Pythagorean Fuzzy LINMAP Method Based on the Entropy Theory for Railway Project Investment Decision Making Wenting Xue,1 Zeshui Xu,1,2,∗ Xiaolu Zhang,3 Xiaoli Tian2 1 School of Economics and Management, Southeast University, Nanjing, Jiangsu 211189, People’s Republic of China 2 School of Computer and Software, Nanjing University of Information Science and Technology, Nanjing 210044, China 3 The Collaborative Innovation Center, Jiangxi University of Finance and Economics, Nanchang 330013, People’s Republic of China The uncertainty and complexity of the decision-making environment and the subjectivity of the decision makers will lead to the inevitable errors of the decision-making data. A poor decision will be produced with those errors, whereas the linear programming technique for multidimensional analysis of preference (LINMAP) method can adjust such errors through constructing an optimal programming model based on the consistency of the decision-making information, and it has been applied widely in multiple attribute group decision making (MAGDM). Moreover, Pythagorean fuzzy information is useful to simulate the ambiguous and uncertain decision-making environment. Therefore, the LINMAP method under the Pythagorean fuzzy circumstance will be proposed in this paper to solve MAGDM problems. To measure the fuzziness and uncertainty of Pythagorean fuzzy set (PFS) and interval-valued PFS, Pythagorean fuzzy entropy (PFE) and interval-valued PFE (IVPFE) grounded on the similarity and hesitancy parts have been defined, respectively. Then, Pythagorean fuzzy LINMAP (PF LINMAP) methods are constructed on the basis of the PFE and IVPFE correspondingly. Under the given preference relations, the maximum consistency and the amount of knowledge can be realized by the proposed methods. After investigating the relevant indicator system, the decision-making problem concerning railway project investment has been solved through the proposed PF LINMAP method with PFE. Finally, the practicability and effectiveness of the PF LINMAP method has been verified via the comparative analysis with C 2017 Wiley Periodicals, Inc. the existing methods. 1. INTRODUCTION In the process of investment decision making, due to the uncertainty of information, it is difficult for the decision makers (DMs) to provide precise evaluations with respect to some attributes, such as the automobile performance, the dynamic ∗ Author to whom all correspondence should be addressed; e-mail: xuzeshui@263.net INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 00, 1–33 (2017) 2017 Wiley Periodicals, Inc. View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.21941 C 2 XUE ET AL. payback period of investment and the soft power. Instead, the fuzzy set (FS) was developed by Zadeh1 to model these imprecise evaluations in the decision-making process. As an extension of FS,1 the intuitionistic fuzzy set (IFS)2 is characterized by the membership degree and the nonmembership degree satisfying the condition that their sum is less than or equal to 1. The IFS is more convenient to depict the vague and imperfect information than the FS, which has been applied widely in multiple attribute group decision making (MAGDM).3–5 To capture more useful information under imprecise and uncertain circumstances, Yager6 recently proposed the concept of Pythagorean fuzzy set (PFS), which is characterized by the membership and the nonmembership degree satisfying the condition that their sum of square is not larger than 1. The PFS is more general than the IFS because the space of PFS’s membership degree is greater than the space of IFS’s membership degree. For example, when a DM gives the evaluation information whose support degree is 0.7 and against degree is 0.5, it can be known that the intuitionistic fuzzy number (IFN) fails to address this issue because 0.7 + 0.5 > 1. However, it is easily observed that 0.72 + 0.52 < 1, that is to say, the Pythagorean fuzzy number (PFN) is capable of representing this evaluation information. For this case, the PFS shows its wider applicability than the IFS. Following the pioneering work of Yager,6 Zhang and Xu7 introduced the mathematical form of the PFS and presented the concept of PFN. Meanwhile, they also presented the basic operation laws and the score function for PFNs. Afterwards, the division and subtraction of PFNs,8 the continuity, derivability and differentiability of PFNs9 were developed. Meanwhile, a series of different types of Pythagorean fuzzy aggregation operators were proposed, such as the Pythagorean fuzzy weighted average operator,6 Pythagorean fuzzy Choquet integral operators,10 the Pythagorean fuzzy Einstein operators,11 the symmetric Pythagorean fuzzy operators,12 etc. What’s more, the measurement studies on PFSs have also drawn great attention from many scholars, such as the correlation coefficient of PFSs13 similarity measure of PFSs14 and projection model of PFSs15 . Considering some situations that it is not easy for the DMs to employ crisp values for expressing their preferences about the membership function and nonmembership function, Zhang16 proposed the intervalvalued PFN (IVPFN) whose membership degree and nonmembership degree are interval values instead of crisp values and presented some basic operations for IVPFNs. Some recent studies have involved the interval-valued Pythagorean fuzzy aggregation operators17 and the corresponding interval-valued Pythagorean fuzzy decision-making methods.16 Additionally, Liang and Xu18 extended the technique for order preference by similarity to ideal solution (TOPSIS) method to deal with hesitant PFS in multiple criteria decision making. Although these aforementioned works have greatly enriched the theory of PFSs, to our best knowledge, there is few study about the entropy in the setting of PFSs. Entropy is a vital measure of fuzzy or uncertain information in the IFS theory, which has received great attention.19–25 Burille and Bustince19 proposed the definition and the formula of intuitionistic fuzzy entropy (IFE) based on how far the IFS is from being a FS to describe the fuzziness degree of IFS. From the viewpoint of geometrical interpretation of IFS and the definition of entropy, Szmidt and Kacprzyk20 defined a nonprobabilistic-type IFE. To distinguish the difference International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 3 when the membership degree is equal to the nonmembership degree, Guo and Qi21 proposed a novel IFE based on the distance between the IFS and its complement and the hesitancy degree considering the amount of knowledge. Inspired by the axiomatic definition of Ref. 20 and the IFE of Ref. 21, in this paper we will develop the Pythagorean fuzzy entropy (PFE) to measure the fuzziness and uncertainty of PFSs. The PFE consists of the similarity part between a PFS and its complement and the hesitancy part. The similarity part and the hesitancy part reflect the fuzziness and uncertainty of PFSs, respectively. To effectively deal with the MAGDM problems with Pythagorean fuzzy information, various Pythagorean fuzzy decision-making methods, such as Pythagorean fuzzy TOPSIS,7 Pythagorean fuzzy an acronym in Portuguese for interactive multicriteria decision making (TODIM),26 Pythagorean fuzzy qualitative flexible multiple criteria (QUALIFLEX)16 and Pythagorean fuzzy elimination and choice translating reality (ELECTRE)17 have been developed. However, the Pythagorean fuzzy MAGDM problems under different situations or with different characteristics may require different decision-making approaches to solve. For example, the Pythagorean fuzzy TODIM26 is especially suitable to deal with the MAGDM problems with consideration of the DM’s psychological behavior, and the Pythagorean fuzzy QUALIFLEX16 is suitable to handle the MAGDM problems under the situations that the number of attributes markedly exceeds the number of alternatives. However, it is still a challenge for us to effectively handle the Pythagorean fuzzy MAGDM problems with respect to the preference information over alternatives. The linear programming technique for multidimensional analysis of preference (LINMAP), initiated by Srinivasan and Shocker,27 is the most representative method for handling this kind of MAGDM problems with crisp values.28 The key feature of LINMAP is the preference relation over alternatives provided by the DMs. The weights of attributes and the positive ideal solution (PIS) are unknown in the LINMAP method. Based on the given preference