Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

Transcription

1 Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets Xiaolu Zhang, 1 Zeshui Xu, 1 School of Economics and Management, Southeast University, Nanjing 11189, People s Republic of China Business School, Sichuan University, Chengdu, Sichuan , People s Republic of China Recently, a new model based on Pythagorean fuzzy set PFS has been presented to manage the uncertainty in real-world decision-making problems. PFS has much stronger ability than intuitionistic fuzzy set to model such uncertainty. In this paper, we define some novel operational laws of PFSs and discuss their desirable properties. For the multicriteria decision-making problems with PFSs, we propose an extended technique for order preference by similarity to ideal solution method to deal effectively with them. In this approach, we first propose a score function based comparison method to identify the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution. Then, we define a distance measure to calculate the distances between each alternative and the Pythagorean fuzzy positive ideal solution as well as the Pythagorean fuzzy negative ideal solution, respectively. Afterward, a revised closeness is introduced to identify the optimal alternative. At length, a practical example is given to illustrate the developed method and to make a comparative analysis. C 014 Wiley Periodicals, Inc. 1. INTRODUCTION Multicriteria decision making MCDM is to find an optimal alternative that has the highest degree of satisfaction from a set of feasible alternatives characterized with multiple criteria, and these kinds of MCDM problems arise in many real-world situations. Considering the inherent vagueness of human preferences as well as the objects being fuzzy and uncertain, Bellman and Zadeh 1 introduced the theory of fuzzy sets in the MCDM problems. They suggested that the decision maker could employ the u Cj x i to express his preference about the membership degree of an alternative x i with respect to a criterion C j, and means the degree to which the alternative x i satisfies the criterion C j. Usually, the decision maker in the practical Author to whom all correspondence should be addressed; xuzeshui@63.net. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 9, C 014 Wiley Periodicals, Inc. View this article online at wileyonlinelibrary.com..1676

2 106 ZHANG AND XU MCDM process may not only provide the degree to which the alternative x i satisfies the criterion C j but also give the degree to which the alternative x i dissatisfies the criterion C j. To this end, Atanassov presented the concept of intuitionistic fuzzy set IFS, which is characterized by a membership degree and a nonmembership degree satisfying the condition that the sum of its membership degree and nonmembership degree is equal to or less than 1. IFSs have been broadly applied in real-life MCDM problems, and the studies of both methods and applications of MCDM problems with IFSs have received extensive attentions. 3 7 Recently, Yager 8,9 introduced Pythagorean fuzzy set PFS characterized by a membership degree and a nonmembership degree satisfying the condition that the square sum of its membership degree and nonmembership degree is equal to or less than 1, which is a generalization of IFS. The motivation of introducing PFSs is that in the real-life decision process, the sum of the support membership degree and the against nonmembership degree to which an alternative satisfying a criterion provided by the decision maker may be bigger than 1 but their square sum is equal to or less than 1. Yager 9 gave an example to illustrate this situation: one expresses his preference about the degree of an alternative x i in a criterion C j, and he may give the degree to which the alternative x i satisfies the criterion C j as 3 and the degree to which the alternative x i dissatisfies the criterion C j as 1. It is easily seen that , thus this situation cannot be described by using the IFS; but , in other words, which is suitable to employ the PFS to capture it. Obviously, PFSs have more powerful ability than IFSs to model the uncertainty in the practical MCDM problems. For further applications of PFSs to decision making, Yager 9 developed a series of aggregation operations for PFSs. On the basis of new aggregation operations and the comparing method of PFSs, he further put forward a useful decision-making approach to handle the MCDM problems with Pythagorean fuzzy information. Moreover, Yager and Abbasov 10 discussed the relation between Pythagorean membership degrees and complex numbers. They showed that the Pythagorean membership degrees are a subclass of complex numbers called i numbers. And they also proposed a decision-making method based on the Pythagorean fuzzy geometric mean and the order weighted geometric operator to solve the MCDM problems with Pythagorean fuzzy information i.e., i numbers. It can be easily seen that these two decision-making methods put their emphasis on the extensions of the aggregation techniques using to solve the MCDM problems under Pythagorean fuzzy scenarios. In this paper, we will extend the technique for order preference by similarity to ideal solution TOPSIS approach to handle the MCDM problem with Pythagorean fuzzy information. The TOPSIS, proposed by Hwang and Yoon, 11 is a kind of simple and useful method to solve the MCDM problems with crisp numbers, which aims at choosing the alternative with the shortest distance from the positive ideal solution PIS and the farthest distance from the negative ideal solution NIS. Some researchers have extended the TOPSIS method for solving the MCDM problems within a variety of different fuzzy environments over the last decades, such as in fuzzy number contexts, 1 interval fuzzy set contexts, 13

