Entropy Measures for Interval-Valued Intuitionistic Fuzzy Sets and Their Application in Group Decision-Making

. Entropy measure is an important topic in the fuzzy set theory and has been investigated by many researchers from different points of view. In this paper, two new entropy measures based on the cosine function are proposed for intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets. According to the features of the cosine function, the general forms of these two kinds of entropy measures are presented. Compared with the existing ones, the proposed entropy measures can overcome some shortcomings and be used to measure both fuzziness and intuitionism of these two fuzzy sets; as a result, the uncertain information of which can be described more sufficiently. These entropy measures have been applied to assess the experts’ weights and to solve multicriteria fuzzy group decision-making problems.

As two important topics in the FS theory, entropy measures and similarity measures of fuzzy sets have been investigated widely by many researchers from different points of view. The entropy of a fuzzy set describes the fuzziness degree of the fuzzy set. de Luca and Termini [12] introduced some axioms which captured people's intuitive comprehension to describe the fuzziness degree of a fuzzy set. Kaufmann [13] proposed a method for measuring the fuzziness degree of a fuzzy set by a metric distance between its membership function and the membership function of its nearest crisp set. Yager [14] suggested the entropy measure being expressed by the distance between a fuzzy set and its complement. Chiu and Wang [15] gave simple calculation for entropies of fuzzy numbers in addition and extension principle. They [16] also investigated the entropy relationship between the original fuzzy set and the image fuzzy set. Hong and Kim [17] introduced a simple method for calculating the entropy of the image fuzzy set without calculating its membership function. Zeng and Li [18] showed that similarity measures and entropies of fuzzy sets can be transformed to each other based on their axiomatic definitions.
Aimed at these important numerical indexes in the fuzzy set theory, some researchers extended these concepts to the IVFS theory and IFS theory and investigated their related topics from different points of view [19][20][21][22][23][24]. We review some generalization study on entropy measure. Burillo and Bustince [25] introduced the notions of entropy on IVFSs and IFSs to measure the degree of intuitionism of an IVFS or an IFS. Szmidt and Kacprzyk [26] proposed a nonprobabilistictype entropy measure with a geometric interpretation of IFSs. Hung and Yang [27] gave their axiomatic definitions of entropies of IFSs and IVFSs by exploiting the concept of probability. Farhadinia [20] generalized some results on the entropy of IVFSs based on the intuitionistic distance and its relationship with similarity measure. After that, many authors also proposed different entropy formulas for IFSs [27][28][29][30][31], IVFSs [32,33], and vague sets [34]. For IVIFSs, Liu et al. in [35] proposed a set of axiomatic requirements for entropy measures, which extended Szmidt and Kacprzyk's axioms formulated for entropy of IFSs [26]. Wei et al. in [36] extended the entropy measure of IFSs proposed in [26] to IVIFSs and gave an approach to construct similarity measures by entropy measures of IVIFSs. By this approach, the proposed entropy measure can yield a similarity measure of IVIFSs, which has been applied in the context of pattern recognition, multiple-criteria fuzzy decision-making, and medical diagnosis. For entropy measures of IVIFSs, we refer to [37][38][39].
In [31], Vlachos and Sergiadis revealed an intuitive and mathematical connection between the notions of entropy for FSs and IFSs in terms of fuzziness and intuitionism. They pointed out that entropy for FSs is indeed a measure of fuzziness, while for IFSs, entropy can measure both fuzziness and intuitionism. Recall that the fuzziness is dominated by the difference between membership degree and nonmembership degree, and the intuitionism is dominated by the hesitation degree. Hence it is very interesting to construct entropy formulas measuring both fuzziness and intuitionism. We propose an entropy measure, as well as its general form, for IFSs and then generalize it to IVIFSs. Our entropy measures are compared with some existing ones in [29][30][31]37]. As an application in multicriteria fuzzy group decision-making, we propose a method to assess the experts' weights by the proposed entropy measures.
The rest of the paper is organized as follows. Section 2 reviews some necessary concepts of IFSs and IVIFSs. In Section 3, we propose a new entropy measure and its general form for IFSs. Then we compare the proposed entropy measure with some existing ones and give some conditions under which these existing entropy measures may not work as desired, while the proposed entropy measure can do well. In Section 4, we extend the entropy measures defined in Section 3 to IVIFSs and propose entropy measures for IVIFSs. These entropy measures are compared with the ones defined by Ye in [37]. Section 5 gives the application of the proposed entropy measures in assessing the weights of experts. Concluding remarks are drawn in Section 6.
Definition 1 (see [2]). Let be a universe of discourse. An intuitionistic fuzzy set in is an object having the form where : with the condition The numbers ( ) and V ( ) denote the degree of membership and nonmembership of to , respectively.
For convenience of notations, we abbreviate "intuitionistic fuzzy set" to IFS and denote by IFS( ) the set of all IFSs in .
For each IFS in , we call ( ) = 1 − ( ) − V ( ) the intuitionistic index of in , which denotes the hesitancy degree of to . The complementary set of is defined as Definition 2 (see [2]). For two IFSs = {⟨ , ( ), V ( )⟩ | ∈ } and = {⟨ , ( ), V ( )⟩ | ∈ }, their relations are defined as follows: Consider that, sometimes, it is not approximate to assume that the membership degrees for certain elements of an IFS are exactly defined, but a value range can be given. In such cases, Atanassov and Gargov [3] introduced the following notion of interval-valued intuitionistic fuzzy sets.
The intervals ( ) and V ( ) denote the degree of membership and nonmembership of to , respectively.
We abbreviate "interval-valued intuitionistic fuzzy set" to IVIFS and denote by IVIFS( ) the set of all IVIFSs in X.
In many practical problems, such as, multicriteria decision-making, the study objects are finite. So in the rest of the paper, we assume that the universe is a finite set, listed by { 1 , 2 , . . . , }.

