 ≻ 1198602 ≻ 1198603

612 Reference Point Approach Ranking of the alternativesin this approach was initialized by determining a referencepoint from the weighted decision matrixThen the deviationof each alternative from the reference point was computed by(28)These deviationswere as follows1198891 = 04681198892 = 1332and 1198893 = 1391 By ranking the alternatives in the descendingorder of deviations the final ranking was determined as1198601 ≻1198602 ≻ 1198603

613 Full Multiplicative Form In applying the third part ofthemethod the full multiplicative approach was used to rankthe alternativesUsing (30) themultiplicative forms of benefitand cost criteria were obtained These multiplicative formswere constructed by sequential multiplication of a set ofIVIFNsThe overall utility of the alternatives was determinedas follows

1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)

Ranking these values according to their score functionsled to ranking of the alternatives as 1198601 ≻ 1198602 ≻ 1198603


Table 2 IVIF-MULTIMOORA ranking of the alternatives

Alternatives IVIF-MOORA ratio system IVIF-MOORA reference point IVIF-full multiplicative form IVIF-MULTIMOORA1198601 1 1 1 11198602 2 2 2 21198603 3 3 3 3

Table 3 Ranking of the alternatives by applying different methods

Alternatives IFOWA TOPSIS-IVIF COPRAS-IVIF WASPAS-IVIF IVIF-MULTIMOORA1198601 1 1 1 1 11198602 2 2 3 2 21198603 3 3 2 3 3

Table 4 IVIF decision matrix for investment alternatives

1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])

Table 5 IVIF-MULTIMOORA ranking of the investment alternatives

Alternatives IVIF-MOORA ratio system IVIF-MOORA reference point IVIF-full multiplicative form IVIF-MULTIMOORA1198601 3 2 3 31198602 4 1 4 41198603 1 4 1 11198604 2 3 2 2

Table 6 Ranking of the investment alternatives by applying different methods

Alternatives Wang et al IFOWA TOPSIS-IVIF COPRAS-IVIF WASPAS-IVIF IVIF-MULTIMOORA1198601 3 4 2 3 3 31198602 4 3 4 4 4 41198603 2 1 1 2 1 11198604 1 2 3 1 2 2

The three sets of rankings are summarized in Table 2Thefinal ranking was obtained by using the dominance theory inthe last column of the table

Table 3 also presents the rankings obtained by somedifferentmethods including IFOWA [62] TOPSIS-IVIF [19]COPRAS-IVIF [31] and WASPAS-IVIF [36]

Kendallrsquos coefficient of concordance in Table 3 is 0840which shows a high degree of concordance Moreover thereis complete coincidence among the IVIF-MULTIMOORAmethod and the other three methods These facts imply thatthere is great homogeneity between IVIF-MULTIMOORAand other methods

62The SecondExample Consider the problemof appraisingfour different investment alternatives 1198601 1198602 1198603 1198604 basedon four criteria of risk (119862

1) growth (119862

2) sociopolitical issues

(1198623) and environmental impacts (119862

4) The vector of criteria

significance is (013 017 039 031) Also 1198621and 119862

3were

considered as cost criteria while 1198622and 119862

4were benefit

criteria The problem was initially formulated by Wang et al[51] as the decision matrix which is given below (Table 4)

The above problem was solved by using the proposedIVIF-MULTIMOORAmethodThe results obtained by usingthe ratio system the reference point approach and the fullmultiplicative form are presented in Table 5 Moreover thefinal ranking obtained based on the dominance theory ispresented in the last column of the table

The results obtained by solving the above problem bydifferent methods including the method of Wang et al [51]IFOWA [62] TOPSIS-IVIF [19] COPRAS-IVIF [31] andWASPAS-IVIF [36] are presented in Table 6

In this case Kendallrsquos coefficient of concordance is 0766which shows a high degree of concordance compared to thecritical value of 0421 [65] Spearmanrsquos rank correlation isused to demonstrate the similarity between different rank-ings Table 7 shows Spearmanrsquos rank correlation between the


Table 7 Ranking similarity of the investment alternatives by applying different methods

Methods Wang et al IFOWA TOPSIS-IVIF CORAS-IVIF WASPAS-IVIF IVIF-MULTIMOORAWang et al [51] mdash 06 (0) 04 (25) 1 (100) 08 (50) 08 (50)IFOWA mdash 04 (25) 06 (0) 08 (50) 08 (50)TOPSIS-IVIF mdash 04 (25) 08 (50) 08 (50)CORAS-IVIF mdash 08 (50) 08 (50)WASPAS-IVIF mdash 1 (100)IVIF-MULTIMOORA mdash

consideredmethodsThepercentages specified in parenthesisin Table 7 show the matched ranks obtained by using theconsidered five methods with ranks of Wang et al [51]method as the percentages of the matched alternatives

Based on the above findings it can be concluded thatexcept for COPRAS-IVIF which completely matches themethod of Wang et al [51] the IVIF-MULTIMOORA alongwith WASPAS-IVIF has the highest similarity (reaching80) to all other methods Moreover the percentage of thematched rankings with those of Wang Li and Wang is 50which is the second high value of congruence These resultshave shown great similarity of IVIF-MULTIMOORA to othermethods which can be considered to be one of its advantages

7 Discussion

Vincke [66] believes that the main difficulty in solvingmultiple criteria decision making problems lies in the factthat usually there is no optimal solution to such a problemwhich could dominate other solutions in all the criteria

As a response to this dilemma variousmethods have beenproposed for solving MCDM problems The selection of themost appropriate and the most robustly and effectively use-able method is a continuous object of scientific discussionsfor decades [67 68]

This means that the experts usually look for Paretooptimal solution Therefore while an optimal solution isobtained by several methods this emphasizes that proposedmethod will achieve the near to optimum answer In thisregard MULTIMOORAmethod has a strong advantage overthe other methods as it summarizes three approaches Therobustness of the aggregated approach has been substantiatedin the previous research [47]

On the other hand the above-mentioned difficulty maybe considered from two additional perspectives First theuncertainty of decision maker in his judgments should betaken into consideration To deal with this challenge severalframeworks have been introduced starting from applicationof Zadeh fuzzy numbers [6] and followed by numerousextensions of decision making methods using type 1 and type2 fuzzy sets involving interval-valued as well as intuitionisticfuzzy sets [69] The extensions of the considered MULTI-MOORAmethod till 2013were summarized in a reviewpaper[13] The most recent extensions not covered by [13] shouldalso be mentioned [14ndash17] Seeking the best imaginationof uncertainty another up-to-date extension namely the

interval-valued intuitionistic fuzzy MULTIMOORAmethodis presented in the current research

Second decisions are usually made by several decisionmakers This makes it necessary to consider a suitableprocedure to aggregate the opinions Therefore this problemis handled as a group decision making problem

As it is not easy to prove that a newly proposed methodis applicable it is useful to test it in solving several multiplecriteria problems To make sure that the method is better orat least it is not worse than the other existing methods it isappropriate to apply several related approaches to comparetheir ranking results for the same problem [70] Accordinglytwo illustrative examples have been presented to fulfill thetask It is encouraging that the results have shown greatsimilarity of IVIF-MULTIMOORA to other methods Thefact can be considered to be one of advantages of the novelapproach

8 Conclusions and Future Work

In the current paper the interval-valued intuitionistic fuzzyMULTIMOORA method is proposed for solving a groupdecision making problem under uncertainty The IVIF is ageneralized form of fuzzy sets which considers the nonmem-bership of objects in a set beside their membership and withrespect to hesitancy of decision makers

TheMULTIMOORAmethod is a decisionmaking proce-dure composed of three parts including the ratio system thereference point approach and full multiplicative form Therankings yielded by this method are finally aggregated and aunique ranking is determined