3 EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING 1063 IFS contexts, 14 hesitant fuzzy set contexts, 15 and hesitant fuzzy linguistic term set contexts. 16 But both the TOPSIS method and its extensions fail to solve the MCDM problems with Pythagorean fuzzy information. Following the pioneering works of Yager, 8 10 we will present some novel concepts and theorems for PFSs. Meanwhile, a novel decision-making method based on the TOPSIS will be developed to deal effectively with the MCDM problems with Pythagorean fuzzy information. To do so, the remainder of this paper is organized as follows: In Section, we review briefly some concepts of IFS and present some novel concepts as well as theorems for PFSs. In Section 3, we develop a Pythagorean fuzzy TOPSIS approach to solve the MCDM problems with PFSs. In Section 4, we provide a practical decision-making problem to demonstrate the implementation process of the proposed method and to conduct a comparison analysis. Section 5 presents our conclusions.. IFS AND PFS In this section, we first review the basic concept of IFSs. Then, we present some novel concepts as well as theorems for PFSs. At length, we make a comparison analysis between IFSs and PFSs..1 Intuitionistic Fuzzy Set DEFINITION.1. Let a set X be a universe of discourse. An IFS I is an object having the form, I ={<x, I μ I x,ν I x > x X.1 where the function μ I : X [0, 1] defines the degree of membership and v I : X [0, 1] defines the degree of nonmembership of the element x X to I, respectively, and for every x X, it holds that 0 μ I x + ν I x 1. For any IFS I and x X, π I x = 1 μ I x ν I x is called the degree of indeterminacy of x to I. For simplicity, Xu 17 denoted α = Iμ α,v α asan intuitionistic fuzzy number IFN, where μ α and ν α is the degree of membership and the degree of nonmembership of the element x X to I, respectively. The IFN α = Iμ α,v α has a physical interpretation, for example, if α = I0.3, 0., then it can be interpreted as the vote for resolution is 3 in favor, against, and 5 abstentions. 18 In the real-life decision-making process, people usually utilize the IFN instead of IFS to express their assessments. The basic operational laws of the IFNs α j = Iμ αj,v αj j = 1, and α = Iμ α,v α, developed by Xu and Yager, 17,19 are introduced as follows:

4 1064 ZHANG AND XU 1. α 1 α = Iμ α1 + μ α μ α1 μ α,v α1 v α ;. α 1 α = Iμ α1 μ α,v α1 + v α v α1 v α ; 3. λα = I1 1 μ α λ,vα λ, λ>0; 4. α λ = Iμ λ α, 1 1 v α λ, λ>0; 5. α c = Iv α,u α. However, the decision makers in many real-world MCDM problems may express their preferences about the degree of an alternative x i with respect to a criterion C j satisfying the condition that the sum of the degree to which the alternative x i satisfies the criterion C j and the degree to which the alternative x i dissatisfies the criterion C j is bigger than 1. Obviously, this situation cannot be described by using the IFS. Instead of requiring the decision makers to change their preference information to suit the constraints of IFSs, Yager 8,9 presented a novel concept of PFS to model this situation. In what follows, we introduce its related concepts.. Pythagorean Fuzzy Sets In literature, 8 10 Yager provided three basic representations for Pythagorean membership grades. The first one is a,b satisfying the conditions that a [0, 1], b [0, 1], and a + b 1. The second one is the polar coordinates r, θ satisfying the conditions that r [0, 1] and θ [0,π / ]. The third one is r, d close to the second one satisfying the conditions that r [0, 1], θ [0,π / ], and d = 1 θ / π. Their relationship is that a + b = r, a = r cosθ, b = r sinθ. He referred to a fuzzy subset having these Pythagorean membership grades as a PFS. Similar to the definition of IFSs, in the following, we introduce the general definition of PFSs. DEFINITION.. Let a set X be a universe of discourse. A PFS P is an object having the form P ={<x, Pμ P x,ν P x > x X}.3 where the function μ P : X [0, 1] defines the degree of membership and v P : X [0, 1] defines the degree of nonmembership of the element x X to P,respectively, and for every x X, it holds that μ P x + ν P x 1.4 For any PFS P and x X, π P x = 1 μ P x ν P x is called the degree of indeterminacy of x to P. For simplicity, we call P μ P x,ν P x a Pythagorean fuzzy number PFN denoted by β = P μ β,v β, where μ β,v β [0, 1], π β = 1 μ β v β, and μ β + v β 1. Given three PFNs β 1 = P u β1,v β1, β = P u β,v β and β = P μ β,v β, Yager 8 10 defined the basic operations on them, which can be described as follows:

5 EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING β 1 β = P max{μ β1,μ β }, min{v β1,v β }.. β 1 β = P min{μ β1,μ β }, max{v β1,v β }. 3. β c = P v β,u β. Remark.1. It is noted that the complement operator of PFN β c = P v β,u β is different from the complement operator of IFN α c = Iv α,u α. The definition of the complement operator of IFN is proposed according to the Sugeno class of complements 0 cx = 1 x / 1 + λx λ 1, when λ = 0 i.e., cx = 1 x, while the definition of the complement operator of PFN is proposed according to the Yager class of complements 1, cx = 1 x σ 1 /σ σ 0, when σ = i.e., cx = 1 x. On the basis of relationship between PFNs and IFNs, we further define some novel operations for PFNs as below: 4. β 1 β = P μ β1 + μ β μ β1 μ β,v β1 v β. 5. β 1 β = P u β1 u β, v β1 + v β v β1 v β. 6. λβ = P 1 1 μ β λ, v β λ, λ>0. 7. β λ = P μ β λ, 1 1 v β λ, λ>0. THEOREM.1. For three PFNs β 1 = P u β1,v β1, β = P u β,v β, and β = P μ β,v β, the following ones are valid: 1. β 1 β = β β 1.. β 1 β = β β λ β 1 + β = λβ 1 λβ, λ>0. 4. λ 1 β λ β = λ 1 + λ β, λ 1,λ > β 1 β λ = β 1 λ β λ, λ>0. 6. β λ 1 β λ = β λ 1+λ, λ 1,λ > 0. Proof. For the three PFNs β, β 1 Definition., we can obtain and β, and λ, λ 1,λ > 0, according to 1. β 1 β = P = β β 1 ;. β 1 β = P = β β 1 ; μ β1 + μ β μ β1 μ β,v β1 v β = P μ β + μ β1 μ β μ β1,v β v β1 u β1 u β, v β1 + v β v β1 v β = P u β u β1, v β + v β1 v β v β1

6 1066 ZHANG AND XU 3. λ β 1 β = λp μ + β1 μ β μ β1 μ,v v β β1 β = P = P 1 1 λβ 1 λβ = P = P = P 1 1 λ λ, 1 μ 1 μ λ vβ1 v β1 β β λ, 1 μ + β1 μ λ β μ β1 μ vβ1 v β β ; 1 λ, 1 μ λ λ, vβ1 P 1 1 μ λ vβ β1 β λ λ λ λ 1 μ β μ β 1 1 μ 1 1 μ, λ v β1 β β1 v β λ λ, 1 μ 1 μ λ vβ1 v β1 β β λ, = P 1 1 μ + β1 μ λ β μ β1 μ vβ1 v β β = λ β 1 β ; 4. λ 1 β λ β = P 1 1 μ λ1 λ1 β, vβ P 1 1 μ λ λ β, vβ =P 1 1 μ λ1 β +1 1 μ λ β 1 1 μ λ1 β 1 1 μ λ β, λ1 λ v β vβ = P 1 1 μ λ1+λ β, λ1+λ vβ = λ 1 + λ β; λ 5. β 1 β λ = P u β1 u β, v β1 + v β v β1 v β = P uβ1 u β λ, 1 1 v β 1 + v β v β 1 v β λ uβ1 = P u λ, β 1 1 vβ λ 1 1 v λ β ; β λ 1 β λ μβ1 λ, = P 1 1 vβ λ μβ λ, 1 P 1 1 vβ λ = P uβ1 u β λ, 1 1 v β 1 λ v β λ 1 v β1 λ 1 1 v β λ uβ1 = P u λ, β 1 1 vβ λ 1 1 v λ β = β 1 β λ ; μβ 6. β λ1 β λ λ1 = P, 1 1 vβ λ1 μβ λ P, 1 1 vβ λ = P = P μβ λ1+λ, μβ λ1+λ, vβ λ vβ λ 1 1 vβ λ1 1 1 vβ λ λ1+λ 1 vβ = β λ1+λ ; which complete the proof of the theorem.

