# Fuzzy Entropy for Pythagorean Fuzzy Sets with Application to Multicriterion Decision Making

Authors: Miin-Shen Yang; Zahid Hussain
Journal: Complexity
 (2018)

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## Abstract

The concept of Pythagorean fuzzy sets (PFSs) was initially developed by Yager in 2013, which provides a novel way to model uncertainty and vagueness with high precision and accuracy compared to intuitionistic fuzzy sets (IFSs). The concept was concretely designed to represent uncertainty and vagueness in mathematical way and to furnish a formalized tool for tackling imprecision to real problems. In the present paper, we have used both probabilistic and nonprobabilistic types to calculate fuzzy entropy of PFSs. Firstly, a probabilistic-type entropy measure for PFSs is proposed and then axiomatic definitions and properties are established. Secondly, we utilize a nonprobabilistic-type with distances to construct new entropy measures for PFSs. Then a min–max operation to calculate entropy measures for PFSs is suggested. Some examples are also used to demonstrate suitability and reliability of the proposed methods, especially for choosing the best one/ones in structured linguistic variables. Furthermore, a new method based on the chosen entropies is presented for Pythagorean fuzzy multicriterion decision making to compute criteria weights with ranking of alternatives. A comparison analysis with the most recent and relevant Pythagorean fuzzy entropy is conducted to reveal the advantages 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.


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## Body

## 1. Introduction

The concept of fuzzy sets was first proposed by Zadeh [1] in 1965. With a widely spread use in various fields, fuzzy sets not only provide broad opportunity to measure uncertainties in more powerful and logical way, but also give us a meaningful way to represent vague concepts in natural language. It is known that most systems based on ‘crisp set theory’ or ‘two-valued logics’ are somehow difficult for handling imprecise and vague information. In this sense, fuzzy sets can be used to provide better solutions for more real world problems. Moreover, to treat more imprecise and vague information in daily life, various extensions of fuzzy sets are suggested by researchers, such as interval-valued fuzzy set [2], type-2 fuzzy 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 or degrees 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 reliability and applicability, Atanassov [5] generalized the concept of ‘fuzzy set theory’ and proposed Intuitionistic fuzzy sets (IFSs) which include both membership degree and nonmembership degree and degree of nondeterminacy or uncertainty where degree of uncertainty = (1- (degree of membership + nonmembership degree)). In IFSs, the pair of membership grades is denoted by μ,ν satisfying the condition of μ+ν≤1. Recently, Yager and Abbasov [8] and Yager [9] extended the condition μ+ν≤1 to μ2+ν2≤1 and then introduced a class of Pythagorean fuzzy sets (PFSs) whose membership values are ordered pairs μ,ν that fulfills the required condition of μ2+ν2≤1 with different aggregation operations and applications in multicriterion decision making. According to Yager and Abbasov [8] and Yager [9], the space of all intuitionistic membership values (IMVs) is also Pythagorean membership values (PMVs), but PMVs are not necessary to be IMVs. For instance, for the situation when the numbers μ=3/2 and ν=1/2, we can use PFSs, but IFSs cannot be used since μ+ν>1, but μ2+ν2≤1. PFSs are wider than IFSs so that they can tackle more daily life problems under imprecision and uncertainty cases.More researchers are actively engaged in the development of PFSs properties. For example, Yager [10] gave Pythagorean membership grades in multicriterion decision making. Extensions of technique for order preference by similarity to an ideal solution (TOPSIS) to multiple criteria decision making with Pythagorean and hesitant fuzzy sets were proposed by Zhang and Xu [11]. Zhang [12] considered a novel approach based on similarity measure for Pythagorean fuzzy multicriteria group decision making. Pythagorean fuzzy TODIM approach to multicriterion decision making was given by Ren et al. [13]. Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making was developed by Peng and Yang [14]. Zhang [15] gave a hierarchical QUALIFLEX approach. Peng et al. [16] investigated Pythagorean fuzzy information measures. Zhang et al. [17] proposed generalized Pythagorean fuzzy Bonferroni mean aggregation operators. Liang and Xu [18] extended TOPSIS to hesitant Pythagorean fuzzy sets. Pérez-Domínguez et al. [19] gave MOORA under Pythagorean fuzzy sets. Recently, Pythagorean fuzzy LINMAP method based on the entropy for railway project investment decision making was proposed by Xue et al. [20]. Zhang and Meng [21] proposed an approach to interval-valued hesitant fuzzy multiattribute group decision making based on the generalized Shapley-Choquet integral. Pythagorean fuzzy (R,S)-norm information measure for multicriteria decision making problem was presented by Guleria and Bajaj [22]. Furthermore, Yang and Hussain [23] proposed distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multicriteria decision making and clustering. Hussain and Yang [24] gave entropy for hesitant fuzzy sets based on Hausdorff metric with construction of hesitant fuzzy TOPSIS.The entropy of fuzzy sets is a measure of fuzziness between fuzzy sets. De Luca and Termini [25] first introduced the axiom construction for entropy of fuzzy sets with reference to Shannon’s probability entropy. Yager [26] defined fuzziness measures of fuzzy sets in terms of a lack of distinction between the fuzzy set and its negation based on Lp norm. Kosko [27] provided a measure of fuzziness between fuzzy sets using a ratio of distance between the fuzzy set and its nearest set to the distance between the fuzzy set and its farthest set. Liu [28] gave some axiom definitions of entropy and also defined a σ-entropy. Pal and Pal [29] proposed exponential entropies. While Fan and Ma [30] gave some new fuzzy entropy formulas. Some extended entropy measures for IFS were proposed by Burillo and Bustince [31], Szmidt and Kacprzyk [32], Szmidt and Baldwin [33], and Hung and Yang [34].In this paper, we propose new entropies of PFS based on probability-type, distance, Pythagorean index, and min–max operation. We also extend the concept toσ-entropy and then apply it to multicriteria decision making. This paper is organized as follows. In Section 2, we review some definitions of IFSs and PFSs. In Section 3, we propose several new entropies of PFSs and then construct an axiomatic definition of entropy for PFSs. Based on the definition of entropy for PFSs, we find that the proposed nonprobabilistic entropies of PFSs are σ-entropy. In Section 4, we exhibit some examples for comparisons and also use structured linguistic variables to validate our proposed methods. In Section 5, we construct a new Pythagorean fuzzy TOPSIS based on the proposed entropy measures. A comparison analysis of the proposed Pythagorean fuzzy TOPSIS with the recently developed entropy of PFS [20] is shown. We then apply the proposed method to multicriterion decision making for ranking China-Pakistan Economic Corridor projects. Finally, we state our conclusion in Section 6.


