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A Fermatean Fuzzy ELECTRE Method for Multi-Criteria Group Decision-Making

Abstract

This paper aims to develop a Fermatean fuzzy ELECTRE method for solving multi-criteria group decision-making problems with unknown weights of decision makers and incomplete weights of criteria. First, a new distance measure between Fermatean fuzzy sets is proposed based on the Jensen–Shannon divergence. The cross entropy for Fermatean fuzzy sets is defined. Three kinds of dominance relationships for Fermatean fuzzy sets are proposed. Then, two optimization models are constructed to obtain positive ideal decision-making information and negative ideal decision-making information, respectively. Accordingly, the credibility degree of each decision maker is calculated. Decision makers’ dynamic weights are determined by their credibility degrees. Besides, to obtain the weights of criteria, an optimization model is constructed based on grey relational analysis for Fermatean fuzzy numbers. Finally, the strong, medium and weak Fermatean fuzzy concordance and discordance sets are identified to construct the Fermatean fuzzy concordance and discordance matrices, respectively. A practical case study is carried out to illustrate the feasibility and applicability of the proposed ELECTRE method. Comparative analyses are performed to demonstrate the superiority and effectiveness of the proposed ELECTRE method.

1Introduction

With the increasing complexity of the socio-economic environment, it is difficult for single Decision Maker (DM) to consider all relevant aspects of a problem, because of the limitation of individual’s knowledge or experience. Multi-Criteria Group Decision-Making (MCGDM) is a widely used efficient method for the complex decision-making problems. DMs or experts express their opinions or preferences about alternatives with respect to different criteria to obtain the best alternative (Wu et al., 2019). For traditional MCGDM, the decision information is represented by crisp numerical values. However, due to the complexity and vagueness of decision-making problems, it is usually challenging for experts to evaluate an object with crisp numerical values. Therefore, various types of fuzzy sets have been applied to MCGDM problems, such as Fuzzy Set (FS) (Zadeh, 1965; Choua and Shen, 2008; Chiclana et al., 2007), Intuitionistic Fuzzy Set (IFS) (Atanassov, 1986; Jiang and Hu, 2021), Pythagorean Fuzzy Set (PFS) (Yager, 2014; Mohagheghi et al., 2017; Zhou and Chen, 2020), etc. Although IFS and PFS are extensively scrutinized by scholars, their applications are relatively limited due to many limitations over the selection of the membership and non-membership grades.

Since Fermatean Fuzzy Set (FFS) proposed by Senapati and Yager (2019c) is able to model the uncertainty in real-life decision-making problems better than IFS and PFS, FFS has received increasing attention. The advantage of FFS is illustrated by an example that an expert may express his/her preference for an alternative over criterion with membership degree 0.8 and non-membership degree 0.9, then it is clearly 0.8+0.91 and 0.82+0.921, but 0.83+0.931. From this point of view, the FFS provides a larger preference domain for experts to express fuzzy information than PFS and IFS. Hence, the space of Fermatean Fuzzy Membership Grades (FFMGs) is greater than the space of Intuitionistic Fuzzy Membership Grades (IFMGs) and Pythagorean Fuzzy Membership Grades (PFMGs), which is shown in Fig. 1. Figure 1 indicates that IFMGs are all points below the line x+y1, PFMGs are all points with x2+y21, and FFMGs are all points with x3+y31. The analysis above suggests that FFS can be used more extensively in MCGDM problems. Therefore, it is necessary to research the theory of FFS.

Fig. 1

Comparison of spaces of FMGS, PMGs and IMGs.

Comparison of spaces of FMGS, PMGs and IMGs.

Since the seminal work of Senapati and Yager (2019c), FFS has been investigated by many scholars. Senapati and Yager (2019c) combined the technique for order preference by similarity to ideal solution (TOPSIS) approach with FFS to handle the multi-criteria decision-making (MCDM) problem. Senapati and Yager (2019b) defined four new weighted aggregated operators including Fermatean fuzzy weighted average operator, Fermatean fuzzy weighted geometric operator, Fermatean fuzzy weighted power average operator, Fermatean fuzzy weighted power geometric operator. Senapati and Yager (2019a) introduced some operations over FFS, then developed a weighted product model based on Fermatean fuzzy information to solve the MCDM problem. Based on Dombi operations, Aydemir and Gunduz (2020) presented a series of aggregation operators for FFS. They then extended TOPSIS with the proposed Fermatean fuzzy Dombi operators. Although some researches are conducted on MCDM methods in the FFS context, there still remain some drawbacks to be handled. The weights of criteria are given by experts in advance. Besides, most of existing decision-making methods for FFSs are aggregation operator-based methods and compromising methods rather than outranking methods. The outranking methods may lead to compensation effect.

Outranking methods are treated as the suitable means for making a successful assessment on the competing criteria. The most widely used outranking method is ELECTRE (elimination and choice translating reality) method, which was proposed by Roy (1991). Since then, numerous studies have been conducted to extend ELECTRE method under fuzzy decision environments, such as triangular fuzzy numbers (Zandi and Roghanian, 2013; Kabak et al., 2012), trapezoidal fuzzy numbers (Hatami and Tavana, 2011), FS (Ferreira et al., 2016), IFS (Shen et al., 2016; Wu and Chen, 2011; Çalı and Balaman, 2019; Mishra et al., 2020; Kilic et al., 2020), interval-valued intuitionistic fuzzy set (Chen, 2014a; Hashemi et al., 2016; Xu and Shen, 2014), hesitant fuzzy set (Chen and Xu, 2015; Mousavi et al., 2017), PFS (Akram et al., 2019, 2021, Chen, 2020) neutrosophic set (Peng et al., 2014; Karasan and Kahraman, 2020; Zhang et al., 2015), etc. However, to the best of our knowledge, no research on ELECTRE method within the context of FFS has yet been conducted.

Recently, many researchers have focused on the construction of outranking relation by using different indices, e.g. the value of score function (Wu and Chen, 2011; Xu and Shen, 2014; Liao et al., 2018), distance measure (Zhang and Yao, 2017; Chen, 2014b), possibility measure (Chen, 2014b, 2015). In general, outranking relation can be sorted as strong dominance and weak dominance. In essence, these two dominance relations are insufficient to describe the degree of superiority and demonstrate superior relation among alternatives. Furthermore, the weights of concordance and discordance sets play an important role in the solution of a MCDM problem with ELECTRE method, which may eventually affect the ranking or selection of alternatives. However, most current ELECTRE methods (Wu and Chen, 2011; Kilic et al., 2020; Chen and Xu, 2015; Akram et al., 2019; Razi, 2015) directly give the weights of concordance and discordance sets on the basis of the subjective judgments of DMs, which lacks the basis of scientific theory and may be unreasonable.

Previous studies on FFS and ELECTRE methods have achieved fruitful research results, some challenging gaps can be identified as follows: Firstly, some methods (Senapati and Yager, 2019a, 2019b, 2019c) under FFSs environment were developed to solve the single expert MCDM problems, which is not suitable for solving group decision-making problems. Literature (Senapati and Yager, 2019a, 2019b, 2019c; Aydemir and Gunduz, 2020) failed to consider the determination of criteria weights, which may lead to unreasonable and unreliable decision-making results. Secondly, distance measure of FFSs in (Senapati and Yager, 2019c) might generate the counter-intuitive results in some cases (see Example 2 in Section 3). Thirdly, there is no research on ELECTRE method with FFS. In addition, due to the computational complexity, many ELECTRE methods (Wu and Chen, 2011; Kilic et al., 2020; Chen and Xu, 2015; Akram et al., 2019; Razi, 2015) provide a priori weights of concordance and discordance sets, which can be easily influenced by DMs’ subjective randomness.

To achieve the aforementioned main objective and fill outlined research gaps, this paper proposes a Fermatean fuzzy ELECTRE method for MCGDM problems. The main contributions and innovations of this paper are outlined below. Firstly, a new distance measure between FFSs is designed by making use of the Jensen–Shannon divergence. The objective weights of concordance and discordance sets are generated by applying the weighted distance measure based on the proposed new distance measure, which is subjectively given by experts in a majority of the studies related to ELECTRE literature (Wu and Chen, 2011; Kilic et al., 2020; Chen and Xu, 2015). Secondly, the weights of DMs are dynamic with respect to each alternative over different criteria, which are generated using the credibility degrees of each DM. As for the DMs’ weights, previous studies usually give them in advance (Wang et al., 2020) or views them as unchangeable for different criteria over different alternatives. Thirdly, the grey relational coefficient and grey relational degree of the FFSs are defined and applied to compute the weights of criteria. Fourthly, in order to show the dominance degree between the pairwise FFNs more exactly, this paper uses membership degree, non-membership degree and indeterminacy degree to compare the outranking relationship for each pair of FFNs. Based on this, the outranking relationship for FFNs can be extended into three situations: strong dominance, medium dominance and weak dominance.

The remainder of this paper is organized as follows: Section 2 introduces some basic concepts associated with FFSs. In Section 3, some information measures for FFSs, including distance measure, cross entropy measure, and grey relational degree, are defined. Outranking relationships for FFSs are introduced in this section, where the related properties of outranking relationships are discussed. An ELECTRE method for MCGDM problems with FFNs is proposed in Section 5. Section 6 illustrates the concrete implementation of the proposed ELECTRE method using a case study on site selection of fangcang shelter hospitals (FSHs), and demonstrates the superiority and effectiveness of the proposed ELECTRE method by comparative analyses. Section 7 gives some conclusions and future research directions

2Preliminaries

This section reviews some concepts, operational rules, comparative methods and aggregation operator of FFSs. The existing Euclidean distance measure of FFNs is also reviewed.

