# A Fuzzy Multicriteria Group Decision-Making Method with New Entropy of Interval-Valued Intuitionistic Fuzzy Sets

Authors: Xiaohong Chen; Li Yang; Pei Wang; Wei Yue
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A new entropy measure of interval-valued intuitionistic fuzzy set (IVIFS) is proposed by using cotangent function, which overcomes several limitations in the existing methods for calculating entropy of IVIFS. The efficiency of the new entropy is demonstrated by comparing it with several classical entropies. Moreover, an entropy weight model is established to determine the entropy weights for fuzzy multicriteria group decision-making (FMCGDMs) problems, which depends on incomplete weight information of criteria in IVIFSs setting. Finally, an illustrative supplier selection problem is used to demonstrate the practicality and effectiveness of the proposed method. It is capable of the handling the FMCGDM problems with incomplete known weights for criteria.


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## Body

## 1. Introduction

The theory of fuzzy sets (FSs) proposed by Zadeh [1] has achieved a great success in various fields. A lot of generalized forms of FSs have been proposed. The classical sets include interval-valued fuzzy sets (IVFSs) [2], intuitionistic fuzzy sets (IFSs) [3, 4], interval-valued intuitionistic fuzzy sets (IVIFSs) [5], R-fuzzy sets [6], and intuitionistic linguistic fuzzy sets (ILFSs) [7]. From [4, 8–10], it turns out that IVFS theory is equivalent to IFS theory, and IVIFS theory extends IFS theory.As an important topic in the theory of fuzzy sets, entropy measures of IFSs have been investigated widely by many researchers from different views. Burillo and Bustince [11] introduced the notion that entropy of IVFSs and IFSs can be used to evaluate the degree of intuitionism of an IVFS or IFS. Szmidt and Kacprzyk [12] proposed a nonprobabilistic-type entropy measure with a geometric interpretation of IFSs. Hung and Yang [13] gave their axiomatic definitions of entropy of IFSs and IVFSs by using the concept of probability. Wei et al. [14] gave a new entropy measure for IVIFSs to overcome the disadvantages of those three entropy measures defined independently by Szmidt and Kacprzyk [12], Wang and Lei [15], and Huang and Liu [16]. Different entropy formulas for IFS [15, 17], IVFS [18, 19], and vague set [16, 20, 21] were also proposed by other researchers.The entropy of IFSs has been applied widely in decision making [22, 23]. On the one hand, due to the increasing complexity of the social-economic environment and a lack of information about the problem domains, the decision information may be provided with IVIFSs, whose membership degree and nonmembership degree are intervals, instead of real numbers. Entropy is concerned as a measure of fuzziness. Therefore, it is highly necessary and significant to study the entropy of IVIFSs. And on the other hand, a proper assessment of attribute weights plays an essential role in the MADM process [24]. In terms of determining weights, the entropy method is one of the most representative approaches, which expresses the relative intensities of attribute importance to signify the average intrinsic information transmitted to the DM [22, 25, 26]. The following are some of the research findings.Ye [27] proposed two entropy measures for IVIFSs and established an entropy weight model, which could be used to determine the criteria weights on alternatives. Zhang et al. [28] proposed a new information entropy measure of IVIFS by using membership interval and nonmembership interval of IVIFSs, which complied with the extended form of De Luca and Termini [29] axioms for fuzzy entropy. Wei et al. [14] also proposed an entropy measure for IVIFSs, and they applied the new entropy measure to solve problem on multicriteria fuzzy decision making.However, the entropy is seldom applied in multiexpert and multicriteria decision making, which is one of the most important branches of decision-making method. Due to limited investigation on multiexpert and multicriteria decision-making issues, this study proposed a novel formula to calculate the entropy of an IVIFS on the basis of the argument on the relationship among the entropies of IFSs given in [27, 30]. For interval-valued intuitionistic fuzzy multicriteria group decision-making problem, in which the information on the weights of criteria is incomplete, a linear fuzzy programming model based on intuitionistic fuzzy entropy is constructed to obtain the criteria weights.The rest of this paper is organized as follows. In Section2, we introduce some basic notions of IFS and IVIFSs. In Section 3, we propose a new entropy measure of interval-valued intuitionistic fuzzy set by using cotangent function. In Section 4, the method and procedure for solving FMCGDM problems with the new entropy measure of interval-valued intuitionistic fuzzy set are developed in detail. An illustrative supplier selection problem was employed to demonstrate how to apply the proposed approach in Section 5. Short conclusion is given in Section 6.


