Fuzzy Entropy for Pythagorean Fuzzy Sets with Application ...downloads.hindawi.com/journals/complexity/2018/2832839.pdf · fuzzy sets. Recently, Pythagorean fuzzy LINMAP method based

Research ArticleFuzzy Entropy for Pythagorean Fuzzy Sets with Application toMulticriterion Decision Making

Miin-Shen Yang 1 and Zahid Hussain12

1Department of Applied Mathematics Chung Yuan Christian University Chung-Li 32023 Taiwan2Department of Mathematics Karakoram International University Gilgit-Baltistan Pakistan

Correspondence should be addressed to Miin-Shen Yang msyangmathcycuedutw

Received 17 June 2018 Revised 1 October 2018 Accepted 16 October 2018 Published 1 November 2018

Academic Editor Diego R Amancio

Copyright copy 2018 Miin-Shen Yang and Zahid Hussain 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

The concept of Pythagorean fuzzy sets (PFSs) was initially developed by Yager in 2013 which provides a novel way to modeluncertainty and vaguenesswith high precision and accuracy compared to intuitionistic fuzzy sets (IFSs)The concept was concretelydesigned to represent uncertainty and vagueness in mathematical way and to furnish a formalized tool for tackling imprecision toreal problems In the present paper we have used both probabilistic and nonprobabilistic types to calculate fuzzy entropy of PFSsFirstly a probabilistic-type entropy measure for PFSs is proposed and then axiomatic definitions and properties are establishedSecondly we utilize a nonprobabilistic-typewith distances to construct new entropymeasures for PFSsThen aminndashmax operationto calculate entropy measures for PFSs is suggested Some examples are also used to demonstrate suitability and reliability of theproposedmethods especially for choosing the best oneones in structured linguistic variables Furthermore a newmethod based onthe chosen entropies is presented for Pythagorean fuzzy multicriterion decision making to compute criteria weights with rankingof alternatives A comparison analysis with the most recent and relevant Pythagorean fuzzy entropy is conducted to reveal theadvantages of our developed methods Finally this method is applied for ranking China-Pakistan Economic Corridor (CPEC)projects These examples with applications demonstrate practical effectiveness of the proposed entropy measures

1 Introduction

The concept of fuzzy sets was first proposed by Zadeh [1] in1965With a widely spread use in various fields fuzzy sets notonly provide broad opportunity to measure uncertainties inmore powerful and logical way but also give us a meaningfulway to represent vague concepts in natural language It isknown that most systems based on lsquocrisp set theoryrsquo or lsquotwo-valued logicsrsquo are somehow difficult for handling impreciseand vague information In this sense fuzzy sets can be usedto provide better solutions for more real world problemsMoreover to treat more imprecise and vague informationin daily life various extensions of fuzzy sets are suggestedby researchers such as interval-valued fuzzy set [2] type-2fuzzy sets [3] fuzzy multiset [4] intuitionistic fuzzy sets [5]hesitant fuzzy sets [6 7] and Pythagorean fuzzy sets [8 9]

Since fuzzy sets were based on membership values ordegrees between 0 and 1 in real life setting it may not

be always true that nonmembership degree is equal to (1-membership) Therefore to get more purposeful reliabilityand applicability Atanassov [5] generalized the concept oflsquofuzzy set theoryrsquo and proposed Intuitionistic fuzzy sets (IFSs)which include bothmembership degree and nonmembershipdegree and degree of nondeterminacy or uncertainty wheredegree of uncertainty = (1- (degree of membership + non-membership degree)) In IFSs the pair ofmembership gradesis denoted by (120583 ]) satisfying the condition of 120583 + ] le 1Recently Yager and Abbasov [8] and Yager [9] extended thecondition 120583+ ] le 1 to 1205832 + ]2 le 1 and then introduced a classof Pythagorean fuzzy sets (PFSs) whose membership valuesare ordered pairs (120583 ]) that fulfills the required conditionof 1205832 + ]2 le 1 with different aggregation operations andapplications in multicriterion decision making Accordingto Yager and Abbasov [8] and Yager [9] the space of allintuitionistic membership values (IMVs) is also Pythagorean

HindawiComplexityVolume 2018 Article ID 2832839 14 pageshttpsdoiorg10115520182832839

2 Complexity

membership values (PMVs) but PMVs are not necessary tobe IMVs For instance for the situation when the numbers120583 = radic32 and ] = 12 we can use PFSs but IFSs cannot beused since 120583 + ] gt 1 but 1205832 + ]2 le 1 PFSs are wider thanIFSs so that they can tackle more daily life problems underimprecision and uncertainty cases

More researchers are actively engaged in the devel-opment of PFSs properties For example Yager [10] gavePythagorean membership grades in multicriterion decisionmaking Extensions of technique for order preference bysimilarity to an ideal solution (TOPSIS) to multiple criteriadecision making with Pythagorean and hesitant fuzzy setswere proposed by Zhang and Xu [11] Zhang [12] considered anovel approach based on similarity measure for Pythagoreanfuzzy multicriteria group decision making Pythagoreanfuzzy TODIM approach to multicriterion decision makingwas given by Ren et al [13] Pythagorean fuzzy Choquetintegral based MABAC method for multiple attribute groupdecisionmakingwas developed by Peng andYang [14] Zhang[15] gave a hierarchical QUALIFLEX approach Peng et al[16] investigated Pythagorean fuzzy information measuresZhang et al [17] proposed generalized Pythagorean fuzzyBonferroni mean aggregation operators Liang and Xu [18]extended TOPSIS to hesitant Pythagorean fuzzy sets Perez-Domınguez et al [19] gave MOORA under Pythagoreanfuzzy sets Recently Pythagorean fuzzy LINMAP methodbased on the entropy for railway project investment deci-sion making was proposed by Xue et al [20] Zhang andMeng [21] proposed an approach to interval-valued hesitantfuzzy multiattribute group decision making based on thegeneralized Shapley-Choquet integral Pythagorean fuzzy(R S) minusnorm information measure for multicriteria decisionmaking problem was presented by Guleria and Bajaj [22]Furthermore Yang and Hussain [23] proposed distance andsimilarity measures of hesitant fuzzy sets based on Hausdorffmetric with applications tomulticriteria decision making andclustering Hussain and Yang [24] gave entropy for hesitantfuzzy sets based on Hausdorff metric with construction ofhesitant fuzzy TOPSIS

The entropy of fuzzy sets is a measure of fuzzinessbetween fuzzy sets De Luca and Termini [25] first intro-duced the axiom construction for entropy of fuzzy setswith reference to Shannonrsquos probability entropy Yager [26]defined fuzziness measures of fuzzy sets in terms of a lackof distinction between the fuzzy set and its negation basedon Lp norm Kosko [27] provided a measure of fuzzinessbetween fuzzy sets using a ratio of distance between the fuzzyset and its nearest set to the distance between the fuzzy setand its farthest set Liu [28] gave some axiom definitionsof entropy and also defined a 120590-entropy Pal and Pal [29]proposed exponential entropies While Fan andMa [30] gavesome new fuzzy entropy formulas Some extended entropymeasures for IFS were proposed by Burillo and Bustince [31]Szmidt and Kacprzyk [32] Szmidt and Baldwin [33] andHung and Yang [34]

In this paper we propose new entropies of PFS based onprobability-type distance Pythagorean index and minndashmaxoperation We also extend the concept to 120590-entropy and

then apply it to multicriteria decision making This paper isorganized as follows In Section 2 we review some definitionsof IFSs and PFSs In Section 3 we propose several newentropies of PFSs and then construct an axiomatic definitionof entropy for PFSs Based on the definition of entropy forPFSs we find that the proposed nonprobabilistic entropies ofPFSs are 120590-entropy In Section 4 we exhibit some examplesfor comparisons and also use structured linguistic variablesto validate our proposed methods In Section 5 we constructa new Pythagorean fuzzy TOPSIS based on the proposedentropy measures A comparison analysis of the proposedPythagorean fuzzy TOPSIS with the recently developedentropy of PFS [20] is shown We then apply the proposedmethod tomulticriterion decisionmaking for rankingChina-Pakistan Economic Corridor projects Finally we state ourconclusion in Section 6

2 Intuitionistic and Pythagorean Fuzzy Sets

In this section we give a brief review for intuitionistic fuzzysets (IFSs) and Pythagorean fuzzy sets (PFSs)

Definition 1 An intuitionistic fuzzy set (IFS) in 119883 isdefined by Atanassov [5] with the following form

= (119909 120583 (119909) ] (119909)) 119909 isin 119883 (1)

where 0 le 120583(119909) + ](119909) le 1 forall119909 isin 119883 and the functions120583(119909) 119883 997888rarr [0 1] denotes the degree of membershipof 119909 in and ](119909) 119883 997888rarr [0 1] denotes the degreeof nonmembership of 119909 in The degree of uncertainty(or intuitionistic index or indeterminacy) of 119909 to isrepresented by 120587(119909) = 1 minus (120583(119909) + ](119909))

For modeling daily life problems carrying imprecisionuncertainty and vagueness more precisely and with highaccuracy than IFSs Yager [9 10] presented Pythagoreanfuzzy sets (PFSs) where PFSs are the generalizations ofIFSs Yager [9 10] also validated that IFSs are contained inPFSs The concept of Pythagorean fuzzy set was originallydeveloped by Yager [8 9] but the general mathematical formof Pythagorean fuzzy set was developed by Zhang and Xu[11]

Definition 2 (Zhang and Xu [11]) A Pythagorean fuzzy set(PFS) in 119883 proposed by Yager [8 9] is mathematicallyformed as

= ⟨119909 120583 (119909) ] (119909)⟩ 119909 isin 119883 (2)

where the functions 120583(119909) 119883 997888rarr [0 1] represent the degreeof membership of 119909 in and ](119909) 119883 997888rarr [0 1] representthe degree of nonmembership of 119909 in For every 119909 isin 119883 thefollowing condition should be satisfied

0 le 1205832(119909) + ]2

(119909) le 1 (3)

Complexity 3

Definition 3 (Zhang and Xu [11]) For any PFS in 119883 thevalue 120587(119909) is called Pythagorean index of the element 119909 in with

120587 (119909) = radic1 minus 1205832(119909) + ]2

(119909)

or 1205872(119909) = 1 minus 1205832

(119909) minus ]2

(119909) (4)

In general 120587(119909) is also called hesitancy (or indetermi-nacy) degree of the element 119909 in It is obvious that 0 le1205872(119909) le 1 forall119909 isin 119883 It is worthy to note for a PFS if1205832(119909) = 0 then ]2

(119909) + 1205872

(119909) = 1 and if 1205832

(119909) = 1 then

]2(119909) = 0 and 1205872

(119909) = 0 Similarly if ]2

(119909) = 0 then1205832

(119909)+1205872

(119909) = 1 If ]2

(119909) = 1 then 1205832

(119909) = 0 and 1205872

(119909) = 0

If 1205872(119909) = 0 then 1205832

(119909) + ]2

(119909) = 1 If 1205872

(119909) = 1 then1205832

(119909) = ]2

(119909) = 0 For convenience Zhang and Xu [11]

denoted the pair (120583(119909) ](119909)) as Pythagorean fuzzy number(PFN) which is represented by 119901 = (120583119901 ]119901)

Since PFSs are a generalized form of IFSs we give thefollowing definition for PFSs

Definition 4 Let be a PFS in 119883 is called a completelyPythagorean if 1205832

(119909) = ]2

(119909) = 0 forall119909 isin 119883

Peng et al [16] suggested various mathematical opera-tions for PFSs as follows

Definition 5 (Peng et al [16]) If and 119876 are two PFSs in 119883then

(i) le 119876 if and only if forall119909 isin 119883 1205832(119909) le 1205832

(119909) and

]2(119909) ge ]2

(119909)

(ii) = 119876 if and only if forall119909 isin 119883 1205832(119909) = 1205832

(119909) and

]2(119909) = ]2

(119909)