2.1Fermatean Fuzzy Sets

In this section, some basic concepts related to FFSs are briefly reviewed.

Definition 1

Definition 1(Senapati and Yager, 2019c).

Let X be a universe of discourse such that X=(x1,x2,,xn). An FFS F in X is an object having the form F={xi,αF(xi),βF(xi)xiX}, where αF(xi):X[0,1] and βF(xi):X[0,1], including the condition 0αF3(xi)+βF3(xi)1, for all xX. The numbers αF(xi) and βF(xi) denote, respectively, the degree of membership and the degree of non-membership of the element xiX in the set F.

For any FFS F and xX, πF(xi)=1αF3(xi)βF3(xi)3 is considered as the degree of indeterminacy.

In addition, F=(αF(xi),βF(xi)) is called a Fermatean fuzzy number (FFN). For convenience, an FFN is denoted as F=(αF,βF).

Definition 2

Definition 2(Senapati and Yager, 2019c).

Let F=(αF,βF), F1=(αF1,βF1), F2=(αF2,βF2) be three FFNs and λ>0, then their operations are defined as follows:

  • (1) F1F2=(min{αF1,αF2},max{βF1,βF2});

  • (2) F1F2=(max{αF1,αF2},min{βF1,βF2});

  • (3) FC=(βF,αF);

  • (4) F1F2=(αF13+αF23αF13αF233,βF13βF23);

  • (5) F1F2=(αF13αF23,βF13+βF23βF13βF233);

  • (6) λF=(1(1αF3)λ3,βFλ);

  • (7) Fλ=(αFλ,1(1βF3)λ3).

Definition 3

Definition 3(Senapati and Yager, 2019c).

Let F=(αF,βF) be an FFN, then the score function of F can be characterized as

(1)
S(F)=αF3βF3.

Definition 4

Definition 4(Senapati and Yager, 2019b).

Let F=(αF,βF) be an FFN, then the accuracy function of F can be narrated as

(2)
A(F)=αF3+βF3.

Definition 5

Definition 5(Senapati and Yager, 2019b).

Let F1=(αF1,βF1) and F2=(αF2,βF2) be two FFNs. The comparative methods of F1 and F2 can be defined as follows:

  • (1) If S(F1)>S(F2), then F1 is bigger than F2, denoted by F1F2;

  • (2) If S(F1)=S(F2), then A(F1)>A(F2)F1F2,A(F1)=A(F2)F1=F2.

Definition 6

Definition 6(Senapati and Yager, 2019b).

Let Fi=(αF(xi),βF(xi)) (i=1,2,,n) be a number of FFNs and w=(w1,w2,,wn)T be a weight vector of Fi with i=1nwi=1. Then, Fermatean fuzzy weighted average (FFWA) operator is defined as follows:

(3)
FFWA(F1,F2,,Fn)=(i=1nwiαFi,i=1nwiβFi).

2.2Distance Measure

This section shortly reviews distance measure related to FFSs.

Definition 7

Definition 7(Senapati and Yager, 2019c).

Let F1=(αF1,βF1) and F2=(αF2,βF2) be two FFSs. The Euclidean distance measure F1 and F2 is defined as follows:

(4)
dE(F1,F2)=12[(αF13αF23)2+(βF13βF23)2+(πF13πF23)2].

3Some New Fermatean Fuzzy Information Measures

In this section, we put forward some new information measures for FFSs, including distance measure, cross entropy measure, and grey relational degree. They will be used later.

3.1A New Distance Measure for FFSs

In this section, we recall Jensen–Shannon divergence measure. Secondly, a distance measure between FFSs is defined based on Jensen–Shannon divergence. Then, some desirable properties of the proposed new distance measure are inferred.

(1) Jensen–Shannon divergence measure

Definition 8

Definition 8(Kullback and Leibler, 1951).

Let X be a discrete random variable, and p1 and p2 be two probability distributions in X. The I directed divergence is defined as

(5)
I(p1,p2)=xXp1(x)logp1(x)p2(x).
It is worthy to note that I(p1,p2) is non-negative, additive, but not symmetric. Hence, the symmetric measure is defined as
(6)
J(p1,p2)=I(p1,p2)+I(p2,p1)=xX(p1(x)p2(x))logp1(x)p2(x),
which is known as J divergence (Jeffreys, 1946). Obviously, J(p1,p2) is undefined if P2(x)=0 and P1(x)0 for xX.

To solve this problem, Lin (1991) proposed a new directed divergence measure as

(7)
K(p1,p2)=xXp1(x)log2p1(x)p1(x)+p2(x).

An obvious relation between K(p1,p2) and I(p1,p2) is that K(p1,p2)=I(p1,p1+p22). It can be observed that K(p1,p2) is not a symmetric measure. Accordingly, a symmetric measure is defined as

(8)
L(p1,p2)=K(p1,p2)+K(p2,p1)=xXp1(x)log2p1(x)p1(x)+p2(x)+xXp2(x)log2p2(x)p1(x)+p2(x).

Jensen–Shannon divergence measure can be derived from Eq. (8) as follows:

(9)
L(p1,p2)=xXp1(x)logp1(x)xXp1(x)logp1(x)+p2(x)2+xXp2(x)logp2(x)xXp2(x)logp1(x)+p2(x)2=2Φ(p1+p22)Φ(p1)Φ(p2),
where Φ(p)=plogp is the Shannon entropy function.

In this paper, we define Fermatean fuzzy distance based on Jensen–Shannon divergence.

(2) A new distance measure for FFSs

Definition 9.

Let X=(x1,x2,,xn) be a universe of discourse, and G and H be two FFSs in X, where G={xi,Gα(xi),Gβ(xi)xiX} and H={xi,Hα(xi),Hβ(xi)xiX}. The Fermatean fuzzy divergence between G and H is defined as

(10)
Ω(G,H)=Φ(G+H2)12Φ(G)12Φ(H)=12{ηGη3(x)log2Gη3(x)Gη3(x)+Hη3(x)+ηHη3(x)log2Hη3(x)Gη3(x)+Hη3(x)},
where Φ(G)=Gη3(x)logGη3(x), η(α,β,π), is the Shannon entropy, Gπ3(x)=1Gα3(x)Gβ3(x).

In the line with Definition 9, a new distance measure for FFSs is given below.

Definition 10.

Let X be a universe of discourse, and G and H be two FFSs. A new distance measure for FFSs, denoted as d¯(G,H), is defined as

(11)
d¯(G,H)=Ω(G,H)=12{ηGη3(x)log2Gη3(x)Gη3(x)+Hη3(x)+ηHη3(x)log2Hη3(x)Gη3(x)+Hη3(x)},
where, η(α,β,π), the base of log is 10.

Some desirable properties of d¯(G,H) can be inferred as follows:

Theorem 1.

Let G, H and M be three arbitrary FFSs in a universe of discourse X, then some properties hold:

  • (P1) d¯(G,H)=0 iff G=H, for G,HX;

  • (P2) d¯(G,H)=d¯(H,G), for G,HX;

  • (P3) d¯(G,H)+d¯(H,M)d¯(G,M), for G,H,MX;

  • (P4) 0d¯(G,H)1, for G,HX.

Proof.

(P1) Let G and H be two FFSs in a universe of discourse X. For the necessity, If G=H, which means Gη3(x)=Hη3(x), then d¯(G,H)=0 can be obtained based on Definition 9. For the sufficiency, if d¯(G,H)=0, then 12{ηGη3(x)log2Gη3(x)Gη3(x)+Hη3(x)+ηHη3(x)log2Hη3(x)Gη3(x)+Hη3(x)}=0. It is evident that Gη3(x)=Hη3(x). Thus, G=H. As a result, d¯(G,H)=0 iff G=H.

(P2) Since

12{ηGη3(x)log2Gη3(x)Gη3(x)+Hη3(x)+ηHη3(x)log2Hη3(x)Gη3(x)+Hη3(x)}=12{ηHη3(x)log2Hη3(x)Gη3(x)+Hη3(x)+ηGη3(x)log2Gη3(x)Gη3(x)+Hη3(x)},
one has d¯(G,H)=d¯(H,G).

(P3) Four hypotheses are formulated as follows:

  • Hypothesis 1: Gα3(x)Hα3(x)Mα3(x),

  • Hypothesis 2: Mα3(x)Hα3(x)Gα3(x),

  • Hypothesis 3: Hα3(x)min{Gα3(x),Mα3(x)},

  • Hypothesis 4: Hα3(x)max{Gα3(x),Mα3(x)}.

According to Hypothesis 1 and Hypothesis 2, it holds that |Gα3(x)Mα3(x)||Gα3(x)Hα3(x)|+|Hα3(x)Mα3(x)|. Due to Hypothesis 3, it has Gα3(x)Hα3(x)0 and Mα3(x)Hα3(x)0. Then, we can obtain

|Gα3(x)Hα3(x)|+|Hα3(x)Mα3(x)||Gα3(x)Mα3(x)|=Gα3(x)Hα3(x)+Mα3(x)Hα3(x)Gα3(x)+Mα3(x),ifGα3(x)Mα3(x)Gα3(x)Hα3(x)+Mα3(x)Hα3(x)+Gα3(x)Mα3(x),ifGα3(x)Mα3(x)=2(min{Gα3(x),Mα3(x)}Hα3(x))0.