## 2. Interval-Valued Intuitionistic Fuzzy Sets

Definition 1 (see [3]).
LetX be an ordinary finite nonempty set. An intuitionistic fuzzy set (IFS) in X is an object of the form:
(1)A={〈x,μA(x),νA(x)〉∣x∈X},
where
(2)μA:X⟶[0,1],νA:X⟶[0,1]
with the condition 0≤μA(x)+νA(x)≤1, for all x∈X.The numbersμA(x), νA(x) denote the degree of membership and nonmembership of the element x in the set A, respectively.For each IFS, we callπA(x)=1-μA(x)-νA(x) the intuitionistic index of the element x in the set A. It also denotes the hesitancy degree of x to A.Definition 2 (see [5]).
LetD[0,1] be the set of all closed subintervals of the interval [0,1], and let X be an ordinary finite nonempty set. An interval-valued intuitionistic fuzzy set (IVIFS) in X is an object of the form:
(3)A={〈x,μ~A(x),ν~A(x)〉∣x∈X},x in the set A, where μ~A:X→D[0,1], ν~A:X→D[0,1], with the condition 0≤sup(μ~A(x))+sup(ν~A(x))≤1, for all x∈X.The intervalsμ~A(x), ν~A(x) denote the degree of membership and nonmembership of the element x in the set A, respectively.For convenience, letμ~A(x)=[μAL(x),μAU(x)], ν~A(x)=[νAL(x),νAU(x)], then
(4)A={〈x,[μAL(x),μAU(x)],[νAL(x),νAU(x)]〉∣x∈X},
where 0≤μAU(x)+νAU(x)≤1, μAL(x)≥0, and νAL(x)≥0.For each elementx, we can compute the intuitionistic index of an intuitionistic fuzzy interval of x∈X in A defined as follows:
(5)π~A(x)=1-μ~A(x)-ν~A(x)=[1-μAU(x)-νAU(x),1-μAL(x)-νAL(x)].For convenience, an IVIFS value is denoted byA=([a,b],[c,d]).Definition 3 (see [5]).
AssumeA,B∈IVIFS(X), then some operations can be defined as follows:
(6)A∪B={〈xi,[μAL(xi)∨μBL(xi),μAU(xi)∨μBU(xi)],[νAL(xi)∧νBL(xi),νAU(xi)∧νBU(xi)]〉},A∩B={〈xi,[μAL(xi)∧μBL(xi),μAU(xi)∧μBU(xi)],[νAL(xi)∨νBL(xi),νAU(xi)∨νBU(xi)]〉}.The following expressions are defined in [5] for all A,B∈IVFSs(X):A⊆B
if and only ifμAL≤μBL, μAU≤μBU, νAL≥νBL, and νAU≥νBU for all x∈X;A=B if and only if A⊆B and B⊆A for all x∈X;AC={〈x,[νAL(x),νAU(x)],[μAL(x),μAU(x)]〉∣x∈X}.In the following, we introduce two weighted aggregation operators related to IVIFSs.Definition 4 (see [31]).
LetAj  (j=1,2,…,n)∈IVIFS(X). The weighted geometric average operator (IVIF-WGA operator) is defined by
(7)Fw(A1,A2,…,An)=([∏j=1nμAjL(x)wj,∏j=1nμAjU(x)wj],[1-∏j=1n(1-νAjL(x))wj,1-∏j=1n(1-νAjU(x))wj]),
where wj is the weight of Aj(j=1,2,…,n),wj∈[0,1] and ∑j=1nwj=1. Particularly, if wj=1/n(j=1,2,…,n), then Fw is called a geometric average operator for IVIFSs.Definition 5 (see [31]).
LetAj(j=1,2,…,n)∈IVIFS(X). The hybrid averaging operator (IVIF-HA operator) is defined by
(8)Hw(A1,A2,…,An)=([∏j=1nμBjL(x)wj,∏j=1nμBjU(x)wj],[1-∏j=1n(1-νBjL(x))wj,1-∏j=1n(1-νBjU(x))wj]),
where wj is the weight of Bj(j=1,2,…,n), wj∈[0,1] and ∑j=1nwj=1. Bj is the jth largest one of all values (nω1A1,nω2A2,…,nωnAn), and ωk is the weight of Ak(k=1,2,…,n), satisfying ωk∈[0,1] and ∑k=1nωk=1. n is a balance factor. Here, W=(w1,w2,…,wn)T can be obtained by the following formula:
(9)wj=Q(jl)-Q((j-1)l),j=1,2,…,n,
where
(10)Q={0,r<a,r-aa-b,a≤r≤b,1,r>b.We utilize the principle of antonym pairsmost,at least half, as many as possible, where the parameters (a,b) are equal to (0.3,0.8), (0,05), and (0.5,1), respectively.Definition 6 (see [32]).
LetA=([a,b],[c,d]) be an interval-valued intuitionistic fuzzy number. A score function S of an interval-valued intuitionistic fuzzy value can be represented as follows:
(11)S(A)=a-c+b-d2.Definition 7 (see [32]).
LetA1=([a1,b1],[c1,d1]) and A2=([a2,b2],[c2,d2]) be two interval-valued intuitionistic fuzzy values, and let S(A1)=(a1-c1+b1-d1)/2 and S(A2)=(a2-c2+b2-d2)/2 be the scores of A1 and A2, respectively, then if S(A1)<S(A2), A1 is smaller than A2, denoted by A1<A2.