The proposed algorithm benefits from the abilities ofIVIFSs to be used in imagination of uncertainty and MUL-TIMOORA method to consider three different viewpointsin analyzing decision alternatives The application of theproposed method is illustrated by two numerical exam-ples and the obtained results are compared to the resultsyielded by other methods of decision making with IVIFinformation The comparison using Kendall coefficient ofconcordance and Spearmanrsquos rank correlation has shown ahigh consistency between the proposedmethod and the otherapproaches that verifies the proposed method can be appliedin case of similar problems in ambiguous and uncertain situ-ation This advantage strengthens the applicability of IVIF-MULTIMOORA as a method for group decision making


under uncertainty when solving engineering technology andmanagement complex problems

Future directions of the research will be focused onsearching modifications of the method the best reflectinguncertain environment and producing the most reliableresults At the moment two novel and up-to-date possi-ble extensions of the MULTIMOORA method could besuggested One of the approaches could be based on theneutrosophic sets [71 72] that are a generalization of the fuzzyand intuitionistic fuzzy sets The other possible approachcould be to apply 119863 numbers that were recently proposed tohandle uncertain information [73 74]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Zeleny Multiple Criteria Decision Making McGraw-HillNew York NY USA 1982

[2] H A Simon Models of Discovery and Other Topics in theMethods of Science D Reidel Boston Mass USA 1977

[3] P L Yu Forming Winning Strategies An Integrated Theory ofHabitual Domains Springer Heidelberg Germany 1990

[4] M C YovitsAdvances in Computers Academic Press OrlandoFla USA 1984

[5] S F Liu and Y Lin Grey Systems Theory and ApplicationSpringer Heidelberg Germany 2011

[6] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[7] W Pedrycz and F Gomide An Introduction to Fuzzy SetsAnalysis and Design MIT Press Cambridge UK 1998

[8] I Grattan-Guinness ldquoFuzzy membership mapped onto inter-vals and many-valued quantitiesrdquo Mathematical Logic Quar-terly vol 22 no 2 pp 149ndash160 1976

[9] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[10] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[11] T Balezentis and S Zeng ldquoGroup multi-criteria decision mak-ing based upon interval-valued fuzzy numbers an extensionof the MULTIMOORA methodrdquo Expert Systems with Applica-tions vol 40 no 2 pp 543ndash550 2013

[12] T Balezentis S Z Zeng and A Balezentis ldquoMULTIMOORA-IFN a MCDM method based on intuitionistic fuzzy numberfor performance managementrdquo Economic Computation andEconomic Cybernetics Studies and Research vol 48 no 4 pp85ndash102 2014

[13] T Balezentis and A Balezentis ldquoA survey on developmentand applications of the multi-criteria decision making methodMULTIMOORArdquo Journal of Multi-Criteria Decision Analysisvol 21 no 3-4 pp 209ndash222 2014

[14] H Liu X Fan P Li and Y Chen ldquoEvaluating the risk of failuremodes with extended MULTIMOORA method under fuzzyenvironmentrdquo Engineering Applications of Artificial Intelligencevol 34 pp 168ndash177 2014

[15] H Liu J X You Ch Lu and Y Z Chen ldquoEvaluating health-care waste treatment technologies using a hybrid multi-criteriadecision making modelrdquo Renewable and Sustainable EnergyReviews vol 41 pp 932ndash942 2015

[16] H Liu J X You C Lu and M M Shan ldquoApplication ofinterval 2-tuple linguistic MULTIMOORA method for health-care waste treatment technology evaluation and selectionrdquoWaste Management vol 34 no 11 pp 2355ndash2364 2014

[17] Z H Li ldquoAn extension of the MULTIMOORA method formultiple criteria group decision making based upon hesitantfuzzy setsrdquo Journal of AppliedMathematics vol 2014 Article ID527836 16 pages 2014

[18] A K Verma R Verma and N C Mahanti ldquoFacility loca-tion selection an interval valued intuitionistic fuzzy TOPSISapproachrdquo Journal of Modern Mathematics and Statistics vol 4no 2 pp 68ndash72 2010

[19] F Ye ldquoAn extended TOPSIS method with interval-valuedintuitionistic fuzzy numbers for virtual enterprise partnerselectionrdquo Expert Systems with Applications vol 37 no 10 pp7050ndash7055 2010

[20] J H Park I Y Park Y C Kwun and X Tan ldquoExtension of theTOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environmentrdquo Applied MathematicalModelling vol 35 no 5 pp 2544ndash2556 2011

[21] D-F Li ldquoTOPSIS-based nonlinear-programmingmethodologyfor multiattribute decision making with interval-valued intu-itionistic fuzzy setsrdquo IEEE Transactions on Fuzzy Systems vol18 no 2 pp 299ndash311 2010

[22] D-F Li ldquoLinear programming method for MADM withinterval-valued intuitionistic fuzzy setsrdquo Expert Systems withApplications vol 37 no 8 pp 5939ndash5945 2010

[23] J H Park H J Cho and Y C Kwun ldquoExtension of theVIKOR method for group decision making with interval-valued intuitionistic fuzzy informationrdquo Fuzzy Optimizationand Decision Making vol 10 no 3 pp 233ndash253 2011

[24] D-F Li ldquoCloseness coefficient based nonlinear programmingmethod for interval-valued intuitionistic fuzzy multiattributedecision making with incomplete preference informationrdquoApplied Soft Computing Journal vol 11 no 4 pp 3402ndash34182011

[25] S-M Chen L-W Lee H-C Liu and S-W Yang ldquoMultiat-tribute decision making based on interval-valued intuitionisticfuzzy valuesrdquo Expert Systems with Applications vol 39 no 12pp 10343ndash10351 2012

[26] D Yu Y Wu and T Lu ldquoInterval-valued intuitionistic fuzzyprioritized operators and their application in group decisionmakingrdquo Knowledge-Based Systems vol 30 pp 57ndash66 2012

[27] H Zhang and L Yu ldquoMADM method based on cross-entropyand extended TOPSIS with interval-valued intuitionistic fuzzysetsrdquo Knowledge-Based Systems vol 30 pp 115ndash120 2012

[28] J Ye ldquoInterval-valued intuitionistic fuzzy cosine similaritymeasures for multiple attribute decision-makingrdquo InternationalJournal of General Systems vol 42 no 8 pp 883ndash891 2013

[29] F Meng Q Zhang and H Cheng ldquoApproaches to multiple-criteria group decision making based on interval-valued intu-itionistic fuzzy Choquet integral with respect to the generalized120582-Shapley indexrdquo Knowledge-Based Systems vol 37 pp 237ndash249 2013

[30] F Meng C Tan and Q Zhang ldquoThe induced generalizedinterval-valued intuitionistic fuzzy hybrid Shapley averagingoperator and its application in decision makingrdquo Knowledge-Based Systems vol 42 pp 9ndash19 2013


[31] S H Razavi Hajiagha S S Hashemi and E K Zavadskas ldquoAcomplex proportional assessment method for group decisionmaking in an interval-valued intuitionistic fuzzy environmentrdquoTechnological and Economic Development of Economy vol 19no 1 pp 22ndash37 2013

[32] J Chai J N K Liu and Z Xu ldquoA rule-based group decisionmodel for warehouse evaluation under interval-valued Intu-itionistic fuzzy environmentsrdquo Expert Systems with Applica-tions vol 40 no 6 pp 1959ndash1970 2013

[33] S Wan and J Dong ldquoA possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision mak-ingrdquo Journal of Computer and System Sciences vol 80 no 1 pp237ndash256 2014

[34] T-Y Chen ldquoThe extended linear assignment method formultiple criteria decision analysis based on interval-valuedintuitionistic fuzzy setsrdquo Applied Mathematical Modelling vol38 no 7-8 pp 2101ndash2117 2014

[35] J Xu and F Shen ldquoA new outranking choice method for groupdecision making under Atanassovrsquos interval-valued intuitionis-tic fuzzy environmentrdquo Knowledge-Based Systems vol 70 pp177ndash188 2014

