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.


Introduction
Fuzzy sets (FSs) [1] 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][3][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][6][7][8][9][10][11][12].Atanassov and Gargov further introduced the concept of interval-valued intuitionistic fuzzy sets (IVIFSs) [13] 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 [14] and Torra [15] 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) [16] 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 [19].Based on the arithmetic aggregation methods [20][21][22][23][24][25][26][27], Xu [28] and Xu and Yager [29] investigated several new intuitionistic fuzzy arithmetic aggregation operators and intuitionistic fuzzy geometric aggregation operators.Wei et al. [30] proposed some intervalvalued 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. [31] 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 [32] 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 intervalvalued 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 [34].Zadeh first introduced the concept of fuzzy entropy [35].Moreover, De Luca and Termini [36] 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 [37] have given the axiomatic requirements of intuitionistic fuzzy entropy measure and proposed a nonprobabilistic-type entropy measure for IFSs.Ye [38] 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. [39] developed an entropy measure for IVIFSs, which generalized three entropy measures for IFSs.Xu and Xia [34] 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.

Preliminaries
In this section, we furnish a brief review on some basic concepts, including IVHFSs and Hamacher t-norm and tconorm.
Chen et al. [17,18] first introduced the concept of IVHFS, which is defined as follows.

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.
(R2) If   ̸ =   , then  = max{  ,   }.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 the following, we recall the triangular norm and conorm, which is an important notion in fuzzy set theory.
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 [42]: In particular, when  = 1, then the Hamacher t-norm and t-conorm are reduced to the Algebraic product   and Algebraic sum   ; when  = 2, then Hamacher t-norm and tconorm are reduced to the Einstein product   and Einstein sum   .
By using Hamacher t-conorm and t-norm, Li and Peng [43] introduced some operational laws for IVHFEs as follows.

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.
The COWA operator was developed by Yager [44], which extends the OWA operator [20].Definition 11.A COWA operator is a mapping  :  →  + associated with a basic unit interval monotonic (BUM) function, , such that where  = [  ,   ] ∈  and  is the set of all nonnegative interval numbers.
Denoting  = ∫ 1 0 (), then we have where  is the attitudinal character of .In what follows, we first present the axiomatic definition of entropy for IVHFEs and then investigate a continuous entropy formula for IVHFEs.
The above conditions are equivalent to the requirements of the axiomatic definition of entropy for hesitant fuzzy elements proposed by Xu and Xia [34].
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 .
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.
then we obtain that From the above analysis, we have () = 0.
On the other hand, assume that () = 0.

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 [43] 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.
Then 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 intervalvalued hesitant fuzzy Hamacher aggregation operators are developed, which satisfy the properties of idempotency and boundedness.
Remark 18.If  = 1, the I-IVHFHOWA operator is reduced to the following one: which is called the improved interval-valued hesitant fuzzy ordered weighted averaging (I-IVHFOWA) operator.
If  = 2, the I-IVHFHOWA operator is reduced to the following one: which is called the improved interval-valued hesitant fuzzy Einstein ordered weighted averaging (I-IVHFEOWA) operator.
In the following, motivated by the geometric mean [45], we investigate the improved IVHFHOWG operator.
Remark 20.If  = 1, the I-IVHFHOWG operator is degenerated to the following: which is called the improved interval-valued hesitant fuzzy ordered weighted geometric (I-IVHFOWG) operator.
If  = 2, the I-IVHFHOWG operator is degenerated to the following operator: which is called the improved interval-valued hesitant fuzzy Einstein ordered weighted geometric (I-IVHFEOWG) operator.
In what follows, we discuss some desirable properties of the I-IVHFHOWA operator and I-IVHFHOWG operator.
It can be easily proved that the I-IVHFHOWA and I-IVHFHOWG operators have the following property.In order to study the relationship among these intervalvalued hesitant fuzzy information aggregation operators, we first introduce the following lemmas.