## 3. Interval-Valued Intuitionistic Fuzzy Entropy

Definition 8 (see [23]).
A real-valued functionE:IVIFS(X)→[0,1] is called an entropy measure on IVIFS(X) if it satisfies the following axiomatic requirements:(P1)
E(A)=0, if and only if A is a crisp set;(P2)
E(A)=1, if and only if μ~A(xi)=ν~A(xi) for all xi∈X;(P3)
E(A)=E(AC) for all xi∈IVIFS(X);(P4)
E(A)≤E(B) if A is less fuzzy than B; that is, μAL(xi)≤νAL(xi),μAU(xi)≤νAU(xi), and A⊆B for all xi∈X or μAL(xi)≥νAL(xi), μAU(xi)≥νAU(xi), and B⊆A for all xi∈X.


### 3.1. The Limitations of the Existing Interval-Valued Intuitionistic Fuzzy Entropy

Let us suppose thatE(Ai) is the entropy of IVIFSs.Vlachos’ entropy measure [30] is as follows
(12)E1(A)=1-12n∑i=1n(|μAL(xi)-νAL(xi)|)2+(|μAU(xi)-νAU(xi)|)2.Example 9.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.1,0.2],[0,0.1]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. If we calculate the E1(A) and E1(B) by (12), then we can obtain
(13)E1(A)=1-12(|0.4-0.3|2+|0.5-0.4|2)=0.9,E1(B)=1-12(|0.1-0|2+|0.2-0.1|2)=0.9,
which indicate that E1(A)=E1(B) and is not consistent with our intuition.
Ye’s entropy measures [27] are as follows
(14)E2(A)={sinπ×[1+μAL(xi)+pWμA(xi)-νAL(xi)-qWνA(xi)]4+sinπ×[1-μAL(xi)-pWμA(xi)+νAL(xi)+qWνA(xi)]4-1}×12-1,
where p,q∈[0,1] are two fixed numbers, WμA(x)=μAU(x)-μAL(x), and WνA(x)=νAU(x)-νAL(x).Example 10.
LetA=([0.5,0.5],[0.1,0.1]) and B=([0.6,0.6],[0.2,0.2]) be two IVIFSs in X.
Intuitively,A is more fuzzy than B. Now the E2(A) and E2(B) can be gained by (14) and take p=q=0.5, and the results are
(15)E2(A)={sinπ×[1+0.5-0.1]4+sinπ×[1-0.5+0.1]4-1}×12-1=0.833,E2(B)={sinπ×[1+0.6-0.2]4+sinπ×[1-0.6+0.2]4-1}×12-1=0.833,
which indicate that E2(A)=E2(B) and are not consistent with our intuition.


