# Interval-Valued Hesitant Fuzzy Multiattribute Group Decision Making Based on Improved Hamacher Aggregation Operators and Continuous Entropy

Authors: Jun Liu; Ning Zhou; Li-Hua Zhuang; Ning Li; Fei-Fei Jin
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

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 Hamachert-norm andt-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.


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## Body

## 1. 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) [2–4], 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) [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 [20–27], 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 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. [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. Hamachert-conorm andt-norm [32] are proposed by Hamacher, which are more general and more flexible [32, 33]. In this paper, we extend the Hamachert-conorm andt-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 [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 Hamachert-norm andt-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 Section2, we briefly review some basic concepts, including IVHFSs, Hamachert-norm, andt-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.


## 2. Preliminaries

In this section, we furnish a brief review on some basic concepts, including IVHFSs and Hamachert-norm andt-conorm.Chen et al. [17, 18] first introduced the concept of IVHFS, which is defined as follows.Definition 1.
LetX=x1,x2,…,xn be a fixed set, let D[0,1] be the set of all closed subintervals of [0,1]. An IVHFS on X is defined as(1)A=xi,αAxi∣xi∈X,where αAxi:X→D0,1 is a set of all possible interval-valued membership degrees of the element xi∈X to the set A. αi=αAxi=γi∣γi∈αAxi=γiL,γiU∣γi∈αAxi is called the IVHFE; here γiL=infγi and γiU=sup⁡γi represent the lower and upper limits of γi, respectively. The complement of the IVHFE α denotes αc=[1-γU,1-γL]∣γ∈α. Let H be the set of all IVHFEs.Remark 2.
Notice that the number of values in different IVHFEs may be different. Suppose thatlα stands for the number of values in α; then the following assumptions are made.
(R1) All the elements in each IVHFEα=γ1L,γ1U,γ2L,γ2U,…,γlαL,γlαU are arranged in decreasing order, and let γtL,γtU be the tth largest interval number in α.
(R2) Iflα≠lβ, then l=max⁡lα,lβ. To have a correct comparison, the two IVHFEs α and β should have the same length. If there are fewer elements in α than in β, an extension of α should be considered optimistically by repeating its maximum element until it has the same length with β.
(R3) For convenience, we assume that all the IVHFEs have the same lengthl.
In order to compare among the different IVHFEs, we first give the properties of interval numbers.Definition 3 (see [40]).
Leta=[aL,aU] and b=[bL,bU] be two interval numbers; then(1)
a=b⇔aL=bL and aU=bU;(2)
a+b=[aL+aU,bL+bU];(3)
ρa=[ρaL,ρaU],ρ≥0.Definition 4 (see [40]).