## 2. Intuitionistic and Pythagorean Fuzzy Sets

In this section, we give a brief review for intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs).Definition 1.
An intuitionistic fuzzy set (IFS)M~ in X is defined by Atanassov [5] with the following form: (1)M~=x,μM~x,νM~x:x∈Xwhere 0≤μM~x+νM~x≤1,∀x∈X, and the functions μM~x:X→0,1 denotes the degree of membership of x in M~ and νM~x:X→0,1 denotes the degree of nonmembership of x in M~. The degree of uncertainty (or intuitionistic index, or indeterminacy) of x to M~ is represented by πM~x=1-μM~x+νM~x.For modeling daily life problems carrying imprecision, uncertainty, and vagueness more precisely and with high accuracy than IFSs, Yager [9, 10] presented Pythagorean fuzzy sets (PFSs), where PFSs are the generalizations of IFSs. Yager [9, 10] also validated that IFSs are contained in PFSs. The concept of Pythagorean fuzzy set was originally developed by Yager [8, 9], but the general mathematical form of Pythagorean fuzzy set was developed by Zhang and Xu [11].Definition 2 (Zhang and Xu [11]).
A Pythagorean fuzzy set (PFS)P~ in X proposed by Yager [8, 9] is mathematically formed as(2)P~=x,μP~x,νP~x:x∈Xwhere the functions  μP~x:X→0,1 represent the degree of membership of x in P~ and νP~x:X→0,1 represent the degree of nonmembership of x in P~. For every x∈X, the following condition should be satisfied:(3)0≤μP~2x+νP~2x≤1.Definition 3 (Zhang and Xu [11]).
For any PFSP~ in X, the value πP~x is called Pythagorean index of the element x in P~ with(4)πP~x=1-μP~2x+νP~2xorπP~2x=1-μP~2x-νP~2x.In general,πP~x is also called hesitancy (or indeterminacy) degree of the element x in P~. It is obvious that 0≤πP~2x≤1,∀x∈X. It is worthy to note, for a PFS P~, if μP~2x=0 then νP~2x+πP~2x=1, and if μP~2x=1 then νP~2x=0 and πP~2x=0. Similarly, if νP~2x=0 then μP~2x+πP~2x=1. If νP~2x=1 then μP~2x=0 and πP~2x=0. If πP~2x=0 then μP~2x+νP~2x=1. If πP~2x=1 then μP~2x=νP~2x=0. For convenience, Zhang and Xu [11] denoted the pair μP~x,νP~x as Pythagorean fuzzy number (PFN), which is represented by p~=μp~,νp~.Since PFSs are a generalized form of IFSs, we give the following definition for PFSs.Definition 4.
LetP~ be a PFS in X. P~ is called a completely Pythagorean if μP~2x=νP~2x=0,∀x∈X.Peng et al. [16] suggested various mathematical operations for PFSs as follows:Definition 5 (Peng et al. [16]).
IfP~ and Q~ are two PFSs in X, then(i)
P~≤Q~ if and only if ∀x∈X, μP~2x≤μQ~2x and νP~2x≥νQ~2x;(ii)
P~=Q~ if and only if ∀x∈X,μP~2x=μQ~2x and νP~2x=νQ~2x;(iii)
P~∪Q~=x,maxμP~2x,μQ~2x,minνP~2x,νQ~2x:x∈X;(iv)
P~∩Q~=x,minμP~2x,μQ~2x,maxνP~2x,νQ~2x:x∈X;(v)
P~c=x,νP~2x,μQ~2x:x∈X.We next define more operations of PFS inX, especially about hedges of “very”, “highly”, “more or less”, “concentration”, “dilation”, and other terms that are needed to represent linguistic variables. We first define then power (or exponent) of PFS as follows.Definition 6.
LetP~=x,μP~x,νP~x:x∈X be a PFS in X. For any positive real numbern, then power (or exponent) of the PFS P~, denoted by P~n, is defined as(5)P~n=x,μP~xn,1-1-νP~2xn:x∈X.It can be easily verified that, for any positive real numbern, 0≤μP~xn+1-1-νP~2xn≤1,∀x∈X.By using Definition6, the concentration and dilation of a PFS P~ can be defined as follows.Definition 7.
The concentrationCONP~ of a PFS P~ in X is defined as(6)CONP~=x,μCONP~x,νCONP~x:x∈Xwhere μCONP~x=μP~x2 and νCONP~x=1-1-νP~2x2.Definition 8.
The dilationDILP~ of a PFS P~ in X is defined as(7)DILP~=x,μDILP~x,νDILP~x:x∈Xwhere μDILP~x=μP~x1/2 and νDILP~x=1-1-νP~2x1/2.In next section, we construct new entropy measures for PFSs based on probability-type, entropy induced by distance, Pythagorean index, and max-min operation. We also give an axiomatic definition of entropy for PFSs.