### 3.2. New Interval-Valued Intuitionistic Fuzzy Entropy Based on Cotangent Function

A new interval-valued intuitionistic fuzzy entropy measure is introduced as follows.Definition 11.
Assuming thatA∈IVIFS(X), then an interval-valued intuitionistic fuzzy entropy measure can be defined as
(16)E(A)=1n∑i=1ncot(14π+(12|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|(12)(12)(12))×(4(1+πA(xi)))-1π14π+((12)|μAL(xi)-νAL(xi))=1n∑i=1ncot(14π+(|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|)×(-μAU(xi)-νAU(xi))-14(μAU(xi)-νAU(xi)4-μAL(xi)-νAL(xi)-μAU(xi)-νAU(xi))-1)π14).Theorem 12.
The mappingE(A), defined by (16), is an entropy measure for IVIFSs.Proof.
In order for (16) to be qualified as a sensible measure of interval-valued intuitionistic fuzzy entropy, it must satisfy the conditions (P1)–(P4) in Definition 8.
Let0≤μ~A(xi), ν~A(xi), and π~A(xi)≤1; we have 0≤|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|≤2. It follows that (0≤|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|/8(1+πA(xi)))π≤(1/4)π. Thus 0≤E(A)≤1.(P1)
LetA be a crisp set. Then we have π~A(xi)=0,μ~A(xi)=1, and ν~A(xi)=0 or π~A(xi)=0, μ~A(xi)=0, and ν~A(xi)=1. So E(A)=0.(P2)
Letμ~A(xi)=ν~B(xi), then E(A)=1.(P3)
It is clear thatAC={〈xi,[νAL(xi),νAU(xi)],[μAL(xi),μAU(xi)]〉∣xi∈X}. By applying (11), we have E(A)=E(AC).(P4)
In order to show that (11) fulfill the requirement of (P4), it is suffice to to prove the following function:
(17)|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|4(4-μAL(xi)-νAL(xi)-μAU(xi)-νAU(xi))≥|μBL(xi)-νBL(xi)+μBU(xi)-νBU(xi)|4(4-μBL(xi)-νBL(xi)-μBU(xi)-νBU(xi)),
where function E is monotonic decreasing.
Suppose thatμBL(xi)≤νBL(xi), μBU(xi)≤νBU(xi), and A⊆B, in order to prove (17); namely, we prove that
(18)(2-μAL(xi)-μAU(xi))(2-νBL(xi)-νBU(xi))≥(2-μBL(xi)-μBU(xi))(2-νAL(xi)-νAU(xi)).
If μAL(xi)≤μBL(xi) and μAU(xi)≤μBU(xi), then 2-μAL(xi)-μAU(xi)≥2-μBL(xi)-μBU(xi)≥0. If νAL(xi)≥νBL(xi) and μAU(xi)≥μBU(xi), then we have 2-νBL(xi)-νBU(xi)≥2-νAL(xi)-νAU(xi)≥0. So we can get that (18) holds. Therefore, E(A)≤E(B).
Similarly, whenμBL(xi)≥νBL(xi), μBU(xi)≥νBU(xi), and B⊆A, we can also prove that E(A)≤E(B).Example 13.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.1,0.2],[0,0.1]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. Now we calculate the E(A) and E(B) by (16), and we can obtain that
(19)E(A)=cot(14π+|0.4-0.3+0.5-0.4|4(4-0.4-0.3-0.5-0.4)π)=0.877,E(B)=cot(14π+|0.1-0+0.2-0.1|4(4-0.1-0-0.2-0.1)π)=0.916,
which indicate that E(A)<E(B) and are consistent with our intuition.Example 14.
LetA=([0.5,0.5],[0.1,0.1]) and B=([0.6,0.6],[0.2,0.2]) be two IVIFSs in X.
Intuitively, we can see thatA is more fuzzy than B. Now we calculate the E(A) and E(B) by (16), and we can obtain that
(20)E(A)=cot(14π+|0.5-0.1+0.5-0.1|4(4-0.5-0.1-0.5-0.1)π)=0.628,E(B)=cot(14π+|0.6-0.2+0.6-0.2|4(4-0.6-0.2-0.6-0.2)π)=0.577,
which indicate that E(A)>E(B) and are consistent with our intuition.


