Research Article | Open Access
Interval-Valued Hesitant Fuzzy Multiattribute Group Decision Making Based on Improved Hamacher Aggregation Operators and Continuous Entropy
Under the interval-valued hesitant fuzzy information environment, we investigate a multiattribute group decision making (MAGDM) method with continuous entropy weights and improved Hamacher information aggregation operators. Firstly, we introduce the axiomatic definition of entropy for interval-valued hesitant fuzzy elements (IVHFEs) and construct a continuous entropy formula on the basis of the continuous ordered weighted averaging (COWA) operator. Then, based on the Hamacher t-norm and t-conorm, the adjusted operational laws for IVHFEs are defined. In order to aggregate interval-valued hesitant fuzzy information, some new improved interval-valued hesitant fuzzy Hamacher aggregation operators are investigated, including the improved interval-valued hesitant fuzzy Hamacher ordered weighted averaging (I-IVHFHOWA) operator and the improved interval-valued hesitant fuzzy Hamacher ordered weighted geometric (I-IVHFHOWG) operator, the desirable properties of which are discussed. In addition, the relationship among these proposed operators is analyzed in detail. Applying the continuous entropy and the proposed operators, an approach to MAGDM is developed. Finally, a numerical example for emergency operating center (EOC) selection is provided, and comparative analyses with existing methods are performed to demonstrate that the proposed approach is both valid and practical to deal with group decision making problems.
Fuzzy sets (FSs)  originally put forward by Zadeh are a very useful tool and have achieved a great success in various fields. Atanassov proposed the intuitionistic fuzzy sets (IFSs) [2–4], which are a generalization of the FSs. The introduction of IFSs proves to be very meaningful and practical and has been found to be highly useful to deal with vagueness [5–12]. Atanassov and Gargov further introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs)  as a generalization of that of IFSs, whose components are intervals rather than exact numbers. Under some conditions, the decision makers (DMs) are usually irresolute and hesitant for one thing or another, which makes it difficult to determine the membership of an element to a set due to doubts among a few different values. In this case, Torra and Narukawa  and Torra  introduced the hesitant fuzzy set (HFS), which permits the membership having a collection of possible values. Due to the fact that the interval-valued fuzzy set (IVFS)  is usually more adequate or sufficient to real-life group decision making (GDM) problems than real numbers, Chen et al. [17, 18] proposed the interval-valued hesitant fuzzy set (IVHFS), which permits the membership having a collection of possible interval-valued numbers.
Since IVHFS was introduced, it has been used to deal with many problems, especially the MAGDM problems, one of the ways is aggregating the DMs’ opinions under each attribute for alternatives and then obtaining the collective of attribute values for each alternative . Based on the arithmetic aggregation methods [20–27], Xu  and Xu and Yager  investigated several new intuitionistic fuzzy arithmetic aggregation operators and intuitionistic fuzzy geometric aggregation operators. Wei et al.  proposed some interval-valued hesitant fuzzy aggregation operators: interval-valued hesitant fuzzy weighted averaging (IVHFWA) operator, interval-valued hesitant fuzzy ordered weighted averaging (IVHFOWA) operator, interval-valued hesitant fuzzy weighted geometric (IVHFWG) operator, interval-valued hesitant fuzzy ordered weighted geometric (IVHFOWG) operator, interval-valued hesitant fuzzy power aggregation operators, interval-valued hesitant fuzzy prioritized weighted average (IVHFPWA) operator, and interval-valued hesitant fuzzy prioritized weighted geometric (IVHFPWG) operator. Then, some of their desirable properties were investigated in detail. Zhu et al.  developed the interval-valued hesitant fuzzy Einstein Choquet ordered averaging (IVHFECOA) operator and the interval-valued hesitant fuzzy Einstein Choquet ordered geometric (IVHFECOG) operator; then they applied these two operators to deal with multiattribute decision making problems. Hamacher t-conorm and t-norm  are proposed by Hamacher, which are more general and more flexible [32, 33]. In this paper, we extend the Hamacher t-conorm and t-norm to interval-valued hesitant fuzzy environment and investigate some improved interval-valued hesitant fuzzy Hamacher operators that allow DMs to have more choice in MAGDM problems.
Entropy is one of the important research topics in the fuzzy theory, which has been widely used in practical applications . Zadeh first introduced the concept of fuzzy entropy . Moreover, De Luca and Termini  presented the axioms with which the fuzzy entropy should comply and defined the entropy for a FS. Based on the ratio of intuitionistic fuzzy cardinalities, Szmidt and Kacprzyk  have given the axiomatic requirements of intuitionistic fuzzy entropy measure and proposed a nonprobabilistic-type entropy measure for IFSs. Ye  constructed two entropy measures for IVIFSs and established an entropy weighted model to determine the entropy weights with respect to a decision matrix provided as IVIFSs. Wei et al.  developed an entropy measure for IVIFSs, which generalized three entropy measures for IFSs. Xu and Xia  introduced the concepts of entropy and cross-entropy for HFS and discussed their desirable properties.