## 3. New Fuzzy Entropies for Pythagorean Fuzzy Sets

We first provide a definition of entropy for PFSs. De Luca and Termini [25] gave the axiomatic definition of entropy measure 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 notions as IFSs to give a definition of entropy for PFSs. Assume that PFSX represents the set of all PFSs inX.Definition 9.
A real functionE:PFSX→0,1 is called an entropy on PFSX if E satisfies the following axioms:(A0)
Nonnegativity0≤EP~≤1;(A1)
MinimalityEP~=0, iffP~isacrispset;(A2)
MaximalityEP~=1, iffμP~x=νP~x,∀x∈X;(A3)
ResolutionEP~≤EQ~, ifP~ is crisper than  Q~,i.e.,∀x∈X,
μP~x≤μQ~xandνP~x≥νQ~xforμQ~x≤νQ~xorμP~x≥μQ~xandνP~x≤νQ~xforμQ~x≥νQ~x;(A4)
SymmetricEP~=EP~c,whereP~cis the complement ofP~;For probabilistic-type entropy, we need to omit the axiom (A0). On the other hand, because we take the three-parameterμP~2,νP~2, and πP~2 as a probability mass function p=μP~2,νP~2,πP~2, the probabilistic-type entropy EP~ should attain a unique maximum at μP~(x)=νP~(x)=πP~(x)=1/3,∀x∈X. Therefore, for probabilistic-type entropy, we replace the axiom A2 with A2′ and the axiom A3 with A3′ as follows:(A2′)
MaximalityEP~attains a unique maximum atμP~x=νP~x=πP~x=1/3, ∀x∈X.(A3′)
ResolutionEP~≤EQ~,ifP~is crisper thanQ~,i.e.,, ∀x∈X, μP~x≤μQ~xandνP~x≤νQ~xformax⁡μQ~x,νQ~x≤1/3andμP~x≥μQ~x,νP~x≥νQ~xformin⁡μQ~x,νQ~x≥1/3.In addition to the five axioms (A0)~(A4) in Definition9, if we add the following axiom (A5), E is called σ- entropy:(A5)
ValuationEP~+EQ~=EP~∪Q~+EP~∩Q~, if∀x∈X
μP~x≤μQ~xandνP~x≥νQ~xforμQ~x≤νQ~x orμP~x≥μQ~xandνP~x≤νQ~xforμQ~x≥νQ~x.We present a property for the axiomA3′ when P~ is crisper than Q~.Property 10.
IfP~ is crisper than Q~ in the axiom A3′, then we have the following inequality:(8)μP~xi-132+νP~xi-132+πP~xi-132≥μQ~xi-132+νQ~xk-132+νQ~xk-132,∀xiProof.
IfP~ is crisper than Q~, then ∀xi, μP~x≤μQ~x and νP~x≤νQ~x for maxμQ~x,νQ~x≤1/3. Therefore, we have that μP~x-1/3≤μQ~x-1/3≤0 and νP~x-1/3≤νQ~x-1/3≤0 and so 1-μP~2x-νP~2x≥1-μQ~2x-νQ~2x≥1/3, i.e., πP~2x≥πQ~2x≥1/3 and πP~x-1/3≥πQ~x-1/3≥0. Thus, we have μP~x-1/32≥μQ~x-1/32, νP~x-1/32≥νQ~x-1/32, and πP~x-1/32≥πQ~x-1/32. This induces the inequality. Similarly, the part of μP~x≥μQ~x and νP~x≥νQ~x for minμQ~x,νQ~x≥1/3 also induces the inequality. Hence, we prove the property.Since PFSs are generalized form of IFSs, the distances between PFSs need to be computed by considering all the three componentsμP~2x,νP~2x and πP~2x in PFSs. The well-known distance between PFSs is Euclidean distance. Therefore, the inequality in Property 10 indicates that the Euclidean distance between μP~x,νP~x,πP~x and 1/3,1/3,1/3 is larger than the Euclidean distance between μQ~x,νQ~x,πQ~x and 1/3,1/3,1/3. This manifests that μQ~x,νQ~x,πQ~x is located more nearby to 1/3,1/3,1/3 than that of μP~x,νP~x,πP~x. From a geometrical perspective, the axiom A3′ is reasonable and logical because the closer PFS to the unique point 1/3,1/3,1/3 with maximum entropy reflects the greater entropy of that PFS.We next construct entropies for PFSs based on a probability-type. To formulate the probability-type of entropy for PFSs, we use the idea of entropyHγpof Havrda and Charavát [35] to a probability mass function p=p1,…,pk with(9)Hγp=1γ-11-∑i=1kpiγ,γ≠1γ>0-∑i=1kpilog⁡pi,γ=1.Let X=x1,x2,…,xn be a finite universe of discourses. Thus, for a PFS P~ in X, we propose the following probability-type entropy for the PFS P~ with(10)eHCγP~=1n∑i=1n1γ-11-μP~2xiγ+νP~2xiγ+πP~2xiγ,γ≠1γ>01n∑i=1n-μP~2xilog⁡μP~2xi+νP~2xilog⁡νP~2xi+πP~2xilog⁡πP~2xi,γ=1.Apparently, one may ask a question: “Are these proposed entropy measures for PFSs are suitable and acceptable?” To answer this question, we present the following theorem.Theorem 11.
LetX=x1,x2,…,xn be a finite universe of discourses. The proposed probabilistic-type entropy eHCγ for a PFS P~ satisfies the axioms (A1),A2′,(A3′), and (A4) in Definition 9.To prove the axiomsA2′ and A3′ for Theorem 11, we need Lemma 12.Lemma 12.
Letψγx,0<x<1 be defined as(11)ψγx=1γ-1x-xγ,γ≠1γ>0-xlog⁡x,γ=1.Then ψγx is a strictly concave function of x.Proof.
By twice differentiatingψγx, we get ψγ′′x=-γxγ-2<0, for γ≠1γ>0. Then ψγx is a strictly concave function of x. Similarly, we can show that ψγ=1x=xlog⁡x is also a strictly concave function of x.Proof of Theorem11.
It is easy to check thateHCγ satisfies the axioms A1 and A4. We need only to prove that eHCγ satisfies the axioms A2′ and A3′. To prove eHCγ for the case γ≠1γ>0 that satisfies the axiom A2′, we use Lagrange multipliers for eHCγ with FμP~2,νP~2,πP~2,λ = 1/n∑i=1n1/γ-11-μP~2xiγ+νP~2xiγ+πP~2xiγ + ∑i=1nλiμP~2xi+νP~2xi+πP~2xi-1. By taking the derivative of FμP~2,νP~2,πP~2,λ with respect to μP~2xi,νP~2xi,πP~2xi, and λi, we obtain(12)∂F∂μP~2xi=-γγ-1μP~2xiγ-1+λi=set0,∂F∂νP~2xi=-γγ-1νP~2xiγ-1+λi=set0,∂F∂πP~2xi=-γγ-1πP~2xiγ-1+λi=set0,∂F∂λi=μP~2xi+νP~2xi+πP~2xi-1=set0.From above PDEs, we get μP~2xiγ-1=λiγ-1/γ and then μP~2xi=λiγ-1/γ1/γ-1; νP~2xiγ-1=λiγ-1/γ and νP~2xi=λiγ-1/γ1/γ-1; πP~2xiγ-1=λiγ-1/γ and πP~2xi=λiγ-1/γ1/γ-1; μP~2xi+νP~2xi+πP~2xi=1 and then λiγ-1/γ1/γ-1=1/3. Thus, we have λi=γ/γ-11/3γ-1. We obtain μP~2xi=γ/γ-11/3γ-1γ-1/γ1/γ-1, and then μP~xi=1/3. We also get νP~xi=1/3 and πP~xi=1/3. That is, μP~xi=νP~xi=πP~xi=1/3,∀i. Similarly, we can show that the equation of eHCγ for the case γ=1 also obtains μP~xi=νP~xi=πP~xi=1/3,∀i. By Lemma 12, we learn that the function ψγx is a strictly concave function of x. We know that eHCγP~=1/n∑i=1nψγμP~2xi+ψγνP~2xi+ψγπP~2xi and so eHCγ is also a strictly concave function. Therefore, it is proved that eHCγ attains a unique maximum at μP~xi=νP~xi=πP~xi=1/3,∀i. We next prove that the probabilistic-type entropy eHCγ satisfies the axiom A3′. If P~ is crisper than Q~, we notice that P~ is far away from 1/3,1/3,1/3 compared to Q~ according