## 3.1. The Limitations of the Existing Interval-Valued Intuitionistic Fuzzy Entropy

Let us suppose thatE(Ai) is the entropy of IVIFSs.Vlachos’ entropy measure [30] is as follows
(12)E1(A)=1-12n∑i=1n(|μAL(xi)-νAL(xi)|)2+(|μAU(xi)-νAU(xi)|)2.Example 9.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.1,0.2],[0,0.1]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. If we calculate the E1(A) and E1(B) by (12), then we can obtain
(13)E1(A)=1-12(|0.4-0.3|2+|0.5-0.4|2)=0.9,E1(B)=1-12(|0.1-0|2+|0.2-0.1|2)=0.9,
which indicate that E1(A)=E1(B) and is not consistent with our intuition.
Ye’s entropy measures [27] are as follows
(14)E2(A)={sinπ×[1+μAL(xi)+pWμA(xi)-νAL(xi)-qWνA(xi)]4+sinπ×[1-μAL(xi)-pWμA(xi)+νAL(xi)+qWνA(xi)]4-1}×12-1,
where p,q∈[0,1] are two fixed numbers, WμA(x)=μAU(x)-μAL(x), and WνA(x)=νAU(x)-νAL(x).Example 10.
LetA=([0.5,0.5],[0.1,0.1]) and B=([0.6,0.6],[0.2,0.2]) be two IVIFSs in X.
Intuitively,A is more fuzzy than B. Now the E2(A) and E2(B) can be gained by (14) and take p=q=0.5, and the results are
(15)E2(A)={sinπ×[1+0.5-0.1]4+sinπ×[1-0.5+0.1]4-1}×12-1=0.833,E2(B)={sinπ×[1+0.6-0.2]4+sinπ×[1-0.6+0.2]4-1}×12-1=0.833,
which indicate that E2(A)=E2(B) and are not consistent with our intuition.


## 3.2. New Interval-Valued Intuitionistic Fuzzy Entropy Based on Cotangent Function

A new interval-valued intuitionistic fuzzy entropy measure is introduced as follows.Definition 11.
Assuming thatA∈IVIFS(X), then an interval-valued intuitionistic fuzzy entropy measure can be defined as
(16)E(A)=1n∑i=1ncot(14π+(12|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|(12)(12)(12))×(4(1+πA(xi)))-1π14π+((12)|μAL(xi)-νAL(xi))=1n∑i=1ncot(14π+(|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|)×(-μAU(xi)-νAU(xi))-14(μAU(xi)-νAU(xi)4-μAL(xi)-νAL(xi)-μAU(xi)-νAU(xi))-1)π14).Theorem 12.
The mappingE(A), defined by (16), is an entropy measure for IVIFSs.Proof.
In order for (16) to be qualified as a sensible measure of interval-valued intuitionistic fuzzy entropy, it must satisfy the conditions (P1)–(P4) in Definition 8.
Let0≤μ~A(xi), ν~A(xi), and π~A(xi)≤1; we have 0≤|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|≤2. It follows that (0≤|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|/8(1+πA(xi)))π≤(1/4)π. Thus 0≤E(A)≤1.(P1)
LetA be a crisp set. Then we have π~A(xi)=0,μ~A(xi)=1, and ν~A(xi)=0 or π~A(xi)=0, μ~A(xi)=0, and ν~A(xi)=1. So E(A)=0.(P2)
Letμ~A(xi)=ν~B(xi), then E(A)=1.(P3)
It is clear thatAC={〈xi,[νAL(xi),νAU(xi)],[μAL(xi),μAU(xi)]〉∣xi∈X}. By applying (11), we have E(A)=E(AC).(P4)
In order to show that (11) fulfill the requirement of (P4), it is suffice to to prove the following function:
(17)|μAL(xi)-νAL(xi)+μAU(xi)-νAU(xi)|4(4-μAL(xi)-νAL(xi)-μAU(xi)-νAU(xi))≥|μBL(xi)-νBL(xi)+μBU(xi)-νBU(xi)|4(4-μBL(xi)-νBL(xi)-μBU(xi)-νBU(xi)),
where function E is monotonic decreasing.
Suppose thatμBL(xi)≤νBL(xi), μBU(xi)≤νBU(xi), and A⊆B, in order to prove (17); namely, we prove that
(18)(2-μAL(xi)-μAU(xi))(2-νBL(xi)-νBU(xi))≥(2-μBL(xi)-μBU(xi))(2-νAL(xi)-νAU(xi)).
If μAL(xi)≤μBL(xi) and μAU(xi)≤μBU(xi), then 2-μAL(xi)-μAU(xi)≥2-μBL(xi)-μBU(xi)≥0. If νAL(xi)≥νBL(xi) and μAU(xi)≥μBU(xi), then we have 2-νBL(xi)-νBU(xi)≥2-νAL(xi)-νAU(xi)≥0. So we can get that (18) holds. Therefore, E(A)≤E(B).
Similarly, whenμBL(xi)≥νBL(xi), μBU(xi)≥νBU(xi), and B⊆A, we can also prove that E(A)≤E(B).Example 13.
LetA=([0.4,0.5],[0.3,0.4]) and B=([0.1,0.2],[0,0.1]) be two IVIFSs in X.
Intuitively, we can see thatB is more fuzzy than A. Now we calculate the E(A) and E(B) by (16), and we can obtain that
(19)E(A)=cot(14π+|0.4-0.3+0.5-0.4|4(4-0.4-0.3-0.5-0.4)π)=0.877,E(B)=cot(14π+|0.1-0+0.2-0.1|4(4-0.1-0-0.2-0.1)π)=0.916,
which indicate that E(A)<E(B) and are consistent with our intuition.Example 14.
LetA=([0.5,0.5],[0.1,0.1]) and B=([0.6,0.6],[0.2,0.2]) be two IVIFSs in X.
Intuitively, we can see thatA is more fuzzy than B. Now we calculate the E(A) and E(B) by (16), and we can obtain that
(20)E(A)=cot(14π+|0.5-0.1+0.5-0.1|4(4-0.5-0.1-0.5-0.1)π)=0.628,E(B)=cot(14π+|0.6-0.2+0.6-0.2|4(4-0.6-0.2-0.6-0.2)π)=0.577,
which indicate that E(A)>E(B) and are consistent with our intuition.