relation, the linear programming model, which aims to reach the maximum consistency, is constructed to obtain the weights of attributes and the PIS. The existing methods of determining the consistency degree are mainly based on (1) the distance between the alternative and the PIS29–33 and (2) the inclusion comparison possibilities.34,35 The optimal alternative is the solution with the shortest distance to the PIS. Although the classical LINMAP method has been extended to various fuzzy circumstances, such as the FS environment,29,30 IFS environment31,32 and IVIF environment,33,34 almost all the studies simply replace the decision-making information of the classical LINMAP method with the uncertain information, and neglect the fuzziness and uncertainty of the incomplete information. Thereby, we extend the classical LINMAP method to propose a new Pythagorean fuzzy LINMAP (PF LINMAP) method considering the fuzzy feature of the incomplete information. Since the PFE can measure the fuzziness and uncertainty of PFNs, we introduce the PFE to quantify the amount of valid knowledge in the PF LINMAP method. The purpose of the objective function of the linear programming model in the PF LINMAP method is to reach the maximum consistency and get an amount of knowledge associated with the preference relation given in advance. The developed PF LINMAP method not only inherits the feature International Journal of Intelligent Systems DOI 10.1002/int 4 XUE ET AL. of classical LINMAP but also absorbs the PFE. The main contributions of the paper are listed in the following aspects: 1. The PFE and the interval-valued PFE (IVPFE) are proposed from the perspective of the similarity part and the hesitancy part of PFN and IVPFE, respectively; 2. The PF LINMAP method that can realize high consistency index and get an amount of knowledge based on the consistency index and the PFE is constructed; 3. A case study involving the railway project investment decision making is provided through the modified evaluation indicators by utilizing the PF LINMAP method. The main structure of this paper is organized as follows: Section 2 recalls some fundamental definitions, such as the IFS, the intuitionistic fuzzy LINMAP (IF LINMAP) method, the PFS and the interval-valued PFS (IVPFS). Section 3 introduces the PFE and the IVPFE. Section 4 proposes the PF LINMAP method and the procedure for MAGDM problems. Section 5 provides a case study about the PF LINMAP method and analyzes the reliability and the sensitivity of the proposed method. The paper ends with some conclusions. 2. PRELIMINARIES In this section, we review some basic concepts, such as the IFS, the IF LINMAP method, the PFS and the IVPFS. 2.1. Intuitionistic Fuzzy Sets DEFINITION 2.1.2 Let X be a fixed set, then an IFS A in X can be defined as A = {x, μA (x), υA (x) |x ∈ X} (1) where the functions μA (x) : X → [0, 1], υA (x) : X → [0, 1], x ∈ X ∈ [0, 1] satisfy the condition: 0 ≤ μA (x) + υA (x) ≤ 1. μA (x), υA (x) denote, respectively, the degree of membership and the degree of nonmembership of the element x ∈ X to the set A. Besides, πA (x) = 1 − μA (x) − νA (x) is the degree of hesitancy of the element x ∈ X to the set A. The basic element of an IFS was called an intuitionistic fuzzy number (IFN) by Xu and Yager,35,36 which can be expressed by an ordered pair of nonnegative real numbers (μA (x), υA (x)). 2.2. Description of the IF LINMAP Method The IF LINMAP method,31 a multidimensional preference analysis method based on the unknown intuitionistic fuzzy PIS (IFPIS) and the attribute weight vector, can resolve intuitionistic fuzzy MCDM problems effectively. The main idea of the IF LINMAP is that according to the part or whole preference relations among alternatives and the decision matrix, the consistency and inconsistency degrees International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 5 of alternatives are defined, respectively. Then, an optimal programming model is constructed on the basis of the consistency and inconsistency degrees of alternatives, and the IFPIS and the attribute weight vector are calculated by solving the optimal programming model. Finally, the ranking of alternatives is obtained by calculating the distances between each alternative and the IFPIS. Let X = (xij )m×n (i = 1, 2, . . . , m, j = 1, 2, . . . , n) be a normalized intuitionistic fuzzy (IF) decision matrix, where xij is the rating of the alternative Aj with respect to the attribute Bi . The DM gives the preference relations among the alternatives by ψ = {(l, j )|Al Aj , (l, j = 1, 2, . . . , n), where “Al Aj ” means the alternative Al is noninferior to the alternative Aj . Then the DM calculates the unknown + + + + + + + + IFPIS A+ = (A+ 1 , A2 , . . . , Am ) = ((μ1 , ν1 ), (μ2 , ν2 ), . . . , (μm , νm )) and the atT tribute weight vector W = (w1 , w2 , . . . , wm ) by the consistency and inconsistency indices. The weighted Euclidean distances between each pair of alternatives (l, j ) m + 2 + 2 and the IFPIS A+ are Sl = m i=1 wi [d(xil , Ai )] and Sj = i=1 wi [d(xij , Ai )] , respectively. According to Sj and Sl , the total inconsistency and consistency indices can be derived as follows: The inconsistency index (Sj − Sl )− is defined as (Sj − Sl )− = Sl − Sj 0 (Sj < Sl ) (Sj ≥ Sl ) (2) Then the total inconsistency index B can be represented by: B= (Sj − Sl )− = (l,j )∈ max{0, Sl − Sj } (3) (l,j )∈ The consistency index (Sj − Sl )+ can be obtained by + (Sj − Sl ) = Sj − Sl 0 (Sj ≥ Sl ) (Sj < Sl ) (4) Meanwhile, the total consistency index G is G= (l,j )∈ (Sj − Sl )+ = max{0, Sj − Sl } (l,j )∈ Then, the IF LINMAP method can be constructed as follows: Model 1 max{G} ⎧ G−B ≥h ⎪ ⎪ ⎪ m ⎨ wi = 1 (i = 1, 2, . . . , m) s.t. ⎪ ⎪ i=1 ⎪ ⎩ wi ≥ ε International Journal of Intelligent Systems DOI 10.1002/int (5) 6 XUE ET AL. By Equations (3) and (5), Model 1 can be transformed into the following model: ⎫ ⎧ ⎬ ⎨ λlj Model 2 max ⎭ ⎩ (l,j )∈ ⎧ ⎪ (Sj − Sl ) ≥ h ⎪ ⎪ ⎪ ⎪ (l,j )∈ ⎪ ⎪S − S + λ ≥ 0 ⎪ ⎪ l j lj ⎪ ⎨ ≥ 0 λ lj s.t. ⎪ ⎪ w i ≥ε ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ wi = 1 ⎩ (i = 1, 2, . . . , m) i=1 where λlj = max{0, Sj − Sl }. The decision-making processes of the classical IF LINMAP approach include the following steps: Step 1. Identify the normalized IF decision matrix X = (xij )m×n and the preference relations = {(l, j )|Al Aj , (l, j = 1, 2, . . . , n)} among the alternatives. Step 2. Solve Model 2, and then compute the attribute weight vector W = (w1 , w2 , . . . , wm )T + + + + + + + + and the IFPIS A+ = (A+ 1 , A2 , . . . , Am ) = ((μ1 , ν1 ), (μ2 , ν2 ), . . . , (μm , νm )). Step 3. Calculate the weighted Euclidean distances Sj (j = 1, 2, . . . , n), and then obtain the decreasing ranking orders of the alternatives Aj (j = 1, 2, . . . , n). Step 4. End. 2.3. Pythagorean Fuzzy Sets DEFINITION 2.2.6 Let X be an arbitrary nonempty set. A PFS P is defined as: P = {x, P (μP (x), νP (x)) |x ∈ X } (6) where the functions μP (x), νP (x) : X → [0, 1] satisfy the condition: 0 ≤ μ2P (x) + υP2 (x) ≤ 1. μP (x), υP (x) denote, respectively, the degree of membership and the degree of nonmembership of the element x ∈ X to the set P . In addition, πP (x) = 1 − μ2P (x) − νP2 (x) denotes the degree of hesitancy of the element x ∈ X to the set P . For simplicity, Zhang and Xu7 defined β = P (μβ (x), νβ (x)) as the PFN, where the functions μβ (x), νβ (x) : X → [0, 1] satisfy the condition: 0 ≤ μ2β (x) + υβ2 (x) ≤ 1. Some basic operation laws for PFNs were introduced as follows: DEFINITION 2.3.6,7 Let β = P (μβ , νβ ), β1 = P (μβ1 , νβ1 ) and β2 = P (μβ2 , νβ2 ) be three PFNs, then International Journal of Intelligent Systems DOI 10.1002/int 7 PYTHAGOREAN FUZZY LINMAP METHOD 1. β c = P (νβ , μβ) 2. β1 ⊕ β2 = P ( μ2β1 + μ2β2 − μ2β1 μ2β2 , νβ1 νβ2 ) 3. β1 ⊗ β2 = P (μβ1 μβ2 , νβ21 + νβ22 − νβ21 νβ22 ) λ 4. λβ = P ( 1 − (1 − μ2β ) , (νβ )λ ), λ > 0 λ 5. β λ = P ((μβ )λ , 1 − (1 − νβ2 ) , λ > 0 In what follows, we introduce the Euclidean distance of PFNs. DEFINITION 2.4.26 Let β1 = P (μβ1 , νβ1 ) and β2 = P (μβ2 , νβ2 ) be two PFNs. The Euclidean distance of PFNs can be defined as follows: 2 2 2 1 2 2 2 2 2 2 μβ1 − μβ2 + νβ1 − νβ2 + πβ1 − πβ2 d(β1 , β2 ) = 2 (7) 2.4. Interval-Valued Pythagorean Fuzzy Set DEFINITION 2.5.16 Let X be an ordinary nonempty set. An IVPFS P̃ can be defined as P̃ = x, P̃ μP̃ (x), νP̃ (x) |x ∈ X (8) where the functions μP̃ (x), νP̃ (x) ⊆ [0, 1] satisfy μP̃ (x) = [μLP̃ (x), μUP̃ (x)], 2 2 (μUP̃ (x)) + (νP̃U (x)) ≤ 1. Besides, πP̃ (x) = νP̃ (x) = [νP̃L (x), νP̃U (x)], U L [πP̃ (x), πP̃ (x)] denotes the degree of hesitancy of the element x ∈ X 2 2 L to the set P̃ , where πP̃ (x) = 1 − (μUP̃ (x)) − (νP̃U (x)) and πP̃U (x) = 2 2 1 − (μLP̃ (x)) − (νP̃L (x)) . If μLP̃ (x) = μUP̃ (x) and νP̃L (x) = νP̃U (x), then the IVPFS reduces to the PFS. For convenience, Zhang16 defined β̃ = P̃ ([μLβ̃ (x), μUβ̃ (x)], [νβ̃L (x), νβ̃U (x)]) as 2 2 the IVPFN, which satisfies (μUβ̃ (x)) + (νβ̃U (x)) ≤ 1. DEFINITION 2.6.16 Let β̃ = P̃ ([μLβ̃ (x), μUβ̃ (x)], [νβ̃L (x), νβ̃U (x)]), β̃1 = P̃ ([μLβ̃ (x), 1 μUβ̃ (x)], [νβ̃L (x), νβ̃U (x)]) and β̃2 = P̃ ([μLβ̃ (x), μUβ̃ (x)], [νβ̃L (x), νβ̃U (x)]) be three 1 1 1 2 2 2 2 IVPFNs, then: 1. β̃ c = P̃ ([νβ̃L (x), νβ̃U (x)], [μLβ̃ (x), μUβ̃ (x)]) 1 1 1 1 2 2 2 2 2 2 2 2 2. β̃1 ⊕ β̃2 = P̃ [ (μLβ̃ ) + (μLβ̃ ) − (μLβ̃ ) (μLβ̃ ) , (μUβ̃ ) + (μUβ̃ ) − (μUβ̃ ) (μUβ̃ ) ], 1 2 1 2 1 2 1 2 [νβ̃L νβ̃L , νβ̃U νβ̃U ] 1 2 1 2 International Journal of Intelligent Systems DOI 10.1002/int 8 XUE ET AL. 2 2 2 2 3. β̃1 ⊗ β̃2 = P̃ [μLβ̃ μLβ̃ , μUβ̃ μUβ̃ ], [ (νβ̃L ) + (νβ̃L ) − (νβ̃L ) (νβ̃L ) , 1 2 1 2 1 2 1 2 2 2 2 2 (νβ̃U ) + (νβ̃U ) − (νβ̃U ) (νβ̃U ) ] 1 2 1 2 2 λ 2 λ λ λ 4. λβ̃ = P̃ ([ 1 − (1 − (μLβ̃ ) ) , 1 − (1 − (μUβ̃ ) ) ], [(νβ̃L ) , (νβ̃U ) ]), λ > 0 λ 2 λ 2 λ λ 5. β̃ λ = P̃ ([(μLβ̃ ) , (μUβ̃ ) ], [ 1 − (1 − (νβ̃L ) ) , 1 − (1 − (νβ̃U ) ) ]), λ > 0 In what follows, we present the Euclidean distance of IVPFNs: DEFINITION 2.7. Let β̃1 = P̃ ([μLβ̃ (x), μUβ̃ (x)], [νβ̃L (x), νβ̃U (x)]) and β̃2 = 1 1 1 1 P̃ ([μLβ̃ (x), μUβ̃ (x)], [νβ̃L (x), νβ̃U (x)]) be two IVPFNs. Then the Euclidean dis2 2 2 2 tance of IVPFNs can be derived as follows: d(β̃1 , β̃2 ) = 1 L 2 L 2 2 U 2 U 2 2 + μβ̃1 (x) − μβ̃2 (x) μβ̃1 (x) − μβ̃2 (x) 2 2 2 2 U 2 U 2 2 + νβ̃L1 (x) − νβ̃L2 (x) + νβ̃1 (x) − νβ̃2 (x) 2 2 2 U 2 U 2 2 1/2 + πβ̃L1 (x) − πβ̃L2 (x) + πβ̃1 (x) − πβ̃2 (x) (9) It is easily observed that d(β̃1 , β̃2 ) possesses the following properties: 1. d(β̃1 , β̃2 ) = d(β̃1 , β̃2 ) 2. d(β̃1 , β̃2 ) = 0 only if β̃1 = β̃2 3. 0 ≤ d(β̃1 , β̃2 ) ≤ 1 The proof is similar to that of the Euclidean distance of PFNs.26 3. PYTHAGOREAN FUZZY ENTROPY AND INTERVAL-VALUED PYTHAGOREAN FUZZY ENTROPY 3.1. Pythagorean Fuzzy Entropy The PFE is based on the similarity part and the hesitancy part reflecting the fuzziness and uncertainty feature of PFN, respectively. Previously, the information entropy was introduced as “quantity of information,” and the thermodynamic entropy describes the degree of chaos or disorder in the system. The similarity part, which delivers the “chaos” of PFN, is represented by the similarity degree between βi and its complement βiC . As the higher value of the similarity part leads to the larger PFE, when the membership degree μβ (xi ) is equal to the nonmembership degree νβ (xi ), the situation is so disordered and chaotic that we barely obtain any credible information from the PFN. Therefore, we assume that E(β) = 1 iff μβ (xi ) = νβ (xi ). The similarity part between βi and its complement βiC is 1 − d(β, β C ) = 1 − |μ2β (xi ) − νβ2 (xi )|. Besides, 1 − |μ2β (xi ) − νβ2 (xi )| is a mapping from the distance function to the entropy function. A geometric interpretation of the similarity part is presented in Figure 1. Given that the maximum entropy satisfies μβ (xi ) = νβ (xi ), International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 9 Figure 1. The distance between β and the function μβ (xi ) = νβ (xi ). therefore the farther distance between β and the function μβ (xi ) = νβ (xi ) the larger PFE is. For example, βA = (1, 0), βB = (0, 1) are the farthest points from the function μβ (xi ) = νβ (xi ) and βO = (0, 0), βC = (0.5, 0.5) are the nearest points from between the function μβ (xi ) = νβ (xi ). The distances √ √ the points A, B, O, C and the function μβ (xi ) = νβ (xi ) are dβA = 2/2, dβB = 2/2, dβO = 0, dβC = 0, respectively. Hence, their PFEs should be E(βA ) = 0, E(βB ) = 0, E(βO ) = 1, E(βC ) = 1, respectively. The distance between β√and the function μβ√(xi ) = νβ (xi ) can be expressed by dβ = |μ2β (xi ) − νβ2 (xi )|/ 2, where dβ ∈ [0, 2/2 ]. Considering that the √ range of the value of entropy is from 0 to 1, we construct a linear mapping [0, 2/2] → [1, 0], and then obtain the part entropy method based on the distance: 1 − |μ2β (xi ) − νβ2 (xi )|. It also verifies the rationality of the similarity part from the distance perspective. In addition, we can hardly learn any valuable information when the degree of hesitancy of the element is equal to 1. The hesitancy part expressed by the hesitancy degree πβ (xi ) directly embodies the amount of knowledge. Therefore, we assume that E(β) = 1 iff πβ (xi ) = 1. DEFINITION 3.1. Let βi (i = 1, 2, . . . , n) be a separate element from β, then the PFE EP (βi ) is defined as follows: EP (βi ) = 1 − μ2β (xi ) − νβ2 (xi ) + πβ2 (xi ) − πβ2 (xi ) 1 − μ2β (xi ) − νβ2 (xi ) = 1 − μ2β (xi ) + νβ2 (xi ) μ2β (xi ) − νβ2 (xi ) (10) International Journal of Intelligent Systems DOI 10.1002/int 10 XUE ET AL. Figure 2. The relationship among EP (xi ), πβ2 (xi ) and 1 − |μ2β (xi ) − νβ2 (xi )|. EP (β) = n 1 1 − μ2β (xi ) + νβ2 (xi ) μ2β (xi ) − νβ2 (xi ) n i=1 (11) The above PFE is mainly affected by two aspects: The similarity part is denoted by 1 − |μ2β (xi ) − νβ2 (xi )|, and the hesitancy part is denoted by πβ2 (xi ). The similarity part 1 − |μ2β (xi ) − νβ2 (xi )| is on the basis of the distance between βi and its complement βiC , and the hesitancy part πβ2 (xi ) is the square of the hesitancy degree. Obviously, the value range of the PFE is [0, 1], which is close to the practical situation and can be applied conveniently. Values of the PFE function are plotted by MATLAB to show the relationships among the PFEs, the similarity part and the hesitancy part in Figures 2–4 vividly. The crisp function E : PFE(X) → [0, 1] is an entropy on PFS(X), if the PFE satisfies the following entropic axiomatic definition:20 1. EP (β) = 0 iff β is a crisp set; 2. EP (β) = 1 iff μβ (xi ) = νβ (xi ) for ∀xi ∈ X; 3. EP (β1 ) ≤ EP (β2 ) if β1 is less fuzzy than β2 , i.e., νβ1 (xi ) ≥ νβ2 (xi ) and μβ1 (xi ) ≤ μβ2 (xi ) for μβ2 (xi ) ≤ νβ2 (xi ),∀xi ∈ X, or νβ1 (xi ) ≤ νβ2 (xi ) and μβ1 (xi ) ≥ μβ2 (xi ) for μβ2 (xi ) ≥ νβ2 (xi ),∀xi ∈ X; 4. EP (β) = EP (β C ). Proof. 1. ⇐For ∀xi ∈ X is a crisp set, i.e., μβ (xi ) = 1, νβ (xi ) = 0 or μβ (xi ) = 0, νβ (xi ) = 1, then EP (β) = 0. ⇒ EP (β) = 0 ⇔ (μ2β (xi ) + νβ2 (xi ))|μ2β (xi ) − νβ2 (xi )| = 1. Given that 0 ≤ μ2β (xi ) + νβ2 (xi ) ≤ 1, 0 ≤ |μ2β (xi ) − νβ2 (xi )| ≤ 1, then μ2β (xi ) + νβ2 (xi ) = 1, International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 11 Figure 3. The relationship between EP (xi ) and πβ2 (xi ). Figure 4. The relationship between EP (xi ) and 1 − |μ2β (xi ) − νβ2 (xi )|. |μ2β (xi ) − νβ2 (xi )| = 1. Since 0 ≤ μβ (xi ) ≤ 1, 0 ≤ νβ (xi ) ≤ 1, we obtain μβ (xi ) = 0, νβ (xi ) = 1 or μβ (xi ) = 0, νβ (xi ) = 1 that is, β is a crisp number. 2. ⇐For ∀xi ∈ X, let μβ (xi ) = νβ (xi ), EP (β) = 1 − (μ2β (xi ) + νβ2 (xi )) 2 |μβ (xi ) − νβ2 (xi )| = 1. ⇒ EP (β) = 1 ⇔ (μ2β (xi ) + νβ2 (xi ))|μ2β (xi ) − νβ2 (xi )| = 0 ⇒ μ2β (xi ) + νβ2 (xi ) = 0 or μ2β (xi )− νβ2 (xi ) = 0 ⇒ μβ (xi ) = νβ (xi ) 3. If β1 is less fuzzy than β2 , then we assume νβ1 (x) ≥ νβ2 (x), μβ1 (x) ≤ μβ2 (x) International Journal of Intelligent Systems DOI 10.1002/int 12 XUE ET AL. for μβ2 (xi ) ≤ νβ2 (xi ). EP (β2 ) − EP (β1 ) = n 1 1 − μ2β2 (xi ) + νβ22 (xi ) μ2β2 (xi ) − νβ22 (xi ) n i=1 1 1 − μ2β1 (xi ) + νβ21 (xi ) μ2β1 (xi ) − νβ21 (xi ) − n i=1 n 1 2 μβ1 (xi ) + νβ21 (xi ) μ2β1 (xi ) − νβ21 (xi ) n i=1 n = − μ2β2 (xi ) + νβ22 (xi ) μ2β2 (xi ) − νβ22 (xi ) 1 2 μβ1 (xi ) + νβ21 (xi ) νβ21 (xi ) − μ2β1 (xi ) n i=1 n = − μ2β2 (xi ) + νβ22 (xi ) νβ22 (xi ) − μ2β2 (xi ) 1 4 νβ1 (xi ) − μ4β1 (xi ) − νβ42 (xi ) + μ4β2 (xi ) n i=1 n = 1 2 νβ1 (xi ) − νβ22 (xi ) · νβ21 (xi ) + νβ22 (xi ) = n i=1 n + μ2β2 (xi ) − μ2β1 (xi ) μ2β2 (xi ) + μ2β1 (xi ) Since 0 ≤ νβ21 (xi ) − νβ22 (xi ) ≤ 1 and 0 ≤ μ2β2 (xi ) − μ2β1 (xi ) ≤ 1, we have EP (β2 ) − EP (β1 ) ≥ 0, i.e., EP (β1 ) ≤ EP (β2 ) holds. Similarly, if β2 is less fuzzy than β1 , we have EP (β2 ) ≤ EP (β1 ). 4. 