In the same way, according to Hypothesis 4, we can get Hα3(x)Gα3(x)0 and Hα3(x)Mα3(x)0. Therefore, it holds that

|Gα3(x)Hα3(x)|+|Hα3(x)Mα3(x)||Gα3(x)Mα3(x)|=Hα3(x)Gα3(x)+Hα3(x)Mα3(x)Gα3(x)+Mα3(x),ifGα3(x)Mα3(x)Hα3(x)Gα3(x)+Hα3(x)Mα3(x)+Gα3(x)Mα3(x),ifGα3(x)Mα3(x)=2(Hα3(x)max{Gα3(x),Mα3(x)})0.

As a result, |Gα3(x)Mα3(x)||Gα3(x)Hα3(x)|+|Hα3(x)Mα3(x)| holds in the contexts of Hypothesis 3 and Hypothesis 4.

Analogously, it follows that |Gβ3(x)Mβ3(x)||Gβ3(x)Hβ3(x)|+|Hα3(x)Mβ3(x)| and |Gπ3(x)Mπ3(x)||Gπ3(x)Hπ3(x)|+|Hπ3(x)Mπ3(x)|.

Hence, this completes the proof of d¯(G,H)+d¯(H,M)d¯(G,M).

(P4) Consider two FFSs G and H in a universe of discourse X, one has

d¯2(G,H)=12{ηGη3(x)log2Gη3(x)Gη3(x)+Hη3(x)+ηHη3(x)log2Hη3(x)Gη3(x)+Hη3(x)}=12(η(Gη3(x)+Hη3(x))){ηGη3(x)Gη3(x)+Hη3(x)log2Gη3(x)Gη3(x)+Hη3(x)+η2Hη3(x)Gη3(x)+Hη3(x)log2Hη3(x)Gη3(x)+Hη3(x)}=12(η(Gη3(x)+Hη3(x))){1H(Gη3(x)Gη3(x)+Hη3(x),Hη3(x)Gη3(x)+Hη3(x))}.

It has been proven in Gallager (1968) that, for and 0θ1, H(θ,1θ)2min(θ,1θ). For min(θ,1θ)=12(1|(θ(1θ)|), it can be obtained that 1H(θ,1θ)|θ(1θ)|. Then, we can get

d¯(G,H)12(η(Gη3(x)+Hη3(x)))|Gη3(x)Gη3(x)+Hη3(x)2Hη3(x)Gη3(x)+Hη3(x)|=12V(G,H),
where V(G,H) is the variational divergence measure (Vajda, 1970). Since the value of V(G,H) ranges from [0,2], which is testified in Toussaint (1975). Therefore, we have 0d¯(G,H)1.

This completes the proof of Theorem 1.  □

Example 1.

Let G and H be two FFSs in the universe of discourse X. These FFSs over X are defined as G=x,η,γ, H=x,γ,η.

The parameters η and γ are the membership and non-membership degrees, respectively, which range from 0 to 1, meeting the condition η3+γ31.

Using Eq. (11), the distance for FFSs can be measured, as shown in Fig. 2.

In consideration of the distance measure results of Example 1, we can verify the non-negativity, symmetry and boundedness properties of distance measure for FFSs.

It can be seen from Fig. 2 that the distance measure for FFSs is always greater than or equal to zero when the parameters η and γ take different values within [0,1]. The non-negativity of distance measure for FFSs is verified.

As shown in Fig. 2, it is obvious that the distance measure for FFSs satisfies symmetry property. Let us cite a concrete instance that η=0.9 and γ=0.3. Based on Eq. (11), the values of d¯(G,H) and d¯(H,G) are 0.4207, thus d¯(G,H)=d¯(H,G).

From Fig. 2, we clearly know the values of distance measure are [0,1]. Specifically, when the parameters η=1 and γ=0, or when η=0 and γ=1, d¯(G,H)=d¯(H,G)=1. The boundedness of distance measure for FFSs is proved.

Fig. 2

The FFSs distance measure in Example 1.

The FFSs distance measure in Example 1.

(3) Comparisons of distance measures for FFNs

In this section, in order to testify the superiority and reasonability of the proposed distance measure, we compare the proposed distance measure with the existing Euclidean distance measure (Senapati and Yager, 2019c) in Example 2.

Example 2.

Let Ai={A1,A2,,A18} and Bi={B1,B2,,B18} be two sets of FFNs under Case i (i=1,2,,18), which are shown in Table 1.

Table 1

Two FFNs Ai and Bi under different cases in Example 2.

FFNsCase 1Case 2Case 3Case 4Case 5Case 6
Ai[0.595,0.690][0.840,0.660][0.629,0.834]0.810,0.673[0.849,0.609][0.827,0.665]
Bi[0.650,0.779][0.726,0.749][0.731,0.755]0.888,0.601[0.912,0.510][0.894,0.600]
FFNsCase 7Case 8Case 9Case 10Case 11Case 12
Ai[0.888,0.601][0.679,0.788][0.804,0.606][0.637,0.833][0.818,0.622][0.755,0.679]
Bi[0.788,0.678][0.606,0.888][0.871,0.607][0.731,0.755][0.894,0.600][0.650,0.779]
FFNsCase 13Case 14Case 15Case 16Case 17Case 18
Ai[0.629,0.834][0.637,0.833][0.881,0.606][0.815,0.628][0.827,0.665][0.818,0.622]
Bi[0.606,0.888][0.606,0.888][0.788,0.678][0.726,0.749]0.728,0.7320.728,0.732

Table 2

Comparisons of Euclidean distance measure and the proposed distance measure.

MethodsCase1Case 2Case 3Case 4Case 5Case 6
dE(A,B)0.1840.1840.1460.1460.1290.129
d¯(A,B)0.10220.09870.07670.08410.07430.0764
MethodsCase 7Case 8Case 9Case 10Case 11Case 12
dE(A,B)0.1830.1830.1410.1410.1580.158
d¯(A,B)0.10630.10810.08870.07330.10380.0855
MethodsCase 13Case 14Case 15Case 16Case 17Case 18
dE(A,B)0.1090.1090.1670.1660.1560.157
d¯(A,B)0.07300.07070.09670.08900.08680.0808

The distance measure results obtained by two methods are shown in Table 2. Carefully observing Table 2, the main conclusions are described as follows.

(1) Compared with the Euclidean distance measure (Senapati and Yager, 2019c), the proposed distance measure has satisfactory performances under Case 1–Case 14. The values of the Euclidean distance measure are equal under Case 1–Case 14. These results seem counter-intuitive, which are highlighted in bold in Table 2. The proposed distance can measure the difference under Case 1–Case 14, which demonstrates the feasibility of the proposed distance.

(2) The discrimination degrees of the proposed distance measure are significantly higher than those of the Euclidean distance measure under Case 15 and Case 16 or Case 17 and Case 18. It can be seen from Table 2 that the discrimination degrees of Euclidean distance measure under Case 15 and Case 16 or Case 17 and Case 18 are only 0.01, while the discrimination degrees of the proposed distance measure under the corresponding cases are over 0.06.

3.2Cross Entropy Measure for FFSs

As one of the most popular information measure, cross entropy is used to measure the divergence information and extensively applied in current literature. However, cross entropy measure for FFSs is rare. Inspired by Song et al. (2019), this paper gives a definition of Fermatean fuzzy cross entropy.

Definition 11.

Let X be a universe of discourse such that X=(x1,x2,,xn), and G and H be two FFSs in X, in which G={xi,Gα(xi),Gβ(xi),Gπ(xi)xiX} and H={xi,Hα(xi),Hβ(xi),Hπ(xi)xiX}. The Fermatean fuzzy cross entropy measure of G against H, denoted as CE(G,H), is defined as:

(12)
CE(G,H)=i=1n(1+Gπ3(xi))ln1+Gπ3(xi)1+Hπ3(xi)+(1+ΔG(xi))ln1+ΔG(xi)1+ΔH(xi),
where ΔG(xi)=|Gα3(xi)Gβ3(xi)| indicates the difference between membership cube and non-membership cube.

Since CE(G,H) is not symmetric, a symmetric cross entropy measure can be given as:

(13)
SCE(G,H)=CE(G,H)+CE(H,G)=i=1n(Gπ3(xi)Hπ3(xi))ln1+Gπ3(xi)1+Hπ3(xi)+(ΔG(xi)ΔH(xi))ln1+ΔG(xi)1+ΔH(xi).

Definition 12.

Let X be a universe of discourse such that X=(x1,x2,,xn), and G and H be two FFSs in X, in which G={xi,Gα(xi),Gβ(xi),Gπ(xi)xiX} and H={xi,Hα(xi),Hβ(xi),Hπ(xi)xiX}. The normalized Fermatean fuzzy cross entropy measure between G and H, denoted as NCE(G,H), is defined as:

(14)
NCE(G,H)=12nln2i=1n(Gπ3(xi)Hπ3(xi))ln1+Gπ3(xi)1+Hπ3(xi)+(ΔG(xi)ΔH(xi))ln1+ΔG(xi)1+ΔH(xi).

Theorem 2.

Let G, H and M be three arbitrary FFSs in a universe of discourse X, then NCE(G,H) satisfies the following properties:

  • (P1) NCE(G,H)=NCE(GC,H)=NCE(G,HC)=NCE(GC,HC);

  • (P2) 0NCE(G,H)1;

  • (P3) NCE(G,H)=0 iff G=H or G=HC, for G,HX.

Proof.