(iii) cup 119876 = ⟨119909max(1205832(119909) 1205832(119909))min(]2

(119909) ]2(119909))⟩ 119909 isin 119883

(iv) cap 119876 = ⟨119909min(1205832(119909) 1205832(119909))max(]2

(119909) ]2(119909))⟩ 119909 isin 119883

(v) 119888 = ⟨119909 ]2(119909) 1205832(119909)⟩ 119909 isin 119883

We next define more operations of PFS in 119883 especiallyabout hedges of ldquoveryrdquo ldquohighlyrdquo ldquomore or lessrdquo ldquoconcentra-tionrdquo ldquodilationrdquo and other terms that are needed to representlinguistic variables We first define the n power (or exponent)of PFS as follows

Definition 6 Let = ⟨119909 120583(119909) ](119909)⟩ 119909 isin 119883 be a PFS in119883 For any positive real number n the n power (or exponent)of the PFS denoted by 119899 is defined as

119899 = ⟨119909 (120583 (119909))119899 radic1 minus (1 minus ]2(119909))119899⟩ 119909 isin 119883 (5)

It can be easily verified that for any positive real number n0 le [120583(119909)]119899 + [radic1 minus (1 minus ]2(119909))119899] le 1 forall119909 isin 119883

By using Definition 6 the concentration and dilation of aPFS can be defined as follows

Definition 7 The concentration 119862119874119873() of a PFS in 119883 isdefined as

119862119874119873() = ⟨119909 120583119862119874119873() (119909) ]119862119874119873() (119909)⟩ 119909 isin 119883 (6)

where 120583119862119874119873()(119909) = [120583(119909)]2 and ]119862119874119873()(119909) =radic1 minus [1 minus ]2(119909)]2

Definition 8 The dilation 119863119868119871() of a PFS in 119883 is definedas

119863119868119871 () = ⟨119909 120583119863119868119871() (119909) ]119863119868119871() (119909)⟩ 119909 isin 119883 (7)


In next section we construct new entropy measures forPFSs based on probability-type entropy induced by distancePythagorean index and max-min operation We also give anaxiomatic definition of entropy for PFSs

3 New Fuzzy Entropies forPythagorean Fuzzy Sets

We first provide a definition of entropy for PFSs De Lucaand Termini [25] gave the axiomatic definition of entropymeasure of fuzzy sets Later on Szmidt and Kacprzyk [32]extended it to entropy of IFS Since PFSs developed by Yager[8 9] are generalized forms of IFSs we use similar notionsas IFSs to give a definition of entropy for PFSs Assume that119875119865119878(119883) represents the set of all PFSs in X

Definition 9 A real function 119864 119875119865119878(119883) 997888rarr [0 1] is calledan entropy on 119875119865119878(119883) if E satisfies the following axioms

(A0) (119873119900119899119899119890119892119886119905119894V119894119905119910) 0 le 119864() le 1(A1) (119872119894119899119894119898119886119897119894119905119910) 119864() = 0 119894119891119891 119894119904 119886 119888119903119894119904119901 119904119890119905(A2) (119872119886119909119894119898119886119897119894119905119910) 119864() = 1 119894119891119891 120583(119909) = ](119909) forall119909 isin 119883(A3) (119877119890119904119900119897119906119905119894119900119899) 119864() le 119864(119876) 119894119891 is crisper than 119876

ie forall119909 isin 119883

120583(119909) le 120583(119909) and ](119909) ge ](119909) for 120583(119909) le](119909) 119900119903120583(119909) ge 120583(119909) and ](119909) le ](119909) 119891119900119903 120583(119909) ge](119909)

(A4) (119878119910119898119898119890119905119903119894119888) 119864() = 119864(119888) where 119888 is the comple-ment of

For probabilistic-type entropy we need to omit the axiom(A0)On the other hand becausewe take the three-parameter1205832 ]2 and 1205872

as a probability mass function 119901 = 1205832

]2 1205872

the probabilistic-type entropy 119864() should attain a unique

4 Complexity

maximum at 120583(119909) = ](119909) = 120587(119909) = 1radic3 forall119909 isin119883 Therefore for probabilistic-type entropy we replace theaxiom (1198602) with (11986021015840) and the axiom (1198603) with (11986031015840) asfollows

(A21015840) (119872119886119909119894119898119886119897119894119905119910) 119864() attains a unique maximum at120583(119909) = ](119909) = 120587(119909) = 1radic3 forall119909 isin 119883(A31015840) (119877119890119904119900119897119906119905119894119900119899) 119864() le 119864(119876) if is crisper than 119876 ieforall119909 isin 119883 120583(119909) le 120583(119909) and ](119909) le ](119909)

for max(120583(119909) ](119909)) le 1radic3 and 120583(119909) ge 120583(119909)](119909) ge ](119909) for min(120583(119909) ](119909)) ge 1radic3

In addition to the five axioms (A0)sim(A4) in Definition 9if we add the following axiom (A5) E is called 120590- entropy(A5) (119881119886119897119906119886119905119894119900119899) 119864() + 119864(119876) = 119864( cup 119876) + 119864( cap 119876)119894119891 forall119909 isin 119883

120583(119909) le 120583(119909) and ](119909) ge ](119909) for 120583(119909) le](119909) or120583(119909) ge 120583(119909) and ](119909) le ](119909) for 120583(119909) ge](119909)

We present a property for the axiom (11986031015840) when iscrisper than 119876Property 10 If is crisper than 119876 in the axiom (11986031015840) thenwe have the following inequality

(120583 (119909119894) minus 1radic3)2 + (] (119909119894) minus 1radic3)

2

+ (120587 (119909119894) minus 1radic3)2

ge (120583 (119909119894) minus 1radic3)2 + (] (119909119896) minus 1radic3)

2

+ (] (119909119896) minus 1radic3)2 forall119909119894

(8)

Proof If is crisper than 119876 then forall119909119894 120583(119909) le 120583(119909) and](119909) le ](119909) for max(120583(119909) ](119909)) le 1radic3 Thereforewe have that 120583(119909) minus 1radic3 le 120583(119909) minus 1radic3 le 0 and](119909) minus 1radic3 le ](119909) minus 1radic3 le 0 and so 1 minus 1205832

(119909) minus ]2

(119909) ge1 minus 1205832

(119909) minus ]2

(119909) ge 13 ie 1205872

(119909) ge 1205872

(119909) ge 13

and 120587(119909) minus 1radic3 ge 120587(119909) minus 1radic3 ge 0 Thus we have(120583(119909) minus 1radic3)2 ge (120583(119909) minus 1radic3)2 (](119909) minus 1radic3)2 ge(](119909) minus 1radic3)2 and (120587(119909) minus 1radic3)2 ge (120587(119909) minus 1radic3)2This induces the inequality Similarly the part of 120583(119909) ge120583(119909) and ](119909) ge ](119909) for min(120583(119909) ](119909)) ge 1radic3 alsoinduces the inequality Hence we prove the property

Since PFSs are generalized form of IFSs the distancesbetween PFSs need to be computed by considering allthe three components 1205832

(119909) ]2(119909) and 1205872

(119909) in PFSs

The well-known distance between PFSs is Euclidean dis-tance Therefore the inequality in Property 10 indicates thatthe Euclidean distance between (120583(119909) ](119909) 120587(119909)) and(1radic3 1radic3 1radic3) is larger than the Euclidean distancebetween (120583(119909) ](119909) 120587(119909)) and (1radic3 1radic3 1radic3) Thismanifests that (120583(119909) ](119909) 120587(119909)) is located more nearbyto (1radic3 1radic3 1radic3) than that of (120583(119909) ](119909) 120587(119909))From a geometrical perspective the axiom (11986031015840) is rea-sonable and logical because the closer PFS to the uniquepoint (1radic3 1radic3 1radic3)withmaximumentropy reflects thegreater entropy of that PFS

We next construct entropies for PFSs based on a probabil-ity-type To formulate the probability-type of entropy forPFSs we use the idea of entropy119867120574(119901)of Havrda and Chara-vat [35] to a probability mass function 119901 = 1199011 119901119896 with

119867120574 (119901) =

1120574 minus 1 (1 minus 119896sum119894=1

119901120574119894 ) 120574 = 1 (120574 gt 0)minus 119896sum119894=1

119901119894 log119901119894 120574 = 1 (9)

Let 119883 = 1199091 1199092 119909119899 be a finite universe of discoursesThus for a PFS in119883 we propose the following probability-type entropy for the PFS with

119890120574119867119862 () =

1119899119899sum119894=1

1120574 minus 1 [1 minus ((1205832(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] 120574 = 1 (120574 gt 0)

1119899119899sum119894=1

minus (1205832(119909119894) log 1205832 (119909119894) + ]2

(119909119894) log ]2 (119909119894) + 1205872

(119909119894) log1205872 (119909119894)) 120574 = 1 (10)

Apparently one may ask a question ldquoAre these pro-posed entropy measures for PFSs are suitable and accept-ablerdquo To answer this question we present the followingtheorem

Theorem 11 Let 119883 = 1199091 1199092 119909119899 be a finite universeof discourses The proposed probabilistic-type entropy 119890120574119867119862 fora PFS satisfies the axioms (1198601) (11986021015840) (11986031015840) and (1198604) inDefinition 9

To prove the axioms (11986021015840) and (11986031015840) for Theorem 11 weneed Lemma 12

Lemma 12 Let 120595120574(119909) 0 lt 119909 lt 1 be defined as120595120574 (119909) =

1120574 minus 1 (119909 minus 119909120574) 120574 = 1 (120574 gt 0)minus119909 log 119909 120574 = 1 (11)

Then 120595120574(119909) is a strictly concave function of 119909

Complexity 5

Proof By twice differentiating 120595120574(119909) we get 12059510158401015840120574 (119909) =minus120574119909120574minus2 lt 0 for 120574 = 1(120574 gt 0)Then 120595120574(119909) is a strictly concavefunction of 119909 Similarly we can show that 120595120574=1(119909) = 119909 log 119909is also a strictly concave function of 119909Proof of Theorem 11 It is easy to check that 119890120574119867119862 satisfiesthe axioms (1198601) and (1198604) We need only to prove that 119890120574119867119862satisfies the axioms (11986021015840) and (11986031015840) To prove 119890120574119867119862 for the case120574 = 1(120574 gt 0) that satisfies the axiom (11986021015840) we use Lagrangemultipliers for 119890120574119867119862 with 119865(1205832

]2 1205872 120582) = (1119899)sum119899119894=1(1(120574 minus1))[1 minus ((1205832

(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] +sum119899119894=1 120582119894(1205832(119909119894) +

]2(119909119894)+1205872

(119909119894)minus1) By taking the derivative of 119865(1205832

]2 1205872 120582)

with respect to 1205832(119909119894) ]2(119909119894) 1205872(119909119894) and 120582119894 we obtain

1205971198651205971205832(119909119894) = minus120574(120574 minus 1) (1205832

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597]2(119909119894) = minus120574(120574 minus 1) (]2

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

1205971198651205971205872(119909119894) = minus120574(120574 minus 1) (1205872

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597120582119894 = 1205832(119909119894) + ]2

(119909119894) + 1205872

(119909119894) minus 1 119904119890119905997904997904997904997904 0

(12)

From above PDEs we get (1205832(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and then1205832

(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (]2

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and

]2(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (1205872

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and1205872

(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) 1205832

(119909119894) + ]2

(119909119894) + 1205872

(119909119894) = 1 and

then (120582119894(120574 minus 1)120574)1(120574minus1) = 13 Thus we have 120582119894 = (120574(120574 minus1))(13)120574minus1 We obtain 1205832(119909119894) = ((120574(120574 minus 1))(13)120574minus1(120574 minus1)120574)1(120574minus1) and then 120583(119909119894) = 1radic3 We also get ](119909119894) =1radic3 and 120587(119909119894) = 1radic3 That is 120583(119909119894) = ](119909119894) = 120587(119909119894) =1radic3 forall119894 Similarly we can show that the equation of 119890120574119867119862 for

the case 120574 = 1 also obtains 120583(119909119894) = ](119909119894) = 120587(119909119894) =1radic3 forall119894 By Lemma 12 we learn that the function 120595120574(119909)is a strictly concave function of 119909 We know that 119890120574119867119862() =(1119899)sum119899119894=1(120595120574(1205832(119909119894))+120595120574(]2((119909119894)))+120595120574(1205872(119909119894))) and so 119890120574119867119862is also a strictly concave function Therefore it is provedthat 119890120574119867119862 attains a unique maximum at 120583(119909119894) = ](119909119894) =120587(119909119894) = 1radic3 forall119894 We next prove that the probabilistic-type entropy 119890120574119867119862 satisfies the axiom (11986031015840) If is crisperthan119876 we notice that is far away from (1radic3 1radic3 1radic3)compared to 119876 according to Property 10 However 119890120574119867119862 is astrictly concave function and 119890120574119867119862 attains a unique maximumat 120583(119909119894) = ](119909119894) = 120587(119909119894) = 1radic3 forall119894 From here weobtain 119890120574119867119862() le 119890120574119867119862(119876) if is crisper than119876Thus we provethe axiom (11986031015840)