[36] E K Zavadskas J Antucheviciene S H R Hajiagha and SS Hashemi ldquoExtension of weighted aggregated sum productassessment with interval-valued intuitionistic fuzzy numbers(WASPAS-IVIF)rdquo Applied Soft Computing vol 24 pp 1013ndash1021 2014

[37] S Geetha V L G Nayagam and R Ponalagusamy ldquoA completeranking of incomplete interval informationrdquo Expert Systemswith Applications vol 41 no 4 pp 1947ndash1954 2014

[38] Y Zhang P Li Y Wang P Ma and X Su ldquoMultiattributedecision making based on entropy under interval-valued intu-itionistic fuzzy environmentrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 526871 8 pages 2013

[39] C Wei and Y Zhang ldquoEntropy measures for interval-valuedintuitionistic fuzzy sets and their application in group decision-makingrdquo Mathematical Problems in Engineering vol 2015Article ID 563745 13 pages 2015

[40] X Tong and L Yu ldquoA novel MADM approach based on fuzzycross entropy with interval-valued intuitionistic fuzzy setsrdquoMathematical Problems in Engineering vol 2015 Article ID965040 9 pages 2015

[41] J Wu and F Chiclana ldquoNon-dominance and attitudinalprioritisation methods for intuitionistic and interval-valuedintuitionistic fuzzy preference relationsrdquo Expert Systems withApplications vol 39 no 18 pp 13409ndash13416 2012

[42] J Wu and F Chiclana ldquoA risk attitudinal ranking method forinterval-valued intuitionistic fuzzy numbers based on novelattitudinal expected score and accuracy functionsrdquoApplied SoftComputing vol 22 pp 272ndash286 2014

[43] Z-S Xu ldquoMethods for aggregating interval-valued intuitionis-tic fuzzy information and their application to decisionmakingrdquoControl and Decision vol 22 no 2 pp 215ndash219 2007

[44] W K M Brauers and E K Zavadskas ldquoThe MOORA methodand its application to privatization in a transition economyrdquoControl and Cybernetics vol 35 no 2 pp 445ndash469 2006

[45] W K M Brauers and E K Zavadskas ldquoProject managementbyMULTIMOORA as an instrument for transition economiesrdquoTechnological and Economic Development of Economy vol 16no 1 pp 5ndash24 2010

[46] W K M Brauers and E K Zavadskas ldquoMultimoora opti-mization used to decide on a bank loan to buy propertyrdquo

Technological and Economic Development of Economy vol 17no 1 pp 174ndash188 2011

[47] W K Brauers and E K Zavadskas ldquoRobustness of MULTI-MOORA a method for multi-objective optimizationrdquo Infor-matica vol 23 no 1 pp 1ndash25 2012

[48] D Streimikiene T Balezentis I Krisciukaitien and A Balezen-tis ldquoPrioritizing sustainable electricity production technolo-gies MCDM approachrdquo Renewable and Sustainable EnergyReviews vol 16 no 5 pp 3302ndash3311 2012

[49] E K Zavadskas J Antucheviciene J Saparauskas and ZTurskis ldquoMCDM methods WASPAS and MULTIMOORAverification of robustness ofmethodswhen assessing alternativesolutionsrdquo Economic Computation and Economic CyberneticsStudies and Research vol 47 no 2 pp 5ndash20 2013

[50] E K Zavadskas J Antucheviciene J Saparauskas and ZTurskis ldquoMulti-criteria assessment of facades alternativespeculiarities of ranking methodologyrdquo Procedia Engineeringvol 57 pp 107ndash112 2013

[51] Z Wang K W Li and W Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[52] W K M Brauers A Balezentis and T Balezentis ldquoMultimoorafor the EU member States updated with fuzzy number theoryrdquoTechnological and Economic Development of Economy vol 17no 2 pp 259ndash290 2011

[53] F Herrera and L Martinez ldquoA 2-tuple fuzzy linguistic representmodel for computing with wordsrdquo IEEE Transactions on FuzzySystems vol 8 no 6 pp 746ndash752 2000

[54] A Balezentis and T Balezentis ldquoAn innovative multi-criteriasupplier selection based on two-tuple MULTIMOORA andhybrid datardquo Economic Computation and Economic CyberneticsStudies and Research vol 45 no 2 pp 37ndash56 2011

[55] A Balezentis and T Balezentis ldquoA novel method for groupmulti-attribute decision making with two-tuple linguistic com-puting supplier evaluation under uncertaintyrdquo Economic Com-putation and Economic Cybernetics Studies and Research vol 45no 4 pp 5ndash30 2011

[56] A Balezentis T Balezentis and W K M Brauers ldquoPersonnelselection based on computing with words and fuzzy MULTI-MOORArdquo Expert Systems with Applications vol 39 no 9 pp7961ndash7967 2012

[57] A Balezentis T Balezentis and W K M Brauers ldquoMULTI-MOORA-FG a multi-objective decision making method forlinguistic reasoning with an application to personnel selectionrdquoInformatica vol 23 no 2 pp 173ndash190 2012

[58] D Stanujkic N Magdalinovic S Stojanovic and R JovanovicldquoExtension of ratio system part of MOORAmethod for solvingdecision-making problems with interval datardquo Informatica vol23 no 1 pp 141ndash154 2012

[59] D Stanujkic N Magdalinovic R Jovanovic and S StojanovicldquoAn objective multi-criteria approach to optimization usingMOORAmethod and interval grey numbersrdquoTechnological andEconomic Development of Economy vol 18 no 2 pp 331ndash3632012

[60] S Karlin and W J Studden Tchebycheff Systems With Appli-cations in Analysis and Statistics Interscience Publishers NewYork NY USA 1966

[61] D-F Li ldquoExtension principles for interval-valued intuitionisticfuzzy sets and algebraic operationsrdquo Fuzzy Optimization andDecision Making vol 10 no 1 pp 45ndash58 2011


[62] H Zhao Z Xu M Ni and S Liu ldquoGeneralized aggregationoperators for intuitionistic fuzzy setsrdquo International Journal ofIntelligent Systems vol 25 no 1 pp 1ndash30 2010

[63] C L Dym Principles of Mathematical Modeling Elsevier Aca-demic Press Boston Mass USA 2nd edition 2001

[64] J C Pomerol and S Barbara-Romero Multicriterion Decisionin Management Principles and Practice Kluwer AcademicPublishers Boston Mass USA 2000

[65] D J Sheskin Handbook of Parametric and NonparametricStatistical Procedures Chapman and Hall Boca Raton FlaUSA 5th edition 2011

[66] P Vincke Multicriteria Decision-Aid John Wiley amp SonsChichester UK 1992

[67] V M Ozernoy ldquoChoosing the best multiple criteria decision-making methodrdquo INFOR vol 30 no 2 pp 159ndash171 1992

[68] M Aruldoss T M Lakshmi and V P Venkatesan ldquoA surveyonmulti criteria decisionmakingmethods and its applicationsrdquoAmerican Journal of Information Systems vol 1 no 1 pp 31ndash432013

[69] A Mardani A Jusoh and E K Zavadskas ldquoFuzzy mul-tiple criteria decision-making techniques and applicationsmdashtwo decades review from 1994 to 2014rdquo Expert Systems withApplications vol 42 no 8 pp 4126ndash4148 2015

[70] PWang Y Li Y-HWang andZ-Q Zhu ldquoAnewmethodbasedon TOPSIS and response surface method for MCDM problemswith interval numbersrdquoMathematical Problems in Engineeringvol 2015 Article ID 938535 11 pages 2015

[71] F Smarandache ldquoA geometric interpretation of the neutro-sophic setmdasha generalization of the intuitionistic fuzzy setrdquo inNeutrosophic Theory and Its Applications F Smarandache EdEuropa Nova Brussels Belgium 2014