1 1 − νβ2 (xi ) + μ2β (xi ) νβ2 (xi ) − μ2β (xi ) n i=1 n EP (β C ) = 1 1 − μ2β (xi ) + νβ2 (xi ) μ2β (xi ) − νβ2 (xi ) = EP (β) = n i=1 n 3.2. Interval-Valued Pythagorean Fuzzy Entropy Let X = {xi |i = 1, 2, . . . , n} be a universe of discourse. The family of all IVPFSs is denoted by IVPFS(X); we next present the axiomatic definition of IVPFE as follows: International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 13 A crisp function E : IVPFE(X) → [0, 1] is an entropy on IVPFS(X) if IVPFE satisfies the axiomatic definition of Ref. 20 as follows: EP (β̃i ) =1− L2 μβ̃ (xi ) + μUβ̃ 2 (xi ) + νβ̃L2 (xi ) + νβ̃U 2 (xi ) μL2 (xi ) + μUβ̃ 2 (xi ) − νβ̃L2 (xi ) + νβ̃U 2 (xi ) β̃ 4 (12) EP (β̃) L2 n μβ̃ (xi ) + μUβ̃ 2 (xi ) + νβ̃L2 (xi ) + νβ̃U 2 (xi ) μL2 (xi ) + μUβ̃ 2 (xi ) − νβ̃L2 (xi ) + νβ̃U 2 (xi ) 1 β̃ 1− = n i=1 4 (13) 1. EP (β̃) = 0 iff β̃ is a crisp set; 2. EP (β̃) = 1 iff μβ̃ (xi ) = νβ̃ (xi ) for ∀xi ∈ X; 3. EP (β̃1 ) ≤ EP (β̃2 ) if β̃1 is less fuzzy than β̃2 , i.e., μLβ̃ (xi ) ≤ νβ̃L (xi ), μUβ̃ (xi ) ≤ 2 2 2 νβ̃U (xi ), ∀xi ∈ X, or μLβ̃ (xi ) ≥ νβ̃L (xi ), μUβ̃ (xi ) ≥ νβ̃U (xi ), ∀xi ∈ X; 2 2 2 2 2 4. EP (β̃) = EP (β̃ C ). Proof. 1. ⇐For ∀xi ∈ X is a crisp set, i.e., μLβ̃ (xi ) = μUβ̃ (xi ) = 1, νβ̃L (xi ) = νβ̃U (xi ) = 0 or μLβ̃ (xi ) = μUβ̃ (xi ) = 0, νβ̃L (xi ) = νβ̃U (xi ) = 1, then EP (β̃) = 0. ⇒ EP (β̃) = 0 ⇔ (μL2 (xi) + μUβ̃ 2 (xi ) + νβ̃L2 (xi ) + νβ̃U 2 (xi ))|(μL2 (xi ) + μUβ̃ 2 (xi )) β̃ β̃ − (νβ̃L2 (xi ) + νβ̃U 2 (xi ))| = 4 Given that 0 ≤ μL2 (xi ) + μUβ̃ 2 (xi ) + νβ̃L2 (xi ) + β̃ νβ̃U 2 (xi ) ≤ 2, |(μL2 (xi ) + μUβ̃ 2 (xi )) − (νβ̃L2 (xi ) + νβ̃U 2 (xi ))| ≤ |μL2 (xi ) + μUβ̃ 2 (xi )| + β̃ β̃ |νβ̃L2 (xi ) + νβ̃U 2 (xi )| ≤ 2μUβ̃ 2 (xi ) + 2νβ̃U 2 (xi ) ≤ 2, we have μL2 (xi ) + μUβ̃ 2 (xi ) + β̃ νβ̃L2 (xi )+ νβ̃U 2 (xi ) = 2, |(μL2 (xi ) + μUβ̃ 2 (xi )) − (νβ̃L2 (xi ) + νβ̃U 2 (xi ))| = 2. Since β̃ μLβ̃ (xi ), μUβ̃ (xi ), νβ̃L (xi ), νβ̃U (xi ) ∈ [0, 1], we obtain μLβ̃ (xi ) = μUβ̃ (xi ) = 1, νβ̃L (xi ) = νβ̃U (xi ) = 0 or μLβ̃ (xi ) = μUβ̃ (xi ) = 0, νβ̃L (xi ) = νβ̃U (xi ) = 1, that is, β̃ is a crisp set. let μβ̃ (xi ) = νβ̃ (xi ), then EP (β̃) = 1 − 2. ⇐For ∀xi ∈ X, 2 (μL2 (xi )+μU (xi )+νβ̃L2 (xi )+νβ̃U 2 (xi )) β̃ β̃ · |(μL2 (xi ) + μUβ̃ 2 (xi )) − (νβ̃L2 (xi ) + νβ̃U 2 (xi ))| = 1. β̃ ⇒ EP (β̃) =1 ⇔ (μL2 (xi) + μUβ̃ 2 (xi ) +νβ̃L2 (xi) + νβ̃U 2 (xi ))|(μL2 (xi ) + μUβ̃ 2 (xi )) β̃ β̃ −(νβ̃L2 (xi ) + νβ̃U 2 (xi ))| = 0 ⇒ μL2 (xi ) + μUβ̃ 2 (xi ) + νβ̃L2 (xi ) + νβ̃U 2 (xi ) = 0 or β̃ (μL2 (xi ) + μUβ̃ 2 (xi )) − (νβ̃L2 (xi ) + νβ̃U 2 (xi )) = 0. ⇒ μβ̃ (xi ) = νβ̃ (xi ) β̃ 3. If β̃1 is less fuzzy than β̃2 , then we assume μLβ̃ (xi ) ≤ νβ̃L (xi ) and 2 2 μUβ̃ (xi ) ≤ νβ̃U (xi ),∀xi ∈ X. Suppose that μLβ̃ (xi ) ≤ μLβ̃ (xi ), μUβ̃ (xi ) ≤ μUβ̃ (xi ) and 2 2 1 2 1 2 νβ̃L (xi ) ≥ νβ̃L (xi ), νβ̃U (xi ) ≥ νβ̃U (xi ), ∀xi ∈ X, then we have μLβ̃ (xi ) ≤ μLβ̃ (xi ) ≤ 1 2 1 2 1 2 νβ̃L (xi ) ≤ νβ̃L (xi ), μUβ̃ (xi ) ≤ μUβ̃ (xi ) ≤ νβ̃U (xi ) ≤ νβ̃U (xi ). 4 2 1 1 2 2 International Journal of Intelligent Systems 1 DOI 10.1002/int 14 XUE ET AL. EP (β̃2 ) − EP (β̃1 ) L2 L2 U2 n μ (xi ) + μU 2 (xi ) − ν L2 (xi ) + ν U 2 (xi ) μβ̃ (xi ) + μUβ˜ 2 (xi ) + νβL2 1 ˜2 (xi ) + νβ˜2 (xi ) β̃2 β̃2 β̃2 β̃2 2 2 = 1− n i=1 4 − = 1L2 n (xi ) + μUβ̃ 2 (xi ) − νβ̃L2 (xi ) + νβ̃U 2 (xi ) μβ̃ (xi ) + μUβ̃ 2 (xi ) + νβ̃L2 (xi ) + νβ̃U 2 (xi ) μL2 1 β̃1 1 1 1 1 1 1 1− n i=1 4 n 1 L2 (xi ) − μUβ̃12 (xi ) μβ̃1 (xi ) + μUβ̃12 (xi ) + νβ̃L21 (xi ) + νβ̃U12 (xi ) νβ̃L21 (xi ) + νβ̃U12 (xi ) − μL2 β̃1 4n i=1 + n 1 L2 (xi ) + μUβ̃22 (xi ) − νβ̃L22 (xi ) − νβ̃U22 (xi ) μβ̃2 (xi )μUβ̃22 (xi ) + νβ̃L22 (xi ) + νβ̃U22 (xi ) μL2 β̃2 4n i=1 = n 2 2 2 2 1 L2 (xi ) + μUβ̃12 (xi ) + μL2 (xi ) + μUβ̃22 (xi ) − νβ̃L22 (xi ) + νβ̃U22 (xi ) νβ̃1 (xi ) + νβ̃U12 (xi ) − μL2 β̃1 β̃2 4n i=1 = n 1 L2 νβ̃1 (xi ) + νβ̃U12 (xi ) + νβ̃L22 (xi ) + νβ̃U22 (xi ) νβ̃L21 (xi ) + νβ̃U12 (xi ) − νβ̃L22 (xi ) − νβ̃U22 (xi ) 4n i=1 (xi ) + μUβ̃22 (xi ) + μL2 (xi ) + μUβ̃12 (xi ) μL2 (xi ) + μUβ̃22 (xi ) − μL2 (xi ) − μUβ̃12 (xi ) + μL2 β̃2 β̃1 β̃2 β̃1 Since μLβ̃ (xi ) ≤ μLβ̃ (xi ) ≤ νβ̃L (xi ) ≤ νβ̃L (xi ), μUβ̃ (xi ) ≤ μUβ̃ (xi ) ≤ νβ̃U (xi ) ≤ 1 2 2 1 1 2 2 νβ̃U (xi ), then EP (β̃2 ) − EP (β̃1 ) ≥ 0, i.e., EP (β̃1 ) ≤ EP (β̃2 ) holds. Similarly, if β̃2 1 is less fuzzy than β̃1 , then EP (β̃2 ) ≤ EP (β̃1 ). 4. EP (β̃ C ) = n 1 1 1 − νβ̃L2 (xi ) + νβ̃U 2 (xi ) + μL2 (xi ) + μUβ̃ 2 (xi ) νβ̃L2 (xi ) + νβ̃U 2 (xi ) β̃ n i=1 4 2 (xi ) + μUβ̃ (xi ) − μL2 β̃ L2 n μβ̃ (xi ) + μUβ̃ 2 (xi ) + νβ̃L2 (xi ) + νβ̃U 2 (xi ) μL2 (xi ) + μUβ̃ 2 (xi ) − νβ̃L2 (xi ) + νβ̃U 2 (xi ) 1 β̃ = 1− n i=1 4 = EP (β̃) 4. PYTHAGOREAN FUZZY LINMAP METHOD In this section, we extend the classical LINMAP method to develop a new PF LINMAP method. The PF LINMAP method not only inherits the merits of traditional LINMAP method but also improves the uncertain LINMAP method by adding the PFE aspect in the objective function. 4.1. Description of the PF LINMAP Method Let Pijk = (pijk )m×n (i = 1, 2, . . . , m; j = 1, 2, . . . , n; k = 1, 2, . . . , K) be the kth DM’s Pythagorean Fuzzy (PF) decision matrix. pijk = P (μkij , νijk ) is a PFN, International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 15 where μkij , νijk indicate the degrees of satisfaction and dissatisfaction of the kth DM for the alternative Aj with respect to the attribute Bi . The MAGDM problem with PFNs can be shown in the PF decision matrix (pijk )m×n : A1k k A2 k k · · · Ank k ⎞ P μ11 , ν11 P μ12 , ν12 · · · P μ1n , ν1n ⎜ k k k k ⎟ k ⎟ (k = 1, 2, . . . K) ⎜ P μ ,ν P μ22 , ν22 · · · P μk2n , ν2n 21 21 ⎟ ⎜ ⎟ ⎜ . .. .. . . . ⎠ ⎝ k. k k. k . k. k P μm1 , νm1 P μm2 , νm2 · · · P μmn , νmn ⎛ pijk m×n B1 = B2 .. . Bm The kth DM identifies the preference relations among the alternatives by ψ k = {(l, j )|Al Aj , (l, j = 1, 2, . . . , n), where “Al Aj ” means that the alternative Al is noninferior to the alternative Aj . The Pythagorean fuzzy PIS (PF+ + + + + PIS), denoted by P + = (P1+ , P2+ , . . . , Pm+ ) = ((μ+ 1 , ν1 ), (μ2 , ν2 ), . . . , (μm , νm )), T and the attribute weight vector, denoted by W = (w1 , w2 , . . . , wm ) , is unknown in advance. Consequently, the weighted Euclidean distance between the alternative Aj and P + is Sjk = m 2 wi d pijk , Pi+ i=1 2 2 1 k 2 2 2 2 2 2 2 wi μij − (u+ + νijk − (νi+ ) + πijk − (πi+ ) i ) 2 i=1 m = (14) which can also be expressed by the following equation: Sjk = m i=1 wi Rijk + m i=1 ui Cijk + m i=1 vi Hijk + m 4 + 4 + 4 wi (u+ i ) + (νi ) + (πi ) /2 i=1 (15) where ⎧ 4 4 4 2 % ⎪ 2 Rijk = μkij + νijk + πijk − 2 πijk ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ C k = πijk − μkij ⎪ ⎪ ⎨ ij 2 2 Hijk = πijk − νijk ⎪ ⎪ ⎪ 2 ⎪ ⎪ ui = wi (u+ ⎪ i ) ⎪ ⎪ ⎪ ⎩ 2 vi = wi (vi+ ) International Journal of Intelligent Systems DOI 10.1002/int 16 XUE ET AL. Associated with the distances between Sjk and Slk of the pair of alternatives (l, j ) and the PFPIS, we give the total inconsistency index B and the total consistency index G as follows, respectively: B= K − (16) K K + max 0, Sjk − Slk Sjk − Slk = (17) k=1 (l,j )∈ k G= K max 0, Slk − Sjk Sjk − Slk k=1 = k=1 (l,j )∈ k k=1 (l,j )∈ k (l,j )∈ k Considering that the PFE can measure the knowledge of PFS, we introduce the PFE into the PF LINMAP method and denote the total PFE of the proposed method as E. To avoid the situation that the value of E is too large or small compared to the inconsistency index B, we derive the normalized E by the following method: n eijk , where eijk is The total PFE of each attribute is denoted by Ei = K k=1 j =1 the PFE of pijk . The normalized PFE of each attribute is ei = (Ei / m i=1 Ei ) · N(ψ), k = {(l, j )|A where N(ψ) is the number of preference relations ψ l Aj , (l, j = 1, 2, . . . , n). Accordingly, E = m (e · w ) is the total PFE of the PF LINMAP i i i=1 method. The larger value of E implies the less knowledge the PF decision matrix has. Hence, min{E} is one of the objective functions and E ≤ h2 is a constraint function, where h2 is a parameter in the PF LINMAP method. The PF LINMAP method minimizes both the total inconsistency index and the total fuzziness and uncertainty to some degree. The PF LINMAP method is constructed as follows: Model 3 s.t. min{B} min{E} ⎧ G − B ≥ h1 ⎪ ⎪ ⎪ ⎪ ⎪ E ≤ h2 ⎪ ⎪ ⎪ ⎪ ⎨(μ+ )2 + (ν + )2 ≤ 1 i i (i = 1, 2, . . . , m) m ⎪ ⎪ ⎪ ⎪ wi = 1 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩ + wi ≥ ε, μ+ i ≥ 0, νi ≥ 0 The bi-objective nonlinear programming Model 3 can be transformed into the linear programming Model 4 by the parameter θ(θ ∈ [0, 1)). The parameter International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 17 θ can coordinate the PFE part E and the total inconsistency index B. Given that Model 3 is developed based on the LINMAP method, the value of θ would not approach 1. Model 4 min{θE + (1 − θ)B} ⎧ G − B ≥ h1 ⎪ ⎪ ⎪ ⎪ E ≤ h2 ⎪ ⎪ ⎪ 2 + 2 ⎪ ⎨(μi ) + (νi+ ) ≤ 1 (i = 1, 2, . . . , m) s.t. m ⎪ ⎪ ⎪ wi = 1 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩ + wi ≥ ε, μ+ i ≥ 0, νi ≥ 0 By Equations 16 