From the above analysis, we can see that IVHFS is a very useful tool to cope with uncertainty. More and more decision making methods and theories have been developed on the basis of IVHFSs. On the one hand, just as the HFSs, introducing the axiomatic definition of entropy and investigating some entropy formulas for IVHFSs are the important issues. On the other hand, more and more multiattribute GDM methods and theories have been developed with the Hamacher t-norm and t-conorm. However, there are few aggregation techniques to use the Hamacher operations on IVHFSs. Therefore, it is necessary and meaningful to study some issues. For example, what is it like the expression of the interval-valued hesitant fuzzy continuous entropy formula on the basis of COWA operator? What is the relationship among the new improved interval-valued hesitant fuzzy Hamacher aggregation operators?
In this paper, the axiomatic definition of entropy and an entropy formula for IVHFEs are investigated, and then some new improved Hamacher aggregation operators are proposed to aggregate interval-valued hesitant fuzzy information. A MAGDM approach is developed, which is based on the entropy weights and the Hamacher information aggregation operators.
To do this, the rest of the paper is organized as follows. In Section 2, we briefly review some basic concepts, including IVHFSs, Hamacher t-norm, and t-conorm. Section 3 gives the axiomatic definition of entropy for IVHFEs and constructs an interval-valued hesitant fuzzy continuous entropy formula. In Section 4, the new adjusted operations for IVHFEs are presented, and we investigate some improved interval-valued hesitant fuzzy Hamacher information aggregation operators, which are followed by the discussion of the relationship among the proposed operators. Section 5 develops an approach to MAGDM with the continuous entropy formula and the proposed operators. In Section 6, we provide a numerical example of EOCs evaluation to illustrate the application of the developed method. Finally, we end the paper by summarizing the main conclusions in Section 7.
In this section, we furnish a brief review on some basic concepts, including IVHFSs and Hamacher t-norm and t-conorm.
Definition 1. Let be a fixed set, let be the set of all closed subintervals of . An IVHFS on is defined aswhere is a set of all possible interval-valued membership degrees of the element to the set . is called the IVHFE; here and represent the lower and upper limits of , respectively. The complement of the IVHFE denotes . Let be the set of all IVHFEs.
Remark 2. Notice that the number of values in different IVHFEs may be different. Suppose that stands for the number of values in ; then the following assumptions are made.
(R1) All the elements in each IVHFE are arranged in decreasing order, and let be the th largest interval number in .
(R2) If , then . To have a correct comparison, the two IVHFEs and should have the same length. If there are fewer elements in than in , an extension of should be considered optimistically by repeating its maximum element until it has the same length with .
(R3) For convenience, we assume that all the IVHFEs have the same length .
In order to compare among the different IVHFEs, we first give the properties of interval numbers.
Definition 3 (see ). Let and be two interval numbers; then(1) and ;(2);(3).
Definition 4 (see ). Let and be two interval numbers, and let and ; then the degree of possibility of is defined as
Definition 5 (see ). For an IVHFE , is called the score function of . For two IVHFEs and , if , then .
In the following, we recall the triangular norm and conorm, which is an important notion in fuzzy set theory.
Definition 6 (see ). A function is a t-norm if and only if it is commutative, associative, and nondecreasing and .
The corresponding t-conorm of is the function defined by .
For many t-norms and t-conorms, there are some basic t-norms and t-conorms, including Algebraic product , Algebraic sum , Einstein product , and Einstein sum . Hamacher given the following generalized t-norm and -conorm denoted the Hamacher t-norm and t-conorm :
In particular, when , then the Hamacher t-norm and t-conorm are reduced to the Algebraic product and Algebraic sum ; when , then Hamacher t-norm and t-conorm are reduced to the Einstein product and Einstein sum .
By using Hamacher t-conorm and t-norm, Li and Peng  introduced some operational laws for IVHFEs as follows.
Definition 8 (see ). Let be a collection of IVHFEs, and is the associated weight vector of , with and . Then an interval-valued hesitant fuzzy Hamacher ordered weighted averaging (IVHFHOWA) operator is a mapping , such thatwhere is a permutation such that .
Definition 9 (see ). Let be a collection of IVHFEs, and is the associated weight vector of , with and . Then an interval-valued hesitant fuzzy Hamacher ordered weighted geometric (IVHFHOWG) operator is a mapping , such thatwhere is a permutation such that .