7 EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING 1067 DEFINITION.3. 9 Let β j = P u βj,v βj j = 1, be two PFNs, a nature quasiordering on the PFNs is defined as follows: β 1 β if and only if u β1 u β and v β1 v β. In what follows, we develop a score function for PFNs and apply it to compare the magnitudes of two PFNs. DEFINITION.4. Let β = P u β,v β be a PFN, then the score function of β can be defined as follows: sβ = u β v β.5 It is noted that the score function sβ has some desirable properties as below. PROPOSITION.1. For any PFN β = P u β,v β, the proposed score function sβ [ 1, 1]. Proof. Because 0 u β + v β 1, then u β v β u β 1 and u β v β v β 1. Thus, 1 u β v β 1, namely, sβ [ 1, 1]. In particular, if β = P 0, 1, then sβ = 1; if β = P 1, 0, then sβ = 1. PROPOSITION.. For two PFNs β j = P u βj,v βj j = 1,,ifβ 1 β, then sβ 1 sβ. Proof. If β 1 β, then according to Definition.3, we can clearly know that u β1 u β and v β1 v β. Obviously, u β1 v β1 u β v β. Thus, sβ 1 sβ, which completes the proof of the proposition. On the basis of score function of PFNs, we define the following laws to compare two PFNs. DEFINITION.5. Let β j = P u βj,v βj j = 1, be two PFNs, sβ 1 and sβ be the scores of β 1 and β, respectively, then 1. If sβ 1 <sβ,thenβ 1 β ;. If sβ 1 >sβ,thenβ 1 β ; 3. If sβ 1 = sβ,thenβ 1 β Example.1. Let β 1 = P 3, 1 and β = P.4-.6, we have s β 1 = 3 1 = 1,sβ =,, according to Definitions = 0. Apparently, sβ 1 >sβ, thus β 1 β. Additionally, to aggregate PFNs, Yager 9 introduced the following weighted averaging aggregation operator.

8 1068 ZHANG AND XU Figure 1. Comparison of spaces of the PFNs and the IFNs. DEFINITION.6. 9 Let β j = P u βj,v βj j = 1,,...,n be a collection of PFNs and w = w 1,w,...,w n T be the weight vector of β j j = 1,,...,n, where w j indicates the importance degree of β j, satisfying w j 0 j = 1,,...,n and n j=1 w j = 1, and let Pythagorean fuzzy weighted averagingpfwa: n if n PFWAβ 1,β,...,β n = P w j u βj, j=1 then the function PFWA is called the PFWA operator. n w j v βj.6 j=1.3 Comparison Analysis between PFNs and IFNs The main difference between PFN and IFN is their different constraint conditions. According to their definitions introduced in Sections.1 and., we know that the constraint condition of IFN is 0 μ α + ν α 1, whereas the constraint condition of PFN is μ β + ν β 1. Because the fact that for any point a,b a,b [0, 1], if a + b 1 then a + b 1; Yager 9 showed that the space of the Pythagorean membership grade is greater than the space of the intuitionistic membership grade. In other words, if one is an IFN, then it must also be a PFN, but not all PFNs are the IFNs. This result can be easily shown in Figure 1. 9 From the above comparison analysis, we can clearly know that the main advantage of PFN is that it cannot only model the decision situations which the IFN can capture that the sum of the degree provided by the decision maker to which an alternative satisfies a criterion and the degree to which an alternative dissatisfies a criterion is equate to or less than 1, but also model some other situations which the