and 17, Model 4 can be transformed into Model 5. Model 5 ⎧ ⎫ m K ⎨ ⎬ min θ ei · wi + (1 − θ) λklj ⎩ ⎭ k i=1 s.t. k=1 (l,j )∈ ⎧ K ⎪ ⎪ ⎪ ⎪ Sjk − Slk ≥ h1 ⎪ ⎪ ⎪ ⎪ k=1 (l,j )∈ k ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ei · wi ≤ h2 ⎪ ⎪ ⎪ ⎪ ⎨ i=1 Sjk − Slk + λklj ≥ 0 ⎪ ⎪ ⎪ ⎪ ui + vi ≤ wi ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ wi = 1 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩w ≥ ε, μ ≥ 0, ν ≥ 0, λk ≥ 0 i i i lj (i = 1, 2, . . . , m) (k = 1, 2, . . . , K; j = 1, 2, . . . , n) + where λklj = max{0, Slk − Sjk }, ui = wi (μ+ i ) , vi = wi (νi ) . For convenience, Model 5 can be represented by Model 6: 2 Model 6 2 ⎫ ⎧ m K ⎬ ⎨ ei · wi + (1 − θ) λklj min θ ⎭ ⎩ k i=1 k=1 (l,j )∈ International Journal of Intelligent Systems DOI 10.1002/int 18 XUE ET AL. ⎧ m m m ⎪ ⎪ ⎪ w R + u C + vi Hi ≥ h1 ⎪ i i i i ⎪ ⎪ ⎪ i=1 i=1 i=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ei · wi ≤ h2 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ m m m ⎪ ⎪ ⎪ k k ⎨ wi Rij l + ui Cij l + vi Hijk l + λklj ≥ 0 s.t. i=1 i=1 i=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ui + vi ≤ wi ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ wi = 1 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩w ≥ ε, μ ≥ 0, ν ≥ 0, λk ≥ 0 i i i lj (i = 1, 2, . . . , m) (k = 1, 2, . . . , K; j = 1, 2, . . . , n) where ⎧ k R = Rijk − Rilk ⎪ ⎪ ⎨ ij l Cijk l = Cijk − Cilk ; ⎪ ⎪ ⎩H k = H k − H k ij l ij il 4.2. ⎧ k 4 k 4 k 4 k 2 % k ⎪ 2 ⎪Rij = μij + νij + πij − 2 πij ⎪ ⎨ 2 2 Cijk = πijk − μkij ⎪ ⎪ ⎪ ⎩H k = π k 2 − ν k 2 ij ij ij Procedure for the MAGDM Problems with the PF LINMAP Method In this subsection, we first introduce the data preprocessing approach and discuss how to determine the DMs’ weight vector. Then, we present some detailed steps of the PF LINMAP approach. 4.2.1. Data Preprocessing Generally, the attributes of the PF decision matrix may include the cost part and the benefit part. The larger value of the cost attribute means worse performance. However, the situation of the benefit attribute is opposite. The original datum p̄ijk can be normalized by the following equation: & p̄ k , for benefit attribute Bi pijk = ijk C (18) p̄ij , for cost attribute Bi C where (p̄ijk ) is the complement of p̄ijk . 4.2.2. Determine the DMs’ Weight Vector Inspired by the idea of TOPSIS, the DM’s weight is given based on the absolute optimal solution (the largest value of the decision-making information), International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 19 denoted by A+ , and the absolute worst solution (the smallest value of the decisionmaking information), denoted by A− . For example, if the decision-making information is P (0.8, 0.4),P (0.7, 0.4),P (0.9, 0.2),P (0.8, 0.4), then A+ = P (1, 0) and A− = P (0, 1). The weights of experts can be derived as follows: (k) (k) First, we calculate each DM’s hybrid distances D+ , D− between pijk and A+ , A− : (k) = D+ n n m m (k) d pijk , A+ , D− = d pijk , A− (k = 1, 2, . . . , K) i=1 j =1 (19) i=1 j =1 Second, we determine the closeness index: c(k) = (k) D− (20) (k) (k) D+ + D− Third, we obtain the weight of experts: c(k) (k) = K (k) k=1 c 4.2.3. (21) Steps of the PF LINMAP Approach The PF LINMAP approach contains the following steps: Step 1. Identify the DMs’ PF decision matrices P̄ijk = (p̄ijk )m×n (k = 1, 2, . . . , K) of the alternatives (A1 , A2 , . . . , An ) with respect to m attributes (B1 , B2 , . . . , Bm ) and the preference relations k = {(l, j )|Plk Pjk , (k = 1, 2, . . . , K; l, j = 1, 2, . . . , n)} among the alternatives. Step 2. Normalize the PF decision matrix P̄ = (p̄ijk )m×n into P = (pijk )m×n by Equation 18. Step 3. Calculate the PFE eijk of each pijk by Equation 10. Step 4. Solve Model 6 according to the simplex method of the linear programming and then obtain the attribute weight vector W = (w1 , w2 , . . . , wm )T and the PFPIS Pi+ . Step 5. Compute Sjk and then aggregate the comprehensive scores of the alternatives Aj (j = (k) k 1, 2, . . . , n) by the equation Sj = K k=1 Sj . Step 6. Rank the alternatives by the decreasing orders of the values of Sj (j = 1, 2, . . . , n). Step 7. End. 4.3. IVPF LINMAP Method To deal with the MAGDM problems under the IVPFS circumstance, we further develop an IVPF LINMAP approach, which is motived by the PF LINMAP approach. Let P̃ijk = (p̃ijk )m×n (i = 1, 2, . . . , m; j = 1, 2, . . . , n; k = 1, 2, . . . , K) be the kth DM’s interval-valued Pythagorean fuzzy (IVPF) decision matrix. p̃ijk = International Journal of Intelligent Systems DOI 10.1002/int 20 XUE ET AL. P̃ (μkij , νijk ) is an IVPFN, where μkij , νijk indicate the degrees of satisfaction and dissatisfaction of the kth DM for the alternative Aj with respect to the attribute Bi . The kth DM provides the preference relations among the alternatives by ψ k = {(l, j )|Al Aj , (l, j = 1, 2, . . . , n), where “Al Aj ” means that the alternative Al is noninferior to the alternative Aj . Then the U+ DMs obtain the unknown IVPFPIS P̃ + = (P̃1+ , P̃2+ , . . . , P̃m+ ) = (([uL+ 1 , u1 ], L+ U+ L+ U+ L+ U+ L+ U+ L+ U+ [ν1 , ν1 ]), ([u2 , u2 ], [ν2 , ν2 ]), . . . , ([um , um ], [νm , νm ])) and the attribute weight vector W = (w1 , w2 , . . . , wm )T by the consistency and inconsistency indices and the IVPFE. The weighted Euclidean distance between alternative Aj and P̃ + is S̃jk = m 2 wi d(p̃ijk , P̃ + ) i=1 1 Lk 2 L+ 2 2 U k 2 U + 2 2 = wi μij − ui + μij − ui 4 i=1 m 2 2 U k 2 U + 2 2 Lk 2 L+ 2 2 2 + νijLk − νiL+ + νij − νi + πij − πi 2 2 2 + πijU k − πiU + (22) which can be derived by the following equation: 1 L 2 Lk U 2 U k 1 Lk ui Cij + ui Cij wi Rij + RijU k + 2 i=1 2 i=1 m S̃jk = m 1 L 2 Lk U 2 U k L+ 4 L+ 4 vi Hij + vi Hij + wi ui + νi 2 i=1 i=1 m + m 4 4 4 4 % + πiL+ + uUi + + νiU + + πiU + 4 (23) Then, the optimal model to derive the weights of attributes and the IVPFPIS is constructed for MAGDM problems as follows: Model 7 ⎫ ⎧ m K ⎬ ⎨ λklj min θ ẽi · wi + (1 − θ ) ⎭ ⎩ k i=1 k=1 (l,j )∈ International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 21 ⎧ m m m L L L L ⎪ ⎪ L U U U ⎪ R + u + vi Hi + viU HiU ≥ h1 w + R C + u C ⎪ i i i i i i i ⎪ ⎪ ⎪ i=1 i=1 i=1 ⎪ ⎪ ⎪ (i = 1, 2, . . . , m) ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ (k = 1, 2, . . . , K; j = 1, 2, . . . , n) ẽi · wi ≤ h2 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎨ m m m L Lk L Lk s.t. wi RijLkl + RijUlk + ui Cij l + uUi CijUlk + vi Hij l + viU HijUlk ⎪ ⎪ ⎪ i=1 i=1 i=1 ⎪ ⎪ k ⎪ ≥ 0 +2λ ⎪ lj ⎪ ⎪ ⎪ uLi + uUi + viL + viU ≤ 2wi ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ wi = 1 ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩ wi ≥ ε, μLi ≥ 0, νiL ≥ 0, λklj ≥ 0 where λklj = max{0, S̃lk − S̃jk } ⎧ Lk R = RijLk − RilLk , RijUlk = RijU k − RilU k ⎪ ⎪ ⎨ ij l CijLkl = CijLk − CilLk , CijUlk = CijU k − CilU k ⎪ ⎪ ⎩H Lk = H Lk − H Lk , H U k = H U k − H U k ij l ij il ij l ij il ⎧ 4 Lk 4 Lk 4 2 % ⎪ 2, + νij + πij − 2 πijLk RijLk = μLk ⎪ ij ⎪ ⎪ ⎪ % ⎪ 4 4 4 2 ⎪ ⎪ 2 RijU k = μUij k + νijU k + πijU k − 2 πijU k ⎪ ⎪ ⎪ ⎪ ⎪ ⎨C Lk = π Lk 2 − μLk 2 , C U k = π U k 2 − μU k 2 ij ij ij ij ij ij 2 2 2 ⎪H Lk = π Lk − ν Lk , H U k = π U k − ν U k 2 ⎪ ⎪ ij ij ij ij ij ij ⎪ ⎪ ⎪ 2 2 ⎪ L+ L ⎪ , uUi = wi uUi + ⎪ ⎪ui = wi ui ⎪ ⎪ ⎪ 2 2 ⎩ L vi = wi viL+ , viU = wi viU + The procedure with the IVPF LINMAP method for MAGDM problems is omitted, since it is similar to the procedure of the PF LINMAP method. 