[72] J J Peng J Q Wang H Y Zhang and X H Chen ldquoAn out-ranking approach for multi-criteria decision-making problemswith simplified neutrosophic setsrdquo Applied Soft Computing vol25 pp 336ndash346 2014

[73] Y Deng ldquoD numbers theory and applicationsrdquo Journal ofInformation and Computational Science vol 9 no 9 pp 2421ndash2428 2012

[74] XDeng X Lu F T Chan R Sadiq SMahadevan andYDengldquoD-CFPR D numbers extended consistent fuzzy preferencerelationsrdquo Knowledge-Based Systems vol 73 pp 61ndash68 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


exactness is an unrealistic assumption Different frameworkshave been proposed to handle uncertainty in practice Liuand Lin [5] classified these frameworks into probability andstatistics (1) and grey system theory and fuzzy set theory(2) Fuzzy set theory introduced by Zadeh [6] is widelyapplied to decision making problems In other words thefuzzy set theory is a generalization of ordinal sets wherea membership degree is assigned to each element of a setEach type of uncertainty has its own characteristics and isappropriate for special cases While probability is concernedwith the occurrence of well-defined events fuzzy sets dealwith gradual concepts and describe their boundaries [7] Infact the fuzzy set theory is appropriate for recognition-baseduncertainty that is common in decision making

Determining a single membership degree is a difficulttask for decision makers and therefore Grattan-Guinness[8] believes that presentation of a linguistic expression inthe form of a fuzzy set is not sufficient In 1986 Atanassov[9] introduced the notion of intuitionistic fuzzy sets (IFSs)as an extension of ordinal fuzzy sets In addition to amembership degree of each element IFS assigns a degree ofnonmembership to each element Later Atanassov and Gar-gov [10] extended the interval-valued intuitionistic fuzzy sets(IVIFSs) where membership and nonmembership degreesare stated as closed intervals

When it became obvious that the type 1 fuzzy sets werenot always sufficient for MADM under uncertain environ-ment type 2 fuzzy sets involving interval-valued as wellas intuitionistic fuzzy sets were proposed As a result theMULTIMOORA method was updated by using generalizedinterval-valued trapezoidal fuzzy numbers [11] or intuition-istic fuzzy numbers [12] The method has been successfullyapplied to making economic technological or managementdecisions The applications of various approaches in 2006ndash2013 including crisp and extended methods were summa-rized by T Balezentis and A Balezentis [13] The most recentapplications cover the extended versions of the method inparticular the fuzzy MULTIMOORA [14 15] or MULTI-MOORA based on the interval 2-tuple linguistic variables[16] Also Li [17] extended MULTIMOORA with hesitantfuzzy numbers where membership degree of elements isdefined over a set of different values

The current research aims to extend the MULTIMOORAmethod using the interval-valued intuitionistic fuzzy setsTaking into account the advantages of IVIFS the interval-valued intuitionistic fuzzy MULTIMOORA (IVIF-MULTI-MOORA) method for group decision making is presentedin the paper For this reason three parts of MULTIMOORAwere extended under IVIF information and the appropriatealgorithm was proposed

The paper is organized as follows In Section 2 theliterature review is presented and then the crisp version ofMULTIMOORA is briefly overviewed in Section 3 Section 4includes a brief description of the interval-valued intuitionis-tic fuzzy setsThe proposed IVIF-MULTIMOORA algorithmis presented in Section 5 The solution of two numericalexamples of civil engineering problems is given in Section 6while the results obtained are compared with the data yieldedby some other methods of IVIF decision making Finally the

discussion and the conclusions are presented in Sections 7and 8

2 Literature Review

Taking into account the flexibility and general characterof IVIFS researchers developed decision making methodsunder the IVIFS environment Some researchers developedthe well-known MADM methods in the IVIF form Someauthors [18ndash20] proposed different frameworks of TOPSISmethod under the IVIF environment Li [21 22] developedsomemathematical programming-basedmethod to solve theMADM problems with both ratings of the alternatives onattributes and weights of attributes expressed with the helpof IVIFS Park et al [23] presented an IVIF version of theVIKOR method Li [24] proposed a closeness coefficientbased on the nonlinear programming method for solvingIVIF MADM problems Chen et al [25] solved the MADMproblems based on the interval-valued intuitionistic fuzzyweighted average operator and newly defined fuzzy rank-ing method for intuitionistic fuzzy values Yu et al [26]introduced some IVIF aggregation operators and appliedthem to solving various decision making problems Zhangand Yu [27] presented an optimization model to determinethe attribute weights in an IVIF decision making problemand then used these weights in an extended TOPSIS torank the alternatives Ye [28] introduced a cosine similaritymeasure and a weighted cosine similarity measure toMADMproblemsMeng et al [29] also proposed twonew aggregationoperators that is the arithmetical interval-valued intuition-istic fuzzy generalized 120582-Shapley Choquet operator and thegeometric interval-valued intuitionistic fuzzy generalized 120582-Shapley Choquet operator and investigated their applicationto solving MADM problems Meng et al [30] used Shapleyfunction in extending a generalized IVIF hybrid Shapleyaveraging operator and proposed its application to solvingMADM problems Razavi Hajiagha et al [31] developedComplex Proportional Assessment (COPRAS) method withIVIF data Chai et al [32] proposed a rule-based decisionmodelwhendecision information in a groupdecisionmakingproblem is provided as IVIF values Wan and Dong [33]defined the possibility degree of comparison between twoIVIF numbers and introduced two ordered averaging opera-tors based on the Karnik-Mendel algorithm to solve MADMproblems with IVIF information Chen [34] developed amethod based on the traditional linear assignment methodfor solving decision making problems in the IVIF contextXu and Shen [35] presented the IVIF outranking choicemethod to solve MADM problems while Zavadskas et al[36] extended the IVIF weighted aggregated sum productassessment (WASPAS) method Geetha et al [37] proposeda new ranking method of IVIF numbers and extended itsapplication in decision making problems Zhang et al [38]proposed a new definition of IVIF entropy an entropy-based MADMmethod Later Wei and Zhang [39] proposedapplying an entropy measure for IFSs and IVIFSs to assessthe expertsrsquo weights for multicriteria fuzzy group decisionmaking Tong and Yu [40] introduced a novel approach for





3 MULTIMOORA



119909119894119895=

119909119894119895

radicsum119898

119894=1 1199092

119894119895

(1)








119910119894=

119895=119892

sum

119895=1119909119894119895119908119895minus

119895=119899

sum

119895=119892+1119909119894119895119908119895 (2)







119895 It







Min(119894)

max(119895)

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (3)



119880119894=119860119894

119861119894

(4)

where 119860119894and 119861



119894and 119861

119894are


119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(5)








119860(119909) V119860(119909)⟩ | 119909 isin 119883 where

120583119860 119883 rarr 119863[0 1] and V


0 ≺ sup119909120583119860(119909) + sup


119860(119909) and





119860119880(119909)


119860 = ⟨119909 [120583119860119871 (119909) 120583119860119880 (119909)] [V119860119871 (119909) V119860119880 (119909)]⟩ | 119909

isin119883

(6)

where 0 ≺ 120583119860119880(119909) + V

119860119880(119909) le 1 120583

119860119871(119909) V119860119871(119909) ge 0 For con-




1198601 +1198602

= ([1198861 + 1198862 minus 11988611198862 1198871 + 1198872 minus 11988711198872] [11988811198882 11988911198892]) (7)

1198601 sdot 1198602

= ([11988611198862 11988711198872] [1198881 + 1198882 minus 11988811198882 1198891 +1198892 minus11988911198892]) (8)

1205821198601


120582] [119888120582

1 119889120582

1])

120582 ge 0

(9)


1198601 divide1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (10)

1198601 minus1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (11)


119904 (119860) =12(119886 minus 119888 + 119887 minus 119889) (12)


ℎ (119860) =12(119886 + 119888 + 119887 + 119889) (13)