9 EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING 1069 IFN cannot describe that the sum of the degree to which an alternative satisfies a criterion and the degree to which an alternative dissatisfies a criterion is bigger than 1, but their square sum is equal to or less than 1. Nevertheless, this advantage of PFN comes at the cost that the operations involving PFNs are generally more complex than those involving IFNs TOPSIS APPROACH TO MCDM PROBLEM WITH PYTHAGOREAN FUZZY INFORMATION This section first introduces the MCDM problem under Pythagorean fuzzy environment. Then, an effective decision-making approach is proposed to deal with such MCDM problems. At length, an algorithm of the proposed method is also presented. 3.1 Description of the MCDM Problem with PFNs A MCDM problem can be expressed as a decision matrix whose elements indicate the evaluation values of all alternatives with respect to each criterion. For a given MCDM problem under Pythagorean fuzzy environment, let X ={x 1,x,...,x m } m be a discrete set of m feasible alternatives, C ={C 1,C,...,C n } be a finite set of criteria, and w = w 1,w,...,w n T be the weight vector of all criteria, which satisfy 0 w j 1 and n j=1 w j = 1. We denote the evaluation values of the alternative x i i = 1,,...,m with respect to the criterion C j j = 1,,...,nby C j x i = P u ij,v ij, and R = Cj x i is a Pythagorean fuzzy decision matrix. m n Therefore, the MCDM problem with PFNs can be represented as the following matrix form: R = C j x i m n = x 1 x. x m C 1 C C n P u 11,v 11 P u 1,v 1 P u 1n,v 1n P u 1,v 1 P u,v P u n,v n.... P u m1,v m1 P u m,v m P u mn,v mn 3.1 where each of elements C j x i = P u ij,v ij is a PFN, which means that the degree to which the alternative x i satisfies the criterion C j is the value u ij and the degree to which the alternative x i dissatisfies the criterion C j is the value v ij. 3. The Proposed Decision Approach To effectively solve the aforementioned MCDM problem with PFNs, in the following we propose a Pythagorean fuzzy TOPSIS method. The proposed method is based on the principle that the optimal alternative should have the shortest distance from the PIS and the farthest distance from the NIS.

10 1070 ZHANG AND XU Therefore, this approach starts with the determination of the Pythagorean fuzzy PIS and the Pythagorean fuzzy NIS. Considering that the decision information takes the form of PFNs, we utilize the score function based comparison approach introduced in Definition.4 to identify the Pythagorean fuzzy PIS and the Pythagorean fuzzy NIS. We denote the Pythagorean fuzzy PIS by x +, which can be determined by the following formula: { x + = C j, max i scj x i } j = 1,,,n = { C 1,P u + 1,v+ 1, C,P u +,v+,, Cn,P u + } n,v+ n 3. In the real-life MCDM process, there usually exist no Pythagorean fuzzy PIS. In other words, the Pythagorean fuzzy PIS x + is usual not be the feasible alternative, namely, x + / X. Otherwise, the Pythagorean fuzzy PIS x + is the optimal alternative of the MCDM problem. Then, we proceed to calculate the distance between each alternative and the Pythagorean fuzzy PIS. To this end, we need to define the concept of distance measure for PFNs. DEFINITION 3.1. Let β j = P μ βj,ν βj j = 1, be two PFNs, then we define the distance between β 1 and β as follows: dβ 1,β = 1 μβ1 μ β + vβ1 v β + πβ1 π β 3.3 THEOREM 3.1. Let β j = P μ βj,ν βj j = 1, be two PFNs, then 0 dβ1,β 1. Proof. Because 0 μ β1,v β1,μ β,v β 1, μ β1 + v β1 1, and μ β + v β 1, then dβ 1,β = 1 μ β1 μ β + v β1 v β + 1 μ β1 v β1 1 μ β v β = 1 μβ1 μ β + vβ1 v β + μβ μ β1 + v β v β1 μβ1 + v β1 + μ β + v β = 1 Additionally, according to Definition 3.1, it can be easily seen that dβ 1,β 0. Thus, 0 dβ 1,β 1, which completes the proof of Theorem 3.1. THEOREM 3.. Let β j = P μ βj,ν βj j = 1, be two PFNs, then dβ 1,β = 0, if and only if β 1 = β.