to Property 10. However, eHCγ is a strictly concave function and eHCγ attains a unique maximum at μP~xi=νP~xi=πP~xi=1/3,∀i. From here, we obtain eHCγP~≤eHCγQ~ if P~ is crisper than Q~. Thus, we prove the axiom A3′.Concept to determine uncertainty from a fuzzy set and its complement was first given by Yager [26]. In this section, we first use the similar idea to measure uncertainty of PFSs in terms of the amount of distinction between a PFS P~ and its complement P~c. However, various distance measures are made to express numerically the difference between two objects with high accuracy. Therefore, the distance between two fuzzy sets plays a vital role in theoretical and practical issues. Let X=x1,x2,…,xn be a finite universe of discourses; we first define a Pythagorean normalized Euclidean (PNE) distance between two Pythagorean fuzzy sets P~,Q~∈PFSsX as(13)ζEP~,Q~=12n∑i=1nμP~2xi-μQ~2xi2+νP~2xi-νQ~2xk2+πP~2xi-πQ~2xi21/2We next propose fuzzy entropy induced by the PNE distance ζE between the PFS P~ and its complement P~c. Let P~=xi,μP~xi,νP~xi:xi∈X be any Pythagorean fuzzy set on the universe of discourse X=x1,x2,…,xn with its complement P~c=x,νP~xi,μP~xi:xi∈X. The PNE distance between PFSs P~ and P~c will be ζEP~,P~c=1/n∑i=1nμP~2xi-νP~2xi21/2. Thus, we define a new entropy eE for the PFS P~ as (14)eEP~=1-ζEP~,P~c=1-1n∑i=1nμP~2xi-νP~2xi2Theorem 13.
LetX=x1,x2,…,xn be a finite universe of discourse. The proposed entropy eE for a PFS P~ satisfies the axioms (A0)~(A5) in Definition 9, and so eE is a σ-entropy.Proof.
We first prove the axiom (A0). Since the distanceζEP~,P~c is between 0 and 1, 0≤1-ζEP~,P~c≤1, and then 0≤eEP~≤1. The axiom (A0) is satisfied. For the axiom (A1), if P~ is crisp, i.e., μP~2xi=0,νP~2xi=1 or μP~2xi=1,νP~2xi=0, ∀xi∈X, then ζEP~,P~c=1/n∑i=1nμP~2xi-νP~2xi2=1. Thus, we obtain eEP~=1-1=0. Conversely, if eEP~=0, then ζEP~,P~c=1-eEP~=1, i.e., 1/n∑i=1nμP~2xi-νP~2xi2=1. Thus, μP~2xi=1,νP~2xi=0 or μP~2xi=0,νP~2xi=1 and so P~ is crisp. Thus, the axiom (A1) is satisfied. Now, we prove the axiom (A2). μP~2xi=νP~2xi,∀xi∈X, implies that ζEP~,P~c=0,eEP~=1. Conversely, eEP~=1 implies ζEP~,P~c=0. Thus, we have that μP~2xi=νP~2xi,∀xi∈X. For the axiom (A3), since μP~2xi≤μQ~2xi and νP~2xi≥νQ~2xi for μQ~2xi≤νQ~2xi imply that μP~2xi≤μQ~2xi≤νQ~2xi≤νP~2xi, then we have the distance ∀xi∈X,1/n∑i=1nμP~2xi-νP~2xi2≥1/n∑i=1nμQ~2xi-νQ~2xi2. Again from the axiom (A3) of Definition 9, we have μP~2xi≥μQ~2xi and νP~2xi≤νQ~2xi for μQ~2xi≥νQ~2xi implies that νP~2xi≤νQ~2xi≤μQ~2xi≤μP~2xi, and then the distance ∀xi∈X,1/n∑i=1nμP~2xi-νP~2xi2≥1/n∑i=1nμQ~2xi-νQ~2xi2. From inequalities 1/n∑i=1nμP~2xi-νP~2xi2≥1/n∑i=1nμQ~2xi-νQ~2xi2 and 1/n∑i=1nμP~2xi-νP~2xi2≥1/n∑i=1nμQ~2xi-νQ~2xi2, we have ζEP~,P~c≥ζHQ~,Q~c, and then 1-ζEP~,P~c≤1-ζEQ~,Q~c. This concludes that eEP~≤eEQ~. In this way the axiom (A3) is proved. Next, we prove the axiom (A4). Since ∀xi∈X,eEP~=1-ζEP~,P~c=1-1/n∑i=1nμP~2xi-νP~2xi2 = 1-1/n∑i=1nνP~2xi-μP~2xi2,1-ζEP~c,P~=eEP~c. Hence, the axiom (A4) is satisfied. Finally, for proving the axiom (A5), let P~ and Q~ be two PFSs. Then, we have(i)
μP~2xi≤μQ~2xiandνP~2xi≥νQ~2xiforμQ~2xi≤νQ~2xi, ∀xi∈X, or(ii)
μP~2xi≥μQ~2xiandνP~2xi≤νQ~2xiforμQ~2xi≥νQ~2xi, ∀xi∈X.
From (i), we have∀xi∈X,μP~2xi≤μQ~2xi≤νQ~2xi≤νP~2xi, then maxμP~2xi,μQ~2xi=μQ~2xi and minνP~2xi,νQ~2xi=νQ~2xi. That is, P~∪Q~=μQ~2xi,νQ~2xi=Q~ which implies that eEP~∪Q~=eEμQ~2xi,νQ~2xi=eEQ~. Also, ∀xi∈X, minμP~2xi,μQ~2xi=μP~2xi and maxνP~2xi,νQ~2xi=νP~2xi, then P~∩Q~=μP~2xi,νP~2xi=P~ implies eEP~∩Q~=eEμP~2xi,νP~2xi=eEP~. Hence, eEP~+eEQ~=eEP~∩Q~+eEP~∪Q~. Again, from (ii), we have ∀xi∈X, νP~2xi≤νQ~2xi≤μQ~2xi≤μP~2xi. Then, maxμP~2xi,μQ~2xi=μP~2xi and minνP~2xi,νQ~2xi=νP~2xi, and so P~∪Q~=μP~2xi,νP~2xi=P~ which implies that eEP~∪Q~=eEμP~2xi,νP~2xi=eEP~. Also, ∀xi∈X,minμP~2xi,μQ~2xi=μQ~2xi and maxνP~2xi,νQ~2xi=νQ~2xi, then P~∩Q~=μQ~2xi,νQ~2xi=Q~ implies that eEP~∩Q~=eEμQ~2xi,νQ~2xi=eEQ~. Hence, eEP~+eEQ~=eEP~∩Q~+eEP~∪Q~. Thus, we complete the proof of Theorem 13.Burillo and Bustince [31] gave fuzzy entropy of intuitionistic fuzzy sets by using intuitionistic index. Now, we modify and extend the similar concept to construct the new entropy measure of PFSs by using Pythagorean index as follows.LetP~ be a PFS on X=x1,x2,…,xn. We define an entropy ePI of P~ using Pythagorean index as(15)ePIP~=1n∑k=1n1-μP~2xk-νP~2xk=1n∑k=1nπP~2xkTheorem 14.
LetX=x1,x2,…,xn be a finite universe of discourse. The proposed entropy ePI for the PFS P~ using Pythagorean index satisfies the axioms (A0)~(A4) and (A5) in Definition 9, and so it is a σ- entropy.Proof.
Similar to Theorem13.We next propose a new and simple method to calculate fuzzy entropy of PFSs by using the ratio of min and max operations. All three componentsμP~2,νP~2,πP~2 of a PFS P~ are given equal importance to make the results more authentic and reliable. The new entropy is easy to be computed. Let P~ be a PFS on X=x1,x2,…,xn, and we define a new entropy for the PFS P~ as (16)emin/maxP~=1n∑i=1nmin⁡μP~2xi,νP~2xi,πP~2xmax⁡μP~2xi,νP~2xi,πP~2xTheorem 15.
LetX=x1,x2,…,xn be a finite universe of discourse. The proposed entropy emin/max for a PFS P~ satisfies the axioms (A0)~(A4) and (A5) in Definition 9, and so it is a σ-entropy.Proof.
Similar to Theorem13.Recently, Xue at el. [20] developed the entropy of Pythagorean fuzzy sets based on the similarity part and the hesitancy part that reflect fuzziness and uncertainty features, respectively. They defined the following Pythagorean fuzzy entropy for a PFS in a finite universe of discourses X=x1,x2,…,xn. Let P~=xi,μP~xi,νP~xi:xi∈X be a PFS in X. The Pythagorean fuzzy entropy, EXueP~, proposed by Xue at el. [20], is defined as(17)EXueP~=1n∑i=1n1-μP~2xi+νP~2xiμP~2xi-νP~2xi.The entropy EXueP~ will be compared and exhibited in next section.