(P1) For two FFSs in X defined as G={x,Gα(x),Gβ(x)xX} and H={x,Hα(x),Hβ(x)xX}, we have GC={x,Gβ(x),Gα(x)xX}, HC={x,Hβ(x),Hα(x)xX}.

Then, we can get that Gπ3(x)=GπC3(x)=1Gα3(x)Gβ3(x), Hπ3(x)=HπC3(x)=1Hα3(x)Hβ3(x), ΔG(x)=ΔGC(x), ΔH(x)=ΔHC(x), xX.

Therefore, NCE(G,H)=NCE(GC,H)=NCE(G,HC)=NCE(GC,HC).

(P2) If Gπ3(x)Hπ3(x), then it has Gπ3(x)Hπ3(x)0, 1+Gπ3(x)1+Hπ3(x)1, (Gπ3(x)Hπ3(x))ln1+Gπ3(x)1+Hπ3(x)0. If Gπ3(x)>Hπ3(x), one gets that Gπ3(x)Hπ3(x)>0, 1+Gπ3(x)1+Hπ3(x)>1, (Gπ3(x)Hπ3(x))ln1+Gπ3(x)1+Hπ3(x)>0. Hence, (Gπ3(x)Hπ3(x))ln1+Gπ3(x)1+Hπ3(x)0 always holds for xX. Analogously, we can get (ΔG(xi)ΔH(xi))ln1+ΔG(xi)1+ΔH(xi)0, xX.

Since 0Gπ3(x)1, 0Hπ3(x)1, 0ΔG(x)1, 0ΔH(x)1, NCE(G,H) gets its maximum when Gπ3(x)=1, Hπ3(x)=0, ΔG(x)=0, ΔH(x)=1 or Gπ3(x)=0, Hπ3(x)=1, ΔG(x)=1, ΔH(x)=0, xX. Hence, it holds that 0NCE(G,H)1.

(P3) In this case of G=H or G=HC, obviously, NCE(G,H)=0. For the sufficiency, if NCE(G,H)=0, we have G=H or G=HC, due to (Gπ3(x)Hπ3(x))ln1+Gπ3(x)1+Hπ3(x)0 and (ΔG(xi)ΔH(xi))ln1+ΔG(xi)1+ΔH(xi)0. So, NCE(G,H)=0 iff G=H or G=HC, for G,HX.

Therefore, Theorem 2 is proved.  □

3.3Grey Relation Analysis Between FFSs

In this section, grey relational theory is extended to FFSs environment, and the FFSs grey relational coefficient and grey relational degree are defined for the first time.

(1) Grey relation analysis

The grey system theory first created by Deng (1982) is a useful method to study the problems with insufficient, poor and uncertain information. As an indispensable part of grey system theory, the basic idea of grey relational analysis (GRA) is to judge whether the geometric shapes of sequence curves are closely related according to their similarity degrees. The closer the curve is, the greater the correlation between the corresponding sequences is, and vice versa. GRA has been widely applied in addressing different kinds of MCMD problems (Li et al., 2020; Wu, 2009; Hamzaçebi and Pekkaya, 2011), due to being computationally simple, robust and practical. In the following, GRA is introduced.

Definition 13

Definition 13(Deng, 1989).

Let X0=(x0(1),x0(2),,x0(j)) and Xi=(xi(1),xi(2),,xi(j)) (i=1,2,,m; j=1,2,,k) be sets of the sequences. The grey relational coefficient is defined by

(15)
γ(x0(j),xi(j))=miniminj|x0(j)xi(j)|+ρmaximaxj|x0(j)xi(j)||x0(j)xi(j)|+ρmaximaxj|x0(j)xi(j)|,
where ρ(0,1) is the distinguished coefficient.

The grey relational degree is defined as:

(16)
γ(X0,Xi)=1kj=1kγ(x0(j),xi(j)).

(2) Grey relational analysis between FFSs
Definition 14.

Let X be a universe of discourse such that X=(x1,x2,,xn), and G and H be two FFSs in X, in which G={xi,Gα(xi),Gβ(xi)xiX} and Hj={xi,Hαj(xi),Hβj(xi)xiX}, i=1,2,,n, j=1,2,,m. The grey relational coefficient between G and H is defined as:

(17)
γ(G(xi),Hj(xi))=miniminjd¯(G,Hj)+ρmaximaxjd¯(G,Hj)d¯(G,Hj)+ρmaximaxjd¯(G,Hj),
where d¯(G,Hj) is the distance between FFSs G and Hj, which is calculated by Eq. (11).

In view of grey relational coefficient between FFSs, the grey relational degree between FFSs G and Hj is defined as:

(18)
γ(G,Hj)=1ni=1nγ(G(xi),Hj(xi)).

4Outranking Relationships for FFSs

In this section, outranking relationships for FFSs are introduced on account of classic outranking model. In addition, the related properties of outranking relationships are discussed.

Although the current two kinds of outranking relationships are widely applied in ELECTRE method, they are inadequate to differentiate between each Fermatean fuzzy pair. In line with the concept of score function, accuracy function and degree of indeterminacy, it is obtained that a better alternative has larger score degree or larger accuracy with the condition that score degrees of alternatives are the same. A larger score degree alludes to a larger degree of membership or smaller degree of non-membership; a larger accuracy degree alludes to a smaller degree of indeterminacy. In order to investigate a proper outranking method under an FFSs environment, we propose three kinds of dominance relationships for FFSs to show their interrelationships more comprehensively and explain their dominance degrees more specifically.

Definition 15.

Let F1=(αF1,βF1,πF1) and F2=(αF2,βF2,πF2) be two FFNs, their outranking relationships can be represented as follows:

(1) Strong dominance: If αF1αF2,βF1<βF2 and πF1<πF2, then F1 strongly dominates F2 or F2 is strongly dominated by F1. This can be denoted as F1>sF2 or F2<sF1.

(2) Medium dominance: If αF1αF2,βF1<βF2 and πF1πF2, then F1 moderately dominates F2 or F2 is moderately dominated by F1. This can be denoted as F1>mF2 or F2<mF1.

(3) Weak dominance: If αF1αF2 and βF1βF2, then F1 weak dominates F2 or F2 is weakly dominated by F1. This can be denoted as F1>wF2 or F2<wF1.

(4) Indifference: If αF1=αF2 and βF1=βF2, then F1 is indifferent to F2. This can be denoted as F1F2.

Theorem 3.

Let F=(αF,βF,πF), F1=(αF1,βF1,πF1), F2=(αF2,βF2,πF2) and F3=(αF3,βF3,πF3) be four FFNs. We can obtain the following conclusions:

  • (1) There are the following properties for the strong dominance:

    • (i) Irreflexivity: FsF, where s shows non-strong dominance.

    • (ii) Asymmetry: F1>sF2F2>sF1.

    • (iii) Transitivity: F1>sF2 and F2>sF3F1>sF3.

  • (2) There are the following properties for the medium dominance:

    • (i) Irreflexivity: FmF, where m shows non-medium dominance.

    • (ii) Asymmetry: F1>mF2F2>mF1.

    • (iii) Transitivity: F1>mF2 and F2>mF3F1>mF3.

  • (3) There are the following properties for the weak dominance:

    • (i) Irreflexivity: FwF, where w shows non-weak dominance.

    • (ii) Asymmetry: F1>wF2F2>wF1.

    • (iii) Transitivity: F1>wF2 and F2>wF3F1>wF3.

  • (4) There are the following properties for the indifference relationship:

    • (i) Reflexivity: FF.

    • (ii) Symmetry: F1F2F2F1.

    • (iii) Transitivity: F1F2 and F2F3F1F3.

Proof.

The transitivity property for strong dominance relationship can be testified as follows:

Let F1=(αF1,βF1,πF1), F2=(αF2,βF2,πF2) and F3=(αF3,βF3,πF3) be three FFNs. When F1>sF2, we obtain αF1αF2,βF1<βF2 and πF1<πF2 according to Definition 15(1). If F2>sF3, we obtain αF2αF3,βF2<βF3 and πF2<πF3 according to Definition 15(1). Thus, one gets αF1αF3,βF1<βF3 and πF1<πF3. It holds that F1>sF3. Therefore, if F1>sF2 and F2>sF3, then F1>sF3. The transitivity properties for medium dominance relationship and weak dominance relationship can be demonstrated in the same manner.

The proof of other properties for strong dominance, medium dominance, and weak dominance are straightforward.  □

5A FermateanFuzzy ELECTRE Method for MCGDM

This section develops a Fermatean fuzzy ELECTRE method for MCGDM.

5.1Problem Description of MCGDM Using FFNs

Assuming there are m non-inferior alternatives Ai (i=1,2,,m), each alternative is evaluated on n criteria Cj (j=1,2,,n) by t experts El (l=1,2,,t). To be specific, the set Cj is classified into two different types of sets, namely, CI and CII. Here, CI and CII show a collection of benefit criteria and a collection of cost criteria, respectively. Let λ=(λ1,λ2,,λt)T be the weight vector of DMs, in which λl[0,1] and l=1tλt=1; w=(w1,w2,,wn)T be the weight vector of criteria where wj[0,1] and j=1nwj=1. λ=(λ1,λ2,,λt)T, w=(w1,w2,,wn)T are completely unknown and incomplete, respectively. The evaluation information for Ai (i=1,2,,m) with respect of Cj (j=1,2,,n) is given by t experts in terms of linguistic assessment. These linguistic assessments can be transformed into FFNs. Let Rl=(aijl)m×n be the Fermatean fuzzy decision matrix, where aijl is the performance of alternative Ai with respect to criterion Cj provided by DM El, and aijl=(αijl,βijl,πijl) is an FFN, where αijl, βijl, πijl are membership degree, non-membership degree and indeterminacy degree, respectively. The MCGDM considered in this paper is how to select the best alternative according to Fermatean fuzzy decision matrices Rl=(aijl)m×n (l=1,2,,t).