Concept to determine uncertainty from a fuzzy set andits complement was first given by Yager [26] In this sectionwe first use the similar idea to measure uncertainty of PFSsin terms of the amount of distinction between a PFS and its complement 119888 However various distance measures

are made to express numerically the difference between twoobjects with high accuracy Therefore the distance betweentwo fuzzy sets plays a vital role in theoretical and practicalissues Let 119883 = 1199091 1199092 119909119899 be a finite universe of dis-courses we first define a Pythagorean normalized Euclidean(PNE) distance between two Pythagorean fuzzy sets 119876 isin119875119865119878119904(119883) as120577119864 ( 119876) = [ 12119899

119899sum119894=1

((1205832(119909119894) minus 1205832

(119909119894))2

+ (]2(119909119894) minus ]2

(119909119896))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(13)

We next propose fuzzy entropy induced by the PNE distance120577119864 between the PFS and its complement 119888 Let =⟨119909119894 120583(119909119894) ](119909119894)⟩ 119909119894 isin 119883 be any Pythagorean fuzzyset on the universe of discourse 119883 = 1199091 1199092 119909119899 withits complement 119888 = ⟨119909 ](119909119894) 120583(119909119894)⟩ 119909119894 isin 119883 ThePNE distance between PFSs and 119888 will be 120577119864( 119888) =[(1119899)sum119899119894=1(1205832(119909119894) minus ]2

(119909119894))2]12 Thus we define a new

entropy 119890119864 for the PFS as

119890119864 () = 1 minus 120577119864 ( 119888)= 1 minus radic 1119899

119899sum119894=1

(1205832(119909119894) minus ]2

(119909119894))2 (14)

Theorem 13 Let 119883 = 1199091 1199092 119909119899 be a finite universe ofdiscourse The proposed entropy 119890119864 for a PFS satisfies theaxioms (A0)sim(A5) in Definition 9 and so 119890119864 is a 120590-entropyProof We first prove the axiom (A0) Since the distance120577119864( 119888) is between 0 and 1 0 le 1 minus 120577119864( 119888) le 1 andthen 0 le 119890119864() le 1 The axiom (A0) is satisfied For theaxiom (A1) if is crisp ie 1205832

(119909119894) = 0 ]2

(119909119894) = 1

or 1205832(119909119894) = 1 ]2

(119909119894) = 0 forall119909119894 isin 119883 then 120577119864(119888) = radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2

(119909119894))2 = 1 Thus we obtain119890119864() = 1 minus 1 = 0 Conversely if 119890119864() = 0 then 120577119864( 119888) =1 minus 119890119864() = 1 ie radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2

(119909119894))2 = 1

Thus 1205832(119909119894) = 1 ]2

(119909119894) = 0 or 1205832

(119909119894) = 0 ]2

(119909119894) = 1

and so is crisp Thus the axiom (A1) is satisfied Nowwe prove the axiom (A2) 1205832

(119909119894) = ]2

(119909119894) forall119909119894 isin 119883

implies that 120577119864( 119888) = 0 119890119864() = 1 Conversely119890119864() = 1 implies 120577119864( 119888) = 0 Thus we have that1205832(119909119894) = ]2

(119909119894) forall119909119894 isin 119883 For the axiom (A3) since1205832

(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le ]2

(119909119894)

imply that 1205832(119909119894) le 1205832

(119909119894) le ]2

(119909119894) le ]2

(119909119894) then we

have the distance forall119909119894 isin 119883radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2(119909119894))2 ge

radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2(119909119894))2 Again from the axiom

(A3) of Definition 9 we have 1205832(119909119894) ge 1205832

(119909119894) and

6 Complexity

]2(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge ]2

(119909119894) implies that

]2(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) and then

the distance forall119909119894 isin 119883radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2(119909119894))2

ge radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2(119909119894))2 From inequalities

radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2(119909119894))2 ge radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2

(119909119894))2

and radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2(119909119894))2 ge

radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2(119909119894))2 we have 120577119864( 119888) ge 120577119867(119876

119876119888) and then 1 minus 120577119864( 119888) le 1 minus 120577119864(119876 119876119888) This concludesthat 119890119864() le 119890119864(119876) In this way the axiom (A3) is provedNext we prove the axiom (A4) Since forall119909119894 isin 119883 119890119864() =1 minus 120577119864( 119888) = 1 minus radic(1119899)sum119899119894=1(1205832(119909119894) minus ]2

(119909119894))2 =

1 minus radic(1119899)sum119899119894=1(]2(119909119894) minus 1205832(119909119894))2 1 minus 120577119864(119888 ) = 119890119864(119888)

Hence the axiom (A4) is satisfied Finally for provingthe axiom (A5) let and 119876 be two PFSs Then wehave

(i) 1205832(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le

]2(119909119894) forall119909119894 isin 119883 or

(ii) 1205832(119909119894) ge 1205832

(119909119894) and ]2

(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge

]2(119909119894) forall119909119894 isin 119883

From (i) we have forall119909119894 isin 119883 1205832(119909119894) le 1205832

(119909119894) le ]2

(119909119894) le

]2(119909119894) then max(1205832

(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) That is ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = 119876

which implies that 119890119864( cup 119876) = 119890119864(1205832(119909119894) ]2(119909119894)) = 119890119864(119876)Also forall119909119894 isin 119883 min(1205832

(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and max(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) then ( cap 119876) = (1205832

(119909119894) ]2(119909119894)) =

implies 119890119864( cap 119876) = 119890119864(1205832(119909119894) ]2(119909119894)) = 119890119864() Hence119890119864() + 119890119864(119876) = 119890119864( cap 119876) + 119890119864( cup 119876) Again from (ii)we have forall119909119894 isin 119883 ]2

(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) Then

max(1205832(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) and so ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = which implies

that 119890119864( cup 119876) = 119890119864(1205832(119909119894) ]2(119909119894)) = 119890119864() Also forall119909119894 isin119883min(1205832(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and max(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) then ( cap 119876) = (1205832

(119909119894) ]2(119909119894)) = 119876 implies that119890119864( cap 119876) = 119890119864(1205832(119909119894) ]2(119909119894)) = 119890119864(119876) Hence 119890119864() +119890119864(119876) = 119890119864( cap 119876) + 119890119864( cup 119876) Thus we complete the proof

of Theorem 13

Burillo and Bustince [31] gave fuzzy entropy of intu-itionistic fuzzy sets by using intuitionistic index Now wemodify and extend the similar concept to construct the newentropy measure of PFSs by using Pythagorean index asfollows

Let be a PFS on 119883 = 1199091 1199092 119909119899 We define anentropy 119890119875119868 of using Pythagorean index as

119890119875119868 () = 1119899119899sum119896=1

(1 minus 1205832(119909119896) minus ]2

(119909119896))

= 1119899119899sum119896=1

1205872(119909119896)

(15)

Theorem 14 Let 119883 = 1199091 1199092 119909119899 be a finite universeof discourse The proposed entropy 119890119875119868 for the PFS usingPythagorean index satisfies the axioms (A0)sim(A4) and (A5) inDefinition 9 and so it is a 120590- entropyProof Similar to Theorem 13

We next propose a new and simple method to calculatefuzzy entropy of PFSs by using the ratio of min and maxoperations All three components (1205832

]2 1205872) of a PFS are

given equal importance to make the results more authenticand reliable The new entropy is easy to be computed Let be a PFS on119883 = 1199091 1199092 119909119899 and we define a new entropyfor the PFS as

119890minmax () = 1119899119899sum119894=1

min (1205832(119909119894) ]2 (119909119894) 1205872 (119909))

max (1205832(119909119894) ]2 (119909119894) 1205872 (119909)) (16)

Theorem 15 Let 119883 = 1199091 1199092 119909119899 be a finite universe ofdiscourse The proposed entropy 119890minmax for a PFS satisfiesthe axioms (A0)sim(A4) and (A5) in Definition 9 and so it is a120590-entropyProof Similar to Theorem 13

Recently Xue at el [20] developed the entropy ofPythagorean fuzzy sets based on the similarity part and thehesitancy part that reflect fuzziness and uncertainty featuresrespectively They defined the following Pythagorean fuzzyentropy for a PFS in a finite universe of discourses 119883 =1199091 1199092 119909119899 Let = ⟨119909119894 120583(119909119894) ](119909119894)⟩ 119909119894 isin 119883 be aPFS in119883 The Pythagorean fuzzy entropy 119864119883119906119890() proposedby Xue at el [20] is defined as

119864119883119906119890 ()= 1119899119899sum119894=1

[1 minus (1205832(119909119894) + ]2

(119909119894)) 100381610038161003816100381610038161205832 (119909119894) minus ]2

(119909119894)10038161003816100381610038161003816 ] (17)

The entropy 119864119883119906119890() will be compared and exhibited in nextsection

4 Examples and Comparisons

In this section we present simple examples to observebehaviors of our proposed fuzzy entropies for PFSs To makeit mathematically sound and practically acceptable as well aschoose better entropy by comparative analysis we give anexample involving linguistic hedges By considering linguisticexample we use various linguistic hedges like ldquomore or lesslargerdquo ldquoquite largerdquo ldquovery largerdquo ldquovery very largerdquo etc inthe problems under Pythagorean fuzzy environment to select

Complexity 7

Table 1 Comparison of the degree of fuzziness with different entropy measures of PFSs

119875119865119878119904 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax 04378 02368 02576 00126 00146119876 08131 05310 05700 05041 00643 10363 06276 07225 03527 03999

Table 2 Degree of fuzziness from entropies of different PFSs

PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax12 06407 03923 03490 02736 02013 06066 03762 03386 03100 0095732 05375 03311 02969 03053 005422 04734 02908 02638 02947 0023352 04231 02625 02414 02843 001083 03848 02425 02267 02763 00050

better entropy We check the performance and behaviors ofthe proposed entropy measures in the environment of PFSsby exhibiting its simple intuition as follows

Example 1 Let 119876 and be singleton element PFSs inthe universe of discourse 119883 = 1199091 defined as =⟨1199091 093 035 01122⟩ 119876 = ⟨1199091 068 018 07100⟩ and = ⟨1199091 068 043 05939⟩ The numerical simulationresults of entropy measures 1198901119867119862 1198902119867119862 119890119864 119890119875119868 and 119890minmax areshown in Table 1 for the purpose of numerical comparisonFrom Table 1 we can see the entropy measures of almosthave larger entropy than and119876without any conflict except119890119875119868That is the degree of uncertainty of is greater than thatof and119876 Furthermore the behavior and performance of allentropies are analogous to each other except 119890119875119868 Apparentlyall entropies 1198901119867119862 1198902119867119862 119890119864 and 119890minmax behavewell except 119890119875119868

However in Example 1 it seems to be difficult forchoosing appropriate entropy that may provide a better wayto decide the fuzziness of PFSs To overcome it we givean example with structured linguistic variables to furtheranalyze and compare these entropy measures in Pythagoreanfuzzy environment Thus the following example with lin-guistic hedges is presented to further check behaviors andperformance of the proposed entropy measures

Example 2 Let be a PFS in a universe of discourse 119883 =1 3 5 7 9 defined as = ⟨1 01 08⟩ ⟨3 04 07⟩⟨5 05 03⟩ ⟨7 09 00⟩ ⟨9 10 00⟩ By Definitions 6 7 and8 where the concentration and dilation of are definedas Concentration 119862119874119873() = 2 Dilation 119863119868119871()12 Byconsidering the characterization of linguistic variables weuse the PFS to define the strength of the structural linguisticvariable119875 in119883 = 1 3 5 7 9 Using above defined operatorswe consider the following12 is regarded as ldquoMore or less LARGErdquo is

regarded as ldquoLARGErdquo32 is regarded as ldquoQuite LARGErdquo 2 is regarded asldquoVery LARGErdquo