## 4. Examples and Comparisons

In this section, we present simple examples to observe behaviors of our proposed fuzzy entropies for PFSs. To make it mathematically sound and practically acceptable as well as choose better entropy by comparative analysis, we give an example involving linguistic hedges. By considering linguistic example, we use various linguistic hedges like “more or less large”, “quite large” “very large”, “very very large”, etc. in the problems under Pythagorean fuzzy environment to select better entropy. We check the performance and behaviors of the proposed entropy measures in the environment of PFSs by exhibiting its simple intuition as follows.Example 1.
LetP~,Q~, and R~ be singleton element PFSs in the universe of discourse X=x1 defined as P~=x1,0.93,0.35,0.1122, Q~=x1,0.68,0.18,0.7100, and R~=x1,0.68,0.43,0.5939. The numerical simulation results of entropy measures eHC1,eHC2,eE,ePI, and emin/max are shown in Table 1 for the purpose of numerical comparison. From Table 1, we can see the entropy measures of R~ almost have larger entropy than P~ and Q~ without any conflict, except ePI. That is, the degree of uncertainty of R~ is greater than that of P~ and Q~. Furthermore, the behavior and performance of all entropies are analogous to each other, except ePI. Apparently, all entropies eHC1, eHC2, eE, and emin/max behave well, except ePI.Table 1
Comparison of the degree of fuzziness with different entropy measures of PFSs.
P F S s e H C 1 e H C 2 e E e P I e m i n / m a x P ~ 0.4378 0.2368 0.2576 0.0126 0.0146 Q ~ 0.8131 0.5310 0.5700 0.5041 0.0643 R ~ 1.0363 0.6276 0.7225 0.3527 0.3999However, in Example1, it seems to be difficult for choosing appropriate entropy that may provide a better way to decide the fuzziness of PFSs. To overcome it, we give an example with structured linguistic variables to further analyze and compare these entropy measures in Pythagorean fuzzy environment. Thus, the following example with linguistic hedges is presented to further check behaviors and performance of the proposed entropy measures.Example 2.
LetP~ be a PFS in a universe of discourse X=1,3,5,7,9 defined as P~=1,0.1,0.8,3,0.4,0.7,5,0.5,0.3,7,0.9,0.0, 9,1.0,00. By Definitions 6, 7, and 8 where the concentration and dilation of P~ are defined as Concentration: CONP~=P~2, Dilation: DILP~1/2. By considering the characterization of linguistic variables, we use the PFS P~ to define the strength of the structural linguistic variable P~ in X=1,3,5,7,9. Using above defined operators, we consider the following:P~1/2 is regarded as “More or less LARGE”; P~ is regarded as “LARGE”;P~3/2 is regarded as “Quite LARGE”; P~2 is regarded as “Very LARGE”;P~5/2 is regarded as “Quite very LARGE”; P~3 is regarded as “Very very LARGE”.
We use above linguistic hedges for PFSs to compare the entropy measureseHC1,eHC2,eE,ePI, and emin/max, respectively. From intuitive point of view, the following requirement of (18) for a good entropy measure should be followed:(18)eP~1/2>eP~>eP~3/2>eP~2>eP~5/2>eP~3.After calculating these entropy measures eHC1,eHC2,eE,ePI, and emin/max for these PFSs, the results are shown in Table 2. From Table 2, it can be seen that these entropy measures eHC1,eHC2,eE, and emin/max satisfy the requirement of (18), but ePI fails to satisfy (18) that has ePIP~>ePIP~3/2>ePIP~2>ePIP~5/2>ePIP~3>ePIP~1/2. Therefore, we say that the behaviors of eHC1,eHC2,eE, and emin/max are good, but ePI is not.
In order to make more comparisons of entropy measures, we shake the degree of uncertainty of the middle value “5” inX. We decrease the degree of uncertainty for the middle point in X, and then we observe the amount of changes and also the impact of entropy measures when the degree of uncertainty of the middle value in X is decreasing. To observe how different PFS “LARGE” in X affects entropy measures, we modify P~ as(19)“LARGE”=P~1=1,0.1,0.8,3,0.4,0.7,5,0.6,0.5,7,0.9,0.0,9,1.0,00Again, we use PFSs P~11/2,P~1,P~13/2,P~12,P~15/2, and P~13 with linguistic hedges to compare and observe behaviors of entropy measures. The results of degree of fuzziness for different PFSs from entropy measures are shown in Table 3. From Table 3, we can see that these entropies eHC1,eE, and emin/max satisfy the requirement of (18), but eHC2 and ePI could not fulfill the requirement of (18) with eHC2P~>eHC2P~1/2>eHC2P~3/2>eHC2P~2>eHC2P~5/2>eHC2P~3 and ePIP~>P~3/2>ePIP~1/2>ePIP~2>ePIP~5/2>ePIP~3. Therefore, the performance of eHC1,eE, and emin/max is good, and eHC2 is not good, but ePI presents very poor. We see that a little change in uncertainty for the middle value in X did not affect entropies eHC1,eE, and emin/max, and it brings a slight change in eHC2, but it gives an absolute big effect for entropy ePI.
In viewing the results from Tables1, 2, and 3, we may say that entropy measures eHC1,eE, and emin/max present better performance. On the other hand, from the viewpoint of structured linguistic variables, we see that entropy measures eHC1,eE, and emin/max are more suitable, reliable, and well suited in Pythagorean fuzzy environment for exhibiting the degree of fuzziness of PFS. We, therefore, recommend these entropies eHC1,eE, and emin/max in a subsequent application involving multicriteria decision making.
In the following example, we conduct the comparison analysis of proposed entropieseHC1,eE, and emin/max with the entropy EXueP~, developed by Xue at el. [20], to demonstrate the advantages of our developed entropies eHC1,eE, and emin/max.Table 2
Degree of fuzziness from entropies of different PFSs.
PFSs e H C 1 e H C 2 e E e P I e m i n / m a x P ~ 1 / 2 0.6407 0.3923 0.3490 0.2736 0.2013 P ~ 0.6066 0.3762 0.3386 0.3100 0.0957 P ~ 3 / 2 0.5375 0.3311 0.2969 0.3053 0.0542 P ~ 2 0.4734 0.2908 0.2638 0.2947 0.0233 P ~ 5 / 2 0.4231 0.2625 0.2414 0.2843 0.0108 P ~ 3 0.3848 0.2425 0.2267 0.2763 0.0050Table 3
Degree of fuzziness from entropies of different PFSs.
PFSs e H C 1 e H C 2 e E e P I e m i n / m a x P ~ 1 1 / 2 0.6569 0.3952 0.3473 0.2360 0.2276 P ~ 1 1 0.6554 0.4086 0.3406 0.2560 0.1966 P ~ 1 3 / 2 0.5997 0.3757 0.2944 0.2440 0.1201 P ~ 1 2 0.5351 0.3356 0.2526 0.2281 0.0662 P ~ 1 5 / 2 0.4753 0.2993 0.2209 0.2145 0.0329 P ~ 1 3 0.4233 0.2682 0.1976 0.2038 0.0171Example 3.
LetP~,Q~, and R~ be PFSs in the singleton universe set X=x1 as (20)P~=x1,0.305,856,0.4174,Q~=x1,0.1850,0.8530,0.4880,R~=x1,0.4130,0.8640,0.2880.The degrees of entropy for different PFSs between the proposed entropies eHC1,eE,emin/max and the entropy EXueP~ by Xue at el. [20] are shown in Table 4. As can be seen from Table 4, we find that despite having three different PFSs, the entropy measure EXue could not distinguish the PFSs P~,Q~, and R~. However, the proposed entropy measures eHC1,eE, and emin/max can actually differentiate these different PFSs P~,Q~, and R~.Table 4
Degree of fuzziness from entropies of different PFSs.
PFSs E X u e e H C 1 e E e m i n / m a x P ~ 0.4718 0.7532 0.3603 0.1270 Q ~ 0.4718 0.6886 0.3066 0.0470 R ~ 0.4718 0.7264 0.4241 0.1111