1198602

(2) if 119904(1198601) = 119904(1198602) then

(a) if ℎ(1198601) = ℎ(1198602) then 1198601 = 1198602



119895=



119908(1198601 1198602 119860119899) was defined as

GIIFWA119908(1198601 1198602 119860119899)

= (1199081119860120582

1 +1199082119860120582

2 + sdot sdot sdot +119908119899119860120582

119899)1120582

(14)


with119908119895ge 0 119895 = 1 2 119899 andsum119899

119895=1 119908119895 = 1 It can be shown



GIIFWA119908(1198601 1198602 119860119899)

= ([

[

(1minus119899

prod

119895=1(1 minus 119886120582

119895)119908119895)

1120582

(1minus119899

prod

119895=1(1 minus 119887120582

119895)119908119895)

1120582

]

]

[

[

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119888

119895)120582

)

119908119895

)

1120582

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119889

119895)120582

)

119908119895

)

1120582

]

]

)

(15)




119895ge 0 119895 =

1 2 119899 and sum119899



119883119896=

[[[[[[[

[

119909119896

11 119909119896


1119899

119909119896

21 119909119896


2119899

sdot sdot sdot

119909119896

1198981 119909119896

1198982 sdot sdot sdot 119909119896

119898119899

]]]]]]]

]

(16)








119909119894119895= ([120583

119897

119894119895 120583119906

119894119895] [V119897119894119895 V119906119894119895]) = IIFWA (1199091

119894119895 119909

2119894119895 119909

119870

119894119895) (17)



119894119895] was





119892+1 119862119892+2 119862119899 then

119910119894=

119892

sum

119895=1119908119895119909119894119895minus

119899

sum

119895=119892+1119908119895119909119894119895 (18)


119895=1 119908119895 and119882minus= sum119899

119895=119892+1 119908119895Then1199081015840

119895= 119908119895119882+ for each 119895 isin 1 2 119892 and 11990810158401015840

119895= 119908119895119882minus


0 119895 = 1 2 119892 and sum119892119895=1 1199081015840

119895= 1 Similarly 11990810158401015840

119895ge 0 119895 =

1 2 119892 and sum119892119895=1 11990810158401015840

119895= 1 Then

1199101015840

119894=

119910119894

119882+119882minus (19)


1199101015840

119894=

1119882minus

119892

sum

119895=11199081015840

119895119909119894119895minus

1

119882+

119899

sum

119895=119892+111990810158401015840

119895119909119894119895 (20)


1199101015840

119894=

1119882minus

IIFWA119882+ (1199091198941 1199091198942 119909119894119892)

minus1119882+

IIFWA119882minus (119909119894119892+1 119909119894119892+2 119909119894119899)

(21)


119894 119894 = 1 2 119898 is









119899119894119895= ([120578119897

119894119895 120578119906

119894119895] [120575119897

119894119895 120575119906

119894119895]) = 119908

119895119909119894119895 (22)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([max1le119894le119898

(120578119897

119894119895) max

1le119894le119898(120578119906

119894119895)]

[min1le119894le119898

(120575119897

119894119895) min

1le119894le119898(120575119906

119894119895)])

(23)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([min1le119894le119898

(120583119897

119894119895) min

1le119894le119898(120583119906

119894119895)]

[max1le119894le119898

(V119897119894119895) max

1le119894le119898(V119906119894119895)])

(24)


Min1le119894le119898

max1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (25)




119878 (119899119894119895) = 120578119897

119894119895minus 120575119897

119894119895+ 120578119906

119894119895minus 120575119906

119894119895 (26)


119878 (119903119895) = 120578lowast119897

119894119895minus 120575lowast119897

119894119895+ 120578lowast119906

119894119895minus 120575lowast119906

119894119895 (27)


119889119894= max

1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816= max

1le119895le119899

10038161003816100381610038161003816119878 (119903119895) minus 119878 (119899

119894119895)10038161003816100381610038161003816 (28)


1 2 119898


119894=119860119894

119861119894

(29)


119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(30)


119894119895 119895 = 1 2 119892 was


=



119860119894=

119892

prod

119895=1119899119894119895= ([120578

119897

119894119895 120578119897

119894119895] [120575119897

119894119895 120575119906

119894119895])

= ([

[

119892

prod

119895=1120578119897

119894119895

119892

prod

119895=1120578119906

119894119895]

]

[[[

[

119892

sum

119895=1120575119897

119894119895minus sum

1198951lt11989521198951 1198952isin12119892

120575119897

1198941198951sdot 120575119897

1198941198952



(120575119897

1198941198951sdot 120575119897

1198941198952sdot sdot sdot 120575119897

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751198971198941 sdot 120575119897

1198942 sdot sdot sdot 120575119897

119894119892)

119892

sum

119895=1120575119906

119894119895

minus sum

1198951lt11989521198951 1198952isin12119892

120575119906

1198941198951sdot 120575119906


+ (minus1)119896+1 sum


(120575119906

1198941198951sdot 120575119906

1198941198952sdot sdot sdot 120575119906

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751199061198941 sdot 120575119906

1198942 sdot sdot sdot 120575119906

119894119892)]]]

]

)

(31)




119894) 119894 = 1 2 119898 and




Full

mul

tiplic

ativ

e for

m

Ratio

syste

m

Refe

renc

e poi

nt ap

proa

ch

Initi

aliz

atio

n


Start






Eq (17)


Eq (22)


Eq (23) Eq (24)


reference point

Eq (28)



alternatives

Eq (21)


Eq (11)


Eq (29)


Eq (11)

Agg

rega

tion




Find y998400

i i = 1 2 m

S(y998400

i) i = 1 2 m

Find Ui i = 1 2 m

di i = 1 2 m

S(Ui) i = 1 2 m












1) as well as their





2








10()





landscape) 11990915




1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15




119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])





1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)








1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













3 MULTIMOORA



119909119894119895=

119909119894119895

radicsum119898

119894=1 1199092

119894119895

(1)








119910119894=

119895=119892

sum

119895=1119909119894119895119908119895minus

119895=119899

sum

119895=119892+1119909119894119895119908119895 (2)







119895 It







Min(119894)

max(119895)

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (3)



119880119894=119860119894

119861119894

(4)

where 119860119894and 119861



119894and 119861

119894are


119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(5)








119860(119909) V119860(119909)⟩ | 119909 isin 119883 where

120583119860 119883 rarr 119863[0 1] and V


0 ≺ sup119909120583119860(119909) + sup


119860(119909) and





119860119880(119909)


119860 = ⟨119909 [120583119860119871 (119909) 120583119860119880 (119909)] [V119860119871 (119909) V119860119880 (119909)]⟩ | 119909

isin119883

(6)

where 0 ≺ 120583119860119880(119909) + V

119860119880(119909) le 1 120583

119860119871(119909) V119860119871(119909) ge 0 For con-




1198601 +1198602

= ([1198861 + 1198862 minus 11988611198862 1198871 + 1198872 minus 11988711198872] [11988811198882 11988911198892]) (7)

1198601 sdot 1198602

= ([11988611198862 11988711198872] [1198881 + 1198882 minus 11988811198882 1198891 +1198892 minus11988911198892]) (8)

1205821198601


120582] [119888120582

1 119889120582

1])

120582 ge 0

(9)


1198601 divide1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (10)

1198601 minus1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (11)


119904 (119860) =12(119886 minus 119888 + 119887 minus 119889) (12)


ℎ (119860) =12(119886 + 119888 + 119887 + 119889) (13)




1198602

(2) if 119904(1198601) = 119904(1198602) then

(a) if ℎ(1198601) = ℎ(1198602) then 1198601 = 1198602



119895=



119908(1198601 1198602 119860119899) was defined as

GIIFWA119908(1198601 1198602 119860119899)