5.2Determine Dynamic Weights of DMs with Respect to Each Criterion over Different Alternatives

Aggregating all individual decisions into a collective decision is regarded as a key part of MCGDM process. Therefore, how to determine the weights of DMs is one of the main activities for MCGDM problems, because different weights of DMs may generate different collective decision matrices and then can have significant impact on the final result. Hence, methods for determining the weights of DMs have received much attention by researchers, however, most of existing methods usually assume that DMs’ weights for all alternatives and criteria are changeless (Yue, 2012; Lin and Chen, 2020; Wan et al., 2013; Ju, 2014; Wan et al., 2015). In the actual decision-making process, it is unlikely that each DM is expected to be good at commenting on all alternatives and criteria due to differences in educational background, knowledge, experience, preference, and title, the weights of each DM may change with different criteria and different alternatives. Hence, distributing different weights to each DM with respect to different alternatives under different criteria is more reasonable and in line with the actual decision-making situation. Therefore, the study of dynamic DM weights is of some practical significance. According to Geng et al. (2017), dynamic weights refer to assigning different weights to each DM with respect to different criteria over different alternatives. DM’s weights will vary with different criteria and different alternatives. However, to the best of our knowledge, there are only several scholars (Geng et al., 2017; Wu et al., 2019) involving this issue up to now. In particular, determining the objectively dynamic weights of DMs in the context of Fermatean fuzzy information remains an unexplored area. To fill the research gap, this paper proposes a new method for determining dynamic weights of DMs. Inspired by Geng et al. (2017), based on the proposed the cross entropy, dynamic weights of DMs are determined as follows:

(1) Determine the positive ideal decision matrix (PIDM) and the negative ideal decision matrix (NIDM)

Let R+=(aij+)m×n and R=(aij)m×n be PIDM and NIDM for all individual decision matrices given by DMs, in which {aij+=(αij+,βij+),1im,1jn} and {aij=(αij,βij),1im,1jn} are positive ideal decision-making information (PIDMI) and negative ideal decision-making information (NIDMI) with respect to i-th alternative over j-th criterion. We construct the following optimization model to determine (αij+,βij+) and (αij,βij), respectively.

(19)
minχij=l=1t(|αij+αijl|+|βij+βijl|),s.t.0αij+1,0βij+1,0(αij+)3+(βij+)31.
(20)
maxηij=l=1t(|αijαijl|+|βijβijl|),s.t.0αij1,0βij1,0(αij)3+(βij)31.

(2) Determine the credibility degree

The credibility degree is determined in line with cross entropy. If αijl has a larger difference from αij+ or aij, it has more credibility. So, the credibility of αijl, denoted by τijl, is defined as

(21)
τijl=|SCE(αijl,αij)SCE(αijl,αij+)|.

(3) Determine the dynamic weights of DMs

It is obvious that the DM El who gives decision-making information with respect to different criteria with larger credibility should be allocated a bigger weight. Accordingly, objective and dynamic weights of DMs, denoted by λijl, are determined as follows:

(22)
λijl=τijll=1tτijl.

Using Eq. (3), Rl (l=1,2,,t) can be integrated into a collective matrix R=(aij)m×n, in which aij can be derived by

(23)
aij=(l=1tλijlαijl,l=1tλijlβijl).

5.3Obtain Criteria Weights Based on the Proposed GRA

Criteria weights play a pivotal role in MCDM problems, because they have an important and direct influence on ranking results. Due to the increasing complexity, time pressure or lack of data in practical situations, the weights of criteria are usually unknown. Therefore, it is an interesting research topic to deduce plausible weights for criteria by selecting suitable methods in the real-life MCDM process, since plausible weights can ensure scientific and plausible decision-making results. In the current literature, methods for deriving criteria weights can be divided into two categories: the subjective weight-determining methods, the objective weight-determining methods.

The subjective weight-determining methods, such as the Delphi method (Dalkey and Helmer, 1963), the AHP method (Saaty, 1987; Kaya and Kahraman, 2011) and SRF method (Figueira and Roy, 2002), determine the weights of criteria based on experiences and subjective judgments. The subjective weight-determining methods are impacted by subjective randomness of the DM’s preference. In addition, when there are a great number of assessment criteria, the subjective weight-determining methods are not suitable for identification of weights of these criteria (Çalı and Balaman, 2019). Different from the subjective methods, the objective methods are capable of eliminating man-made instabilities and obtaining more realistic weights according to mathematical model. A majority of them have focused on calculation of entropy value so as to derive the criteria weights. Entropy weight method, which is a straightforward method for weight determination, has been extensively applied to diverse decision-making fields (Zhang and Yao, 2017; Xu and Shen, 2014; Ye, 2010; Liu and Zhang, 2011), however, it can deduce irrational weight values in some cases (Das et al., 2015).

Since its inception in Deng (Wang, 1997), GRA method has been widely employed for obtaining objective weights of criteria (Wei, 2011a, 2010; Luo et al., 2019; Meng et al., 2015), because its greatest strength is that it is computationally simple, robust and practical (Wei, 2011b). This is a discerning evidence that the GRA method is deemed to be a more feasible method to obtain criteria weights in this study.

In the following, we utilize GRA method to determine the criteria weights with incomplete information.

Firstly, the Fermatean Fuzzy Positive Ideal Point (FF-PIP) aj+ and the Fermatean Fuzzy Negative Ideal Point (FF-NIP) aj can be defined as:

(24)
aj+=(αj+,βj+,πj+)=(max1imαij,min1imβij,min1imπij),for benefit criterion,(min1imαij,max1imβij,max1imπij),for cost criterion,
(25)
aj=(αj,βj,πj)=(min1imαij,max1imβij,max1imπij),for benefit criterion,(max1imαij,min1imβij,min1imπij),for cost criterion.

Then, based on the proposed distance measure for FFSs in Section 3.2, the distances of the rating values aij to the FF-PIP aj+ and FF-NIP aj can be computed respectively by:

(26)
d¯ij+=d¯(aij,aj+)=12{η(aijη)3log2(aijη)3(aijη)3+(aj+η)3+η(aj+η)3log2(aj+η)3(aijη)3+(aj+η)3},
(27)
d¯ij=d¯(aij,aj)=12{η(aijη)3log2(aijη)3(aijη)3+(ajη)3+η(aj+η)3log2(ajη)3(aijη)3+(ajη)3},
where η(α,β,π) in Eqs. (26) and (27).

Next, the grey relational coefficients of the rating values aij from PIP and NIP are computed using the following equations, respectively:

(28)
ξij+=miniminjd¯(αij,αj+)+δmaximaxjd¯(αij,αj+)d¯(αij,αj+)+δmaximaxjd¯(αij,αj+),
(29)
ξij=miniminjd¯(αij,αj)+δmaximaxjd¯(αij,αj)d¯(αij,αj)+δmaximaxjd¯(αij,αj),
where d¯(αij,αj+) and d¯(αij,αj) are the distance of the rating values aij to the FF-PIP aj+ and FF-NIP aj, respectively, i=1,2,m, j=1,2,,n; δ is considered as an identification coefficient to mitigate the effect of the maxjd¯(αij,αj+) or maxjd¯(αij,αj) on the relational coefficient, with a value ranging from 0 to 1. In this paper, δ is equal to 0.5.

Subsequently, the degrees of grey relational coefficient of the alternative Ai from PIP and NIP are computed using the following equations, respectively:

(30)
ϑi+=j=1nwjξij+,i=1,2,,m,
(31)
ϑi=j=1nwjξij,i=1,2,,m,
where wj (j=1,2,,n) are the criteria weights.

According to the GRA method, the optimal alternative should have the “largest degree of grey relation” from the positive-ideal solution and the “smallest degree of grey relation” from the negative-ideal solution. Based on this idea, a multiple objective optimization model (M-1) is established to get criteria weights with incomplete weight information with respect to alternative Ai.

(M-1)
maxϑi+=j=1nwjiξij+,minϑi=j=1nwjiξij,s.t.wi=(w1i,w2i,,wni)TΛ,j=1nwji=1,wji0,i=1,2,,m,j=1,2,,n,
where Λ is set of incomplete information about weight criteria given by DMs. Incomplete information structures of criteria weights are constructed in the following five forms (Wan et al., 2015; Xu, 2007; Xu and Da, 2008), for hj:

  • Form 1. A weak ranking: whwj;

  • Form 2. A strict ranking: whwjεj,εj>0;

  • Form 3. A ranking with multiples: whεjwj, 0εj1;

  • Form 4. An interval form: κhwhκh+ιh, 0κhwκh+ιh;

  • Form 5. A ranking of differences: whwjwkwlεj, jkl.

Since each alternative is non-inferior, there exists no preference relation on all alternatives. We may aggregate the above multiple objective optimization model with equal weights into the following single-objective optimization model (M-2):

(M-2)
maxΘi=j=1nwjiξij+j=1nwjiξij,s.t.wi=(w1i,w2i,,wni)TΛ,j=1nwji=1,wji0,i=1,2,,m,j=1,2,,n.