Theorem 10 (idempotency, see , Theorems 19 and 24). Let be a collection of IVHFEs, if all , are equal: that is, for all ; then
3. Interval-Valued Hesitant Fuzzy Continuous Entropy
In this section, we introduce the axiomatic definition of entropy for IVHFEs and then construct an interval-valued hesitant fuzzy continuous entropy formula on the basis of COWA operator.
Definition 11. A COWA operator is a mapping associated with a basic unit interval monotonic (BUM) function, , such thatwhere and is the set of all nonnegative interval numbers.
Denoting , then we havewhere is the attitudinal character of . Thus, is the weighted average of the end points of the closed interval with attitudinal character parameter, and it is called the attitudinal expected value of .
In what follows, we first present the axiomatic definition of entropy for IVHFEs and then investigate a continuous entropy formula for IVHFEs.
Definition 12. An entropy on IVHFE is an interval-valued function , which satisfies the following axiomatic requirements:(E1), if and only if or .(E2), if and only if , for .(E3).(E4), if and when , for ,or and when , for .
Note that if the IVHFE is reduced to a hesitant fuzzy element , that is, all the interval numbers in are reduced to crisp values , then the requirements of axiomatic definition of entropy for IVHFE are reduced to the following conditions:(1), if and only if or .(2), if and only if , for .(3).(4), if for , or for .
The above conditions are equivalent to the requirements of the axiomatic definition of entropy for hesitant fuzzy elements proposed by Xu and Xia .
Based on the COWA operator, we construct an information measure formula for IVHFE as follows:
In what follows, we prove that is an entropy measure of IVHFE .
Theorem 13. The mapping , defined by (10), is an entropy measure for IVHFE .
Proof. In order for (10) to be qualified as a sensible measure of interval-valued hesitant fuzzy entropy, it must satisfy conditions (E1)–(E4) in Definition 9.
Let , ; then we have .
If , then , which means that is an increasing function of , for . If , then , which means that is a decreasing function of , for . Since , then , if and only if or ; , if and only if .
(E1) If or , then we obtain thatFrom the above analysis, we have .
On the other hand, assume that . Since , then . For all , we haveIt follows that ; then . Therefore, every term in the summation of is nonnegative. Since , then we deduce that every term should be zero in ; that is, for , we haveFrom the above analysis, we know that (13) holds for all , if and only if or , for .
(i) If , for .
Let ; then we obtain thatTherefore, for , we havethen and . As , thus . Hence, .
(ii) If , for . Since and , for , and then we obtain that , for . According to (9), we have , for ; then . Hence, .
(E2) Suppose that , for ; then , and we haveFrom (10), we obtain thatOn the other hand, from the above analysis, we have and , and it is obvious that . If , thenTherefore,it follows that and ; that is,(E3) As , then , and we haveSince , and then(E4) Assume that and when , for ; thenIt follows thatThus,Notice that is an increasing function of , for ; thus, we obtain that .
Similarly, if and when , for , one can also prove that .
Definition 14. Suppose that is a COWA operator associated with BUM function and is an IVHFE; then , defined by (10), is called the continuous entropy on IVHFE .
4. Improved Interval-Valued Hesitant Fuzzy Hamacher Information Aggregation Operators and the Relationship among the Proposed Operators
In this section, we first point out that the IVHFHOWA operator and IVHFHOWG operator proposed by Li and Peng  do not satisfy the property of idempotency by employing an illustrative example, and then some new operational laws for IVHFEs are defined. Two improved interval-valued hesitant fuzzy Hamacher information aggregation operators are further investigated, including the I-IHHOWA operator and the I-IHHOWG operator. In addition, we analyze the relationship among these proposed operators.
4.1. Improved Interval-Valued Hesitant Fuzzy Hamacher Information Aggregation Operators
Example 15. Suppose that and are two IVHFEs, and . Without loss of generality, take . According to Definition 8, it follows thatLet ; then the score values of and can be calculated as , . Therefore, the degree of possibility of isThen we have ; thus ; that is,On the other hand, by the IVHFHOWG operator in Definition 9, it is obtained thatLet ; then the score value of is . Therefore, the degree of possibility of isThen we have ; thus, ; that is,
Example 15 demonstrates that Theorem 10 of the IVHFHOWA operator and IVHFHOWG operator cannot be tenable, which suffer from serious drawbacks. In this case, the operations on the IVHFEs need to be improved. In the following, some adjusted operations for IVHFEs are presented, and then two new improved interval-valued hesitant fuzzy Hamacher aggregation operators are developed, which satisfy the properties of idempotency and boundedness.
Definition 16. Let , , and be three IVHFEs; then