5. CASE STUDY 5.1. The Application of the PF LINMAP Method On March 5, 2017, the National People’s Congress and the Chinese People’s Political Consultative Conference were convened in Beijing. During the conference, the railway investment became a hot topic and Chinese Premier Keqiang Li International Journal of Intelligent Systems DOI 10.1002/int 22 XUE ET AL. mentioned about the railway construction many times and reported that China will invest over 11.5 billion dollars to railway construction in this year. At present, as the international economic situation is in a deep transformation, only by further strengthening the close collaboration between countries, can we overcome the current difficulties and achieve win–win cooperation. China’s “the belt-and-road” strategy was presented from a global perspective and won wide recognition of many countries. The operation of central European multinational international trains not only promotes the development of the China–EU economic and trade exchanges but also fully demonstrates the capability of Chinese railway. In recent years, with the technological progress and engineering innovation, China’s high-speed railway is developed rapidly and accounts for over 65% of the world’s high-speed rail operating system. In “the belt-and-road” strategy, the railway becomes the pioneers of Chinese output advanced technology. More and more countries have expressed their willingness to cooperate with China in the railway field. China’s high-speed rail has three advantages: (1). advanced, safe and reliable technology, (2). better price/performance ratio, and (3). operating experience. Owning to these outstanding merits, China’s railway has obtained many railway or high-speed rail projects in the other countries, such as Russia, Mexico, America and Indonesia. After browsing the web page of China Railway,37 we found that some countries, including Germany, Singapore and Russia, have intentions to cooperate with China in the railway field. Hence, take Germany, Russia as the alternatives A1 and A2 , respectively. Since China will participate in the competitive bidding of Kuala Lumpur–Singapore rail project and Malaysia–Singapore rail project, we take them as the alternatives A3 and A4 , respectively. We surveyed the indicator system of railway project selection after information consultation on the basis of the financial evaluation and the noneconomic evaluation. The financial evaluation indicators mainly include profitability analysis and solvency analysis, which can be conducted in five aspects: • Financial Internal Rate of Return (FIRR). The FIRR on an investment is defined as the rate of return that sets the net present value (PV) of total cash flows from the investment equal to 0.38 FIRR, the main dynamic evaluation index of investigating project’s profitability, reflects the profitability of capital occupied by the project to some degree. The DMs can use the FIRR to compare the profitability of capital projects in capital budgeting in terms of the rate of return. • Net present value (NPV). The NPV is a measurement of the profitability of an undertaking computed by subtracting the PVs of cash outflows from the PVs of cash inflows over a period of time.39 • Investment recovery period. The investment recovery period can be divided into the static payback period, which does not consider the time value of money, and the dynamic payback period, which considers the time value of money. • Debt ratio and current ratio. The debt ratio is a financial ratio that indicates the percentage of a corporation’s assets provided via debt.40 The current ratio is a liquidity ratio that measures whether or not a corporation has enough capacity to repay its short-term obligations.41 • Repayment period of loan. The diplomatic factor plays an important role in the international cooperation, such as American’s economic aid to Japan after World War II. The Tanzania– International Journal of Intelligent Systems DOI 10.1002/int 23 PYTHAGOREAN FUZZY LINMAP METHOD Table I. The decision matrix P 1 = (pij1 )m×n B1 B2 B3 B4 B5 B6 A1 A2 A3 A4 P (0.7,0.6) P (0.8,0.6) P (0.5,0.5) P (0.4,0.7) P (0.9,0.4) P (0.4,0.9) P (0.9,0.4) P (0.8,0.6) P (0.7,0.7) P (0.9,0.3) P (0.8,0.2) P (0.9,0.4) P (0.7,0.5) P (0.4,0.3) P (0.9,0.1) P (0.8,0.2) P (0.7,0.4) P (0.6,0.6) P (0.9,0.3) P (0.7,0.2) P (0.4,0.3) P (0.9,0.4) P (0.5,0.4) P (0.6,0.7) Table II. The decision matrix P 2 = (pij2 )m×n B1 B2 B3 B4 B5 B6 A1 A2 A3 A4 P (0.5,0.4) P (0.3,0.9) P (0.4,0.3) P (0.9,0.4) P (0.3,0.7) P (0.4,0.5) P (0.9,0.3) P (0.8,0.5) P (0.7,0.6) P (0.9,0.2) P (0.9,0.2) P (0.8,0.3) P (0.7,0.6) P (0.5,0.3) P (0.8,0.1) P (0.9,0.1) P (0.7,0.3) P (0.5,0.5) P (0.3,0.9) P (0.2,0.3) P (0.3,0.7) P (0.5,0.8) P (0.7,0.6) P (0.4,0.7) Zambia railway is the first railway project that China built in overseas in 1970. During a period of China’s economic hardship, it is more accurate to define the project of Tanzania–Zambia railway a diplomatic tactic rather than an economy investment. Hence, apart from the profitability analysis and solvency analysis, we also present the noneconomic factors, such as public benefit and diplomatic influence. • Public benefit and diplomatic influence. In the case, the above six attributes are denoted by the attributes Bi (i = 1, 2, . . . , 6). We invited three DMs to identify the PF decision matrices P̄ k = (p̄ijk )m×n (k = 1, 2, . . . , 3) and the preference relations ψ 1 = {(3, 2), (4, 1), (3, 1)}, ψ 2 = {(2, 1), (4, 3), (2, 4), (3, 1)}, ψ 3 = {(3, 1), (3, 4), (2, 3), (2, 4)}. Next, we employed the developed PF LINMAP approach to solve the above decision-making problem. First, we normalize P̄ k = (p̄ijk )m×n into P k = (pijk )m×n shown in Tables I–III by using Equation 18. According to the decision matrices, we know A+ = P (1, 0), A− = P (0, 1). (k) (k) Second, we calculate D+ , D− and the closeness index c(k) by using Equations (19) and (20), respectively, and then the DMs’ weights (k) (k = 1, 2, 3) are determined by using Equation (21). All the above-calculated results are shown in Table IV. Third, by Model 6, we construct the linear programming model as follows: min θ(1.7447w 6 ) 1 + 2.0833w2 + 2.0792w3 + 1.3268w4 + 1.8960w5 + 1.8699w +(1 − θ) λ132 + λ141 + λ131 + λ221 + λ243 + λ224 + λ231 + λ331 + λ334 + λ323 + λ324 International Journal of Intelligent Systems DOI 10.1002/int 24 XUE ET AL. Table III. The decision matrix P 3 = (pij3 )m×n B1 B2 B3 B4 B5 B6 A1 A2 A3 A4 P (0.4,0.3) P (0.3,0.4) P (0.5,0.4) P (0.6,0.6) P (0.7,0.7) P (0.3,0.7) P (0.8,0.3) P (0.7,0.1) P (0.8,0.2) P (0.9,0.1) P (0.8,0.3) P (0.8,0.1) P (0.2,0.5) P (0.3,0.4) P (0.8,0.6) P (0.9,0.1) P (0.3,0.1) P (0.6,0.4) P (0.6,0.8) P (0.4,0.1) P (0.2,0.4) P (0.7,0.1) P (0.6,0.2) P (0.8,0.1) Table IV. The overall values of each region Region (k) D+ (k) D− (k) c (k) k=1 k=2 k=3 5.8455 12.7155 0.6851 0.3640 8.5402 11.8302 0.5808 0.3086 9.1739 14.7339 0.6163 0.3274 ⎧ −0.3392w1 + 1.7257w2 − 2.7253w3 − 1.1311w4 + 0.4993w5 + 0.4953w6 + 1.7500u1 − 0.6800u2 + 5.7000u3 ⎪ ⎪ ⎪ ⎪ +2.6800u4 − 0.0200u5 + 1.1400u6 − 1.1800v1 − 3.3100v2 + 2.2800v3 − 0.7900v4 − 1.7500v5 − 1.26v6 ≥ h1 ⎪ ⎪ ⎪ ⎪ 1.7447w1 + 2.0833w2 + 2.0792w3 + 1.3268w4 + 1.8960w5 + 1.8699w6 ≤ h2 ⎪ ⎪ ⎪0.3862w1 + 0.7215w2 + 0.0560w3 + 0.3003w4 +0.0927w5 + 0.4225w6 − 0.5500u1 − 1.2300u2 + 0.1600u3 ⎪ ⎪ ⎪ ⎪ ⎪ −0.3900u4 − 0.1800u5 − 0.7000u6 − 0.1400v1 − 1.0200v2 − 0.6400v3 − 0.27v4 + 0.09v5 − 0.05v6 + λ132 ≥ 0 ⎪ ⎪ ⎪−0.1910w + 0.5083w + 0.1394w − 0.4672w + 0.6832w + 0.2652w + 0.3700u − 0.6200u − 0.3400u ⎪ 1 2 3 4 5 6 1 2 3 ⎪ ⎪ ⎪ 1 ⎪ ⎪+0.9700u4 − 1.1200u5 + 0.0800u6 − 0.2200v1 − 0.7900v2 − 0.4100v3 − 0.01v4 − 0.56v5 − 0.44v6 + λ41 ≥ 0 ⎪ ⎪ ⎪ ⎪0.1210w1 + 0.7215w2 − 0.4768w3 − 0.0927w4 + 0.4672w5 + 0.4225w6 − 0.1100u1 − 1.2300u2 + 0.8800u3 ⎪ ⎪ ⎪ +0.5100u4 − 0.6400u5 − 0.0500u6 − 0.2200v1 − 1.0200v2 + 0.0800v3 − 0.42v4 − 0.32v5 − 0.7v6 + λ131 ≥ 0 ⎪ ⎪ ⎪ ⎪ −0.6090w1 + 0.1050w2 − 0.4980w3 + 0.1212w4 − 0.3978w5 − 0.3472w6 + 1.0500u1 + 0.5400u2 + 0.9300u3 ⎪ ⎪ ⎪ ⎪ −0.1200u4 + 0.9900u5 + 0.8000u6 + 0.4200v1 − 0.5700v2 + 0.8700v3 − 0.24v4 − 0.18v5 + 0.16v6 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪+λ21 ≥ 0 − 0.1910w1 + 0.0798w2 + 0.1238w3 + 0.0322w4 − 0.2538w5 − 0.1566w6 − 0.3500u1 − 0.4200u2 ⎪ ⎪ −0.62u ⎪ 3 − 0.4900u4 + 0.2700u5 + 0.0600u6 + 0.5000v1 − 0.2100v2 + 0.4100v3 + 0.7v4 + 0.54v5 + 0.39v6 ⎪ ⎪ 2 ⎪ ⎪ ⎨+λ43 ≥ 0 − 0.00w1 − 0.6188w2 − 0.2538w3 − 0.0580w4 − 0.1440w5 − 0.1312w6 + 0.7200u1 + 1.3600u2 + 2 s.t. 0.6700u3 + 0.5200u4 + 0.3200u5 + 0.5600u6 − 0.7200v1 + 0.92v2 + 0.14v3 − 0.64v4 − 0.32v5 − 0.32v6 + λ24 ≥ 0 ⎪ ⎪−0.4180w1 + 0.6440w2 − 0.3680w3 + 0.1470w4 + 0.0000w5 − 0.0594w6 + 0.6800u1 − 0.40u2 + 0.88u3 ⎪ ⎪ ⎪ ⎪−0.1500u4 + 0.4000u5 + 0.1800u6 + 0.6400v1 − 1.2800v2 + 0.32v3 − 0.3v4 − 0.4v5 + 0.09v6 + λ231 ≥ 0 ⎪ ⎪ ⎪ ⎪−0.0260w1 − 0.6415w3 − 0.2755w4 + 0.7112w5 + 0.0795w6 − 0.0800u1 + 0.9800u3 + 0.5500u4 ⎪ ⎪ ⎪ ⎪−1.2800u5 + 0.2100u6 + 0.2000v1 + 0.7900v3 − 0.2500v4 − 1.3600v5 − 0.3900v6 + λ331 ≥ 0 ⎪ ⎪ ⎪ 0.6955w1 − 0.0208w2 − 0.7360w3 − 0.4192w4 + 0.1365w5 + 0.2033w6 − 1.0300u1 + 0.0100u2 + 1.40u3 ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎪+0.6400u4 − 0.5700u5 − 0.4100u6 − 1.10v1 + 0.2300v2 + 1.00v3 + 0.32v4 − 0.33v5 + 0.02v6 + λ34 ≥ 0 ⎪ ⎪ ⎪ −0.4012w − 0.1970w + 0.3328w − 0.4662w − 0.2033w + 1.0400u + 0.6500u − 0.3200u 1 2 3 5 6 1 2 3 ⎪ ⎪ 3 ⎪ ⎪ ⎪+1.1800u5 + 0.4100u6 + 0.2800v1 + 0.1000v2 − 0.6400v3 + 0.7100v5 − 0.0200v6 + λ23 ≥ 0 ⎪ ⎪ ⎪ 0.2943w1 − 0.2178w2 − 0.4032w3 − 0.4192w4 − 0.3297w5 + 0.0100u1 + 0.6600u2 + 1.0800u3 ⎪ ⎪ ⎪ ⎪ +0.6400u4 + 0.6100u5 − 0.8200v1 + 0.3300v2 + 0.3600v3 + 0.3200v4 + 0.3800v5 + λ324 ≥ 0 ⎪ ⎪ ⎪ ⎪ui + vi ≤ wi (i = 1, 2, . . . , m) ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ w = 1 i ⎪ ⎪ ⎪ ⎪ ⎩ i=1 wi ≥ ε, μi ≥ 0,νi ≥ 0,λklj ≥ 0 In this case, let θ = 0.4, h1 = 0.1, h2 = 3, ε = 0.001, the above model can be solved by the simplex method of the linear programming and the attribute weight + vector and the PFPIS Pi+ = (μ+ i , νi ) are obtained as given below: W = (0.0807, 0.0010, 0.0175, 0.8413, 0.0010, 0.0585)T , μ+ i = (0, 0, 0.0001, 0.7294, 1, 1), νi+ = (1, 0, 1, 0.4984, 0, 0). International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 25 Table V. The weighted Euclidean distances Sjk Sjk j =1 j =2 j =3 j =4 k=1 k=2 k=3 0.1086 0.0634 0.0249 0.0599 0.0643 0.0693 0.0342 0.0693 0.0696 0.0627 0.1229 0.0696 Table VI. The ranking results of alternatives with different values of θ (h1 = 0.1, ε = 0.001) θ (h1 = 0.1, ε = 0.001) 0.1 0.2 0.3 0.4 0.5 Ranking results A2 A2 A3 A3 A2 A3 A3 A2 A2 A3 A4 A4 A4 A4 A4 θ (h1 = 0.1, ε = 0.001) A1 A1 A1 A1 A1 0.6 0.7 0.8 0.9 0.9999 Ranking results A3 A3 A3 A3 A3 A2 A2 A2 A2 A2 A1 A1 A1 A1 A1 A4 A4 A4 A4 A4 Then, the weighted Euclidean distances Sjk (k = 1, 2, 3; j = 1, 2, 3, 4) between the alternatives Aj and A+ are calculated by using Equation 7 and the calculated results are listed in Table V. (k) k The comprehensive scores Sj = K k=1 Sj (j = 1, 2, 3, 4) of the alternatives are obtained as follows: S1 = 0.0567, S2 = 0.0643, S3 = 0.0673, S4 = 0.0835 Thus, the ranking orders of the alternatives is A1 A2 A3 A4 , so A1 is the optimal alternative. 5.2. 5.2.1. Decision Support for the PF LINMAP Method The Reliability Analysis of the PF LINMAP Method According to the description of Model 6, θ is the parameter that adjusts the proportion of PFE part and the inconsistency index. If θ = 0, then the PF LINMAP method is reduced to the IF LINMAP. Since the PF LINMAP method is developed based on the LINMAP method, θ = 1. In the following, we study the ranking variation tendency of alternatives with different values of θ and h1 = 0.1, ε = 0.001. In the three-dimensional Figures 5, 7 and 9, to view the figure conveniently, we let the relative score be (Sjmax − Sj )/(Sjmax − Sjmin ). The highest relative score of the alternative implies that it works better than others. The three-dimensional Figure 5 shows the results of alternative with θ varying from 0 to 0.9999. Figures 5 and 6 and Table VI show the rough regulations: (1) If the value of θ is close to 0, then the optimal alternative is A2 ; (2) If the value of θ is close to 1, then the optimal alternative is A3 ; (3) If the value of θ is around 0.5, then the optimal alternative is A3 . Obviously, different values of θ may lead to different ranking results. If θ = 0, then the PF LINMAP method reduces to the previous LINMAP method. Especially, International Journal of Intelligent Systems DOI 10.1002/int 26 XUE ET AL. Figure 5. The ranking order results of alternatives θ (h1 = 0.1, ε = 0.001) varying from 0 to 0.9999. Figure 6. The comparative analysis results of 1 − Sj with different values of θ (h1 = 0.1, ε = 0.001). International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 27 Figure 7. The ranking order results of alternatives with ε(θ = 0.5, h1 = 0.1) varying from 10−1 to 10−6 . the situations that θ = 0 and θ = 0 obtain quite different ranking results. It can be concluded that the adding of PFE would have great impact on the ranking orders. Since the PFE part and the inconsistency index part of the PF LINMAP method are adjusted by the linear programming model, the relative score is not linear. Therefore, in Table VI, the optimal alternative is A3 when θ = 0.5 and the ranking order is different with others. In addition, the DMs can flexibly adjust the size of the parameter θ. The larger the value of θ, the more important the PFE part in the PF LINMAP method. 5.2.2. The Sensitivity Analysis on the Parameters ε and h1 In general, the value of ε should be given in advance. Given that six attributes are adopted in the case, the value of ε is less than 1/6. Thus, we study the results of alternative with ε varying from 10−1 to 10−6 in Figure 7. From Figures 7 and 8 and Table VII, it is easy to find that (1) The PF LINMAP method shows strong robustness when ε ≤ 10−2 ; (2) the relative scores of alternatives vary dramatically if ε ∈ (10−1 , 10−2 ), which causes different ranking orders of alternatives. The model has weak robustness when ε = 0.1, that is, ∀wi ≥ 0.1, i = (1, 2, . . . , n). Thus, if the decision-making problem has many attributes, the value of ε should diminish accordingly. When the number of attributes is six, we suggest ε < 0.01 so as to keep the robustness of the model. In the actual operation, the linear programming model appears to be infeasible when h1 ≥ 3. We investigate the results of alternative with h1 (θ = 0.5, ε = 0.001) International Journal of Intelligent Systems DOI 10.1002/int 28 XUE ET AL. Figure 8. The comparative analysis results of 1 − Sj with different values of ε(θ = 0.5, h1 = 0.1). Table VII. The ranking orders of alternatives with different values of the parameters ε(θ = 0.5, h1 = 0.1) 10−1 10−2 10−2.5 10−3 10−3.5 10−4 10−4.5 10−5 10−5.5 10−6 Ranking results A2 A3 A2 A2 A2 A2 A2 A2 A2 A2 A3 A2 A3 A3 A3 A3 A3 A3 A3 A3 A4 A4 A4 A4 A4 A4 A4 A4 A4 A4 h1 (θ = 0.5, ε = 0.001) A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ranking results A2 A2 A2 A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 A2 A2 A2 A2 A2 A2 A2 A4 A4 A4 A4 A4 A4 A4 A4 A4 A4 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 varying from 0 to 1 in Figures 9 and 10 and Table VII roughly show that (1) If h1 ∈ [0, 0.4), then the optimal alternative is A2 ; (2) if h1 ∈ [0.4, 1), then the optimal alternative is A3 . Theoretically, the value of h1 is unnecessarily too large, because h1 represents the lowest threshold value of the consistency index G minus the inconsistency index B. We suggest that the appropriate range of h1 is from 0 to 0.4. Note that the value of h2 should be larger than 1.4, or the linear programming model appears to be infeasible. When h2 ≥ 1.4, the ranking order results of alternatives are almost unchangeable. Therefore, we do not specifically analyze the parameter h2 . International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 29 Figure 9. The relative scores of alternatives with h1 (θ = 0.5, ε = 0.001) varying from 0 to 1. Figure 10. The comparative analysis results of 1 − Sj with different values of h1 (θ = 0.5, ε = 0.001). International Journal of Intelligent Systems DOI 10.1002/int 30 XUE ET AL. Table VIII. The comparison analysis of methods Methods Types of attribute values The method of determining DMs’ weight The main idea of the method Types of problems IF LINMAP28 IFN Average MAGDM IVIF LINMAP29 IVIFN \ PF TOPSIS7 PF TODIM15 PFN PFN \ Average PF LINMAP PFN, IVPFN The closeness index The inconsistency index and consistency index The inclusion comparison possibilities The revised closeness The dominance of the alternative The PFE, the inconsistency index and consistency index MADM MADM MAGDM MAGDM Table IX. The ranking results of alternatives derived by different methods Methods Ranking results A4 A2 A5 A1 A3 A2 A5 A1 A4 A3 A2 A4 A5 A1 A3 PF TOPSIS PF TODIM PF LINMAP The DMs can flexibly select the appropriate values of θ, h1 , ε according to the actual application. 