By solving the model (M-2), we obtain the optimal weight vector of criteria wi=(w1i,w2i,,wni) with respect to alternative Ai.

Then, we will substitute wk=(w1k,w2k,,wnk) (k=1,2,,m) into the objective function Θi in model (M-2). The value of objective function Θi can be calculated as

(32)
Θik=j=1nwjkξij+j=1nwjkξij.

Then, let Θi+=max{Θikk=1,2,,m} and Θi=min{Θikk=1,2,,m} be maximum and minimum values of Θi, respectively. It is easy to know from model (M-2) that Θi+=Θii.

The weight vector wt (t=1,2,,m) corresponding to the value Θi is defined as the worst weight vector for Ai and denoted by wi. The weight vector ws (s=1,2,,m) corresponding to the value Θi+ is defined as the optimal weight vector for Ai and denoted by wi+.

In the following, an optimization model motivated by the ideal of TOPSIS method is constructed to determine the weight of each criterion in incomplete weight information context. The main steps are described as follows:

(1) Determine the Positive Ideal Weight Vector (PIWV) and Negative Ideal Weight Vector (NIWV) of the criterion weight for each alternative.

According to the above analysis, PIWV and NIWV are defined as

(33)
wi+=(w1i,w2i,,wni)T,
(34)
wi=(w1t,w2t,,wjt)T.

(2) Calculate the distance of each criterion weight from the PIWV and NIWV respectively.

(35)
di+=j=1n|wjwji|,
(36)
di=j=1n|wjwjt|,
where wj is the j-th criterion weight. wji and wjt are ij-th element of j-th criteria for i-th alternative in PIWV and NIWV, respectively.

According to the ideal of TOPSIS method, a multiple objective optimization model (M-3) is constructed to derive criteria weights with incomplete weight information,

(M-3)
maximdi=i=1mj=1n|wjwjt|,minimdi+=i=1mj=1n|wjwji|,s.t.w=(w1,w2,,wn)TΛ,j=1nwj=1,wj0,j=1,2,,n.

The above multiple objective optimization model is equal to the following single objective optimization model (M-4) by using equal weight linear weighting method:

(M-4)
minim(di+di)=i=1mj=1n|wjwji|i=1mj=1n|wjwjt|,s.t.w=(w1,w2,,wn)TΛ,j=1nwj=1,wj0,j=1,2,,n.

The optimal solution w=(w1,w2,,wn) can be obtained by the model (M-4).

5.4Construct the Concordance and Discordance Sets

For each Fermatern fuzzy pair of Af and Ag, the set of criteria is classified into two distinct subsets: concordance set and discordance set. The concordance set consists of all criteria for which Af is preferred to alternative Ag. The discordance set, the complement set of concordance set, contains all criteria for which Af is worse than Ag. On account of Definition 15, the concordance set for any two alternatives Af and Ag can be partitioned into three categories.

  • (1) Strong concordance set is portrayed as:

    (37)
    JCfg={jAfj>sAgj}={jαfjαgj,βfj<βgj,πfj<πgj}.

  • (2) Medium concordance set is portrayed as:

    (38)
    JCfg={jAfj>mAgj}={jαfjαgj,βfj<βgj,πfjπgj}.

  • (3) Weak concordance set is portrayed as:

    (39)
    JCfg={jAfj>wAgj}={jαfjαgj,βfjβgj}.

The discordance set JDfg of Af and Ag consists of all criteria for which Af is not superior to Ag. The discordance set JDfg can also divided into three categories in the same way.

  • (1) Strong discordance set is portrayed as:

    (40)
    JDfg={jAfj<sAgj}={jαfj<αgj,βfjβgj,πfjπgj}.

  • (2) Medium discordance set is portrayed as:

    (41)
    JDfg={jAfj<mAgj}={jαfj<αgj,βfjβgj,πfj<πgj}.

  • (3) Weak discordance set is portrayed as:

    (42)
    JDfg={jAfj<wAgj}={jαfj<αgj,βfj<βgj}.

5.5Identify the Weights of Concordance and Discordane Sets

This paper applies objective weighting method based on the proposed distance measure to identify the weights of concordance and discordance sets. The weights of strong, medium, and weak concordance sets are computed by Eqs. (43), (44) and (45), respectively.

The weight of strong concordance set wC is computed with Eq. (43) as follows:

(43)
wC=f,g=1fgmjCfgwj×d¯(Afj,Agj)×{f,g=1fgmjCfgwj×d¯(Afj,Agj)+f,g=1fgmjCfgwj×d¯(Afj,Agj)+f,g=1fgmjCfgwj×d¯(Afj,Agj)}1.

The weight of medium concordance set wC is computed with Eq. (44) as follows:

(44)
wC=f,g=1fgmjCfgwj×d¯(Afj,Agj)×{f,g=1fgmjCfgwj×d¯(Afj,Agj)+f,g=1fgmjCfgwj×d¯(Afj,Agj)+f,g=1fgmjCfgwj×d¯(Afj,Agj)}1.

The weight of weak concordance set wC is computed with Eq. (45) as follows:

(45)
wC=f,g=1fgmjCfgwj×d¯(Afj,Agj)×{f,g=1fgmjCfgwj×d¯(Afj,Agj)+f,g=1fgmjCfgwj×d¯(Afj,Agj)+f,g=1fgmjCfgwj×d¯(Afj,Agj)}1.

The weights of strong, medium, and weak discordance sets are computed by Eqs. (46), (47) and (48), respectively.

The weight of strong discordance set wD is computed with Eq. (46) as follows:

(46)
wD=f,g=1fgmjDfgwj×d¯(Afj,Agj)×{f,g=1fgmjDfgwj×d¯(Afj,Agj)+f,g=1fgmjDfgwj×d¯(Afj,Agj)+f,g=1fgmjDfgwj×d¯(Afj,Agj)}1.

The weight of strong discordance set wD is computed with Eq. (47) as follows:

(47)
wD=f,g=1fgmjDfgwj×d¯(Afj,Agj)×{f,g=1fgmjDfgwj×d¯(Afj,Agj)+f,g=1fgmjDfgwj×d¯(Afj,Agj)+f,g=1fgmjDfgwj×d¯(Afj,Agj)}1.

The weight of strong discordance set wD is computed with Eq. (48) as follows:

(48)
wD=f,g=1fgmjDfgwj×d¯(Afj,Agj)×{f,g=1fgmjDfgwj×d¯(Afj,Agj)+f,g=1fgmjDfgwj×d¯(Afj,Agj)+f,g=1fgmjDfgwj×d¯(Afj,Agj)}1.

In Eqs. (43)–(48), d¯(Afj,Agj) is the distance between αf and αg under Cj, wj is the weight of Cj.

5.6Construction of Fermatern Fuzzy Concordance Matrix and Discordance Matrix

The concordance matrix and discordance matrix are constructed based on the concordance and discordance index, respectively. In order to specify an outranking relationship between Af and Ag, it is essential to compute two main indices called concordance index and discordance index. The concordance index for a pair of alternative Af and Ag, which shows the degree of superiority of alternative Af to alternative Ag, is related to the weights of the concordance sets and the corresponding criteria weights. Therefore, the concordance index Vfg between two alternatives Af and Ag is defined as

(49)
Vfg=wC×jCfgwj+wC×jCfgwj+wC×jCfgwj,
where wC, wC and wC are the weights of strong concordance set, medium concordance set and weak discordance set, respectively; wj is the weight of the corresponding criterion.

After determination of all concordance indices, the concordance matrix V is generated as follows:

V=V12V1nV21V23V2nV(n1)1V(n1)nVn1Vn2Vn(n1).

The discordance index for a pair of alternative Af and Ag shows the degree of inferiority of alternative Af to alternative Ag according to criteria in the discordance sets. The discordance index Dfg between Af and Ag is represented as

(50)
Dfg=max{wD×max{d¯jJDfg(Afj,Agj)},wD×max{d¯jJDfg(Afj,Agj)},wD×max{d¯jJDfg(Afj,Agj)}}×{maxjJd¯(Afj,Agj)}1,
where wD, wD and wD are the weights of three kinds of Fermatern fuzzy discordance sets, and d¯(Afj,Agj) stands for the distance measure between alternative Af and Ag with reference to criterion Cj.

Based on the discordance index, the discordance matrix D is defined as follows:

D=D12D1nD21D2nD(n1)1D(n1)nDn1Dn(n1).

5.7Computation of the Net Superiority Index and the Net Inferiority Index

As mentioned above, the concordance index Vfg reveals the degree of superiority of alternative Af to alternative Ag, the bigger the value of Vfg, the more superior Af is to Ag. Likewise, the discordance index Dfg shows the degree to which alternative Af is inferior to alternative Ag, the bigger the value of Dfg, the more inferior Af is to Ag. That is to say, Vfg, to some degree, represents the inferiority degree of alternative Ag to alternative Af, and Dfg, shows superiority degree of alternative Ag to alternative Af. Hence, both Vfg and Dgf display the superiority degree of Af to Ag, and both Vgf and Dfg display the inferiority degree of Af to Ag. Therefore, the net superiority index of alternative Af can be computed as

(51)
NSf=f=1,fgnVfg+f=1,fgnDgf.

Thus, it may be known that the net superiority index NSf shows the relative superiority degree of alternative Af over all the other alternatives.

Furthermore, the net inferiority index of alternative Af can be computed as

(52)
NIf=f=1,fgnVgf+f=1,fgnDfg.

The net inferiority index NIf shows the relative inferiority degree of alternative Af to all the other alternatives.