11 EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING 1071 THEOREM 3.3. Let β j = P μ βj,ν βj j = 1, be two PFNs, then dβ 1,β = dβ,β 1. THEOREM 3.4. Let β j = P μ βj,ν βj j = 1,, 3 be three PFNs, if β 1 β β 3, then dβ 1,β dβ 1,β 3 and dβ,β 3 dβ 1,β 3. Proof. If β 1 β β 3, according to Definition.3, we obtain μ β1 μ β μ β3 and v β1 v β v β3. Then, μ β1 μ β μ β1 μ β3, v β1 v β vβ1 v β3 and πβ1 π β = 1 μβ1 v β1 1 μ β v β = μ β μ β1 + v β v β1 1 μ β1 v β1 1 μ β3 v β3 = μβ3 μ β1 + v β3 v β1 = πβ1 π β3 Thus, dβ 1,β = 1 μβ1 μ β + vβ1 v β + πβ1 π β μβ1 μ β3 + vβ1 v β3 + πβ1 π β3 = dβ 1,β 3 1 Analogously, we can also prove dβ,β 3 dβ 1,β 3, which completes the proof of this theorem. Thus, the distance between the alternative x i and the Pythagorean fuzzy PIS x + can be calculated by using Equation 3.3 as follows: D x i,x + = = 1 n w j d C j x i,c j x + j=1 n j=1 w j μij μ + j + vij v + j + πij π + j, i = 1,,...,n 3.4 Usually, the smaller D x i,x + the better the alternative x i, and let D min xi,x + = min 1 i m D x i,x However, the alternative with the closest distance to Pythagorean fuzzy PIS may be not the farthest from Pythagorean fuzzy NIS. We denote the Pythagorean

12 107 ZHANG AND XU fuzzy NIS by x, which can be determined by the following formula: { x = C j, min i scj x i } j = 1,,...,n = { C 1,Pu 1,v 1, C,Pu,v,..., C n,pu n,v n } 3.6 It is easily seen from Equation 3.6 that the obtained value of Pythagorean fuzzy NIS under each criterion is minimal among all the alternatives. Usually, in the practical MCDM process, there may not exist the Pythagorean fuzzy NIS. In other words, the Pythagorean fuzzy NIS x is usually an unfeasible alternative, namely, x / X. Otherwise, the Pythagorean fuzzy NIS x is the worst alternative of the MCDM problem, which should be deleted in the decision process. Using Equation 3.3, the distance between the alternative x i and the Pythagorean fuzzy NIS x can be obtained as follows: D x i,x = = 1 n w j d C j x i,c j x j=1 n μij w j μ j vij + v j πij + π j, j=1 i = 1,,...,m 3.7 In general, the bigger D x i,x the better the alternative x i, and let D max xi,x = max 1 i m D x i,x 3.8 In the classical TOPSIS method, we usually need to calculate the relative closeness of the alternative x i with respect to the Pythagorean fuzzy PIS x + as below: RC x i = D x i,x D x i,x + + D x i,x 3.9 According to the closeness index RC x i, the ranking orders of all alternatives and the optimal alternatives can be determined. However, Hadi-Vencheh and Mirjaberi 3 showed that in some situations, the relative closeness cannot achieve the aim that the optimal solution should have the shortest distance from the PIS and the farthest distance from the NIS, simultaneously. Thus, they suggested that one may use the following formula instead of the relative closeness index i.e., Equation

13 3.9: EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING 1073 ζ x i = D x i,x D max x i,x D xi,x + D min x i,x which is called the revised closeness used to measure the extent to which the alternative x i is close to the Pythagorean fuzzy PIS x + and is far away from the Pythagorean fuzzy NIS x, simultaneously. It can be easily seen that ζ x i 0i = 1,,...,m and the bigger ζ x i, the better the alternative x i. If there exists an alternative x satisfying the conditions that Dx,x = D max x i,x and Dx,x + = D min x i,x +, simultaneously, then ζ x = 0 and, obviously, the alternative x is the best alternative that is closest to the Pythagorean fuzzy PIS x + and farthest away from the Pythagorean fuzzy NIS x, simultaneously. 3.3 The Algorithm of the Proposed Method On the basis of above analysis, in what follows, we present a practical algorithm of the Pythagorean fuzzy TOPSIS approach. The algorithm involves the following steps: Step 1. For a MCDM problem with PFNs, we construct the decision matrix R = C j x i m n where the elements C j x i i = 1,,...,m,j = 1,,...,n are the assessments of the alternative x i X with respect to the criterion C j C. Step. Employ Equations 3. and 3.6 to identify the Pythagorean fuzzy PIS = { C 1 x +,C x +,...,C n x + } and the Pythagorean fuzzy NIS x = { C1 x,c x,...,c n x }, respectively. Step 3. Use Equations 3.4 and 3.7 to calculate the distances between the alternative x i and the Pythagorean fuzzy PIS x + as well as the Pythagorean fuzzy NIS x, respectively. Step 4. Utilize Equation 3.10 to calculate the revised closeness ζ x i of the alternative x i i = 1,,...,m. Step 5. Determine the optimal ranking order of the alternatives and identify the optimal alternative. On the basis of the revised closeness ζ x i obtained from Step 4, we put the alternatives into orders with respect to the decreasing values of ζ x i i = 1,,...,m,and the alternative with the maximal revised closeness ζ x i is the best alternative, namely, x := {x i :i : ζ x i = max 1 l m ζ x l} ILLUSTRATION EXAMPLE In this section, we consider a decision-making problem that concerns the evaluation of the service quality among domestic airlines adapted from the literature 4,5 to illustrate the proposed approach and conduct a comparison analysis.