5.3. Comparative Analysis In this section, we compare the PF LINMAP method with some related methods, such as the IF LINMAP method,32 the PF TOPSIS method7 and the PF TODIM method26 in the method analysis and the specific data analysis. We adopt the decision metrics of Ref. 26, and then aggregate the decisionmaking information by the PF TODIM approach (θ = 2.5) and the PF TOPSIS approach with the same experts’ weight. In the PF LINMAP method, suppose that the preference relations provided by the experts are ψ 1 = {(2, 5), (5, 1), (1, 4)}, ψ 2 = {(2, 1), (4, 3), (2, 4), (1, 3)}, ψ 3 = {(2, 4), (4, 3), (1, 5)}, and the values of the related parameter are θ = 0.4, h1 = 0.1, h2 = 3, ε = 0.001, then the result comparison is listed in Table IX. Table IX shows that the optimal alternative is A4 derived by the PF TOPSIS method, whereas the optimal alternative is A2 by utilizing the PF TODIM method and the PF LINMAP method. Although the ranking orders of alternatives are remarkably different by employing different methods, the worst alternative is still A3 in the above three decision-making approaches. From Tables VIII and IX, we find that the PF TOPSIS, the PF TODIM and the PF LINMAP are distinctly different in both the method analysis and the ranking order. This difference is mainly because International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 31 the PF TOPSIS method and the PF TODOM method purely depend on the data provided by the DMs. Owing to the subjectivity of the DMs and the uncertainty, enormous quantity and complexity of the data, some data are very likely incredible and inconsistent with others. It is necessary to give some preference relations to verify the rationality and consistency of the data. Meanwhile, the PF LINMAP model absorbs the PFE, which considers the uncertainty and fuzziness of the uncertain information, so as to improve the quality of decision. Combining the preference relations, the decision-making matrices and the PFE, we construct the linear programming model, which can realize the maximum consistency and reliability of data. Hence, the ranking results based on the PF LIMAP method are more reliable and reasonable than the ranking results derived by the PF TOPSIS method and the PF TODIM method. Based on these analyses, the merits of the PF LINMAP method can be summarized as follows: 1. The PF LINMAP method considers the consistency degrees of the preference relations and decision-making matrices; 2. The fuzzy and uncertain feature of decision-making information is captured by means of PFE in the proposed method; 3. The proposed method is robust, reliable and flexible with the proper parameters. 6. CONCLUSIONS Owing to the complexity and uncertainty of the decision-making environment, the data given subjectively by the DMs may be incredible or inconsistent with others. Therefore, it is quite necessary to study the consistency and reliability of the decisionmaking information. Fortunately, the LINMAP method was developed to adjust the consistency of the decision-making data represented by crisp numbers according to the preference relations over alternatives provided in advance but it fails to deal with the PFNs. However, to deal with the MAGDM problems with PFNs, it may be not suitable to study the LINMAP with crisp numbers simply or just introduce the Pythagorean fuzzy information to the classical LINMAP method. Thus, we have defined a PFE to measure the amount of knowledge of the Pythagorean fuzzy information. Then, based on the PFE, the PF LINMAP method has been constructed, and it has been used to deal with Pythagorean fuzzy information in the decision making of railway project investment. Later, the feasibility and necessity of the PFE part which measures the fuzziness and the amount of knowledge in the PF LINMAP model has been demonstrated via the reliability analysis. The maximum consistency and reliability of the data can be obtained by the modified linear programming model, i.e., the PF LINMAP method. Hence, the ranking of results based on the PF LIMAP method considering the uncertainty and inconsistency of the decisionmaking information is quite reasonable and reliable. Acknowledgments The work was partly supported by the National Natural Science Foundation of China (nos. 71571123, 71771155, 71661010). International Journal of Intelligent Systems DOI 10.1002/int 32 XUE ET AL. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. Zadeh LA. Fuzzy sets. Inform Control 1965;8:338–356. Atanassov KT. Intuitionistic fuzzy set. Fuzzy Sets Syst 1986;20:87–96. Xu ZS, Cai XQ. Recent advances in intuitionistic fuzzy information aggregation. Fuzzy Optim Decis Mak 2010;9 (4):359–381. Xu ZS. Intuitionistic fuzzy preference modeling and interactive decision making. New York: Springer; 2013. Merigó JM, Gil-Lafuente AM. New decision making techniques and their application in the selection of financial products. Inform Sci 2010;180(11):2085–2094. Yager RR. Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 2014;22:958–965. Zhang XL, Xu ZS. Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 2014;29:1061–1078. Peng XD, Yang Y. Some results for Pythagorean fuzzy sets. Int J Intell Syst 2015;30(11):1133–1160. Gou XJ, Xu ZS, Ren PJ. The properties of continuous Pythagorean fuzzy information. Int J Intell Syst 2016;31(5):401–424. Peng XD, Yang Y. Pythagorean fuzzy Choquet integral based MABAC method for multiple attribute group decision making. Int J Intell Syst 2016;31(10):989–1020. Garg H. Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process. Int J Intell Syst 2017;32(6):597–630. https://doi.org/10.1002/int.21860. Ma ZM, Xu ZS. Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multicriteria decision-making problems. Int J Intell Syst 2016;31(12):1198–1219. Garg H. A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision-making processes. Int J Intell Syst 2016;31(12):1234–1252. Zhang XL. A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int J Intell Syst 2016;31(6):593–611. Liang DC, Xu ZS, Darko AP. Projection model for fusing the information of Pythagorean fuzzy multicriteria group decision making based on Geometric Bonferroni mean. Int J Intell Syst 2017;32(9):966–987. Zhang XL. Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Inform Sci 2016;330:104–124. Peng XD, Yang Y. Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators. Int J Intell Syst 2016;31(5):444–487. Liang DC, Xu ZS. The new extension of TOPSIS method for MCDM with hesitant Pythagorean fuzzy sets. Appl Soft Comput 2017;60:167–179. Burillo P, Bustince H. Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 1996;78(3):305–316. Szmidt E, Kacprzyk J. Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 2001;118(3):467–477. Guo KH, Song Q. On the entropy for Atanassovs intuitionistic fuzzy sets: an interpretation from the perspective of amount of knowledge. Appl Soft Comput 2014;24:328–340. Hung WL, Yang MS. Fuzzy entropy on intuitionistic fuzzy sets. Int J Intell Syst 2006;21(4):443–451. Xia MM, Xu ZS. Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment. Inform Fusion 2012;13(1):31–47. Li JQ, Deng GN, Li HX, Zeng WY. The relationship between similarity measure and entropy of intuitionistic fuzzy sets. Inform Sci 2012;188(4):314–321. Zhao N, Xu ZS. Entropy measures for interval-valued intuitionistic fuzzy information from a comparative perspective and their application to decision making. Informatica 2016;27(1):203–228. International Journal of Intelligent Systems DOI 10.1002/int PYTHAGOREAN FUZZY LINMAP METHOD 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 33 Ren PJ, Xu ZS, Gou XJ. Pythagorean fuzzy TODIM approach to multi-criteria decision making. Appl Soft Comput 2016;42:246–259. Srinivasan V, Shocker AD. Linear programming techniques for multidimensional analysis of preference. Psychometrika 1973;38 (3):337–342. Zhang XL, Xu ZS, Xing XM. Hesitant fuzzy programming technique for multidimensional analysis of hesitant fuzzy preferences. OR Spectrum 2016;38(3):789–817. Bereketli I, Genevois ME, Albayrak YE, Ozyol M. WEEE treatment strategies’ evaluation using fuzzy LINMAP method. Exp Syst Appl 2011;38 (1):71–79. Li DF, Yang JB. Fuzzy linear programming technique for multiattribute group decision making in fuzzy environments. Inform Sci 2004;158(1):263–275. Li DF. Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment. Fuzzy Optim Decis Makg 2008;7(1):17–34. Li DF, Chen GH, Huang ZG. Linear programming method for multiattribute group decision making using IF sets. Inform Sci 2010;180(9):1591–1609. Chen TY. An interval-valued intuitionistic fuzzy LINMAP method with inclusion comparison possibilities and hybrid averaging operations for multiple criteria group decision making. Knowl-Based Syst 2013;45:134–146. Chen TY. The inclusion-based LINMAP method for multiple criteria decision analysis within an interval-valued Atanassov’s intuitionistic fuzzy environment. Int J Inform Technol Decis Mak 2014;13(6):1325–1360. Xu ZS, Yager RR. Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 2006;35(4):417–433. Xu ZS. Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 2007;15(6):1179– 1187. China Railway. Available at http://www.china-railway.com.cn/gjhz/gtjl/, accessed April 16, 2017. Internal rate of return – Wikipedia. Available at https://en.wikipedia.org/wiki/Internal_rate_ of_return, accessed April 26, 2017. Net present value – Wikipedia. Available at https://en.wikipedia.org/wiki/Net_present_value, accessed April 26, 2017. Debt ratio – Wikipedia. Available at https://en.wikipedia.org/wiki/Debt_ratio, accessed April 27, 2017. Current ratio – Wikipedia. Available at https://en.wikipedia.org/wiki/Current_ratio, accessed April 27, 2017. 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