To rank alternatives an overall evaluation index is defined as

(53)
Zf=NSfNIf.

The overall evaluation index Zf stands for the overall superiority degree of alternative Af over all the other alternatives. If an alternative has the biggest value of the net superiority index and the smallest value of the net inferiority index, it is the optimal alternative.

On the basis of the overall evaluation index, the optimal alternative can be selected as follows:

(54)
A=max1fn{Zf}.

5.8A Fermatean Fuzzy ELECTRE Method

On the basis of the above analyses, the steps of the proposed Fermatean fuzzy ELECTRE method are summarized as follows:

  • Step 1. Form the group decision matrices. DMs give their evaluations of all alternatives regarding to each criterion with linguistic terms. Then, these linguistic assessments can be transformed into FFNs, and thus build up the group decision matrices.

  • Step 2. Determine dynamic weights of DMs using Eqs. (19)–(22).

  • Step 3. Aggregate all individual decision matrices into a collective one using Eq. (23).

  • Step 4. Obtain criteria weights using Eqs. (24)–(36) and models (M-1)–(M-4).

  • Step 5. Construct strong, medium and weak concordance sets and discordance sets based on Eqs. (37)–(39) and Eqs. (40)–(42), respectively.

  • Step 6. Identify the weights of strong, medium and weak concordance and discordance sets by using Eqs. (43)–(45) and Eqs. (46)–(48), respectively.

  • Step 7. Construct concordance matrix and discordance matrix using Eq. (49) and Eq. (50), respectively.

  • Step 8. Compute the net superiority index and the net inferiority index using Eq. (51) and Eq. (52), respectively.

  • Step 9. Compute the overall evaluation indices for all alternatives by Eq. (53).

  • Step 10. Choose the optimal alternative based on Eq. (54).

6Case Study Concerning Site Selection of FSHs for COVID-19 Patients in Wuhan

In this section, a practical case concerning site selection of FSHs for COVID-19 in Wuhan is provided to show the implementation process of the proposed ELECTRE method. Then, some comparisons are carried out to verify the superiority and effectiveness of the proposed ELECTRE method.

6.1Description of Site Selection of FSHs

Nevertheless the spread of the COVID-19 around the world, it was unfortunately detected at the end of 2019 in Wuhan, the capital city of Hubei Province, China. By February 26, 2020, there have been 47824 confirmed cases in Wuhan, accounting for 60.9% of the total confirmed cases in China. Owing to the lack of medical resources, especially the number of beds for patients with confirmed COVID-19 is seriously insufficient, a large number of confirmed patients failed to be isolated and treated in time, causing cross infection in the community and accelerating the spread of the epidemic. In order to collect and treat patients with mild COVID-19, the Chinese government launched an emergency construction of FSHs. The rapid establishment and operation of FSHs have played an irreplaceable role in COVID-19 prevention and control.

Site selection is the first and most critical step in the construction of FSHs. Without loss of generality, this paper only considers the site selection for the first FSHs in Wuhan. Site selection takes three aspects into consideration: first, it should be far away from residential areas and densely populated places, and be in the downwind position of this area; second, it should be convenient for transportation of patients and medical staff; third, the internal structure of the site is convenient for rapid transformation and has certain functionality.

There exist five candidate buildings (alternatives) suitable for being reconstructed to FSHs in Wuhan. They are Wuhan Sports Center (A1), Hongshan Gymnasium (A2), Wuhan International Conference and Exhibition Center (A3), Wuhan Gymnasium (A4), and China (Wuhan) Cultural Exhibition Center (A5). A1 is located in the southwest of Wuhan and situated in Wuhan economic and technological development zone. A2 is located in Hongshan Square, the centre of Wuchang District. A3 is located in Jianghan District, which is the most prosperous business center in Wuhan. A4 is located at No. 612, Jiefang Avenue in downtown Hankou. A5 is located in Jinyintan, Dongxihu District.

The experts (or DMs) panel consists of five experts. They were selected from the areas of disease control and prevention, scientific research institution, public health education, architectural design and research institute, etc. They had more than ten-year working experience and high-level academic titles.

In light of technical requirements for design and reconstruction of FSHs issued by Department of Housing and Urban-Rural Development of Hubei Province (see http://zjt.hubei.gov.cn/), eight main technical requirements (i.e., criteria) for candidate buildings (alternatives) are extracted as follows: traffic convenience (C1), environmental protection (C2), geographical position (C3), infrastructure (C4), regional communication convenience (C5), capacity (C6), reconstruction difficulty (C7) and reconstruction cost (C8). Here, C1, C2, C3, C4, C5 and C6 are benefit criteria, but C7 and C8 are cost criteria. After further discussion and negotiation, the information regarding to criteria weights given by the group of DMs is incomplete as follows:

Λ=wΛ00.07w80.1,w8w70.02,0.15w10.15,0.1w40.12,w4w30.02,w30.1,0.12w50.15,w5w40.03,w11.5w2,0.13w60.16,w1w60.03,w2w80.02.

The hierarchical structure of this group decision-making problem is shown in Fig. 3.

Fig. 3

Hierarchical structure of case study.

Hierarchical structure of case study.

6.2Application of the Proposed Fermatean Fuzzy ELECTRE Method

The main steps of the proposed ELECTRE method can be described as follows:

Step 1: Form Fermatean fuzzy group decision matrices.

Each DM is required to present his/her evaluations of alternative Ai (i=1,2,3,4,5) with respect to criterion Cj (j=1,2,,8). Five DMs evaluate five alternatives under the given criteria using the linguistic variables defined in Table 3. Table 4 describes the linguistic values for alternatives over different criteria given by five DMs. The linguistic evaluations shown in Table 4 are transformed into FFNs by using the mapping relations given by in Table 3. Consequently, Fermatean fuzzy group decision matrices Rl=(aijl)m×n are constructed and shown in Table 5.

Table 3

Performance ratings of alternatives as linguistic values.

Linguistic variablesFFNsIFNsPFNs
Absolutely Good (AG)F(0.98, 0.02)I(1.0, 0.0)P(0.98, 0.1)
Very Good (VG)F(0.9, 0.6)I(0.90, 0.05)P(0.87, 0.35)
Good (G)F(0.8, 0.65)I(0.75, 0.2)P(0.7, 0.4)
Medium Good (MG)F(0.75, 0.6)I(0.65, 0.3)P(0.65, 0.45)
Average (A)F(0.5, 0.5)I(0.55, 0.4)P(0.5, 0.55)
Medium Bad (MB)F(0.6, 0.7)I(0.4, 0.5)P(0.4, 0.7)
Bad (B)F(0.7, 0.8)I(0.36, 0.6)P(0.36, 0.8)
Very Bad (VB)F(0.6, 0.9)I(0.2, 0.7)P(0.25, 0.87)
Absolutely Bad (AB)F(0.02, 0.98)I(0.1, 0.8)P(0.1, 0.98)
Table 4

Evaluation values described as linguistic variables by five DMs.

DMAlternativeC1C2C3C4C5C6C7C8
E1A1MGVBAVGGAMGG
A2MBVGAMGAMGVGMG
A3VGVGMBMGVGVGVBG
A4MBVGMGAMGMBMGMB
A5VGVGMBMGAGGGMG
E2A1VGMGMBMGMBMGAG
A2MGGMBVBMBGMGA
A3GMGVGVBVBVGVGB
A4MGVGVGBMGMGAVB
A5AGBMGMBVBGVBA
E3A1MGVGABMBMGAB
A2VGVGMBVBVBVGMBMB
A3VGMBGBBVGVBVB
A4VBMGBVBVBVGVBVB
A5VGVGMBVBBMBMGMB
E4A1GVGAVBMBGBMB
A2VGGMBVBBVGVBVB
A3VGVGBMGVBBVBVB
A4MGGMBVBVBVGBVB
A5VGBBBVGBVGB
E5A1VGMGMBBBVGVBB
A2VGVGMBVBMBVGBVB
A3MGVGVBBVBVGVBB
A4GVGMBBBBVBMB
A5VGMGVBVBBVGBVB

Step 2: Normalize the decision matrix.

Table 5

Evaluation values described as FFNs by five DMs.