= (1199081119860120582

1 +1199082119860120582

2 + sdot sdot sdot +119908119899119860120582

119899)1120582

(14)


with119908119895ge 0 119895 = 1 2 119899 andsum119899

119895=1 119908119895 = 1 It can be shown



GIIFWA119908(1198601 1198602 119860119899)

= ([

[

(1minus119899

prod

119895=1(1 minus 119886120582

119895)119908119895)

1120582

(1minus119899

prod

119895=1(1 minus 119887120582

119895)119908119895)

1120582

]

]

[

[

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119888

119895)120582

)

119908119895

)

1120582

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119889

119895)120582

)

119908119895

)

1120582

]

]

)

(15)




119895ge 0 119895 =

1 2 119899 and sum119899



119883119896=

[[[[[[[

[

119909119896

11 119909119896


1119899

119909119896

21 119909119896


2119899

sdot sdot sdot

119909119896

1198981 119909119896

1198982 sdot sdot sdot 119909119896

119898119899

]]]]]]]

]

(16)








119909119894119895= ([120583

119897

119894119895 120583119906

119894119895] [V119897119894119895 V119906119894119895]) = IIFWA (1199091

119894119895 119909

2119894119895 119909

119870

119894119895) (17)



119894119895] was





119892+1 119862119892+2 119862119899 then

119910119894=

119892

sum

119895=1119908119895119909119894119895minus

119899

sum

119895=119892+1119908119895119909119894119895 (18)


119895=1 119908119895 and119882minus= sum119899

119895=119892+1 119908119895Then1199081015840

119895= 119908119895119882+ for each 119895 isin 1 2 119892 and 11990810158401015840

119895= 119908119895119882minus


0 119895 = 1 2 119892 and sum119892119895=1 1199081015840

119895= 1 Similarly 11990810158401015840

119895ge 0 119895 =

1 2 119892 and sum119892119895=1 11990810158401015840

119895= 1 Then

1199101015840

119894=

119910119894

119882+119882minus (19)


1199101015840

119894=

1119882minus

119892

sum

119895=11199081015840

119895119909119894119895minus

1

119882+

119899

sum

119895=119892+111990810158401015840

119895119909119894119895 (20)


1199101015840

119894=

1119882minus

IIFWA119882+ (1199091198941 1199091198942 119909119894119892)

minus1119882+

IIFWA119882minus (119909119894119892+1 119909119894119892+2 119909119894119899)

(21)


119894 119894 = 1 2 119898 is









119899119894119895= ([120578119897

119894119895 120578119906

119894119895] [120575119897

119894119895 120575119906

119894119895]) = 119908

119895119909119894119895 (22)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([max1le119894le119898

(120578119897

119894119895) max

1le119894le119898(120578119906

119894119895)]

[min1le119894le119898

(120575119897

119894119895) min

1le119894le119898(120575119906

119894119895)])

(23)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([min1le119894le119898

(120583119897

119894119895) min

1le119894le119898(120583119906

119894119895)]

[max1le119894le119898

(V119897119894119895) max

1le119894le119898(V119906119894119895)])

(24)


Min1le119894le119898

max1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (25)




119878 (119899119894119895) = 120578119897

119894119895minus 120575119897

119894119895+ 120578119906

119894119895minus 120575119906

119894119895 (26)


119878 (119903119895) = 120578lowast119897

119894119895minus 120575lowast119897

119894119895+ 120578lowast119906

119894119895minus 120575lowast119906

119894119895 (27)


119889119894= max

1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816= max

1le119895le119899

10038161003816100381610038161003816119878 (119903119895) minus 119878 (119899

119894119895)10038161003816100381610038161003816 (28)


1 2 119898


119894=119860119894

119861119894

(29)


119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(30)


119894119895 119895 = 1 2 119892 was


=



119860119894=

119892

prod

119895=1119899119894119895= ([120578

119897

119894119895 120578119897

119894119895] [120575119897

119894119895 120575119906

119894119895])

= ([

[

119892

prod

119895=1120578119897

119894119895

119892

prod

119895=1120578119906

119894119895]

]

[[[

[

119892

sum

119895=1120575119897

119894119895minus sum

1198951lt11989521198951 1198952isin12119892

120575119897

1198941198951sdot 120575119897

1198941198952



(120575119897

1198941198951sdot 120575119897

1198941198952sdot sdot sdot 120575119897

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751198971198941 sdot 120575119897

1198942 sdot sdot sdot 120575119897

119894119892)

119892

sum

119895=1120575119906

119894119895

minus sum

1198951lt11989521198951 1198952isin12119892

120575119906

1198941198951sdot 120575119906


+ (minus1)119896+1 sum


(120575119906

1198941198951sdot 120575119906

1198941198952sdot sdot sdot 120575119906

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751199061198941 sdot 120575119906

1198942 sdot sdot sdot 120575119906

119894119892)]]]

]

)

(31)




119894) 119894 = 1 2 119898 and




Full

mul

tiplic

ativ

e for

m

Ratio

syste

m

Refe

renc

e poi

nt ap

proa

ch

Initi

aliz

atio

n


Start






Eq (17)


Eq (22)


Eq (23) Eq (24)


reference point

Eq (28)



alternatives

Eq (21)


Eq (11)


Eq (29)


Eq (11)

Agg

rega

tion




Find y998400

i i = 1 2 m

S(y998400

i) i = 1 2 m

Find Ui i = 1 2 m

di i = 1 2 m

S(Ui) i = 1 2 m












1) as well as their





2








10()





landscape) 11990915




1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15




119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])





1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)








1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119860(119909) V119860(119909)⟩ | 119909 isin 119883 where

120583119860 119883 rarr 119863[0 1] and V


0 ≺ sup119909120583119860(119909) + sup


119860(119909) and





119860119880(119909)


119860 = ⟨119909 [120583119860119871 (119909) 120583119860119880 (119909)] [V119860119871 (119909) V119860119880 (119909)]⟩ | 119909

isin119883

(6)

where 0 ≺ 120583119860119880(119909) + V

119860119880(119909) le 1 120583

119860119871(119909) V119860119871(119909) ge 0 For con-




1198601 +1198602

= ([1198861 + 1198862 minus 11988611198862 1198871 + 1198872 minus 11988711198872] [11988811198882 11988911198892]) (7)

1198601 sdot 1198602

= ([11988611198862 11988711198872] [1198881 + 1198882 minus 11988811198882 1198891 +1198892 minus11988911198892]) (8)

1205821198601


120582] [119888120582

1 119889120582

1])

120582 ge 0

(9)


1198601 divide1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (10)

1198601 minus1198602 = [min (1198861 1198862) min (1198871 1198872)]

[max (1198881 1198882) max (1198891 1198892)] (11)


119904 (119860) =12(119886 minus 119888 + 119887 minus 119889) (12)


ℎ (119860) =12(119886 + 119888 + 119887 + 119889) (13)




1198602

(2) if 119904(1198601) = 119904(1198602) then

(a) if ℎ(1198601) = ℎ(1198602) then 1198601 = 1198602



119895=



119908(1198601 1198602 119860119899) was defined as

GIIFWA119908(1198601 1198602 119860119899)

= (1199081119860120582

1 +1199082119860120582

2 + sdot sdot sdot +119908119899119860120582

119899)1120582

(14)


with119908119895ge 0 119895 = 1 2 119899 andsum119899

119895=1 119908119895 = 1 It can be shown



GIIFWA119908(1198601 1198602 119860119899)

= ([

[

(1minus119899

prod

119895=1(1 minus 119886120582

119895)119908119895)

1120582

(1minus119899

prod

119895=1(1 minus 119887120582

119895)119908119895)

1120582

]

]

[

[

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119888

119895)120582

)

119908119895

)

1120582

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119889

119895)120582

)

119908119895

)

1120582

]

]

)

(15)




119895ge 0 119895 =

1 2 119899 and sum119899



119883119896=

[[[[[[[

[

119909119896

11 119909119896


1119899

119909119896

21 119909119896


2119899

sdot sdot sdot

119909119896

1198981 119909119896

1198982 sdot sdot sdot 119909119896

119898119899

]]]]]]]