The Approaches to Determine the Attribute Weights.
In the following, we propose two approaches to determine the weight vector of attributes based on the continuous entropy.
Considering the entropy of the attribute   , the averaging entropy (  ) of the attribute   is given as and each (  ) can be calculated by (10).
According to the entropy theory, the entropy of an attribute is smaller across alternatives, which implies that it can provide decision makers with the effective information, and the attribute should be assigned a bigger weight.
In the process of decision making, sometimes, the information about attribute weights is completely unknown or incompletely known because of lack of knowledge or data, the influence of the decision environment, and the expert's limited expertise about the problem domain.
If the information about weight   of the attribute   ,  = 1, 2, . . .,  is completely unknown, we can establish the following equations for determining attribute weights: ,  = 1, 2, . . ., . (66) If the information about weight   of the attribute   ,  = 1, 2, . . .,  is incompletely known, in such a case, let Ω be the set of incomplete information about attribute weights; in order to get the optimal weight vector, the following model can be constructed: ≥ 0,  = 1, 2, . . ., . (67)

The Procedure of Interval-Valued Hesitant Fuzzy MAGDM.
From the above analysis, we investigate a MAGDM method under interval-valued hesitant fuzzy environment; the following steps are involved.
Step 6. Select the priority of the alternatives according to the ranking of   ( = 1, 2, . . ., ).Note that the bigger the   , the better the alternative   .

Illustrative Example
Emergency risk management (ERM) is a process which involves dealing with risks to the community arising from emergency events.Emergency management evaluation as one of the important parts of ERM aims at evaluating and improving social preparedness and organizational ability of an emergency operating center (EOC) in identifying, analyzing, and treating emergency risks to the community arising from emergency events [47,48].Fuzzy MAGDM method is widely used for emergency management evaluation, and there are some predetermined alternatives, associated with the attribute; based on these attributes, the evaluation result is to be made.Suppose that there are four EOCs  = { 1 ,  2 ,  3 ,  4 } to be evaluated by evaluators, and the emergency management evaluation task has the following three features: (1) There are five attributes to evaluate four EOCs, including  1 : energy,  2 : health and medical services, (2) The attribute weight vector  = ( 1 ,   1.
The proposed MAGDM method is used to rank the EOCs and get the most desirable EOC.The following steps are involved.
Step 2. Without loss of generality, take the BUM function () = , and then  = 0.5 (medium value).For each attribute weight is completely unknown, we calculate the weight vector of attribute by the following equations: By (10) and (65), we obtain the attribute weights as follows:  2.
According to the ordering of the overall IVHFEs   ( = 1, 2, 3, 4), we can get the ranking of the EOCs, which are shown in Table 5.As can be seen, when we utilized the I-IVHFHOWA operator and the I-IVHFHOWG operator, respectively, the ranking of the EOCs is the same and the most desirable EOC is  3 .
Moreover, we can analyze how the different parameter value  affects the rankings of the EOCs; in this case, we consider different values of , which are provided by the decision makers.The results of collective overall preference values are shown in Figures 1-6.Figures 1 and 2 give the lower and upper limits of scores for EOCs, respectively, obtained by the I-IVHFHOWA operator, from which we can find that the lower and upper limits of scores for EOCs decrease as  increases from 0 to 10.If we use the I-IVHFHOWG operator instead of the I-IVHFHOWA operator to aggregate the attribute values, then the lower and upper limits of scores for EOCs can be found in Figures 3 and 4. As the parameter value  increases from 0 to 10, we find that the lower and upper limits of scores for EOCs increase.Figure 5 illustrates the deviation values of lower limits of scores for EOCs obtained by I-IVHFHOWA operator and I-IVHFHOWG operator.Figure 6 illustrates the deviation values of upper limits of scores for EOCs obtained by I-IVHFHOWA operator and I-IVHFHOWG operator.Figures 5 and 6 show that the scores for EOCs obtained by the I-IVHFHOWA operator are bigger than the ones obtained by the I-IVHFHOWG operator, and as the parameter value  increases, we can find that the deviation decreases.
As can be seen, depending on the different parameter values  used, the ranking of the EOCs does not change, and  5 is the best one.
In what follows, we apply the IVHFHOWG operator (6) proposed by Li and Peng [43] to deal with the aforementioned problem, and the ranking result and effectiveness will be compared with our proposed MAGDM approach.The following steps are involved.
called the IVHFE; here    = inf   and    = sup   represent the lower and upper limits of   , respectively.The complement of the IVHFE  denotes   = {[1 −   , 1 −   ] |  ∈ }.Let  be the set of all IVHFEs.

Step 4 Figure 1 :Figure 2 :
Figure 1: Lower limits of scores for EOCs obtained by I-IVHFHOWA operator.
2 ,  1 ⊕  2 , and  1 ⊗  2 have the same length .Based on the adjusted operational principle for IVHFEs, we develop the improved interval-valued hesitant fuzzy Hamacher aggregation operators as follows.

Table 3 :
The score functions obtained by the I-IVHFHOWA operator and I-IVHFHOWG operator.