DMAlternativeC1C2C3C4C5C6C7C8
E1A1(0.75,0.6)(0.6,0.9)(0.5,0.5)(0.9,0.6)(0.8,0.65)(0.5,0.5)(0.75,0.6)(0.8,0.65)
A2(0.6,0.7)(0.9,0.6)(0.5,0.5)(0.75,0.6)(0.5,0.5)(0.75,0.6)(0.9,0.6)(0.75,0.6)
A3(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.75,0.6)(0.9,0.6)(0.9,0.6)(0.6,0.9)(0.8,0.65)
A4(0.6,0.7)(0.9,0.6)(0.75,0.6)(0.5,0.5)(0.75,0.6)(0.6,0.7)(0.75,0.6)(0.6,0.7)
A5(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.75,0.6)(0.98,0.02)(0.8,0.65)(0.8,0.65)(0.75,0.6)
E2A1(0.9,0.6)(0.75,0.6)(0.6,0.7)(0.75,0.6)(0.6,0.7)(0.75,0.6)(0.5,0.5)(0.8,0.65)
A2(0.75,0.6)(0.8,0.65)(0.6,0.7)(0.6,0.9)(0.6,0.7)(0.8,0.65)(0.75,0.6)(0.5,0.5)
A3(0.8,0.65)(0.75,0.6)(0.9,0.6)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.9,0.6)(0.7,0.8)
A4(0.75,0.6)(0.9,0.6)(0.9,0.6)(0.7,0.8)(0.75,0.6)(0.75,0.6)(0.5,0.5)(0.6,0.9)
A5(0.98,0.02)(0.7,0.8)(0.75,0.6)(0.6,0.7)(0.6,0.9)(0.8,0.65)(0.6,0.9)(0.5,0.5)
E3A1(0.75,0.6)(0.9,0.6)(0.5,0.5)(0.7,0.8)(0.6,0.7)(0.75,0.6)(0.5,0.5)(0.7,0.8)
A2(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.6,0.7)(0.6,0.7)
A3(0.9,0.6)(0.6,0.7)(0.8,0.65)(0.7,0.8)(0.7,0.8)(0.9,0.6)(0.6,0.9)(0.6,0.9)
A4(0.6,0.9)(0.75,0.6)(0.7,0.8)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.6,0.9)(0.6,0.9)
A5(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.6,0.9)(0.7,0.8)(0.6,0.7)(0.75,0.6)(0.6,0.7)
E4A1(0.8,0.65)(0.9,0.6)(0.5,0.5)(0.6,0.9)(0.6,0.7)(0.8,0.65)(0.7,0.8)(0.6,0.7)
A2(0.9,0.6)(0.8,0.65)(0.6,0.7)(0.6,0.9)(0.7,0.8)(0.9,0.6)(0.6,0.9)(0.6,0.9)
A3(0.9,0.6)(0.9,0.6)(0.7,0.8)(0.75,0.6)(0.6,0.9)(0.7,0.8)(0.6,0.9)(0.6,0.9)
A4(0.75,0.6)(0.8,0.65)(0.6,0.7)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.7,0.8)(0.6,0.9)
A5(0.9,0.6)(0.7,0.8)(0.7,0.8)(0.7,0.8)(0.9,0.6)(0.7,0.8)(0.9,0.6)(0.7,0.8)
E5A1(0.9,0.6)(0.75,0.6)(0.6,0.7)(0.7,0.8)(0.7,0.8)(0.9,0.6)(0.6,0.9)(0.7,0.8)
A2(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.6,0.9)(0.6,0.7)(0.9,0.6)(0.7,0.8)(0.6,0.9)
A3(0.75,0.6)(0.9,0.6)(0.6,0.9)(0.7,0.8)(0.6,0.9)(0.9,0.6)(0.6,0.9)(0.7,0.8)
A4(0.8,0.65)(0.9,0.6)(0.6,0.7)(0.7,0.8)(0.7,0.8)(0.7,0.8)(0.6,0.9)(0.6,0.7)
A5(0.9,0.6)(0.75,0.6)(0.6,0.9)(0.6,0.9)(0.7,0.8)(0.9,0.6)(0.7,0.8)(0.6,0.9)

Since criteria are classified into cost criteria and benefit criteria in this paper, it is necessary to convert raw data into comparable value by a normalization procedure. During the normalization, cost criteria must be converted into benefit criteria. The mathematical expression of the decision matrix Rl=(aijl)m×n normalized into R˜l=(a˜ijl)m×n is given below:

(55)
a˜ijl=(α˜ijl,β˜ijl)=(αijl,βijl),for benefit criteriaCj,(βijl,αijl),for cost criteriaCj,l=1,2,,t.

The normalized decision matrix is presented in Tables 6.

Step 3: Compute the dynamic DMs’ weights for different alternatives and different criteria.

Table 6

Normalized evaluation values described as FFNs from five DMs.

DMAlternativeC1C2C3C4C5C6C7C8
E1A1(0.75,0.6)(0.6,0.9)(0.5,0.5)(0.9,0.6)(0.8,0.65)(0.5,0.5)(0.6,0.75)(0.65,0.8)
A2(0.6,0.7)(0.9,0.6)(0.5,0.5)(0.75,0.6)(0.5,0.5)(0.75,0.6)(0.6,0.9)(0.6,0.75)
A3(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.75,0.6)(0.9,0.6)(0.9,0.6)(0.9,0.6)(0.65,0.8)
A4(0.6,0.7)(0.9,0.6)(0.75,0.6)(0.5,0.5)(0.75,0.6)(0.6,0.7)(0.6,0.75)(0.7,0.6)
A5(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.75,0.6)(0.98,0.02)(0.8,0.65)(0.65,0.8)(0.6,0.75)
E2A1(0.9,0.6)(0.75,0.6)(0.6,0.7)(0.75,0.6)(0.6,0.7)(0.75,0.6)(0.5,0.5)(0.65,0.8)
A2(0.75,0.6)(0.8,0.65)(0.6,0.7)(0.6,0.9)(0.6,0.7)(0.8,0.65)(0.6,0.75)(0.5,0.5)
A3(0.8,0.65)(0.75,0.6)(0.9,0.6)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.6,0.9)(0.8,0.7)
A4(0.75,0.6)(0.9,0.6)(0.9,0.6)(0.7,0.8)(0.75,0.6)(0.75,0.6)(0.5,0.5)(0.9,0.6)
A5(0.98,0.02)(0.7,0.8)(0.75,0.6)(0.6,0.7)(0.6,0.9)(0.8,0.65)(0.9,0.6)(0.5,0.5)
E3A1(0.75,0.6)(0.9,0.6)(0.5,0.5)(0.7,0.8)(0.6,0.7)(0.75,0.6)(0.5,0.5)(0.8,0.7)
A2(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.7,0.6)(0.7,0.6)
A3(0.9,0.6)(0.6,0.7)(0.8,0.65)(0.7,0.8)(0.7,0.8)(0.9,0.6)(0.9,0.6)(0.9,0.6)
A4(0.6,0.9)(0.75,0.6)(0.7,0.8)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.9,0.6)(0.9,0.6)
A5(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.6,0.9)(0.7,0.8)(0.6,0.7)(0.6,0.75)(0.7,0.6)
E4A1(0.8,0.65)(0.9,0.6)(0.5,0.5)(0.6,0.9)(0.6,0.7)(0.8,0.65)(0.8,0.7)(0.7,0.6)
A2(0.9,0.6)(0.8,0.65)(0.6,0.7)(0.6,0.9)(0.7,0.8)(0.9,0.6)(0.9,0.6)(0.9,0.6)
A3(0.9,0.6)(0.9,0.6)(0.7,0.8)(0.75,0.6)(0.6,0.9)(0.7,0.8)(0.9,0.6)(0.9,0.6)
A4(0.75,0.6)(0.8,0.65)(0.6,0.7)(0.6,0.9)(0.6,0.9)(0.9,0.6)(0.8,0.7)(0.9,0.6)
A5(0.9,0.6)(0.7,0.8)(0.7,0.8)(0.7,0.8)(0.9,0.6)(0.7,0.8)(0.6,0.9)(0.8,0.7)
E5A1(0.9,0.6)(0.75,0.6)(0.6,0.7)(0.7,0.8)(0.7,0.8)(0.9,0.6)(0.9,0.6)(0.8,0.7)
A2(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.6,0.9)(0.6,0.7)(0.9,0.6)(0.8,0.7)(0.9,0.6)
A3(0.75,0.6)(0.9,0.6)(0.6,0.9)(0.7,0.8)(0.6,0.9)(0.9,0.6)(0.9,0.6)(0.8,0.7)
A4(0.8,0.65)(0.9,0.6)(0.6,0.7)(0.7,0.8)(0.7,0.8)(0.7,0.8)(0.9,0.6)(0.7,0.6)
A5(0.9,0.6)(0.75,0.6)(0.6,0.9)(0.6,0.9)(0.7,0.8)(0.9,0.6)(0.8,0.7)(0.9,0.6)

(i) Determine PIDM and NIDM by using Models (19)–(20), respectively. For example, PIDM is presented in Table 7.

Table 7

PIDM.

AlternativeC1C2C3C4C5C6C7C8
A1(0.8,0.6)(0.75,0.6)(0.5,0.5)(0.7,0.8)(0.6,0.7)(0.75,0.6)(0.6,0.6)(0.7,0.7)
A2(0.9,0.6)(0.9,0.6)(0.6,0.7)(0.6,0.9)(0.6,0.7)(0.8,0.6)(0.7,0.7)(0.7,0.6)
A3(0.9,0.6)(0.9,0.6)(0.7,0.7)(0.7,0.8)(0.6,0.9)(0.9,0.6)(0.9,0.6)(0.8,0.7)
A4(0.75.0.65)(0.9,0.6)(0.7,0.7)(0.6,0.8)(0.7,0.8)(0.75,0.6)(0.8,0.6)(0.9,0.6)
A5(0.9,0.6)(0.75,0.6)(0.6,0.7)(0.6,0.8)(0.7,0.8)(0.8,0.65)(0.65,0.75)(0.7,0.6)

(ii) Calculate credibility degree for DM El based on Eq. (21), which are shown in Table 8.

Table 8

Credibility degree for El.

DMAlternativeC1C2C3C4C5C6C7C8
E1A10.19540.48240.02410.51990.38170.07380.15580.2778
A20.02570.58920.03020.09770.03020.19540.39500.1975
A30.58920.58920.12370.17650.58920.58920.58920.3144
A40.13290.58920.17550.14130.17650.13610.19540.0257
A50.10020.48240.14310.19540.59480.32000.31050.1975
E2A10.53570.20450.08890.17650.14310.20450.00970.2778
A20.09770.26410.14310.58920.14310.31610.17550.0302
A3