]

(16)








119909119894119895= ([120583

119897

119894119895 120583119906

119894119895] [V119897119894119895 V119906119894119895]) = IIFWA (1199091

119894119895 119909

2119894119895 119909

119870

119894119895) (17)



119894119895] was





119892+1 119862119892+2 119862119899 then

119910119894=

119892

sum

119895=1119908119895119909119894119895minus

119899

sum

119895=119892+1119908119895119909119894119895 (18)


119895=1 119908119895 and119882minus= sum119899

119895=119892+1 119908119895Then1199081015840

119895= 119908119895119882+ for each 119895 isin 1 2 119892 and 11990810158401015840

119895= 119908119895119882minus


0 119895 = 1 2 119892 and sum119892119895=1 1199081015840

119895= 1 Similarly 11990810158401015840

119895ge 0 119895 =

1 2 119892 and sum119892119895=1 11990810158401015840

119895= 1 Then

1199101015840

119894=

119910119894

119882+119882minus (19)


1199101015840

119894=

1119882minus

119892

sum

119895=11199081015840

119895119909119894119895minus

1

119882+

119899

sum

119895=119892+111990810158401015840

119895119909119894119895 (20)


1199101015840

119894=

1119882minus

IIFWA119882+ (1199091198941 1199091198942 119909119894119892)

minus1119882+

IIFWA119882minus (119909119894119892+1 119909119894119892+2 119909119894119899)

(21)


119894 119894 = 1 2 119898 is









119899119894119895= ([120578119897

119894119895 120578119906

119894119895] [120575119897

119894119895 120575119906

119894119895]) = 119908

119895119909119894119895 (22)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([max1le119894le119898

(120578119897

119894119895) max

1le119894le119898(120578119906

119894119895)]

[min1le119894le119898

(120575119897

119894119895) min

1le119894le119898(120575119906

119894119895)])

(23)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([min1le119894le119898

(120583119897

119894119895) min

1le119894le119898(120583119906

119894119895)]

[max1le119894le119898

(V119897119894119895) max

1le119894le119898(V119906119894119895)])

(24)


Min1le119894le119898

max1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (25)




119878 (119899119894119895) = 120578119897

119894119895minus 120575119897

119894119895+ 120578119906

119894119895minus 120575119906

119894119895 (26)


119878 (119903119895) = 120578lowast119897

119894119895minus 120575lowast119897

119894119895+ 120578lowast119906

119894119895minus 120575lowast119906

119894119895 (27)


119889119894= max

1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816= max

1le119895le119899

10038161003816100381610038161003816119878 (119903119895) minus 119878 (119899

119894119895)10038161003816100381610038161003816 (28)


1 2 119898


119894=119860119894

119861119894

(29)


119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(30)


119894119895 119895 = 1 2 119892 was


=



119860119894=

119892

prod

119895=1119899119894119895= ([120578

119897

119894119895 120578119897

119894119895] [120575119897

119894119895 120575119906

119894119895])

= ([

[

119892

prod

119895=1120578119897

119894119895

119892

prod

119895=1120578119906

119894119895]

]

[[[

[

119892

sum

119895=1120575119897

119894119895minus sum

1198951lt11989521198951 1198952isin12119892

120575119897

1198941198951sdot 120575119897

1198941198952



(120575119897

1198941198951sdot 120575119897

1198941198952sdot sdot sdot 120575119897

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751198971198941 sdot 120575119897

1198942 sdot sdot sdot 120575119897

119894119892)

119892

sum

119895=1120575119906

119894119895

minus sum

1198951lt11989521198951 1198952isin12119892

120575119906

1198941198951sdot 120575119906


+ (minus1)119896+1 sum


(120575119906

1198941198951sdot 120575119906

1198941198952sdot sdot sdot 120575119906

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751199061198941 sdot 120575119906

1198942 sdot sdot sdot 120575119906

119894119892)]]]

]

)

(31)




119894) 119894 = 1 2 119898 and




Full

mul

tiplic

ativ

e for

m

Ratio

syste

m

Refe

renc

e poi

nt ap

proa

ch

Initi

aliz

atio

n


Start






Eq (17)


Eq (22)


Eq (23) Eq (24)


reference point

Eq (28)



alternatives

Eq (21)


Eq (11)


Eq (29)


Eq (11)

Agg

rega

tion




Find y998400

i i = 1 2 m

S(y998400

i) i = 1 2 m

Find Ui i = 1 2 m

di i = 1 2 m

S(Ui) i = 1 2 m












1) as well as their





2








10()





landscape) 11990915




1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15




119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])





1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)








1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











GIIFWA119908(1198601 1198602 119860119899)

= ([

[

(1minus119899

prod

119895=1(1 minus 119886120582

119895)119908119895)

1120582

(1minus119899

prod

119895=1(1 minus 119887120582

119895)119908119895)

1120582

]

]

[

[

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119888

119895)120582

)

119908119895

)

1120582

1

minus(1minus119899

prod

119895=1(1 minus (1 minus 119889

119895)120582

)

119908119895

)

1120582

]

]

)

(15)




119895ge 0 119895 =

1 2 119899 and sum119899



119883119896=

[[[[[[[

[

119909119896

11 119909119896


1119899

119909119896

21 119909119896


2119899

sdot sdot sdot

119909119896

1198981 119909119896

1198982 sdot sdot sdot 119909119896

119898119899

]]]]]]]

]

(16)








119909119894119895= ([120583

119897

119894119895 120583119906

119894119895] [V119897119894119895 V119906119894119895]) = IIFWA (1199091

119894119895 119909

2119894119895 119909

119870

119894119895) (17)



119894119895] was





119892+1 119862119892+2 119862119899 then

119910119894=

119892

sum

119895=1119908119895119909119894119895minus

119899

sum

119895=119892+1119908119895119909119894119895 (18)


119895=1 119908119895 and119882minus= sum119899

119895=119892+1 119908119895Then1199081015840

119895= 119908119895119882+ for each 119895 isin 1 2 119892 and 11990810158401015840

119895= 119908119895119882minus


0 119895 = 1 2 119892 and sum119892119895=1 1199081015840

119895= 1 Similarly 11990810158401015840

119895ge 0 119895 =

1 2 119892 and sum119892119895=1 11990810158401015840

119895= 1 Then

1199101015840

119894=

119910119894

119882+119882minus (19)


1199101015840

119894=

1119882minus

119892

sum

119895=11199081015840

119895119909119894119895minus

1

119882+

119899

sum

119895=119892+111990810158401015840

119895119909119894119895 (20)


1199101015840

119894=

1119882minus

IIFWA119882+ (1199091198941 1199091198942 119909119894119892)

minus1119882+

IIFWA119882minus (119909119894119892+1 119909119894119892+2 119909119894119899)

(21)


119894 119894 = 1 2 119898 is









119899119894119895= ([120578119897

119894119895 120578119906

119894119895] [120575119897

119894119895 120575119906

119894119895]) = 119908

119895119909119894119895 (22)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([max1le119894le119898

(120578119897

119894119895) max

1le119894le119898(120578119906

119894119895)]

[min1le119894le119898

(120575119897

119894119895) min

1le119894le119898(120575119906

119894119895)])

(23)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([min1le119894le119898

(120583119897

119894119895) min

1le119894le119898(120583119906

119894119895)]

[max1le119894le119898

(V119897119894119895) max

1le119894le119898(V119906119894119895)])

(24)


Min1le119894le119898

max1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (25)




119878 (119899119894119895) = 120578119897

119894119895minus 120575119897

119894119895+ 120578119906

119894119895minus 120575119906

119894119895 (26)


119878 (119903119895) = 120578lowast119897

119894119895minus 120575lowast119897

119894119895+ 120578lowast119906

119894119895minus 120575lowast119906

119894119895 (27)