## 5. Pythagorean Fuzzy Multicriterion Decision Making Based on New Entropies

In this section, we construct a new multicriterion decision making method. Specifically, we extend the technique for order preference by similarity to an ideal solution (TOPSIS) to multicriterion decision making, based on the proposed entropy measures for PFSs. Impreciseness and vagueness is a reality of daily life which requires close attentions in the matters of management and decision. In real life setting with decision making process, information available is often uncertain, vague, or imprecise. PFSs are found to be a powerful tool to solve decision making problems involving uncertain, vague, or imprecise information with high precision. To display practical reasonability and validity, we apply our proposed new entropieseHC1,eE, and emin/max in a multicriteria decision making problem, involving unknown information about criteria weights for alternatives in Pythagorean fuzzy environment.We formalize the problem in the form of decision matrix in which it lists various project alternatives. We assume that there arem project alternatives and we want to compare them onn various criteria Cj,j=1,2,…,n. Suppose, for each criterion, we have an evaluation value. For instance, the first project on the first criterion has an evaluation x11. The first project on the second criterion has an evaluation x12, and the first project onnth criterion has an evaluation x1n. Our objective is to have these evaluations on individual criteria and come up with a consolidated value for the project 1 and do something similar to the project 2 and so on. We then ultimately obtain a value for each of the projects. Finally, we can rank the projects with selecting the best one among all projects.LetA=A1,A2,…,Am be the set of alternatives, and let the set of criteria for the alternatives Ai,i=1,2,…,m be represented by Cj,j=1,2,…,n. The aim is to choose the best alternative out of then alternatives. The construction steps for the new Pythagorean fuzzy TOPSIS based on the proposed entropy measures are as follows.Step 1 (construction of Pythagorean fuzzy decision matrix).
Consider that the alternativeAi acting on the criteria Cj is represented in terms of Pythagorean fuzzy value b~ij=μij,νij,πijp,i=1,2,…,m,j=1,2,…,n, where μij denotes the degree of fulfillment, νij represents the degree of not fulfillment, and πij represents the degree of hesitancy against the alternative Ai to the criteria Cj with the following conditions: 0≤μij2≤1,0≤νij2≤1,0≤πij2≤1, and μij2+νij2+πij2=1. The decision matrix D~=b~ijm×n is constructed to handle the problems involving multicriterion decision making, where the decision matrix D~=b~ijm×n can be constructed as follows:(21)D~=b~ijm×n=C1A1A2⋮Amμ11,ν11,π11pμ21,ν21,π21p⋮μm1,νm1,πm1pC2μ12,ν12,π12pμ22,ν22,π22p⋮μm2,νm2,πm2p⋯⋯⋯⋮⋯Cnμ1n,ν1n,π1npμ2n,ν2n,π2np⋮μmn,νmn,πmnpStep 2 (determination of the weights of criteria).
In this step, the crux to the problem is that weights to criteria have to be identified. The weights or priorities can be obtained by different ways. Suppose that the criteria information weights are unknown and therefore, the weightswj,j=1,2,3,…,n of criteria for Pythagorean fuzzy entropy measures can be obtained by using eHC1,eE, and emin/max, respectively. Suppose the weights of criteria Cj,j=1,2,…,n are wj,j=1,2,3,…,n with 0≤wj≤1 and ∑j=1nwj=1. Since weights of criteria are completely unknown, we propose a new entropy weighting method based on the proposed Pythagorean fuzzy entropy measures as follows:(22)wj=E~j∑j=1nE~jwhere the weights of the criteria Cj is calculated with E~j=1/m∑i=1mb~ij.Step 3 (Pythagorean fuzzy positive-ideal solution (PFPIS) and Pythagorean fuzzy negative-ideal solution (PFNIS)).
In general, it is important to determine the positive-ideal solution (PIS) and negative-ideal solution (NIS) in a TOPSIS method. Since the evaluation criteria can be categorized into two categories, benefit and cost criteria in TOPSIS, letM1 and M2 be the sets of benefit criteria and cost criteria in criteria Cj, respectively. According to Pythagorean fuzzy sets and the principle of a TOPSIS method, we define a Pythagorean fuzzy PIS (PFPIS) as follows:(23)A+=Cj,μj+,νj+,πj+,whereμj+,νj+,πj+=1,0,0,μj-,νj-,πj-=0,1,0,j∈M1Similarly, a Pythagorean fuzzy NIS (PFNIS) is defined as(24)A-=Cj,μj-,νj-,πj-,whereμj-,νj-,πj-=0,1,0,μj+,νj+,πj+=1,0,0,j∈M2Step 4 (calculation of distance measures from PFPIS and PFNIS).
In this step, we need to use a distance between two PFSs. Following the similar idea from the previously defined PNE distanceζE between two PFSs, we define a Pythagorean weighted Euclidean (PWE) distance for any two PFSs P~,Q~∈PFSX as(25)ζwEP~,Q~=12∑i=1nwiμP~2xi-μQ~2xi2+νP~2xi-νQ~2xi2+πP~2xi-πQ~2xi21/2We next use the PWE distance ζwE to calculate the distances D~+Ai and D~-Ai of each alternative Ai from PFPIS and PFNIS, respectively, as follows:(26)D~+Ai=d~EAi,A+=12∑j=1nwj1-μij22+νij22+1-μij2-νij22D~-Ai=d~EAi,A-=12∑j=1nwjμij22+1-νij22+1-μij2-νij22Step 5 (calculation of relative closeness degree and ranking of alternatives).
The relative closeness degreeN~Ai of each alternative Ai with respect to PFPIS and PFNIS is obtained by using the following expression: (27)N~Ai=D~-AiD~-Ai+D~+AiFinally, the alternatives are ordered according to the relative closeness degrees. The larger value of the relative closeness degrees reflects that an alternative is closer to PFPIS and farther from PFNIS, simultaneously. Therefore, the ranking order of all alternatives can be determined according to ascending order of the relative closeness degrees. The most preferred alternative is the one with the highest relative closeness degree.In the next example, we present a comparison between the proposed entropieseHC1 and emin/max with the entropy EXue by Xue at el. [20] based on PFSs for multicriteria decision making problem. The prime objective of decision makers is to select a best alternative from a set of available alternatives according to some criteria in multicriteria decision making process. Corruption is the misuse and mishandle of public power and resources for private and individual interest and benefits, usually in the form of bribery and favouritism. In addition, corruption twists and manipulates the basis of competitions by misallocating resources and slowing economic activity (Wikipedia).Example 4.