14 1074 ZHANG AND XU Table I. Criteria for Evaluating Domestic Airlines Criterion Description of criterion Booking and ticketing service C 1 Check-in and boarding process C Cabin service C 3 Responsiveness C 4 Booking and ticketing service involves conveniences of booking or buying ticket, promptness of booking or buying ticket, courtesy of booking or buying ticket Check-in and boarding process consists of convenience check-in, efficient check-in, courtesy of employee, clarity of announcement, and so on Cabin service can be divided into cabin safety demonstration, variety of newspapers and magazines, courtesy of flight attendants, flight attendant willing to help, clean and comfortable interior, in-flight facilities, and captain s announcement Responsiveness consists of fair waiting-list call, handing of delayed flight, complaint handing, and missing baggage handling Table II. Pythagorean Fuzzy Decision Matrix C 1 C C 3 C 4 x 1 P 0.9, 0.3 P 0.7, 0.6 P 0.5, 0.8 P 0.6, 0.3 x P 0.4, 0.7 P 0.9, 0. P 0.8, 0.1 P 0.5, 0.3 x 3 P 0.8, 0.4 P 0.7, 0.5 P 0.6, 0. P 0.7, 0.4 x 4 P 0.7, 0. P 0.8, 0. P 0.8, 0.4 P 0.6, Description Owing to the development of high-speed railroad, the domestic airline marketing has faced a stronger challenge in Taiwan, People s Republic of China. More and more airlines have attempted to attract customers by reducing price. Unfortunately, they soon found that there was a no-win situation and only service quality is the critical and fundamental element to survive in this highly competitive domestic market. To improve the service quality of domestic airline, the civil aviation administration of Taiwan CAAT wants to know which airline is the best in Taiwan and then calls for the others to learn from it. So the CAAT constructs a committee to investigate the four major domestic airlines, which are UNI Air x 1, Transasia x, Mandarin x 3, and Daily Air x 4, according to the following four major criteria: Booking and ticketing service C 1, check-in and boarding process C, cabin service C 3, and responsiveness C 4. The detailed description of the four criteria is given in Table I. 6 The weight vector of the criteria is given by the committee as W = 0.15, 0.5, 0.35, 0.5 T. Assume that the assessment values of the alternatives with respect to each criteria provided by the committee are represented by PFNs as shown in the Pythagorean fuzzy decision matrix given in Table II. The element C 1 x 1 = P 0.9, 0.3 in Table II can be explained that the degree to which the alternative x 1 UNI Air satisfies the criterion C 1 Booking and ticketing service is 0.9

15 EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING 1075 Table III. Results Obtained by the Pythagorean Fuzzy TOPSIS Approach Dx i,x + Dx i,x ζ x i Ranking x x x x and the degree to which the alternative x 1 dissatisfies the criterion C 1 is 0.3, and the others in Table II have the similar meanings. 4. Decision Process In the following, we use the Pythagorean fuzzy TOPSIS approach to solve the decision problem mentioned in Section 4.1. First, we utilize Equations 3. and 3.6 to determine the Pythagorean fuzzy PIS x + and the Pythagorean fuzzy NIS x, respectively, and the results are obtained as follows: x + = {P 0.9, 0.3,P0.9, 0.,P0.8, 0.1,P0.7, 0.4} x = {P 0.4, 0.7,P0.7, 0.6,P0.5, 0.8,P0.6, 0.6} Then, we employ Equations 3.4 and 3.7 to calculate the distances between the alternative x i and the Pythagorean fuzzy PIS x + as well as the Pythagorean fuzzy NIS x, respectively. The results are shown in Table III. Moreover, we utilize Equation 3.10 to calculate the revised closeness ζ x i of the alternative x i, and the results are also listed in Table III. According to ζ x i, we can obtain the ranking of all alternatives as shown in Table III. It is shown in Table III that the optimal ranking order of these four major domestic airlines is x x 3 x 4 x 1, and thus the best alternative is x, namely, Transasia. 4.3 Comparative Analysis In the literature, 9 Yager proposed a useful method based on the PFWA aggregation operator to solve the MCDM problems with Pythagorean fuzzy information. We call this approach Yager s method. To compare Yager s method with the proposed method, here we also utilize it to solve the decision problem mentioned in Section 4.1. Using Equation.6, we 4 j=1 w jv 1j = P , can calculate Cx 1 = P 4 j=1 w ju 1j, Analogously, we can also obtain Cx = P , 0.650,Cx 3 = P , , Cx 4 = P ,