119889119894= max

1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816= max

1le119895le119899

10038161003816100381610038161003816119878 (119903119895) minus 119878 (119899

119894119895)10038161003816100381610038161003816 (28)


1 2 119898


119894=119860119894

119861119894

(29)


119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(30)


119894119895 119895 = 1 2 119892 was


=



119860119894=

119892

prod

119895=1119899119894119895= ([120578

119897

119894119895 120578119897

119894119895] [120575119897

119894119895 120575119906

119894119895])

= ([

[

119892

prod

119895=1120578119897

119894119895

119892

prod

119895=1120578119906

119894119895]

]

[[[

[

119892

sum

119895=1120575119897

119894119895minus sum

1198951lt11989521198951 1198952isin12119892

120575119897

1198941198951sdot 120575119897

1198941198952



(120575119897

1198941198951sdot 120575119897

1198941198952sdot sdot sdot 120575119897

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751198971198941 sdot 120575119897

1198942 sdot sdot sdot 120575119897

119894119892)

119892

sum

119895=1120575119906

119894119895

minus sum

1198951lt11989521198951 1198952isin12119892

120575119906

1198941198951sdot 120575119906


+ (minus1)119896+1 sum


(120575119906

1198941198951sdot 120575119906

1198941198952sdot sdot sdot 120575119906

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751199061198941 sdot 120575119906

1198942 sdot sdot sdot 120575119906

119894119892)]]]

]

)

(31)




119894) 119894 = 1 2 119898 and




Full

mul

tiplic

ativ

e for

m

Ratio

syste

m

Refe

renc

e poi

nt ap

proa

ch

Initi

aliz

atio

n


Start






Eq (17)


Eq (22)


Eq (23) Eq (24)


reference point

Eq (28)



alternatives

Eq (21)


Eq (11)


Eq (29)


Eq (11)

Agg

rega

tion




Find y998400

i i = 1 2 m

S(y998400

i) i = 1 2 m

Find Ui i = 1 2 m

di i = 1 2 m

S(Ui) i = 1 2 m












1) as well as their





2








10()





landscape) 11990915




1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15




119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])





1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)








1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119899119894119895= ([120578119897

119894119895 120578119906

119894119895] [120575119897

119894119895 120575119906

119894119895]) = 119908

119895119909119894119895 (22)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([max1le119894le119898

(120578119897

119894119895) max

1le119894le119898(120578119906

119894119895)]

[min1le119894le119898

(120575119897

119894119895) min

1le119894le119898(120575119906

119894119895)])

(23)


119903119895= ([120578lowast119897

119894119895 120578lowast119906

119894119895] [120575lowast119897

119894119895 120575lowast119906

119894119895])

= ([min1le119894le119898

(120583119897

119894119895) min

1le119894le119898(120583119906

119894119895)]

[max1le119894le119898

(V119897119894119895) max

1le119894le119898(V119906119894119895)])

(24)


Min1le119894le119898

max1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816 (25)




119878 (119899119894119895) = 120578119897

119894119895minus 120575119897

119894119895+ 120578119906

119894119895minus 120575119906

119894119895 (26)


119878 (119903119895) = 120578lowast119897

119894119895minus 120575lowast119897

119894119895+ 120578lowast119906

119894119895minus 120575lowast119906

119894119895 (27)


119889119894= max

1le119895le119899

10038161003816100381610038161003816119903119895minus119909119894119895119908119895

10038161003816100381610038161003816= max

1le119895le119899

10038161003816100381610038161003816119878 (119903119895) minus 119878 (119899

119894119895)10038161003816100381610038161003816 (28)


1 2 119898


119894=119860119894

119861119894

(29)


119860119894=

119892

prod

119895=1119908119895119909119894119895

119861119894=

119899

prod

119895=119892+1119908119895119909119894119895

(30)


119894119895 119895 = 1 2 119892 was


=



119860119894=

119892

prod

119895=1119899119894119895= ([120578

119897

119894119895 120578119897

119894119895] [120575119897

119894119895 120575119906

119894119895])

= ([

[

119892

prod

119895=1120578119897

119894119895

119892

prod

119895=1120578119906

119894119895]

]

[[[

[

119892

sum

119895=1120575119897

119894119895minus sum

1198951lt11989521198951 1198952isin12119892

120575119897

1198941198951sdot 120575119897

1198941198952



(120575119897

1198941198951sdot 120575119897

1198941198952sdot sdot sdot 120575119897

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751198971198941 sdot 120575119897

1198942 sdot sdot sdot 120575119897

119894119892)

119892

sum

119895=1120575119906

119894119895

minus sum

1198951lt11989521198951 1198952isin12119892

120575119906

1198941198951sdot 120575119906


+ (minus1)119896+1 sum


(120575119906

1198941198951sdot 120575119906

1198941198952sdot sdot sdot 120575119906

119894119895119896) + sdot sdot sdot

+ (minus1)119892+1 (1205751199061198941 sdot 120575119906

1198942 sdot sdot sdot 120575119906

119894119892)]]]

]

)

(31)




119894) 119894 = 1 2 119898 and




Full

mul

tiplic

ativ

e for

m

Ratio

syste

m

Refe

renc

e poi

nt ap

proa

ch

Initi

aliz

atio

n


Start






Eq (17)


Eq (22)


Eq (23) Eq (24)


reference point

Eq (28)



alternatives

Eq (21)


Eq (11)


Eq (29)


Eq (11)

Agg

rega

tion




Find y998400

i i = 1 2 m

S(y998400

i) i = 1 2 m

Find Ui i = 1 2 m

di i = 1 2 m

S(Ui) i = 1 2 m












1) as well as their





2








10()





landscape) 11990915




1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15




119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])





1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)








1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Full

mul

tiplic

ativ

e for

m

Ratio

syste

m

Refe

renc

e poi

nt ap

proa

ch

Initi

aliz

atio

n


Start






Eq (17)


Eq (22)


Eq (23) Eq (24)


reference point

Eq (28)



alternatives

Eq (21)


Eq (11)


Eq (29)


Eq (11)

Agg

rega

tion




Find y998400

i i = 1 2 m

S(y998400

i) i = 1 2 m

Find Ui i = 1 2 m

di i = 1 2 m

S(Ui) i = 1 2 m












1) as well as their





2








10()





landscape) 11990915




1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15




119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])





1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)








1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1) as well as their





2








10()





landscape) 11990915




1 1199092 1199093 1199094 1199095 1199096 1199098 1199099 11990910 11990911 11990912 and 119909

15




119894 119894 = 1 2 3 were


1199101015840

1= ([0285 0363] [0531 0602])

1199101015840

2= ([0110 0136] [0787 0833])

1199101015840

3= ([0077 0110] [0824 0858])





1 = ([6010119864minus 14 2028119864minus 12] [9999999825119864

minus 1 99999999998119864minus 1])

2 = ([4127119864minus 20 2310119864minus 17]

[999999999999909119864minus 1 999999999999998119864

minus 1])

3 = ([0 0] [1 1])

(33)








1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1198621 1198622 1198623 1198624

1198601 ([042 048] [04 05]) ([06 07] [005 0025]) ([04 05] [02 05]) ([055 075] [015 025])1198602 ([04 05] [04 05]) ([05 08] [01 02]) ([03 06] [03 04]) ([06 07] [01 03])1198603 ([03 05] [04 05]) ([01 03] [02 04]) ([07 08] [01 02]) ([05 07] [01 02])1198604 ([02 04] [04 05]) ([06 07] [02 03]) ([05 06] [02 03]) ([07 08] [01 02])









1) growth (119862





3were


4were benefit










7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














7 Discussion

















References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














References





















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article The Interval-Valued Intuitionistic Fuzzy …downloads.hindawi.com/journals/mpe/2015/560690.pdf · 2019. 7. 31. · Multiple criteria decision making methods have