52 is regarded as ldquoQuite very LARGErdquo 3 isregarded as ldquoVery very LARGErdquo

We use above linguistic hedges for PFSs to compare theentropy measures 1198901119867119862 1198902119867119862 119890119864 119890119875119868 and 119890minmax respectivelyFrom intuitive point of view the following requirement of (18)for a good entropy measure should be followed

119890 (12) gt 119890 () gt 119890 (32) gt 119890 (2) gt 119890 (52)gt 119890 (3) (18)

After calculating these entropy measures 1198901119867119862 1198902119867119862 119890119864 119890119875119868and 119890minmax for these PFSs the results are shown in Table 2From Table 2 it can be seen that these entropy measures1198901119867119862 1198902119867119862 119890119864 and 119890minmax satisfy the requirement of (18) but119890119875119868 fails to satisfy (18) that has 119890119875119868() gt 119890119875119868(32) gt 119890119875119868(2) gt119890119875119868(52) gt 119890119875119868(3) gt 119890119875119868(12) Therefore we say that thebehaviors of 1198901119867119862 1198902119867119862 119890119864 and 119890minmax are good but 119890119875119868 is not

In order to make more comparisons of entropy measureswe shake the degree of uncertainty of the middle value ldquo5rdquo in119883We decrease the degree of uncertainty for the middle pointin X and then we observe the amount of changes and also theimpact of entropymeasureswhen the degree of uncertainty ofthe middle value in X is decreasing To observe how differentPFS ldquoLARGErdquo in X affects entropy measures we modify as

ldquoLARGErdquo = 1 = ⟨1 01 08⟩ ⟨3 04 07⟩ ⟨5 06 05⟩ ⟨7 09 00⟩ ⟨9 10 00⟩ (19)

Again we use PFSs 121 1 321 21 521 and 31 with lin-guistic hedges to compare and observe behaviors of entropymeasuresThe results of degree of fuzziness for different PFSsfrom entropy measures are shown in Table 3 From Table 3we can see that these entropies 1198901119867119862 119890119864 and 119890minmax satisfythe requirement of (18) but 1198902119867119862 and 119890119875119868 could not fulfill therequirement of (18) with 1198902119867119862() gt 1198902119867119862(12) gt 1198902119867119862(32) gt1198902119867119862(2) gt 1198902119867119862(52) gt 1198902119867119862(3) and 119890119875119868() gt (32) gt119890119875119868(12) gt 119890119875119868(2) gt 119890119875119868(52) gt 119890119875119868(3) Therefore the

8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111

performance of 1198901119867119862 119890119864 and 119890minmax is good and 1198902119867119862 is notgood but 119890119875119868 presents very poorWe see that a little change inuncertainty for the middle value in X did not affect entropies1198901119867119862 119890119864 and 119890minmax and it brings a slight change in 1198902119867119862 butit gives an absolute big effect for entropy 119890119875119868

In viewing the results from Tables 1 2 and 3 we maysay that entropy measures 1198901119867119862 119890119864 and 119890minmax present betterperformance On the other hand from the viewpoint ofstructured linguistic variables we see that entropy measures1198901119867119862 119890119864 and 119890minmax are more suitable reliable and wellsuited in Pythagorean fuzzy environment for exhibiting thedegree of fuzziness of PFS We therefore recommend theseentropies 1198901119867119862 119890119864 and 119890minmax in a subsequent applicationinvolving multicriteria decision making

In the following example we conduct the comparisonanalysis of proposed entropies 1198901119867119862 119890119864 and 119890minmax with theentropy 119864119883119906119890() developed by Xue at el [20] to demonstratethe advantages of our developed entropies 1198901119867119862 119890119864 and119890minmax

Example 3 Let 119876 and be PFSs in the singleton universeset119883 = 1199091 as

= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)

The degrees of entropy for different PFSs between the pro-posed entropies 1198901119867119862 119890119864 119890minmax and the entropy 119864119883119906119890() byXue at el [20] are shown in Table 4 As can be seen fromTable 4 we find that despite having three different PFSs theentropy measure 119864119883119906119890 could not distinguish the PFSs 119876and However the proposed entropy measures 1198901119867119862 119890119864 and119890minmax can actually differentiate these different PFSs 119876and

5 Pythagorean Fuzzy Multicriterion DecisionMaking Based on New Entropies

In this section we construct a new multicriterion decisionmaking method Specifically we extend the technique fororder preference by similarity to an ideal solution (TOPSIS)to multicriterion decision making based on the proposedentropy measures for PFSs Impreciseness and vagueness isa reality of daily life which requires close attentions in thematters of management and decision In real life settingwith decision making process information available is oftenuncertain vague or imprecise PFSs are found to be a power-ful tool to solve decision making problems involving uncer-tain vague or imprecise information with high precision Todisplay practical reasonability and validity we apply our pro-posed new entropies 1198901119867119862 119890119864 and 119890minmax in a multicriteriadecision making problem involving unknown informationabout criteria weights for alternatives in Pythagorean fuzzyenvironment

We formalize the problem in the form of decision matrixin which it lists various project alternatives We assume thatthere arem project alternatives and wewant to compare themon n various criteria 119862119895 119895 = 1 2 119899 Suppose for eachcriterion we have an evaluation value For instance the firstproject on the first criterion has an evaluation 11990911 The firstproject on the second criterion has an evaluation 11990912 andthe first project on nth criterion has an evaluation 1199091119899 Ourobjective is to have these evaluations on individual criteriaand come up with a consolidated value for the project 1 anddo something similar to the project 2 and so on We thenultimately obtain a value for each of the projects Finally wecan rank the projects with selecting the best one among allprojects

Let 119860 = 1198601 1198602 119860119898 be the set of alternatives andlet the set of criteria for the alternatives 119860 119894 119894 = 1 2 119898be represented by 119862119895 119895 = 1 2 119899 The aim is to choosethe best alternative out of the n alternatives The constructionsteps for the new Pythagorean fuzzy TOPSIS based on theproposed entropy measures are as follows

Complexity 9

Step 1 (construction of Pythagorean fuzzy decision matrix)Consider that the alternative 119860 119894 acting on the criteria 119862119895 isrepresented in terms of Pythagorean fuzzy value 119894119895 = (120583119894119895 ]119894119895120587119894119895)119901 119894 = 1 2 119898 119895 = 1 2 119899 where 120583119894119895 denotes thedegree of fulfillment ]119894119895 represents the degree of not fulfill-ment and 120587119894119895 represents the degree of hesitancy against the

alternative 119860 119894 to the criteria119862119895 with the following conditions0 le 1205832119894119895 le 1 0 le ]2119894119895 le 1 0 le 1205872119894119895 le 1 and1205832119894119895+]2119894119895+1205872119894119895 = 1Thedecisionmatrix119863 = (119894119895)119898times119899 is constructed to handle the prob-lems involving multicriterion decision making where thedecision matrix 119863 = (119894119895)119898times119899 can be constructed as follows

119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901

sdot sdot sdotsdot sdot sdotsdot sdot sdotsdot sdot sdot

119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)

Step 2 (determination of the weights of criteria) In this stepthe crux to the problem is that weights to criteria have tobe identified The weights or priorities can be obtained bydifferent ways Suppose that the criteria information weightsare unknown and therefore the weights 119908119895 119895 = 1 2 3 119899of criteria for Pythagorean fuzzy entropy measures can beobtained by using 1198901119867119862 119890119864 and 119890minmax respectively Supposethe weights of criteria 119862119895 119895 = 1 2 119899 are 119908119895 119895 =1 2 3 119899 with 0 le 119908119895 le 1 and sum119899119895=1119908119895 = 1 Sinceweights of criteria are completely unknown we proposea new entropy weighting method based on the proposedPythagorean fuzzy entropy measures as follows

119908119895 = 119864119895sum119899119895=1 119864119895 (22)

where the weights of the criteria 119862119895 is calculated with 119864119895 =(1119898)sum119898119894=1 119894119895Step 3 (Pythagorean fuzzy positive-ideal solution (PFPIS)and Pythagorean fuzzy negative-ideal solution (PFNIS))In general it is important to determine the positive-idealsolution (PIS) and negative-ideal solution (NIS) in a TOPSISmethod Since the evaluation criteria can be categorized intotwo categories benefit and cost criteria in TOPSIS let1198721 and1198722 be the sets of benefit criteria and cost criteria in criteria119862119895 respectively According to Pythagorean fuzzy sets and theprinciple of a TOPSISmethod we define a Pythagorean fuzzyPIS (PFPIS) as follows

119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)

(120583minus119895 ]minus119895 120587minus119895 ) = (0 1 0) 119895 isin 1198721

(23)

Similarly a Pythagorean fuzzy NIS (PFNIS) is defined as

119860minus = ⟨119862119895 (120583minus119895 ]minus119895 120587minus119895 )⟩

119908ℎ119890119903119890 (120583minus119895 ]minus119895 120587minus119895 ) = (0 1 0) (120583+119895 ]+119895 120587+119895 ) = (1 0 0)

119895 isin 1198722(24)

Step 4 (calculation of distance measures from PFPIS andPFNIS) In this step we need to use a distance between twoPFSs Following the similar idea from the previously definedPNE distance 120577119864 between two PFSs we define a Pythagoreanweighted Euclidean (PWE) distance for any two PFSs 119876 isin119875119865119878(119883) as120577119908119864 ( 119876) = [12

119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)

We next use the PWE distance 120577119908119864 to calculate the distances119863+(119860 119894) and 119863minus(119860 119894) of each alternative 119860 119894 from PFPIS andPFNIS respectively as follows

119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1

119908119895 [(1 minus 1205832119894119895)2 + (]2119894119895)2 + (1 minus 1205832119894119895 minus ]2119894119895)2]119863minus (119860 119894) = 119889119864 (A119894Aminus)

= radic 12119899sum119895=1

119908119895 [(1205832119894119895)2 + (1 minus ]2119894119895)2 + (1 minus 1205832119894119895 minus ]2119894119895)2]

(26)

Step 5 (calculation of relative closeness degree and rankingof alternatives) The relative closeness degree (119860 119894) of each

10 Complexity

Table 5 Criteria to evaluate an audit company

Criteria Description of criteria

Required experience and capability tomake independent decision (1198621)

Certification and required knowledge on accounting business and taxation lawunderstanding of management system auditor should not be trembled or influenceby anyone actions decision and report should be based on careful analysis

The capability to comprehend differentbusiness needs (1198622) Ability to work with different companies setups analytical ability planning and

strategy

Effective communication skills (1198623) Mastered excellent communication skills well versed in compelling reportconvincing skills to present their reports should be patient enough to elaboratepoints to the entire satisfaction of the auditee

Table 6 Pythagorean fuzzy decision matrices

Alternatives Criteria Evalutionlowast1198621 1198622 11986231198601 (05500 04130 07259) (06030 051400 06101) (05000 01000 08602)1198602 (04000 02000 08944) (04000 02000 08944) (04500 05050 07365)1198603 (05000 01000 08602) (06030 051400 06101) (05500 04130 07259)Table 7 Weight of criteria

1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151

alternative 119860 119894 with respect to PFPIS and PFNIS is obtainedby using the following expression

(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)

Finally the alternatives are ordered according to therelative closeness degrees The larger value of the relativecloseness degrees reflects that an alternative is closer toPFPIS and farther from PFNIS simultaneously Thereforethe ranking order of all alternatives can be determinedaccording to ascending order of the relative closeness degreesThe most preferred alternative is the one with the highestrelative closeness degree

In the next example we present a comparison between theproposed entropies 1198901119867119862 and 119890minmax with the entropy 119864119883119906119890by Xue at el [20] based on PFSs for multicriteria decisionmaking problem The prime objective of decision makers isto select a best alternative from a set of available alternativesaccording to some criteria in multicriteria decision makingprocess Corruption is the misuse and mishandle of publicpower and resources for private and individual interest andbenefits usually in the form of bribery and favouritism Inaddition corruption twists andmanipulates the basis of com-petitions by misallocating resources and slowing economicactivity (Wikipedia)

