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

Anewentropymeasureofinterval-valuedintuitionisticfuzzyset(IVIFS)isproposedbyusingcotangentfunction,whichovercomes severallimitationsintheexistingmethodsforcalculatingentropyofIVIFS.Theefficiencyofthenewentropyisdemonstratedby comparingitwithseveralclassicalentropies.Moreover,anentropyweightmodelisestablishedtodeterminetheentropyweightsfor fuzzymulticriteriagroupdecision-making(FMCGDMs)problems,whichdependsonincompleteweightinformationofcriteria inIVIFSssetting.Finally,anillustrativesupplierselectionproblemisusedtodemonstratethepracticalityandeffectivenessofthe proposedmethod.ItiscapableofthehandlingtheFMCGDMproblemswithincompleteknownweightsforcriteria.

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 nonprobabilistictype 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 Section 2, we introduce some basic notions of IFS and IVIFSs.In Section 3, we propose a new entropy measure of intervalvalued 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.
The numbers   (), ]  () denote the degree of membership and nonmembership of the element  in the set , respectively.
The intervals μ (), ] () denote the degree of membership and nonmembership of the element  in the set , respectively.
For each element , we can compute the intuitionistic index of an intuitionistic fuzzy interval of  ∈  in  defined as follows: For convenience, an IVIFS value is denoted by  = ([, ], [, ]).

Interval-Valued Intuitionistic Fuzzy Entropy
Definition 8 (see [23]).A real-valued function  : Vlachos' entropy measure [30] is as follows Intuitively, we can see that  is more fuzzy than .If we calculate the  1 () and  1 () by ( 12), then we can obtain which indicate that  1 () =  1 () and is not consistent with our intuition.Ye's entropy measures [27] are as follows Intuitively,  is more fuzzy than .Now the  2 () and  2 () can be gained by (14) and take  =  = 0.5, and the results are which indicate that  2 () =  2 () and are not consistent with our intuition.

New Interval-Valued Intuitionistic Fuzzy Entropy Based on Cotangent Function.
A new interval-valued intuitionistic fuzzy entropy measure is introduced as follows.
Proof.In order for (16)  (P4) In order to show that (11) fulfill the requirement of (P4), it is suffice to to prove the following function: where function  is monotonic decreasing.
For the criteria   , the entropy of the alternative   of the th decision maker can be given as And the entropy for the alternative    of the kth decision maker is given as 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 alternative   is given as 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.
Let  be the set of incomplete information about criteria weights; to get the optimal weight vector, the following model can be constructed: By solving model (25) with Lingo software, we get the optimal solution ( 1 ,  2 , . . .,   )  .
In summary, the main procedure of the decision method proposed is listed in the following.

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 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 decisionmaking 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 suppliers  1 ,  2 ,  3 , and  4 which are chosen as candidates.A committee of three decision makers  1 ,  2 , and  3 is established, which are an engineering expert, financial expert, and quality control expert, respectively.Four evaluated criteria are considered, including finance ( 1 ), performance ( 2 ), technique ( 3 ), and organizational culture ( 4 ).The expert weight vector is given by  = (0.35, 0.35, 0.3)  .The interval-valued intuitionistic fuzzy decision matrices of criterion values are constructed as follows: ] . ( 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]):  = {0.228≤  1 ≤ 0.8758, 0.2285 ≤  2 ≤ 0.8789, 0.1642 ≤  3 ≤ 0.7979, 0.1419 ≤  4 ≤ 0.7824} . ( Step 1. Calculate the weight vector  = ( 1 ,

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.