## 4. Fuzzy Multicriteria Group Decision-Making Method Based on the New Interval-Valued Intuitionistic Fuzzy Entropy

In this section, we propose a method for fuzzy group decision-making problems based on the interval-valued intuitionistic fuzzy entropy.Consider an interval-valued intuitionistic fuzzy multicriteria group decision-making problem. Assume that there aren alternatives X={x1,x2,…,xn} and m decision criteria C={c1,c2,…,cm} with weight vector W=(w1,w2,…,wm)T associated with C, where wj∈[0,1] and ∑j=1mwj=1. Assume that there are t decision makers D={d1,d2,…,dt} whose corresponding weight vector is λ=(λ1,λ2,…,λt)T, where λk∈[0,1] and ∑k=1tλk=1. In this case, the characteristic of the alternative Ai of the kth decision maker dk is represented by the following IVIFS:
(21)Aik={〈Cjk,[μAikL(Cjk),μAikU(Cjk)],[νAikL(Cjk),νAikU(Cjk)]〉∣Cjk∈C},
where 0≤μAikU(Cjk)+νAikU(Cjk)≤1, μAikL(Cjk)≥0, μAikL(Cjk)≥0, i=1,2,…,n, j=1,2,…,m, and k=1,2,…,t. The IVIFS value that is the pair of intervals μAik(Cjk)=[aijk,bijk], νAik(Cjk)=[cijk,dijk] for Cjk∈C is denoted by rijk=([aijk,bijk],[cijk,dijk]). Here, we can elicit the interval-valued intuitionistic fuzzy decision matrix Rk=(rijk)n×m.If the information about weightwj of the criterion Cj(j=1,2,…,m) is incomplete, for determining the criterion weight from the decision matrix we can establish a model of interval-valued intuitionistic fuzzy entropy weights.For the criteriaCj, the entropy of the alternative Ai of the kth decision maker can be given as
(22)E(rijk)=cot(14π+(|μAL(rijk)-νAL(rijk)+μAU(rijk)-νAU(rijk)|)×(-μAU(rijk)-νAU(rijk))4(4-μAL(rijk)-νAL(rijk)-μAU(rijk)-νAU(rijk)))-1π14).And the entropy for the alternativeAik of the kth decision maker is given as
(23)E(Aik)=∑j=1mcot(14π+(|μAL(rijk)-νAL(rijk)+μAU(rijk)-νAU(rijk)|)×(-μAU(rijk)-νAU(rijk)))-1π4(νAU(rijk)4-μAL(rijk)-νAL(rijk)-μAU(rijk)-νAU(rijk)))-1π14).As each alternative is made in a fair competitive environment, and the fuzzy entropy of each alternative is from a same criteria weight coefficient, the alternatives should be combined. The overall entropy for the alternativeAi is given as
(24)E(Ai)=∑k=1t∑j=1mcot(14π+(|μAL(rijk)-νAL(rijk)+μAU(rijk)-νAU(rijk)|)×(-μAU(rijk)-νAU(rijk))4(4-μAL(rijk)-νAL(rijk)-μAU(rijk)-νAU(rijk)))-1π14).According to the entropy theory, if the entropy value for an alternative is smaller across alternatives, it can provide decision makers with the useful information. Therefore, the criteria should be assigned to a bigger weight value. Then the smaller the value of (24) is, the better the weight we should assign to the criteria.LetH be the set of incomplete information about criteria weights; to get the optimal weight vector, the following model can be constructed:
(25)minE(A)=∑i=1n∑k=1t∑j=1mwj×cot(14π+(|μAL(rijk)-νAL(rijk)+μAU(rijk)-νAU(rijk)|)×(-μAU(rijk)-νAU(rijk))4((rijk)4-μAL(rijk)-νAL(rijk)-μAU(rijk)-νAU(rijk)))-1π),s.t.w∈H,∑j=1mwj=1,wj≥0,j=1,2,…,m.By solving model (25) with Lingo software, we get the optimal solution (w1,w2,…,wm)T.In summary, the main procedure of the decision method proposed is listed in the following.Step 1.
Calculate the weight vectorW=(w1,w2,…,wm)T by solving model (25).Step 2.
Aggregate the values of each decision maker, and utilize the IVIF-WGA operator to derive the valueszik (i=1,2,…,n) of each decision maker as follows:
(26)zik=Fw(ri1k,ri2k,…,rimk),i=1,2,…,n,k=1,2,…,t.Step 3.
Aggregate the values of all decision makers, and utilize the IVIF-HA operator to derive the collective valueszi of the alternative xi as follows
(27)zi=Hλ(zi1,zi2,…,zit),i=1,2,…,n,
where the associated weighting vector λ=(λ1,λ2,…,λt) of the IVIF-HA operator.Step 4.
Calculate the scoresS(zi)(i=1,2,…,n) of the collective overall values to rank all the alternatives Ai(i=1,2,…,n) and then to select the best one(s).