Example 4 In this example a real world problemon selectionof well renowned national or international audit company istaken into account to demonstrate the comparison analysis

among the proposed probabilistic entropy 1198901119867119862 and non-probabilistic entropy 119890minmax with the entropy 119864119883119906119890 [20]To ensure the transparency and accountability of state runintuitions the ministry of finance of a developing countryoffers quotations to select a renowned audit company toget unbiased and fair audit report to keep on tract theeconomic development of a state The quotations which aregone through scrutiny process and found successful by acommittee are comprised of experts The quotations whichare found successfully by the committee are called eligiblewhile the rest are rejected A commission of experts isinvited to rank the audit companies 1198601 1198602 1198603 and toselect the best one on the basis of set criteria 1198621 1198622 1198623The descriptions about criteria are given in Table 5 andPythagorean fuzzy decision matrices are presented in Table 6The obtained weights of criteria from the entropies 1198901119867119862119890minmax and 119864119883119906119890 are shown in Table 7 From Table 7 itis seen that the weights of criteria obtained by the entropy119864119883119906119890 are always the same despite having different alternativesHowever the proposed entropies 1198901119867119862 and 119890minmax correctlydifferentiate the weights of criteria for each alternative 119860 119894The weights of criteria in Table 7 are also used to calculatethe distances 119863+(119860 119894) and119863minus(119860 119894) of each alternative 119860 119894 fromPFPIS and PFNIS respectively where the results are shownin Table 8 Furthermore the relative closeness degrees ofeach alternative to ideal solution are shown in Table 9 It canbe seen that the relative closeness degrees obtained by theproposed entropies 1198901119867119862 and 119890minmax are different for differentalternatives119860 119894 but the relative closeness degrees obtainedby the entropy 119864119883119906119890 [20] could not differentiate different

Complexity 11

Table 8 Distance for each alternative

119864119883119906119890 119863minus (119860 119894) 119863+ (119860 119894) 1198901119867119862 119863minus (119860 119894) 119863+ (119860 119894) 119890minmax 119863minus (119860 119894) 119863+ (119860 119894)1198601 07593 06476 1198601 07587 06468 1198601 07455 063631198602 08230 07842 1198602 08208 07830 1198602 08269 078621198603 07593 06476 1198603 07493 06403 1198603 07284 06244

Table 9 Degree of relative closeness

119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384

Table 10 Ranking of alternatives by different methods

Method Ranking Best alternative119864119883119906119890 1198602 ≺ 1198601 = 1198603 None1198901119867119862 1198602 ≺ 1198603 ≺ 1198601 1198601119890minmax 1198602 ≺ 1198603 ≺ 1198601 1198601alternatives so that it gives bias ranking of alternatives Theseranking results of different alternatives by the entropies 1198901119867119862119890minmax and 119864119883119906119890 are shown in Table 10 As can be seen theranking of alternatives by the proposed entropies 1198901119867119862 and119890minmax is well however the entropy119864119883119906119890 [20] could not rankthe alternative 1198601 and 1198603 It is found that there is no conflictin ranking alternatives by using the proposed Pythagoreanfuzzy TOPSIS method based on the proposed entropies 1198901119867119862and 119890minmax Totally the comparative analysis shows that thebest alternative is 1198601

We next apply the constructed Pythagorean fuzzy TOP-SIS in multicriterion decision making for China-PakistanEconomic Corridor projects

Example 5 A case study in ranking China-Pakistan Eco-nomic Corridor projects on priorities basis in the light ofrelated expertsrsquo opinions is used in order to demonstrate theefficiency of the proposed Pythagorean fuzzy TOPSIS beingapplied to multicriterion decision making China-PakistanEconomic Corridor (CPEC) is a collection of infrastructureprojects that are currently under construction throughoutin Pakistan Originally valued at $46 billion the value ofCPEC projects is now worth $62 billion CPEC is intendedto rapidly modernize Pakistani infrastructure and strengthenits economy by the construction of modern transportationnetworks numerous energy projects and special economiczones CPEC became partly operational when Chinese cargowas transported overland to Gwadar Port for onward mar-itime shipment to Africa and West Asia (see Wikipedia) Itis not only to benefit China and Pakistan but also to havepositive impact on other countries and regions Under CPECprojects it will have more frequent and free exchanges ofgrowth people to people contacts and integrated region ofshared destiny by enhancing understanding through aca-demic cultural regional knowledge and activity of higher

volume of flows in trades and businesses The enhancementof geographical linkages and cooperation by awin-winmodelwill result in improving the life standard of people roadrail and air transportation systems and also sustainable andperpetual development in China and Pakistan

Now suppose the concern and relevant experts areallowed to rank the CPEC projects according to needs of bothcountries on priorities basis Assume that initially there arefivemega projectswhich areGwadar Port (A1 ) Infrastructure(A2) Economic Zones (A3) Transportation and Energy(A4) and Social Sector Development (A5) according tothe following four criteria time frame and infrastructuralimprovement (C1) maintenance and sustainability (C2)socioeconomic development (C3) and eco-friendly (C4) Adetailed description of such criteria is displayed in Table 11Consider a decision organization with the five concernswhere relevant experts are authorized to rank the satisfactorydegree of an alternative with respect to the given criterionwhich is represented by a Pythagorean fuzzy value (PFV)Theevaluation values of the five alternatives 119860 119894 119894 = 1 2 5with Pythagorean fuzzy decision matrix are given in Table 12

We next use entropymeasures 1198901119867119862 119890119864 and 119890minmax basedon better performance in numerical analysis to calculate thecriteria weights 119908119895 using (22) These results are shown inTable 13 From Table 13 we can see that the criteria weightsand ranking of weights obtained by each entropy measure aredifferent We find the distances119863+ and119863minus of each alternativefrom PFPIS and PFNIS using (23) and (24) The results areshown in Table 14 We also calculate the relative closenessdegrees of alternatives using (27) The results are shownin Table 15 From Table 15 it can be clearly seen that underdifferent entropy measures the relative closeness degreesof alternatives obtained are different but the gap betweenthese values are considerably small Thus the ranking ofalternatives is almost the sameThe final ranking results fromdifferent entropies are shown in Table 16 From Table 16 itcan be identified that there is no conflict in selecting thebest alternative among alternatives by using the proposedPythagorean fuzzy TOPSIS method based on the entropies1198901119867119862 119890119864 and 119890minmaxThere is only one conflict to be found indeciding the preference ordering of alternatives 1198604 and1198605 in119890119864 Hence the results of ranking of alternatives according to

12 Complexity

Table 11 Criteria to assess CPEC projects

Criterion Description of criterion

Time frame and infrastructuralimprovement 1198621

Roadmap to ensure timely completion of project without any interruption andhurdle and play a vital role in making infrastructural improvement anddevelopment

Maintenance and sustainability 1198622 Themaintenance repair reliability and sustainability of the project

Socioeconomic development 1198623 Bring visible development and improvement in GDP economy stability andprosperity life expectancy education health employment personal dignitypersonal safety and freedom

Eco minus friendly 1198624 Not harmful to the environment contributes to green living practices that helpconserve natural resources and prevent contribution to air water and land pollution

Table 12 Pythagorean fuzzy decision matrix

Alternatives Criteria Evaluation1198621 1198622 1198623 11986241198601 (060 050 06245) (065 045 06124) (035 070 06225) (050 070 05099)1198602 (080 040 04472) (080 040 04472) (070 030 06481) (060 030 07416)1198603 (060 050 06245) (070 050 05099) (070 035 06225) (040 020 08944)1198604 (090 030 03162) (080 035 04873) (050 030 08124) (020 050 08426)1198605 (080 040 04472) (050 030 08124) (070 050 05099) (060 050 06245)Table 13 Entropies and weights of the criteria

1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288


1198901119867119862 119863minus (119860 119894) 119863+ (119860 119894) 119890119864 119863minus (119860 119894) 119863+ (119860 119894) 119890minmax 119863minus (119860 119894) 119863+ (119860 119894)1198601 05732 06297 1198601 05608 06354 1198601 05825 061961198602 07757 04381 1198602 07776 04557 1198602 07741 042941198603 07445 05848 1198603 07592 06055 1198603 07377 058651198604 08012 05819 1198604 07944 06163 1198604 08049 055821198605 07252 05268 1198605 07180 05331 1198605 07302 05195

Table 15 Degree of relative closeness for each alternative

1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843

the closeness degrees are made in an increasing orderThere-fore our analysis shows that the most feasible alternative is1198602 which is unanimously chosen by all proposed entropymeasures

6 Conclusions

In this paper we have proposed new fuzzy entropy measuresfor PFSs based on probabilistic-type distance Pythagorean

Complexity 13

Table 16 Ranking of alternative for different entropies

Method Ranking Best alternative1198901119867119862 1198601 ≺ 1198603 ≺ 1198605 ≺ 1198604 ≺ 1198602 1198602119890119864 1198601 ≺ 1198603 ≺ 1198604 ≺ 1198605 ≺ 1198602 1198602119890minmax 1198601 ≺ 1198603 ≺ 1198605 ≺ 1198604 ≺ 1198602 1198602index and minndashmax operator We have also extended non-probabilistic entropy to120590-entropy for PFSsThe entropymea-sures are considered especially for PFSs on finite universesof discourses As these are not only used in purposes ofcomputing environment but also used in more general casesfor large universal sets Structured linguistic variables areused to analyze and compare behaviors and performanceof the proposed entropies for PFSs in different Pythagoreanfuzzy environments We have examined and analyzed thesecomparison results obtained from these entropy measuresand then selected appropriate entropies which can be usefuland also be helpful to decide fuzziness of PFSs more clearlyand efficiently We have utilized our proposed methods toperform comparison analysis with the most recently devel-oped entropy measure for PFSs In this connection we havedemonstrated a simple example and a problem involvingMCDM to show the advantages of our suggested methodsFinally the proposed entropy measures of PFSs are appliedin an application to multicriterion decision making forranking China-Pakistan Economic Corridor projects Basedon obtained results we conclude that the proposed entropymeasures for PFSs are reasonable intuitive and well suitedin handling different kinds of problems involving linguisticvariables and multicriterion decision making in Pythagoreanfuzzy environment

Data Availability

All data are included in the manuscript

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] I B Turksen ldquoInterval valued fuzzy sets based on normalformsrdquo Fuzzy Sets and Systems vol 20 no 2 pp 191ndash210 1986

[3] J M Mendel and R I B John ldquoType-2 fuzzy sets made simplerdquoIEEE Transactions on Fuzzy Systems vol 10 no 2 pp 117ndash1272002

[4] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems Methodology Applications Education vol 13no 1 pp 23ndash37 1987

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

[6] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[7] V Torra andY Narukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju-do Republic of Korea August 2009

[8] R R Yager and A M Abbasov ldquoPythagorean membershipgrades complex numbers and decision makingrdquo InternationalJournal of Intelligent Systems vol 28 no 5 pp 436ndash452 2013

[9] R R Yager ldquoPythagorean fuzzy subsetsrdquo in Proceedings of the9th Joint World Congress on Fuzzy Systems and NAFIPS AnnualMeeting IFSANAFIPS 2013 pp 57ndash61 Edmonton CanadaJune 2013

[10] R R Yager ldquoPythagorean membership grades in multicriteriondecision makingrdquo IEEE Transactions on Fuzzy Systems vol 22no 4 pp 958ndash965 2014

[11] X L Zhang and Z S Xu ldquoExtension of TOPSIS to multiplecriteria decision making with pythagorean fuzzy setsrdquo Interna-tional Journal of Intelligent Systems vol 29 no 12 pp 1061ndash10782014

[12] X L Zhang ldquoA novel approach based on similarity measurefor Pythagorean fuzzymulti-criteria group decision makingrdquoInternational Journal of Intelligence Systems vol 31 pp 593ndash6112016

[13] P Ren Z Xu and X Gou ldquoPythagorean fuzzy TODIMapproach to multi-criteria decision makingrdquo Applied Soft Com-puting vol 42 pp 246ndash259 2016

[14] X D Peng and Y Yang ldquoPythagorean fuzzy Choquet integralbased MABAC method for multiple attribute group decisionmakingrdquo International Journal of Intelligent Systems vol 31 no10 pp 989ndash1020 2016