Leta=[aL,aU] and b=[bL,bU] be two interval numbers, and let La=aU-aL>0 and Lb=bU-bL>0; then the degree of possibility of a≥b is defined as(2)pa≥b=max⁡1-max⁡bU-aLLa+Lb,0,0.Definition 5 (see [17]).
For an IVHFEα, sα=1/l∑tlγt is called the score function of α. For two IVHFEs α1 and α2, if sα1≥sα2, then α1≥α2.In the following, we recall the triangular norm and conorm, which is an important notion in fuzzy set theory.Definition 6 (see [41]).
A functionT:[0,1]2→[0,1] is at-norm if and only if it is commutative, associative, and nondecreasing and Tx,1=x,∀x∈0,1.The correspondingt-conorm of S is the function S:0,12→[0,1] defined by Sx,y=1-T1-x,1-y.For manyt-norms andt-conorms, there are some basict-norms andt-conorms, including Algebraic product TA, Algebraic sum SA, Einstein product TE, and Einstein sum SE. Hamacher given the following generalizedt-norm and t-conorm denoted the Hamachert-norm andt-conorm [42]:(3)Tx,y=xyξ+1-ξx+y-xy,ξ>0,Sx,y=x+y-xy-1-ξxy1-1-ξxy,ξ>0.In particular, whenξ=1, then the Hamachert-norm andt-conorm are reduced to the Algebraic product TA and Algebraic sum SA; when ξ=2, then Hamachert-norm andt-conorm are reduced to the Einstein product TE and Einstein sum SE.By using Hamachert-conorm andt-norm, Li and Peng [43] introduced some operational laws for IVHFEs as follows.Definition 7 (see [43]).
Letα,α1, and α2 be three IVHFEs; then(4)1α1⊕α2=γ1L+γ2L-γ1Lγ2L-1-ξγ1Lγ2L1-1-ξγ1Lγ2L,γ1U+γ2U-γ1Uγ2U-1-ξγ1Uγ2U1-1-ξγ1Lγ2L∣γ1∈α1,γ2∈α2;2α1⊗α2=γ1Lγ2Lξ+1-ξγ1L+γ2L-γ1Lγ2L,γ1Uγ2Uξ+1-ξγ1U+γ2U-γ1Uγ2U∣γ1∈α1,γ2∈α2;3ρα=1+ξ-1γLρ-1-γLρ1+ξ-1γLρ+ξ-11-γLρ,1+ξ-1γUρ-1-γUρ1+ξ-1γUρ+ξ-11-γUρ∣γ∈α,ρ>0;4αρ=ξγLρ1+ξ-11-γLρ+ξ-1γLρ,ξγUρ1+ξ-11-γUρ+ξ-1γUρ∣γ∈α,ρ>0.
Based on the above operations, Li and Peng [43] developed a series of specific aggregation operators for IVHFEs.Definition 8 (see [43]).
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs, and W=w1,w2,…,wnT is the associated weight vector of αjj=1,2,…,n, with 0≤wj≤1,j=1,2,…,n and ∑j=1nwj=1. Then an interval-valued hesitant fuzzy Hamacher ordered weighted averaging (IVHFHOWA) operator is a mapping IVHFHOWA:Hn→H, such that(5)IVHFHOWAα1,α2,…,αn=⨁j=1nwjαj=∏j=1n1+ξ-1γjLwj-∏j=1n1-γjLwj∏j=1n1+ξ-1γjLwj+ξ-1∏j=1n1-γjLwj,∏j=1n1+ξ-1γjUwj-∏j=1n1-γjUwj∏j=1n1+ξ-1γjUwj+ξ-1∏j=1n1-γjUwj∣γ1∈α1,γ2∈α2,…,γn∈αn,where j:1,2,…,n→1,2,…,n is a permutation such that α1≥α2≥⋯≥αn.Definition 9 (see [43]).
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs, and W=w1,w2,…,wnT is the associated weight vector of αjj=1,2,…,n, with 0≤wj≤1,j=1,2,…,n and ∑j=1nwj=1. Then an interval-valued hesitant fuzzy Hamacher ordered weighted geometric (IVHFHOWG) operator is a mapping IVHFHOWG:Hn→H, such that(6)IVHFHOWGα1,α2,…,αn=⨂j=1nαjwj=ξ∏j=1nγjLwj∏j=1n1+ξ-11-γjLwj+ξ-1∏j=1nγjLwj,ξ∏j=1nγjUwj∏j=1n1+ξ-11-γjUwj+ξ-1∏j=1nγjUwj∣γ1∈α1,γ2∈α2,…,γn∈αn,where j:1,2,…,n→1,2,…,n is a permutation such that α1≥α2≥⋯≥αn.Theorem 10 (idempotency, see [43], Theorems  19 and  24).
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs, if all αj,j=1,2,…,n, are equal: that is, αj=α for all j; then(7)IVHFHOWAα1,α2,…,αn=α,IVHFHOWGα1,α2,…,αn=α.