16 1076 ZHANG AND XU Figure. Pictorial representation of the rankings of all alternatives. Using the score function based comparison method, we can further obtain scx = >scx 4 = >scx 3 = >scx 1 = It is easily seen that the optimal ranking order of these four major domestic airlines is x x 4 x 3 x 1, and thus the best alternative is x, namely, Transasia. To provide a better view of the comparison results, we show the results of the rankings of the alternatives obtained by the proposed method and Yager s method, 9 respectively, in Figure. It can be easily seen from Figure that the ranking of the four potential alternatives obtained by the proposed method is quite similar to the result by Yager s method. The best alternative obtained by these two methods is the same, namely, x, and the difference is just the ranking order between x 3 and x 4, that is, x 3 x 4 for the former, whereas x 3 x 4 for the latter. In the real-life decision process, any preference between the two results may be a subjective choice of the decision maker. 5. CONCLUSIONS TOPSIS is one of the classical decision-making methods for solving the MCDM problems with crisp numbers, which has a simple computation process, systematic procedure, and a sound logic that represent the rationale of human choice. 7 In this paper, we have extended the TOPSIS method to deal effectively with the MCDM problems with PFSs. The main contributions of this paper are summarized as follows:

17 EXTENSION OF TOPSIS TO MULTIPLE CRITERIA DECISION MAKING we have defined some new operational laws of PFNs and discussed their desirable properties; we have defined the score function of PFNs and proposed a score function based comparison method for PFNs; 3 we have defined a distance measure for PFNs and discussed its properties; and 4 we have developed a simple and effective decision method to solve the MCDM problem with PFNs. Acknowledgments The authors are very grateful to the Editor-in-Chief R. R. Yager for his insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China No , the Fundamental Research Funds for the Central Universities No.CXZZ , and the Scientific Research Foundation of Graduate School of Southeast University No.YBJJ1339. References 1. Bellman RE, Zadeh LA. Decision-making in a fuzzy environment. Manage Sci 1970;17: B-141 B Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;0: Yager RR. Level sets and the representation theorem for intuitionistic fuzzy sets. Soft Comput 010;14: Xu ZS, Yager RR. Dynamic intuitionistic fuzzy multi-attribute decision making. Int J Approx Reason 008;48: Atanassov KT, Pasi G, Yager RR. Intuitionistic fuzzy interpretations of multi-criteria multiperson and multi-measurement tool decision making. Int J Syst Sci 005;36: Yager RR. OWA aggregation of intuitionistic fuzzy sets. Int J Gen Syst 009;38: Xu ZS, Yager RR. Intuitionistic fuzzy Bonferroni means. IEEE Trans Syst Man Cybern B 011;41: Yager RR. Pythagorean fuzzy subsets. In Proc. Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada, 013. pp Yager RR. Pythagorean membership grades in multi-criteria decision making. IEEE Trans Fuzzy Syst 014;: Yager RR, Abbasov AM. Pythagorean membership grades, complex numbers, and decision making. Int J Intell Syst 013;8: Hwang CL, Yoon KS. Multiple attibute decision methods and applications. Berlin, Germany: Springer; Chen CT. Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets Syst 000;114: Chen TY, Tsao CY. The interval-valued fuzzy TOPSIS method and experimental analysis. Fuzzy Sets Syst 008;159: Boran FE, Genç S, Kurt M, Akay D. A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Expert Syst Appl 009;36: Xu ZS, Zhang XL. Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl-Based Syst 013;5: Beg I, Rashid T. TOPSIS for hesitant fuzzy linguistic term sets. Int J Intell Syst 013;8: Xu ZS. Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 007;15: Gau WL, Buehrer DJ. Vague sets. IEEE Trans Syst Man Cybern 1993;3: Xu ZS, Yager RR. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 006;35:

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