[15] X Zhang ldquoMulticriteria Pythagorean fuzzy decision analysisA hierarchicalQUALIFLEX approachwith the closeness index-based rankingmethodsrdquo Information Sciences vol 330 pp 104ndash124 2016

[16] X Peng H Yuan and Y Yang ldquoPythagorean fuzzy informationmeasures and their applicationsrdquo International Journal of Intel-ligent Systems vol 32 no 10 pp 991ndash1029 2017

[17] R Zhang J Wang X Zhu M Xia and M Yu ldquoSomeGeneralized Pythagorean Fuzzy Bonferroni Mean AggregationOperators with Their Application to Multiattribute GroupDecision-Makingrdquo Complexity vol 2017 Article ID 5937376 16pages 2017

[18] D Liang and Z Xu ldquoThe new extension of TOPSIS methodfor multiple criteria decision making with hesitant Pythagoreanfuzzy setsrdquo Applied Soft Computing vol 60 pp 167ndash179 2017

[19] L Perez-Domınguez L A Rodrıguez-Picon A Alvarado-Iniesta D Luviano Cruz and Z Xu ldquoMOORA underPythagorean Fuzzy Set for Multiple Criteria Decision MakingrdquoComplexity vol 2018 Article ID 2602376 10 pages 2018

[20] W Xue Z Xu X Zhang and X Tian ldquoPythagorean fuzzyLINMAP method based on the entropy theory for railwayproject investment decision makingrdquo International Journal ofIntelligent Systems vol 33 no 1 pp 93ndash125 2018

[21] L Zhang and FMeng ldquoAn approach to interval-valued hesitantfuzzy multiattribute group decision making based on thegeneralized Shapley-Choquet integralrdquo Complexity vol 2018Article ID 3941847 19 pages 2018

[22] A Guleria and R K Bajaj ldquoPythagorean fuzzy informationmeasure for multicriteria decision making problemrdquo Advancesin Fuzzy SystemsmdashApplications andTheory vol 2018 Article ID8023013 11 pages 2018

[23] M S Yang andZHussain ldquoDistance and similarity measures ofhesitant fuzzy sets based on Hausdorff metric with applications

14 Complexity

to multi-criteria decision making and clusteringrdquo Soft Comput-ing 2018

[24] Z Hussain and M-S Yang ldquoEntropy for hesitant fuzzy setsbased on Hausdorff metric with construction of hesitant fuzzyTOPSISrdquo International Journal of Fuzzy Systems vol 20 no 8pp 2517ndash2533 2018

[25] A de Luca and S Termini ldquoA definition of a nonprobabilisticentropy in the setting of fuzzy sets theoryrdquo Information andComputation vol 20 pp 301ndash312 1972

[26] R R Yager ldquoOn the measure of fuzziness and negation PartI membership in the unit intervalrdquo International Journal ofGeneral Systems vol 5 no 4 pp 189ndash200 1979

[27] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[28] X C Liu ldquoEntropy distance measure and similarity measure offuzzy sets and their relationsrdquo Fuzzy Sets and Systems vol 52no 3 pp 305ndash318 1992

[29] N R Pal and S K Pal ldquoSome properties of the exponentialentropyrdquo Information Sciences vol 66 no 1-2 pp 119ndash137 1992

[30] J-L Fan and Y-L Ma ldquoSome new fuzzy entropy formulasrdquoFuzzy Sets and Systems vol 128 no 2 pp 277ndash284 2002

[31] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[32] E Szmidt and JKacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[33] E Szmidt and J Baldwin ldquoEntropy for intuitionistic fuzzy settheory and mass assignment theoryrdquo Notes on IFSs vol 10 pp15ndash28 2004

[34] W L Hung and M S Yang ldquoFuzzy entropy on intuitionisticfuzzy setsrdquo International Journal of Intelligent Systems vol 21no 4 pp 443ndash451 2006

[35] J Havrda and F S Charvat ldquoQuantification method of classifi-cation processes Concept of structural120572-entropyrdquoKybernetikavol 3 pp 30ndash35 1967

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2 Complexity

membership values (PMVs) but PMVs are not necessary tobe IMVs For instance for the situation when the numbers120583 = radic32 and ] = 12 we can use PFSs but IFSs cannot beused since 120583 + ] gt 1 but 1205832 + ]2 le 1 PFSs are wider thanIFSs so that they can tackle more daily life problems underimprecision and uncertainty cases

More researchers are actively engaged in the devel-opment of PFSs properties For example Yager [10] gavePythagorean membership grades in multicriterion decisionmaking Extensions of technique for order preference bysimilarity to an ideal solution (TOPSIS) to multiple criteriadecision making with Pythagorean and hesitant fuzzy setswere proposed by Zhang and Xu [11] Zhang [12] considered anovel approach based on similarity measure for Pythagoreanfuzzy multicriteria group decision making Pythagoreanfuzzy TODIM approach to multicriterion decision makingwas given by Ren et al [13] Pythagorean fuzzy Choquetintegral based MABAC method for multiple attribute groupdecisionmakingwas developed by Peng andYang [14] Zhang[15] gave a hierarchical QUALIFLEX approach Peng et al[16] investigated Pythagorean fuzzy information measuresZhang et al [17] proposed generalized Pythagorean fuzzyBonferroni mean aggregation operators Liang and Xu [18]extended TOPSIS to hesitant Pythagorean fuzzy sets Perez-Domınguez et al [19] gave MOORA under Pythagoreanfuzzy sets Recently Pythagorean fuzzy LINMAP methodbased on the entropy for railway project investment deci-sion making was proposed by Xue et al [20] Zhang andMeng [21] proposed an approach to interval-valued hesitantfuzzy multiattribute group decision making based on thegeneralized Shapley-Choquet integral Pythagorean fuzzy(R S) minusnorm information measure for multicriteria decisionmaking problem was presented by Guleria and Bajaj [22]Furthermore Yang and Hussain [23] proposed distance andsimilarity measures of hesitant fuzzy sets based on Hausdorffmetric with applications tomulticriteria decision making andclustering Hussain and Yang [24] gave entropy for hesitantfuzzy sets based on Hausdorff metric with construction ofhesitant fuzzy TOPSIS

The entropy of fuzzy sets is a measure of fuzzinessbetween fuzzy sets De Luca and Termini [25] first intro-duced the axiom construction for entropy of fuzzy setswith reference to Shannonrsquos probability entropy Yager [26]defined fuzziness measures of fuzzy sets in terms of a lackof distinction between the fuzzy set and its negation basedon Lp norm Kosko [27] provided a measure of fuzzinessbetween fuzzy sets using a ratio of distance between the fuzzyset and its nearest set to the distance between the fuzzy setand its farthest set Liu [28] gave some axiom definitionsof entropy and also defined a 120590-entropy Pal and Pal [29]proposed exponential entropies While Fan andMa [30] gavesome new fuzzy entropy formulas Some extended entropymeasures for IFS were proposed by Burillo and Bustince [31]Szmidt and Kacprzyk [32] Szmidt and Baldwin [33] andHung and Yang [34]

In this paper we propose new entropies of PFS based onprobability-type distance Pythagorean index and minndashmaxoperation We also extend the concept to 120590-entropy and

then apply it to multicriteria decision making This paper isorganized as follows In Section 2 we review some definitionsof IFSs and PFSs In Section 3 we propose several newentropies of PFSs and then construct an axiomatic definitionof entropy for PFSs Based on the definition of entropy forPFSs we find that the proposed nonprobabilistic entropies ofPFSs are 120590-entropy In Section 4 we exhibit some examplesfor comparisons and also use structured linguistic variablesto validate our proposed methods In Section 5 we constructa new Pythagorean fuzzy TOPSIS based on the proposedentropy measures A comparison analysis of the proposedPythagorean fuzzy TOPSIS with the recently developedentropy of PFS [20] is shown We then apply the proposedmethod tomulticriterion decisionmaking for rankingChina-Pakistan Economic Corridor projects Finally we state ourconclusion in Section 6

2 Intuitionistic and Pythagorean Fuzzy Sets

In this section we give a brief review for intuitionistic fuzzysets (IFSs) and Pythagorean fuzzy sets (PFSs)

Definition 1 An intuitionistic fuzzy set (IFS) in 119883 isdefined by Atanassov [5] with the following form

= (119909 120583 (119909) ] (119909)) 119909 isin 119883 (1)

where 0 le 120583(119909) + ](119909) le 1 forall119909 isin 119883 and the functions120583(119909) 119883 997888rarr [0 1] denotes the degree of membershipof 119909 in and ](119909) 119883 997888rarr [0 1] denotes the degreeof nonmembership of 119909 in The degree of uncertainty(or intuitionistic index or indeterminacy) of 119909 to isrepresented by 120587(119909) = 1 minus (120583(119909) + ](119909))

For modeling daily life problems carrying imprecisionuncertainty and vagueness more precisely and with highaccuracy than IFSs Yager [9 10] presented Pythagoreanfuzzy sets (PFSs) where PFSs are the generalizations ofIFSs Yager [9 10] also validated that IFSs are contained inPFSs The concept of Pythagorean fuzzy set was originallydeveloped by Yager [8 9] but the general mathematical formof Pythagorean fuzzy set was developed by Zhang and Xu[11]

Definition 2 (Zhang and Xu [11]) A Pythagorean fuzzy set(PFS) in 119883 proposed by Yager [8 9] is mathematicallyformed as

= ⟨119909 120583 (119909) ] (119909)⟩ 119909 isin 119883 (2)

where the functions 120583(119909) 119883 997888rarr [0 1] represent the degreeof membership of 119909 in and ](119909) 119883 997888rarr [0 1] representthe degree of nonmembership of 119909 in For every 119909 isin 119883 thefollowing condition should be satisfied

0 le 1205832(119909) + ]2

(119909) le 1 (3)

Complexity 3


120587 (119909) = radic1 minus 1205832(119909) + ]2

(119909)

or 1205872(119909) = 1 minus 1205832

(119909) minus ]2

(119909) (4)


(119909) + 1205872

(119909) = 1 and if 1205832

(119909) = 1 then

]2(119909) = 0 and 1205872

(119909) = 0 Similarly if ]2

(119909) = 0 then1205832

(119909)+1205872

(119909) = 1 If ]2

(119909) = 1 then 1205832

(119909) = 0 and 1205872

(119909) = 0

If 1205872(119909) = 0 then 1205832

(119909) + ]2

(119909) = 1 If 1205872

(119909) = 1 then1205832

(119909) = ]2





(119909) = ]2

(119909) = 0 forall119909 isin 119883




(119909) and

]2(119909) ge ]2

(119909)


(119909) and

]2(119909) = ]2

(119909)

(iii) cup 119876 = ⟨119909max(1205832(119909) 1205832(119909))min(]2

(119909) ]2(119909))⟩ 119909 isin 119883

(iv) cap 119876 = ⟨119909min(1205832(119909) 1205832(119909))max(]2

(119909) ]2(119909))⟩ 119909 isin 119883

(v) 119888 = ⟨119909 ]2(119909) 1205832(119909)⟩ 119909 isin 119883







119862119874119873() = ⟨119909 120583119862119874119873() (119909) ]119862119874119873() (119909)⟩ 119909 isin 119883 (6)



119863119868119871 () = ⟨119909 120583119863119868119871() (119909) ]119863119868119871() (119909)⟩ 119909 isin 119883 (7)












]2 1205872


4 Complexity








2

+ (120587 (119909119894) minus 1radic3)2


2


(8)


(119909) minus ]2

(119909) ge1 minus 1205832

(119909) minus ]2

(119909) ge 13 ie 1205872

(119909) ge 1205872

(119909) ge 13



(119909) ]2(119909) and 1205872

(119909) in PFSs



119867120574 (119901) =

1120574 minus 1 (1 minus 119896sum119894=1

119901120574119894 ) 120574 = 1 (120574 gt 0)minus 119896sum119894=1

119901119894 log119901119894 120574 = 1 (9)


119890120574119867119862 () =

1119899119899sum119894=1

1120574 minus 1 [1 minus ((1205832(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] 120574 = 1 (120574 gt 0)