## 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.The COWA operator was developed by Yager [44], which extends the OWA operator [20].Definition 11.
A COWA operator is a mappingF:M→R+ associated with a basic unit interval monotonic (BUM) function, Q, such that(8)FQa=FQaL,aU=∫01dQydyaU-yaU-aLdy,where a=[aL,aU]∈M and M is the set of all nonnegative interval numbers.Denotingλ=∫01Qydy, then we have(9)FQa=FQaL,aU=λaU+1-λaL,where λ is the attitudinal character of Q. Thus, FQaL,aU 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 aL,aU.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 IVHFEα=γtL,γtU∣t=1,2,…,l is an interval-valued function E:H→0,1, which satisfies the following axiomatic requirements:(E1)
Eα=0, if and only if α=0,0 or α=1,1.(E2)
Eα=1, if and only if γtL+γl-t+1U=1, for t=1,2,…,l.(E3)
Eα=Eαc.(E4)
Eα≤Eβ, ifγtL≤βtL and γtU≤βtU when βtL+βl-t+1U≤1, for t=1,2,…,l,
orγtL≥βtL and γtU≥βtU when βtL+βl-t+1U≥1, for t=1,2,…,l.Note that if the IVHFEα=γtL,γtU∣t=1,2,…,l is reduced to a hesitant fuzzy element h=γt∣t=1,2,…,l, that is, all the interval numbers γtL,γtU in α are reduced to crisp values γt, then the requirements of axiomatic definition of entropy for IVHFE are reduced to the following conditions:(1)
Eh=0, if and only if h=0 or h=1.(2)
Eh=1, if and only if γt+γl-t+1=1, for t=1,2,…,l.(3)
Eh=Ehc.(4)
Eh≤Eg, if γt≤βt for βt+βl-t+1≤1, or γt≥βt for βt+βl-t+1≥1,t=1,2,…,l.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 IVHFEα as follows:(10)Eα=12-1l∑t=1l2sin⁡FQγt+2-FQγtc4π-1.In what follows, we prove thatEα is an entropy measure of IVHFE α.Theorem 13.
The mappingEα, defined by (10), is an entropy measure for IVHFE α.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.
Letfx=1/2-12sin⁡π/4x-1, x∈[1,3]; then we have dfx/dx=2π/42-1cos⁡π/4x.
Ifx∈[1,2], then dfx/dx≥0, which means that fx is an increasing function of x, for x∈1,2. If x∈2,3, then dfx/dx≤0, which means that fx is a decreasing function of x, for x∈[2,3]. Since fx∈0,1, then fminx=0, if and only if x=1 or x=3; fmaxx=1, if and only if x=2.
(E1) Ifα=[0,0] or α=[1,1], then we obtain that(11)FQγt+2-FQγtc=1orFQγt+2-FQγtc=3,t=1,2,…,l.From the above analysis, we have Eα=0.
On the other hand, assume thatEα=0. Since αc=γ1c,γ2c,…,γlc=1-γlU,1-γlL,1-γl-1U,1-γl-1L,…,1-γ1U,1-γ1L, then γtc=1-γl-t+1U,1-γl-t+1L. For all t=1,2,…,l, we have(12)0≤γtL=λγtL+1-λγtL≤λγtU+1-λγtL≤λγtU+1-λγtU=γtU≤1.It follows that FQγt∈0,1,t=1,2,…,l; then FQγt+2-FQγtc∈1,3,t=1,2,…,l. Therefore, every term in the summation of Eα is nonnegative. Since Eα=0, then we deduce that every term should be zero in Eα; that is, for ∀t∈1,2,…,l, we have(13)12-12sin⁡FQγt+2-FQγtc4π-1=0.