In this example, a real world problem on selection of well renowned national or international audit company is taken into account to demonstrate the comparison analysis among the proposed probabilistic entropyeHC1 and nonprobabilistic entropy emin/max with the entropy EXue [20]. To ensure the transparency and accountability of state run intuitions, the ministry of finance of a developing country offers quotations to select a renowned audit company to get unbiased and fair audit report to keep on tract the economic development of a state. The quotations which are gone through scrutiny process and found successful by a committee are comprised of experts. The quotations which are found successfully by the committee are called eligible while the rest are rejected. A commission of experts is invited to rank the audit companies A1,A2,A3 and to select the best one on the basis of set criteria C1,C2,C3. The descriptions about criteria are given in Table 5, and Pythagorean fuzzy decision matrices are presented in Table 6. The obtained weights of criteria from the entropies eHC1, emin/max, and EXue are shown in Table 7. From Table 7, it is seen that the weights of criteria obtained by the entropy EXue are always the same despite having different alternatives. However, the proposed entropies eHC1 and emin/max correctly differentiate the weights of criteria for each alternative Ai. The weights of criteria in Table 7 are also used to calculate the distances D~+Ai and D~-Ai of each alternative Ai from PFPIS and PFNIS, respectively, where the results are shown in Table 8. Furthermore, the relative closeness degrees of each alternative to ideal solution are shown in Table 9. It can be seen that the relative closeness degrees obtained by the proposed entropies eHC1 and emin/max are different for different alternativesAi, but the relative closeness degrees obtained by the entropy EXue [20] could not differentiate different alternatives so that it gives bias ranking of alternatives. These ranking results of different alternatives by the entropies eHC1, emin/max and EXue are shown in Table 10. As can be seen, the ranking of alternatives by the proposed entropies eHC1 and emin/max is well; however, the entropy EXue [20] could not rank the alternative A1 and A3. It is found that there is no conflict in ranking alternatives by using the proposed Pythagorean fuzzy TOPSIS method based on the proposed entropies eHC1 and emin/max. Totally, the comparative analysis shows that the best alternative is A1.Table 5
Criteria to evaluate an audit company.
Criteria Description of criteria Required experience and capability to make independent decision (C1) Certification and required knowledge on accounting business and taxation law, understanding of management system, auditor should not be trembled or influence by anyone, actions, decision and report should be based on careful analysis. The capability to comprehend different business needs(C2) Ability to work with different companies setups, analytical ability, planning and strategy. Effective communication skills(C3) Mastered excellent communication skills, well versed in compelling report, convincing skills to present their reports, should be patient enough to elaborate points to the entire satisfaction of the auditee.Table 6
Pythagorean fuzzy decision matrices.
Alternatives Criteria Evalution∗ C 1 C 2 C 3 A 1 ( 0.5500,0.4130,0.7259 ) ( 0.6030,0.51400,0.6101 ) ( 0.5000,0.1000,0.8602 ) A 2 ( 0.4000,0.2000,0.8944 ) ( 0.4000,0.2000,0.8944 ) ( 0.4500,0.5050,0.7365 ) A 3 ( 0.5000,0.1000,0.8602 ) ( 0.6030,0.51400,0.6101 ) ( 0.5500,0.4130,0.7259 )Table 7
Weight of criteria.
w 1 w 2 w 3 E X u e 0.3333 0.3333 0.3333 e H C 1 0.2912 0.3646 0.3442 e m i n / m a x 0.2209 0.4640 0.3151Table 8
Distance for each alternative.
E X u e D - A i D + A i e H C 1 D - A i D + A i e m i n / m a x D - A i D + A i A 1 0.7593 0.6476 A 1 0.7587 0.6468 A 1 0.7455 0.6363 A 2 0.8230 0.7842 A 2 0.8208 0.7830 A 2 0.8269 0.7862 A 3 0.7593 0.6476 A 3 0.7493 0.6403 A 3 0.7284 0.6244Table 9
Degree of relative closeness.
E X u e N A i e H C 1 N A i e m i n / m a x N A i A 1 0.5397 A 1 0.5398 A 1 0.5395 A 2 0.5121 A 2 0.5118 A 2 0.5126 A 3 0.5397 A 3 0.5392 A 3 0.5384Table 10
Ranking of alternatives by different methods.
Method Ranking Best alternative E X u e A 2 ≺ A 1 = A 3 None e H C 1 A 2 ≺ A 3 ≺ A 1 A 1 e m i n / m a x A 2 ≺ A 3 ≺ A 1 A 1We next apply the constructed Pythagorean fuzzy TOPSIS in multicriterion decision making for China-Pakistan Economic Corridor projects.Example 5.
A case study in ranking China-Pakistan Economic Corridor projects on priorities basis in the light of related experts’ opinions is used in order to demonstrate the efficiency of the proposed Pythagorean fuzzy TOPSIS being applied to multicriterion decision making. China-Pakistan Economic Corridor (CPEC) is a collection of infrastructure projects that are currently under construction throughout in Pakistan. Originally valued at $46 billion, the value of CPEC projects is now worth $62 billion. CPEC is intended to rapidly modernize Pakistani infrastructure and strengthen its economy by the construction of modern transportation networks, numerous energy projects, and special economic zones. CPEC became partly operational when Chinese cargo was transported overland to Gwadar Port for onward maritime shipment to Africa and West Asia (see Wikipedia). It is not only to benefit China and Pakistan, but also to have positive impact on other countries and regions. Under CPEC projects, it will have more frequent and free exchanges of growth, people to people contacts, and integrated region of shared destiny by enhancing understanding through academic, cultural, regional knowledge, and activity of higher volume of flows in trades and businesses. The enhancement of geographical linkages and cooperation by a win-win model will result in improving the life standard of people, road, rail, and air transportation systems and also sustainable and perpetual development in China and Pakistan.Now suppose the concern and relevant experts are allowed to rank the CPEC projects according to needs of both countries on priorities basis. Assume that initially there are five mega projects which are Gwadar Port (A1), Infrastructure (A2), Economic Zones (A3), Transportation and Energy (A4), and Social Sector Development (A5), according to the following four criteria: time frame and infrastructural improvement (C1), maintenance and sustainability (C2), socioeconomic development (C3), and eco-friendly (C4). A detailed description of such criteria is displayed in Table 11. Consider a decision organization with the five concerns, where relevant experts are authorized to rank the satisfactory degree of an alternative with respect to the given criterion, which is represented by a Pythagorean fuzzy value (PFV). The evaluation values of the five alternatives Ai,i=1,2,…,5 with Pythagorean fuzzy decision matrix are given in Table 12.Table 11