Entropy Measures for IFSs
In this section we will propose a concrete entropy measure for IFSs and demonstrate its efficiency through comparisons with some existing entropy measures in [28][29][30][31]. [26] extended the axioms of de Luca and Termini [12] to propose the following definition of an entropy measure for IFSs.

A New Entropy Measure for IFSs. Szmidt and Kacprzyk
Definition 9 (see [26]). A real-valued function : IFS( ) → [0, 1] is called an entropy measure for IFSs if it satisfies the following axiomatic requirements: In this subsection, we introduce a new entropy measure for IFSs. For each ∈ IFS( ), define ( ) by Then we have the following theorem.

Mathematical Problems in Engineering
Theorem 10. The mapping , defined by (11), is an entropy measure for IFSs.
Analyzing the features of the cosine function, we give the following general form of the entropy measure defined in (11), which is suggested by the referee.
Then is an entropy measure for IFSs.
Proof. The process of the proof is similar to that for Theorem 10. We omit it.

Comparison with Existing Entropy Measures. For an IFS
in , Ye [29] proposed two entropy measures 1 ( ) and 2 ( ): The following proposition shows that these two formulas are the same.
Proof. By the properties of trigonometric functions, we have, for each = {⟨ , ( ), V ( )⟩ | ∈ }, It follows that 1 ( ) = 2 ( ). Next, we can simplify it as follows: The following example shows that the entropy measure can produce some counterintuitive cases. The absolute differences between the membership degrees and the nonmembership degrees of each to 1 , 2 , and 3 are the same; thus, by formula (17), we can obtain that ( 1 ) = ( 2 ) = ( 3 ) = 0.9580. But we can see that the hesitancy degrees of the element to 1 , 2 , and 3 are different. Intuitively, the uncertain information of 1 is more than that of 2 , and the uncertain information of 3 is the least. Obviously, the results obtained by using Ye's formula are not in accordance with our intuition. Now, let us calculate the entropies of 1 , 2 , and 3 by formula (11). We have ( 1 ) = 0.9808, ( 2 ) = 0.9659, and . This is consistent with our intuition according to the above analysis. The following theorem is a straightforward exercise. Comparing the entropy measures and , we find that could measure not only the degree of fuzziness, but also the degree of intuitionism of IFSs, which overcomes the shortcoming that could only measure the degree of fuzziness of IFSs. The entropy measures in [28][29][30][31] could also measure both fuzziness and intuitionism, but in some cases, 6 Mathematical Problems in Engineering some of them may not work well as desired. Next we compare the entropy measure with them.
In [30,31], Vlachos and Sergiadis proposed an entropy measure VS1 according to a cross-entropy measure and an entropy measure VS2 based on the product of two vectors: These three entropy measures satisfy the set of requirements in Definition 9.
The following examples show that the entropy measures VS1 and VS2 may give inconsistent information in some cases.      Table 1, we can see that, for and 1 , the closer the membership degree and the nonmembership degree, or the bigger the hesitation degree, the greater its entropy. Particularly, when the membership degree is equal to the nonmembership degree, the entropy reaches the maximum value 1. The results obtained by using the entropy measures and 1 are in accordance with our intuition.
Hence, compared with the above entropy measures, the entropy measure defined by formula (11) is more effective and reasonable to measure the uncertain information of IFSs.