From the above analysis, we know that (13) holds for all j, if and only if FQγt+2-FQγtc=1 or FQγt+2-FQγtc=3, for ∀λ∈0,1,t=1,2,…,l.
(i) IfFQγt+2-FQγtc=1, for ∀λ∈0,1,t=1,2,…,l.
LetΦtλ=FQγt+2-FQγtc-1,t=1,2,…,l; then we obtain that(14)Φtλ=FQγt+2-FQγtc-1=λγtU+1-λγtL-λ1-γl-t+1L+1-λ1-γl-t+1U+1=γtU-γtL+γl-t+1L-γl-t+1Uλ+γtL+γl-t+1U.Therefore, for ∀λ∈[0,1],t=1,2,…,l, we have(15)γtU-γtL+γl-t+1L-γl-t+1Uλ+γtL+γl-t+1U=0.then γtU-γtL+γl-t+1L-γl-t+1U=0 and γtL+γl-t+1U=0,t=1,2,…,l. As γtU,γtL,γl-t+1L,γl-t+1U∈0,1,t=1,2,…,l, thus γtL=γtU=0,t=1,2,…,l. Hence, α=0,0.
(ii) IfFQγt+2-FQγtc=3, for ∀λ∈0,1,t=1,2,…,l. Since FQγt∈0,1 and FQγt+2-FQγtc∈1,3, for ∀λ∈0,1,t=1,2,…,l, and then we obtain that FQγt=1,FQγtc=0, for ∀λ∈[0,1],t=1,2,…,l. According to (9), we have λγtU+1-λγtL=1, for ∀λ∈0,1,t=1,2,…,l; then γtL=γtU=1,t=1,2,…,l. Hence, α=1,1.
(E2) Suppose thatγtL+γl-t+1U=1, for t=1,2,…,l; then γtU+γl-t+1L=1,t=1,2,…,l, and we have(16)FQγt+2-FQγtc=λγtU+1-λγtL+2-λ1-γl-t+1L+1-λ1-γl-t+1U=λγtU+γl-t+1L+1-λγtL+γl-t+1U+1=λ+1-λ+1=2.From (10), we obtain that(17)E1α=12-1l∑j=1l2sin⁡24π-1=1.
On the other hand, from the above analysis, we haveFQγt∈0,1 and FQγt+2-FQγtc∈1,3, and it is obvious that 0≤Eα≤1. If Eα=1, then(18)FQγt+2-FQγtc=2,for∀λ∈0,1,t=1,2,…,l.Therefore,(19)γtU-γtL+γl-t+1L-γl-t+1Uλ+γtL+γl-t+1U-1=0,for∀λ∈0,1,t=1,2,…,l;it follows that γtU-γtL+γl-t+1L-γl-t+1U=0 and γtL+γl-t+1U-1=0,t=1,2,…,l; that is,(20)γtL+γl-t+1U=1,t=1,2,…,l.
(E3) Asαc=1-γlU,1-γlL,1-γl-1U,1-γl-1L,…,1-γ1U,1-γ1L, then αcc=α, and we have(21)FQγtc+2-FQγctc=FQγtc+2-FQγt,t=1,2,…,l.
Sincesin⁡x+π/2=sin⁡-x+π/2,x∈R, and then(22)Eαc=12-1l∑t=1l2sin⁡FQγtc+2-FQγctc4π-1=12-1l∑t=1l2sin⁡FQγtc+2-FQγt4π-1=12-1l∑t=1l2sin⁡FQγtc-FQγt4π+π2-1=12-1l∑t=1l2sin⁡-FQγtc-FQγt4π+π2-1=12-1l∑t=1l2sin⁡FQγt+2-FQγtc4π-1=Eα.
(E4) Assume thatγtL≤βtL and γtU≤βtU when βtL+βl-t+1U≤1, for t=1,2,…,l; then(23)γtL+γl-t+1U≤βtL+βl-t+1U≤1,γtU+γl-t+1L≤βtU+βl-t+1L≤1,t=1,2,…,l.It follows that(24)1≤FQγt+2-FQγtc=λγtU+γl-t+1L+1-λγtL+γl-t+1U+1≤λβtU+βl-t+1L+1-λβtL+βl-t+1U+1=FQβt+2-FQβtc,FQβt+2-FQβtc=λβtU+βl-t+1L+1-λβtL+βl-t+1U+1≤λ+1-λ+1=2.Thus,(25)1≤FQγt+2-FQγtc≤FQβt+2-FQβtc≤2.Notice that fx=1/2-12sin⁡π/4x-1 is an increasing function of x, for x∈1,2; thus, we obtain that Eα≤Eβ.
Similarly, ifγtL≥βtL and γtU≥βtU when βtL+βl-t+1U≥1, for t=1,2,…,l, one can also prove that Eα≤Eβ.Definition 14.
Suppose thatF is a COWA operator associated with BUM function Q and α is an IVHFE; then Eα, defined by (10), is called the continuous entropy on IVHFE α.