Criteria to assess CPEC projects.
C r i t e r i o n Description of criterion Time frame and infrastructural improvementC1 Roadmap to ensure timely completion of project without any interruption and hurdle and play a vital role in making infrastructural improvement and development Maintenance and sustainabilityC2 The maintenance, repair, reliability and sustainability of the project Socioeconomic developmentC3 Bring visible development and improvement in GDP, economy stability and prosperity, life expectancy, education, health, employment, personal dignity, personal safety and freedom Eco-friendlyC4 Not harmful to the environment, contributes to green living, practices that help conserve natural resources and prevent contribution to air, water and land pollution.Table 12
Pythagorean fuzzy decision matrix.
Alternatives Criteria Evaluation C 1 C 2 C 3 C 4 A 1 ( 0.60,0.50,0.6245 ) ( 0.65,0.45,0.6124 ) ( 0.35,0.70,0.6225 ) ( 0.50,0.70,0.5099 ) A 2 ( 0.80,0.40,0.4472 ) ( 0.80,0.40,0.4472 ) ( 0.70,0.30,0.6481 ) ( 0.60,0.30,0.7416 ) A 3 ( 0.60,0.50,0.6245 ) ( 0.70,0.50,0.5099 ) ( 0.70,0.35,0.6225 ) ( 0.40,0.20,0.8944 ) A 4 ( 0.90,0.30,0.3162 ) ( 0.80,0.35,0.4873 ) ( 0.50,0.30,0.8124 ) ( 0.20,0.50,0.8426 ) A 5 ( 0.80,0.40,0.4472 ) ( 0.50,0.30,0.8124 ) ( 0.70,0.50,0.5099 ) ( 0.60,0.50,0.6245 )We next use entropy measureseHC1,eE, and emin/max based on better performance in numerical analysis to calculate the criteria weights wj using (22). These results are shown in Table 13. From Table 13, we can see that the criteria weights and ranking of weights obtained by each entropy measure are different. We find the distances D~+ and D~- of each alternative from PFPIS and PFNIS using (23) and (24). The results are shown in Table 14. We also calculate the relative closeness degrees N~ of alternatives using (27). The results are shown in Table 15. From Table 15, it can be clearly seen that, under different entropy measures, the relative closeness degrees of alternatives obtained are different, but the gap between these values are considerably small. Thus, the ranking of alternatives is almost the same. The final ranking results from different entropies are shown in Table 16. From Table 16, it can be identified that there is no conflict in selecting the best alternative among alternatives by using the proposed Pythagorean fuzzy TOPSIS method based on the entropies eHC1,eE, and emin/max. There is only one conflict to be found in deciding the preference ordering of alternatives A4 and A5 in eE. Hence, the results of ranking of alternatives according to the closeness degrees are made in an increasing order. Therefore, our analysis shows that the most feasible alternative is A2 which is unanimously chosen by all proposed entropy measures.Table 13
Entropies and weights of the criteria.
w 1 w 2 w 3 w 4 e H C 1 0.2487 0.2563 0.2585 0.2366 e E 0.2063 0.2411 0.2539 0.2987 e m i n / m a x 0.3048 0.2524 0.2141 0.2288Table 14
Distance for each alternative.
e H C 1 D - A i D + A i e E D - A i D + A i e m i n / m a x D - A i D + A i A 1 0.5732 0.6297 A 1 0.5608 0.6354 A 1 0.5825 0.6196 A 2 0.7757 0.4381 A 2 0.7776 0.4557 A 2 0.7741 0.4294 A 3 0.7445 0.5848 A 3 0.7592 0.6055 A 3 0.7377 0.5865 A 4 0.8012 0.5819 A 4 0.7944 0.6163 A 4 0.8049 0.5582 A 5 0.7252 0.5268 A 5 0.7180 0.5331 A 5 0.7302 0.5195Table 15
Degree of relative closeness for each alternative.
e H C 1 N A i e E N A i e m i n / m a x N A i A 1 0.4765 A 1 0.4688 A 1 0.4846 A 2 0.6391 A 2 0.6305 A 2 0.6432 A 3 0.5601 A 3 0.5563 A 3 0.5571 A 4 0.5793 A 4 0.5631 A 4 0.5905 A 5 0.5792 A 5 0.5739 A 5 0.5843Table 16
Ranking of alternative for different entropies.
Method Ranking Best alternative e H C 1 A 1 ≺ A 3 ≺ A 5 ≺ A 4 ≺ A 2 A 2 e E A 1 ≺ A 3 ≺ A 4 ≺ A 5 ≺ A 2 A 2 e m i n / m a x A 1 ≺ A 3 ≺ A 5 ≺ A 4 ≺ A 2 A 2


## 6. Conclusions

In this paper, we have proposed new fuzzy entropy measures for PFSs based on probabilistic-type, distance, Pythagorean index, and min–max operator. We have also extended nonprobabilistic entropy toσ-entropy for PFSs. The entropy measures are considered especially for PFSs on finite universes of discourses. As these are not only used in purposes of computing environment, but also used in more general cases for large universal sets. Structured linguistic variables are used to analyze and compare behaviors and performance of the proposed entropies for PFSs in different Pythagorean fuzzy environments. We have examined and analyzed these comparison results obtained from these entropy measures and then selected appropriate entropies which can be useful and also be helpful to decide fuzziness of PFSs more clearly and efficiently. We have utilized our proposed methods to perform comparison analysis with the most recently developed entropy measure for PFSs. In this connection, we have demonstrated a simple example and a problem involving MCDM to show the advantages of our suggested methods. Finally, the proposed entropy measures of PFSs are applied in an application to multicriterion decision making for ranking China-Pakistan Economic Corridor projects. Based on obtained results, we conclude that the proposed entropy measures for PFSs are reasonable, intuitive, and well suited in handling different kinds of problems, involving linguistic variables and multicriterion decision making in Pythagorean fuzzy environment.



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*Source: 2832839-2018-11-01.xml*