1119899119899sum119894=1

minus (1205832(119909119894) log 1205832 (119909119894) + ]2

(119909119894) log ]2 (119909119894) + 1205872

(119909119894) log1205872 (119909119894)) 120574 = 1 (10)







Complexity 5


]2 1205872 120582) = (1119899)sum119899119894=1(1(120574 minus1))[1 minus ((1205832

(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] +sum119899119894=1 120582119894(1205832(119909119894) +

]2(119909119894)+1205872


]2 1205872 120582)


1205971198651205971205832(119909119894) = minus120574(120574 minus 1) (1205832

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597]2(119909119894) = minus120574(120574 minus 1) (]2

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

1205971198651205971205872(119909119894) = minus120574(120574 minus 1) (1205872

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597120582119894 = 1205832(119909119894) + ]2

(119909119894) + 1205872

(119909119894) minus 1 119904119890119905997904997904997904997904 0

(12)


(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (]2

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and

]2(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (1205872

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and1205872

(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) 1205832

(119909119894) + ]2

(119909119894) + 1205872

(119909119894) = 1 and





119899sum119894=1

((1205832(119909119894) minus 1205832

(119909119894))2

+ (]2(119909119894) minus ]2

(119909119896))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(13)





119899sum119894=1

(1205832(119909119894) minus ]2

(119909119894))2 (14)


(119909119894) = 0 ]2

(119909119894) = 1

or 1205832(119909119894) = 1 ]2



(119909119894))2 = 1

Thus 1205832(119909119894) = 1 ]2

(119909119894) = 0 or 1205832

(119909119894) = 0 ]2

(119909119894) = 1


(119909119894) = ]2

(119909119894) forall119909119894 isin 119883



(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le ]2

(119909119894)

imply that 1205832(119909119894) le 1205832

(119909119894) le ]2

(119909119894) le ]2

(119909119894) then we




(119909119894) and

6 Complexity

]2(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge ]2


]2(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) and then




(119909119894))2




(119909119894))2 =



(i) 1205832(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le

]2(119909119894) forall119909119894 isin 119883 or

(ii) 1205832(119909119894) ge 1205832

(119909119894) and ]2

(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge

]2(119909119894) forall119909119894 isin 119883


(119909119894) le ]2

(119909119894) le

]2(119909119894) then max(1205832

(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) That is ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = 119876


(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and max(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) then ( cap 119876) = (1205832

(119909119894) ]2(119909119894)) =


(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) Then

max(1205832(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) and so ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = which implies


(119909119894) and max(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) then ( cap 119876) = (1205832


of Theorem 13



119890119875119868 () = 1119899119899sum119896=1

(1 minus 1205832(119909119896) minus ]2

(119909119896))

= 1119899119899sum119896=1

1205872(119909119896)

(15)



]2 1205872) of a PFS are


119890minmax () = 1119899119899sum119894=1

min (1205832(119909119894) ]2 (119909119894) 1205872 (119909))

max (1205832(119909119894) ]2 (119909119894) 1205872 (119909)) (16)



119864119883119906119890 ()= 1119899119899sum119894=1

[1 minus (1205832(119909119894) + ]2

(119909119894)) 100381610038161003816100381610038161205832 (119909119894) minus ]2

(119909119894)10038161003816100381610038161003816 ] (17)




Complexity 7


119875119865119878119904 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax 04378 02368 02576 00126 00146119876 08131 05310 05700 05041 00643 10363 06276 07225 03527 03999


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax12 06407 03923 03490 02736 02013 06066 03762 03386 03100 0095732 05375 03311 02969 03053 005422 04734 02908 02638 02947 0023352 04231 02625 02414 02843 001083 03848 02425 02267 02763 00050








119890 (12) gt 119890 () gt 119890 (32) gt 119890 (2) gt 119890 (52)gt 119890 (3) (18)





8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111





= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)






Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3


120587 (119909) = radic1 minus 1205832(119909) + ]2

(119909)

or 1205872(119909) = 1 minus 1205832

(119909) minus ]2

(119909) (4)


(119909) + 1205872

(119909) = 1 and if 1205832

(119909) = 1 then

]2(119909) = 0 and 1205872

(119909) = 0 Similarly if ]2

(119909) = 0 then1205832

(119909)+1205872

(119909) = 1 If ]2

(119909) = 1 then 1205832

(119909) = 0 and 1205872

(119909) = 0

If 1205872(119909) = 0 then 1205832

(119909) + ]2

(119909) = 1 If 1205872

(119909) = 1 then1205832

(119909) = ]2





(119909) = ]2

(119909) = 0 forall119909 isin 119883




(119909) and

]2(119909) ge ]2

(119909)


(119909) and

]2(119909) = ]2

(119909)

(iii) cup 119876 = ⟨119909max(1205832(119909) 1205832(119909))min(]2

(119909) ]2(119909))⟩ 119909 isin 119883

(iv) cap 119876 = ⟨119909min(1205832(119909) 1205832(119909))max(]2

(119909) ]2(119909))⟩ 119909 isin 119883

(v) 119888 = ⟨119909 ]2(119909) 1205832(119909)⟩ 119909 isin 119883







119862119874119873() = ⟨119909 120583119862119874119873() (119909) ]119862119874119873() (119909)⟩ 119909 isin 119883 (6)



119863119868119871 () = ⟨119909 120583119863119868119871() (119909) ]119863119868119871() (119909)⟩ 119909 isin 119883 (7)












]2 1205872


4 Complexity








2

+ (120587 (119909119894) minus 1radic3)2


2


(8)


(119909) minus ]2

(119909) ge1 minus 1205832

(119909) minus ]2

(119909) ge 13 ie 1205872

(119909) ge 1205872

(119909) ge 13



(119909) ]2(119909) and 1205872

(119909) in PFSs



119867120574 (119901) =

1120574 minus 1 (1 minus 119896sum119894=1

119901120574119894 ) 120574 = 1 (120574 gt 0)minus 119896sum119894=1

119901119894 log119901119894 120574 = 1 (9)


119890120574119867119862 () =

1119899119899sum119894=1

1120574 minus 1 [1 minus ((1205832(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] 120574 = 1 (120574 gt 0)

1119899119899sum119894=1

minus (1205832(119909119894) log 1205832 (119909119894) + ]2

(119909119894) log ]2 (119909119894) + 1205872

(119909119894) log1205872 (119909119894)) 120574 = 1 (10)







Complexity 5


]2 1205872 120582) = (1119899)sum119899119894=1(1(120574 minus1))[1 minus ((1205832

(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] +sum119899119894=1 120582119894(1205832(119909119894) +

]2(119909119894)+1205872


]2 1205872 120582)


1205971198651205971205832(119909119894) = minus120574(120574 minus 1) (1205832

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597]2(119909119894) = minus120574(120574 minus 1) (]2

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

1205971198651205971205872(119909119894) = minus120574(120574 minus 1) (1205872

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597120582119894 = 1205832(119909119894) + ]2

(119909119894) + 1205872

(119909119894) minus 1 119904119890119905997904997904997904997904 0

(12)


(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (]2

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and

]2(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (1205872

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and1205872

(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) 1205832

(119909119894) + ]2

(119909119894) + 1205872

(119909119894) = 1 and





119899sum119894=1

((1205832(119909119894) minus 1205832

(119909119894))2

+ (]2(119909119894) minus ]2

(119909119896))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(13)





119899sum119894=1

(1205832(119909119894) minus ]2

(119909119894))2 (14)


(119909119894) = 0 ]2

(119909119894) = 1

or 1205832(119909119894) = 1 ]2



(119909119894))2 = 1

Thus 1205832(119909119894) = 1 ]2

(119909119894) = 0 or 1205832

(119909119894) = 0 ]2

(119909119894) = 1


(119909119894) = ]2

(119909119894) forall119909119894 isin 119883



(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le ]2

(119909119894)

imply that 1205832(119909119894) le 1205832

(119909119894) le ]2

(119909119894) le ]2

(119909119894) then we




(119909119894) and

6 Complexity

]2(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge ]2


]2(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) and then




(119909119894))2




(119909119894))2 =



(i) 1205832(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le

]2(119909119894) forall119909119894 isin 119883 or

(ii) 1205832(119909119894) ge 1205832

(119909119894) and ]2

(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge

]2(119909119894) forall119909119894 isin 119883


(119909119894) le ]2

(119909119894) le

]2(119909119894) then max(1205832

(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) That is ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = 119876


(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and max(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) then ( cap 119876) = (1205832

(119909119894) ]2(119909119894)) =


(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) Then

max(1205832(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) and so ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = which implies


(119909119894) and max(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) then ( cap 119876) = (1205832


of Theorem 13



119890119875119868 () = 1119899119899sum119896=1

(1 minus 1205832(119909119896) minus ]2

(119909119896))

= 1119899119899sum119896=1

1205872(119909119896)

(15)



]2 1205872) of a PFS are


119890minmax () = 1119899119899sum119894=1

min (1205832(119909119894) ]2 (119909119894) 1205872 (119909))

max (1205832(119909119894) ]2 (119909119894) 1205872 (119909)) (16)



119864119883119906119890 ()= 1119899119899sum119894=1

[1 minus (1205832(119909119894) + ]2

(119909119894)) 100381610038161003816100381610038161205832 (119909119894) minus ]2

(119909119894)10038161003816100381610038161003816 ] (17)




Complexity 7


119875119865119878119904 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax 04378 02368 02576 00126 00146119876 08131 05310 05700 05041 00643 10363 06276 07225 03527 03999


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax12 06407 03923 03490 02736 02013 06066 03762 03386 03100 0095732 05375 03311 02969 03053 005422 04734 02908 02638 02947 0023352 04231 02625 02414 02843 001083 03848 02425 02267 02763 00050








119890 (12) gt 119890 () gt 119890 (32) gt 119890 (2) gt 119890 (52)gt 119890 (3) (18)





8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111





= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)






Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity








2

+ (120587 (119909119894) minus 1radic3)2


2


(8)


(119909) minus ]2

(119909) ge1 minus 1205832

(119909) minus ]2

(119909) ge 13 ie 1205872

(119909) ge 1205872

(119909) ge 13



(119909) ]2(119909) and 1205872

(119909) in PFSs



119867120574 (119901) =

1120574 minus 1 (1 minus 119896sum119894=1

119901120574119894 ) 120574 = 1 (120574 gt 0)minus 119896sum119894=1

119901119894 log119901119894 120574 = 1 (9)


119890120574119867119862 () =

1119899119899sum119894=1

1120574 minus 1 [1 minus ((1205832(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] 120574 = 1 (120574 gt 0)

1119899119899sum119894=1

minus (1205832(119909119894) log 1205832 (119909119894) + ]2

(119909119894) log ]2 (119909119894) + 1205872

(119909119894) log1205872 (119909119894)) 120574 = 1 (10)







Complexity 5


]2 1205872 120582) = (1119899)sum119899119894=1(1(120574 minus1))[1 minus ((1205832

(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] +sum119899119894=1 120582119894(1205832(119909119894) +

]2(119909119894)+1205872


]2 1205872 120582)


1205971198651205971205832(119909119894) = minus120574(120574 minus 1) (1205832

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597]2(119909119894) = minus120574(120574 minus 1) (]2

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

1205971198651205971205872(119909119894) = minus120574(120574 minus 1) (1205872

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597120582119894 = 1205832(119909119894) + ]2

(119909119894) + 1205872

(119909119894) minus 1 119904119890119905997904997904997904997904 0

(12)


(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (]2

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and

]2(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (1205872

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and1205872

(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) 1205832

(119909119894) + ]2

(119909119894) + 1205872

(119909119894) = 1 and





119899sum119894=1

((1205832(119909119894) minus 1205832

(119909119894))2

+ (]2(119909119894) minus ]2

(119909119896))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(13)





119899sum119894=1

(1205832(119909119894) minus ]2

(119909119894))2 (14)