An Entropy Measure for Interval-Valued Intuitionistic Fuzzy Sets
In this section we will extend the entropy measure to IVIFSs and define a new entropy measure for IVIFSs which is compared to the entropy measures defined in [37].
Liu et al. [35] and Zhang et al. [38] proposed a set of axiomatic requirements for an entropy measure of IVIFSs, which extends Szmidt and Kacprzyk's axioms formulated for entropy of IFSs [26]. (E4) ( ) ≤ ( ) if is less fuzzy than , which is defined as In this section we will introduce a formula to calculate the entropy of an IVIFS based on the entropy measure for IFSs defined by formula (11). For any ∈ IVIFS( ) we define Then we have the following theorem.
Proof. In order to prove that the mapping is an entropy measure, it is sufficient to show that satisfies the conditions (E1)-(E4) in Definition 17. Suppose that we can easily obtain − ( ) = + ( ) = 1, Applying this condition to (27) In order to prove ( ) ≤ ( ), we need to prove that By the assumption, it is equivalent to prove that which can be simplified to prove that Indeed, since 2 − V − ( ) − V + ( ) ≥ 0 and − ( ) + + ( ) − 2 ≤ 0, we have Then, by computing the sum of the left (resp., right) terms of the above two inequalities, we have (33). Hence ( ) ≤ ( ) holds.
By a similar way, we can also prove that ( ) ≤ ( ) for the other three cases. Since ( ) ≤ ( ) for each , we have ( ) ≤ ( ).
Similar to Theorem 11, we give the following general form of the entropy measure defined in (27).

(35)
Then is an entropy measure for IVIFSs.
Proof. The process of the proof is similar to that for Theorem 18. We omit it.
In the following, we will compare the entropy measure defined by (27) with the entropy measures defined in [37,41]:  [41] proposed a concrete entropy measure: Ye [37] introduced two entropy measures 1 and 2 as follows: where , and , ∈ [0, 1] are two fixed numbers. Proof. The process of the proof is similar to that for Proposition 12. We omit it.

Theorem 20. For each in IVIFS( ), let
By the entropy measure , we get The difference between the membership degrees and nonmembership degrees of 4 and 5 is the same, and the hesitant degree of 4 is bigger than that of 5 , so the entropy of 4 should be bigger than that of 5 . However, by the entropy measure , we have ( 4 ) < ( 5 ).
Using (43), we have We suppose that = = 0.5 in formula (39); then formula (39) reduces to From these results, we can see that the entropy formula defined by (43) has the following two drawbacks.
(2) It only reflects the difference between the membership degree and nonmembership degree. Thus, for any two IVIFS and satisfying | − ( ) for all ∈ , we have ( ) = ( ).
Now by the entropy formula defined by (27), we can obtain The results show that ( 1 ) ̸ = ( 2 ) and ( 4 ) > ( 5 ). The proposed entropy measure can overcome the above shortcomings of the entropy measures and .

The Application of Entropy Measures in Multicriteria Decision-Making
Entropy measures have been applied in many problems such as optimizing the distinguishability of input space partitioning [42] and assessing the weights of experts or criteria in intuitionistic fuzzy decision-making [43][44][45]. In this section we propose a method to determinate experts' weights in multicriteria group decision-making with intervalvalued intuitionistic fuzzy information by using the proposed entropy measures. The group decision-making problem which is considered in this paper can be represented as follows. Let ) is an IVIFV, provided by the decision maker ∈ for the alternative ∈ with respect to the criterion ∈ . Decision maker's goal is to obtain the ranking order of the alternatives.

Determining the Weights of Experts.
In many practical group decision-making problems, it is an important research topic to determine the weights of experts according to experts' evaluation information. In this subsection, we propose a method to derive the weights of experts based on the proposed entropy measures. It is known that entropies can measure the uncertainty degrees of IVIFSs.  (27), the entropy of can be calculated, which is denoted by . indicates the uncertainty degree of assessment information provided by expert . During the practical group decisionmaking process, we usually expect that the uncertainty degree of the assessment information is as small as possible. Thus, the bigger is, the smaller the weight should be given to . Conversely, the smaller is, the bigger the weight should be given to . Therefore, the weights of experts are defined as follows: In the next subsection, we will aggregate the overall assessment information of individual experts to get the evaluations of the group for alternatives.

An Approach to Solve Interval-Valued Intuitionistic Fuzzy
Group Decision-Making Problems. According to the analysis in Section 5.1, we develop the following steps to get the ranking of alternatives.
Step 5. Use Definition 6 to compare the overall assessment values ( = 1, 2, . . . , ) and rank the alternatives ( = 1, 2, . . . , ). , then these interval-valued intuitionistic fuzzy decision matrices reduce to intuitionistic fuzzy decision matrices. So the above method is also suitable to solve intuitionistic fuzzy group decision-making problems with unknown experts' weights.
In the following, we give an example which was adapted from Xu and Cai [47] to illustrate the above approach.