## 5. An Illustration for Solving the Supplier Selection Problem

This section adopts a supplier selection problem in [33, 34] to demonstrate how to apply the proposed approach. With continual business development, globalized markets become increasingly competitive. Establishing effective supply chain management (SCM) becomes a critical activity because a sound SCM system can reduce supply chain risk, maximize revenue, optimize business processes, and allow a company to maintain a dominant position in the market [34, 35]. On the other hand, it is a hard problem since supplier selection is typically a multicriteria group decision-making problem involving several conflicting criteria on which decision maker’s knowledge is usually vague and imprecise [34]. Previous research concerning supplier selection often used exact numbers to measure criterion weights. In this study, considering that the decision maker may have difficulty in eliciting precise criterion weights, the proposed approach is proposed to select appropriate supplier in group decision-making environment. It should be noted that, as suggested and illustrated by Merigo and Gil-Lafuente [36], the proposed approach can be easily applied to a host of practical decision problems that involve choosing an optimal alternative from a list of alternatives when multiple attributes must be considered.Suppose that a high-tech company which manufactures electronic products intends to evaluate and select a supplier of USB connectors. There are four suppliersx1, x2, x3, and x4 which are chosen as candidates. A committee of three decision makers d1, d2, and d3 is established, which are an engineering expert, financial expert, and quality control expert, respectively. Four evaluated criteria are considered, including finance (c1), performance (c2), technique (c3), and organizational culture (c4). The expert weight vector is given by λ=(0.35,0.35,0.3)T. The interval-valued intuitionistic fuzzy decision matrices of criterion values are constructed as follows: (28)R1=[([0.3,0.4],[0.4,0.6])([0.6,0.7],[0.1,0.2])([0.5,0.7],[0.2,0.3])([0.7,0.8],[0.0,0.1])([0.7,0.8],[0.1,0.2])([0.6,0.7],[0.2,0.3])([0.2,0.3],[0.4,0.6])([0.5,0.6],[0.1,0.3])([0.5,0.8],[0.1,0.2])([0.7,0.8],[0.0,0.1])([0.5,0.5],[0.4,0.5])([0.2,0.3],[0.2,0.4])([0.2,0.3],[0.4,0.5])([0.5,0.7],[0.1,0.3])([0.6,0.7],[0.1,0.2])([0.4,0.5],[0.1,0.3])],R2=[([0.4,0.5],[0.3,0.4])([0.5,0.6],[0.2,0.2])([0.6,0.7],[0.2,0.3])([0.7,0.8],[0.1,0.2])([0.6,0.8],[0.1,0.2])([0.5,0.6],[0.3,0.4])([0.4,0.5],[0.3,0.4])([0.4,0.6],[0.3,0.4])([0.5,0.6],[0.3,0.4])([0.5,0.7],[0.1,0.2])([0.5,0.6],[0.3,0.4])([0.3,0.4],[0.2,0.5])([0.5,0.6],[0.3,0.4])([0.7,0.8],[0.0,0.1])([0.4,0.5],[0.2,0.4])([0.5,0.7],[0.1,0.2])],R3=[([0.4,0.6],[0.3,0.4])([0.5,0.7],[0.1,0.2])([0.5,0.6],[0.2,0.4])([0.6,0.8],[0.1,0.2])([0.5,0.8],[0.1,0.2])([0.3,0.5],[0.2,0.3])([0.3,0.6],[0.2,0.4])([0.4,0.5],[0.2,0.4])([0.5,0.6],[0.0,0.1])([0.5,0.8],[0.1,0.2])([0.4,0.7],[0.2,0.3])([0.2,0.4],[0.2,0.3])([0.5,0.7],[0.1,0.3])([0.4,0.6],[0.0,0.1])([0.3,0.5],[0.2,0.4])([0.7,0.9],[0.0,0.1])].The incomplete information about the criterion weights are as follows (in this problem, the criterion weights are incomplete information. The specific weight calculation method can be found in [34]):(29)H={0.228≤w1≤0.8758,0.2285≤w2≤0.8789,0.1642≤w3≤0.7979,0.1419≤w4≤0.7824}.Step 1.