(119909119894) = 0 ]2

(119909119894) = 1

or 1205832(119909119894) = 1 ]2



(119909119894))2 = 1

Thus 1205832(119909119894) = 1 ]2

(119909119894) = 0 or 1205832

(119909119894) = 0 ]2

(119909119894) = 1


(119909119894) = ]2

(119909119894) forall119909119894 isin 119883



(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le ]2

(119909119894)

imply that 1205832(119909119894) le 1205832

(119909119894) le ]2

(119909119894) le ]2

(119909119894) then we




(119909119894) and

6 Complexity

]2(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge ]2


]2(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) and then




(119909119894))2




(119909119894))2 =



(i) 1205832(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le

]2(119909119894) forall119909119894 isin 119883 or

(ii) 1205832(119909119894) ge 1205832

(119909119894) and ]2

(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge

]2(119909119894) forall119909119894 isin 119883


(119909119894) le ]2

(119909119894) le

]2(119909119894) then max(1205832

(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) That is ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = 119876


(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and max(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) then ( cap 119876) = (1205832

(119909119894) ]2(119909119894)) =


(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) Then

max(1205832(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) and so ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = which implies


(119909119894) and max(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) then ( cap 119876) = (1205832


of Theorem 13



119890119875119868 () = 1119899119899sum119896=1

(1 minus 1205832(119909119896) minus ]2

(119909119896))

= 1119899119899sum119896=1

1205872(119909119896)

(15)



]2 1205872) of a PFS are


119890minmax () = 1119899119899sum119894=1

min (1205832(119909119894) ]2 (119909119894) 1205872 (119909))

max (1205832(119909119894) ]2 (119909119894) 1205872 (119909)) (16)



119864119883119906119890 ()= 1119899119899sum119894=1

[1 minus (1205832(119909119894) + ]2

(119909119894)) 100381610038161003816100381610038161205832 (119909119894) minus ]2

(119909119894)10038161003816100381610038161003816 ] (17)




Complexity 7


119875119865119878119904 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax 04378 02368 02576 00126 00146119876 08131 05310 05700 05041 00643 10363 06276 07225 03527 03999


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax12 06407 03923 03490 02736 02013 06066 03762 03386 03100 0095732 05375 03311 02969 03053 005422 04734 02908 02638 02947 0023352 04231 02625 02414 02843 001083 03848 02425 02267 02763 00050








119890 (12) gt 119890 () gt 119890 (32) gt 119890 (2) gt 119890 (52)gt 119890 (3) (18)





8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111





= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)






Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5


]2 1205872 120582) = (1119899)sum119899119894=1(1(120574 minus1))[1 minus ((1205832

(119909119894))120574 + (]2

(119909119894))120574 + (1205872

(119909119894))120574)] +sum119899119894=1 120582119894(1205832(119909119894) +

]2(119909119894)+1205872


]2 1205872 120582)


1205971198651205971205832(119909119894) = minus120574(120574 minus 1) (1205832

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597]2(119909119894) = minus120574(120574 minus 1) (]2

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

1205971198651205971205872(119909119894) = minus120574(120574 minus 1) (1205872

(119909119894))120574minus1 + 120582119894 119904119890119905997904997904997904997904 0

120597119865120597120582119894 = 1205832(119909119894) + ]2

(119909119894) + 1205872

(119909119894) minus 1 119904119890119905997904997904997904997904 0

(12)


(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (]2

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and

]2(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) (1205872

(119909119894))120574minus1 = 120582119894(120574 minus 1)120574 and1205872

(119909119894) = (120582119894(120574 minus 1)120574)1(120574minus1) 1205832

(119909119894) + ]2

(119909119894) + 1205872

(119909119894) = 1 and





119899sum119894=1

((1205832(119909119894) minus 1205832

(119909119894))2

+ (]2(119909119894) minus ]2

(119909119896))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(13)





119899sum119894=1

(1205832(119909119894) minus ]2

(119909119894))2 (14)


(119909119894) = 0 ]2

(119909119894) = 1

or 1205832(119909119894) = 1 ]2



(119909119894))2 = 1

Thus 1205832(119909119894) = 1 ]2

(119909119894) = 0 or 1205832

(119909119894) = 0 ]2

(119909119894) = 1


(119909119894) = ]2

(119909119894) forall119909119894 isin 119883



(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le ]2

(119909119894)

imply that 1205832(119909119894) le 1205832

(119909119894) le ]2

(119909119894) le ]2

(119909119894) then we




(119909119894) and

6 Complexity

]2(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge ]2


]2(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) and then




(119909119894))2




(119909119894))2 =



(i) 1205832(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le

]2(119909119894) forall119909119894 isin 119883 or

(ii) 1205832(119909119894) ge 1205832

(119909119894) and ]2

(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge

]2(119909119894) forall119909119894 isin 119883


(119909119894) le ]2

(119909119894) le

]2(119909119894) then max(1205832

(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) That is ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = 119876


(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and max(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) then ( cap 119876) = (1205832

(119909119894) ]2(119909119894)) =


(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) Then

max(1205832(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) and so ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = which implies


(119909119894) and max(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) then ( cap 119876) = (1205832


of Theorem 13



119890119875119868 () = 1119899119899sum119896=1

(1 minus 1205832(119909119896) minus ]2

(119909119896))

= 1119899119899sum119896=1

1205872(119909119896)

(15)



]2 1205872) of a PFS are


119890minmax () = 1119899119899sum119894=1

min (1205832(119909119894) ]2 (119909119894) 1205872 (119909))

max (1205832(119909119894) ]2 (119909119894) 1205872 (119909)) (16)



119864119883119906119890 ()= 1119899119899sum119894=1

[1 minus (1205832(119909119894) + ]2

(119909119894)) 100381610038161003816100381610038161205832 (119909119894) minus ]2

(119909119894)10038161003816100381610038161003816 ] (17)




Complexity 7


119875119865119878119904 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax 04378 02368 02576 00126 00146119876 08131 05310 05700 05041 00643 10363 06276 07225 03527 03999


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax12 06407 03923 03490 02736 02013 06066 03762 03386 03100 0095732 05375 03311 02969 03053 005422 04734 02908 02638 02947 0023352 04231 02625 02414 02843 001083 03848 02425 02267 02763 00050








119890 (12) gt 119890 () gt 119890 (32) gt 119890 (2) gt 119890 (52)gt 119890 (3) (18)





8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111





= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)






Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

]2(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge ]2


]2(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) and then




(119909119894))2




(119909119894))2 =



(i) 1205832(119909119894) le 1205832

(119909119894) and ]2

(119909119894) ge ]2

(119909119894) for 1205832

(119909119894) le

]2(119909119894) forall119909119894 isin 119883 or

(ii) 1205832(119909119894) ge 1205832

(119909119894) and ]2

(119909119894) le ]2

(119909119894) for 1205832

(119909119894) ge

]2(119909119894) forall119909119894 isin 119883


(119909119894) le ]2

(119909119894) le

]2(119909119894) then max(1205832

(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) That is ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = 119876


(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and max(]2

(119909119894)

]2(119909119894)) = ]2

(119909119894) then ( cap 119876) = (1205832

(119909119894) ]2(119909119894)) =


(119909119894) le ]2

(119909119894) le 1205832

(119909119894) le 1205832

(119909119894) Then

max(1205832(119909119894) 1205832(119909119894)) = 1205832

(119909119894) and min(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) and so ( cup 119876) = (1205832

(119909119894) ]2(119909119894)) = which implies


(119909119894) and max(]2

(119909119894) ]2(119909119894)) =

]2(119909119894) then ( cap 119876) = (1205832


of Theorem 13



119890119875119868 () = 1119899119899sum119896=1

(1 minus 1205832(119909119896) minus ]2

(119909119896))

= 1119899119899sum119896=1

1205872(119909119896)

(15)



]2 1205872) of a PFS are


119890minmax () = 1119899119899sum119894=1

min (1205832(119909119894) ]2 (119909119894) 1205872 (119909))

max (1205832(119909119894) ]2 (119909119894) 1205872 (119909)) (16)



119864119883119906119890 ()= 1119899119899sum119894=1

[1 minus (1205832(119909119894) + ]2

(119909119894)) 100381610038161003816100381610038161205832 (119909119894) minus ]2

(119909119894)10038161003816100381610038161003816 ] (17)




Complexity 7


119875119865119878119904 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax 04378 02368 02576 00126 00146119876 08131 05310 05700 05041 00643 10363 06276 07225 03527 03999


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax12 06407 03923 03490 02736 02013 06066 03762 03386 03100 0095732 05375 03311 02969 03053 005422 04734 02908 02638 02947 0023352 04231 02625 02414 02843 001083 03848 02425 02267 02763 00050








119890 (12) gt 119890 () gt 119890 (32) gt 119890 (2) gt 119890 (52)gt 119890 (3) (18)





8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111





= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)






Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7


119875119865119878119904 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax 04378 02368 02576 00126 00146119876 08131 05310 05700 05041 00643 10363 06276 07225 03527 03999


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax12 06407 03923 03490 02736 02013 06066 03762 03386 03100 0095732 05375 03311 02969 03053 005422 04734 02908 02638 02947 0023352 04231 02625 02414 02843 001083 03848 02425 02267 02763 00050








119890 (12) gt 119890 () gt 119890 (32) gt 119890 (2) gt 119890 (52)gt 119890 (3) (18)





8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111





= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)






Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity


PFSs 1198901119867119862 1198902119867119862 119890119864 119890119875119868 119890minmax121 06569 03952 03473 02360 0227611 06554 04086 03406 02560 01966321 05997 03757 02944 02440 0120121 05351 03356 02526 02281 00662521 04753 02993 02209 02145 0032931 04233 02682 01976 02038 00171


PFSs 119864119883119906119890 1198901119867119862 119890119864 119890minmax 04718 07532 03603 01270119876 04718 06886 03066 00470 04718 07264 04241 01111





= ⟨1199091 0305 856 04174⟩ 119876 = ⟨1199091 01850 08530 04880⟩ = ⟨1199091 04130 08640 02880⟩

(20)






Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9



119863 = (119894119895)119898times119899 =119862111986011198602119860119898

[[[[[[[[

(12058311 ]11 12058711)119901(12058321 ]21 12058721)119901(1205831198981 ]1198981 1205871198981)119901

1198622(12058312 ]12 12058712)119901(12058322 ]22 12058722)119901(1205831198982 ]1198982 1205871198982)119901


119862119899(1205831119899 ]1119899 1205871119899)119901(1205832119899 ]2119899 1205872119899)119901(120583119898119899 ]119898119899 120587119898119899)119901

]]]]]]]](21)


119908119895 = 119864119895sum119899119895=1 119864119895 (22)


119860+ = ⟨119862119895 (120583+119895 ]+119895 120587+119895 )⟩ 119908ℎ119890119903119890 (120583+119895 ]+119895 120587+119895 ) = (1 0 0)


(23)




119895 isin 1198722(24)


119899sum119894=1

119908119894 ((1205832 (119909119894) minus 1205832(119909119894))2

+ (]2(119909119894) minus ]2

(119909119894))2 + (1205872

(119909119894) minus 1205872

(119909119894))2)]12

(25)


119863+ (119860 119894) = 119889119864 (A119894A+)= radic 12

119899sum119895=1


= radic 12119899sum119895=1


(26)


10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity






strategy




1199081 1199082 1199083119864119883119906119890 03333 03333 033331198901119867119862 02912 03646 03442119890minmax 02209 04640 03151


(119860 119894) = 119863minus (119860 119894)119863minus (119860 119894) + 119863+ (119860 119894) (27)





Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11




119864119883119906119890 119873(119860 119894) 1198901119867119862 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 05397 1198601 05398 1198601 053951198602 05121 1198602 05118 1198602 051261198603 05397 1198603 05392 1198603 05384








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity










1199081 1199082 1199083 11990841198901119867119862 02487 02563 02585 02366119890119864 02063 02411 02539 02987119890minmax 03048 02524 02141 02288




1198901119867119862 119873(119860 119894) 119890119864 119873(119860 119894) 119890minmax 119873(119860 119894)1198601 04765 1198601 04688 1198601 048461198602 06391 1198602 06305 1198602 064321198603 05601 1198603 05563 1198603 055711198604 05793 1198604 05631 1198604 059051198605 05792 1198605 05739 1198605 05843


6 Conclusions


Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13



Data Availability




References
























14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity





















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Fuzzy Entropy for Pythagorean Fuzzy Sets with Application ...downloads.hindawi.com/journals/complexity/2018/2832839.pdf · fuzzy sets. Recently, Pythagorean fuzzy LINMAP method based