Calculate the weight vectorW=(w1,w2,w3,w4)T by solving model (25) as follows:
(30)W=(0.228,0.4659,0.1642,0.1419)T.Step 2.
Aggregate the values of each decision maker, andutilize the IVIF-WGA operator to derive the valueszik(i=1,2,…,n) of each decision maker.
The integrated values for alternativesx1, x2, x3 and x4 of decision maker d1 are, respectively,
(31)z11=([0.508,0.628],[0.183,0.320]),z21=([0.506,0.614],[0.203,0.342]),z31=([0.514,0.644],[0.130,0.249]),z41=([0.405,0.550],[0.179,0.337]).
The integrated values for alternativesx1, x2, x3 and x4 of decision maker d2 are, respectively,
(32)z12=([0.514,0.615],[0.211,0.267]),z22=([0.487,0.622],[0.259,0.359]),z32=([0.439,0.584],[0.198,0.331]),z42=([0.546,0.649],[0.124,0.259]).
The integrated values for alternativesx1, x2, x3 and x4 of decision maker d3 are, respectively,
(33)z13=([0.488,0.672],[0.166,0.285]),z23=([0.351,0.573],[0.178,0.312]),z33=([0.423,0.664],[0.111,0.211]),z43=([0.435,0.639],[0.06,0.205]).Step 3.
Aggregate the values of all decision makers, and utilize the IVIF-HA operator to derive the collective valueszi of the alternative xi.
The weighting vector of the IVIL-HA operator is obtained by the principle of antonym pairs many and(a,b) equal to (0.3, 0.8). So we can obtain ω=(0.243,0.514,0.243) and the integrated values as follows:
(34)z1=([0.489,0.617],[0.202,0.297]),z2=([0.436,0.593],[0.236,0.357]),z3=([0.430,0.625],[0.144,0.262]),z4=([0.463,0.606],[0.129,0.278]).Step 4.
Calculate the scoresS(zi)(i=1,2,3,4) of the collective overall values zi(i=1,2,3,4) as follows:
(35)S(z1)=0.607,S(z2)=0.436,S(z3)=0.649,S(z4)=0.663.Step 5.
Rank all the alternativesAi(i=1,2,3,4) in accordance with the score S(zi) of the collective overall interval-valued intuitionistic fuzzy values zi(i=1,2,3,4): A4≻A3≻A1≻A2, and thus the best alternative is A4.


## 6. Conclusion

In this paper, a new entropy measure of IVIFS is proposed by using cotangent function, which can overcome limitations of some existing methods. And we provide several numerical examples to illustrate its validity. For interval-valued intuitionistic fuzzy multicriteria group decision-making problem with incomplete information on the weights of criteria, an entropy weight model is established to determine the entropy weights. In addition, the method and procedure are developed to solve FMCGDM problems. Finally, the supplier selection problem is used as an example to demonstrate how to apply the proposed multicriteria group decision-making approach.



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*Source: 827268-2013-05-29.xml*