## 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 [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.


### 4.1. Improved Interval-Valued Hesitant Fuzzy Hamacher Information Aggregation Operators

Example 15.
Suppose thatα1 and α2 are two IVHFEs, and α1=α2=α=0.1,0.2,0.3,0.5,W=0.5,0.5T. Without loss of generality, take ξ=1. According to Definition 8, it follows that(26)IVHFHOWAα,α=IVHFHOWAα1,α2=1-∏j=121-γjLwj,1-∏j=121-γjUwj∣γ1∈α1,γ2∈α2=1-1-0.10.3×1-0.10.7,1-1-0.20.3×1-0.20.7,1-1-0.10.3×1-0.30.7,1-1-0.20.3×1-0.50.7,1-1-0.30.3×1-0.10.7,1-1-0.50.3×1-0.20.7,1-1-0.30.3×1-0.50.7,1-1-0.30.3×1-0.50.7=0.1000,0.2000,0.2452,0.4243,0.1654,0.3052,0.3000,0.5000.Let β=IVHFHOWAα,α; then the score values of α and β can be calculated as sα=0.2,0.35, sβ=0.2027,0.3574. Therefore, the degree of possibility of sα≥sβ is(27)psα≥sβ=max⁡1-max⁡0.3574-0.20.15+0.1547,0,0=0.4834<0.5.Then we have sβ>sα; thus β>α; that is,(28)IVHFHOWAα,α>α.
On the other hand, by the IVHFHOWG operator in Definition9, it is obtained that(29)IVHFHOWGα,α=IVHFHOWGα1,α2=∏j=12γjLwj,∏j=12γjUwj∣γ1∈α1,γ2∈α2=0.10.3×0.10.7,0.20.3×0.20.7,0.10.3×0.30.7,0.20.3×0.50.7,0.30.3×0.10.7,0.50.3×0.20.7,0.30.3×0.50.7,0.30.3×0.50.7=0.1000,0.2000,0.2158,0.3798,0.1390,0.2633,0.3000,0.5000.Let χ=IVHFHOWGα,α; then the score value of χ is sχ=0.1887,0.3358. Therefore, the degree of possibility of sα≥sχ is(30)psα≥sχ=max⁡1-max⁡0.3358-0.20.15+0.1471,0,0=0.5429>0.5.Then we have sα>sχ; thus, χ<α; that is,(31)IVHFHOWGα,α<α.Example15 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.
Letα, α1, and α2 be three IVHFEs; then(32)1α1⊕˙α2=⋃t=1lγ1tL+γ2tL-γ1tLγ2tL-1-ξγ1tLγ2tL1-1-ξγ1tLγ2tL,γ1tU+γ2tU-γ1tUγ2tU-1-ξγ1tUγ2tU1-1-ξγ1tUγ2tU;2α1⊗˙α2=⋃t=1lγ1tLγ2tLξ+1-ξγ1tL+γ2tL-γ1tLγ2tL,γ1tUγ2tUξ+1-ξγ1tU+γ2tU-γ1tUγ2tU;3ρα=⋃t=1l1+ξ-1γtLρ-1-γtLρ1+ξ-1γtLρ+ξ-11-γtLρ,1+ξ-1γtUρ-1-γtUρ1+ξ-1γtUρ+ξ-11-γtUρ,ρ>0;4αρ=⋃t=1lξγtLρ1+ξ-11-γtLρ+ξ-1γtLρ,ξγtUρ1+ξ-11-γtUρ+ξ-1γtUρ,ρ>0.
According to Definition16, we obtain that α1,α2,α1⊕˙α2, and α1⊗˙α2 have the same length l. Based on the adjusted operational principle for IVHFEs, we develop the improved interval-valued hesitant fuzzy Hamacher aggregation operators as follows.Definition 17.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs; the improved IVHFHOWA (I-IVHFHOWA) operator is a mapping I-IVHFHOWA:Hn→H, such that(33)I-IVHFHOWAα1,α2,…,αn=⨁˙j=1nwjαj=⋃t=1l∏j=1n1+ξ-1γjtLwj-∏j=1n1-γjtLwj∏j=1n1+ξ-1γjtLwj+ξ-1∏j=1n1-γjtLwj,∏j=1n1+ξ-1γjtUwj-∏j=1n1-γjtUwj∏j=1n1+ξ-1γjtUwj+ξ-1∏j=1n1-γjtUwj,where j:1,2,…,n→1,2,…,n is a permutation such that α1≥α2≥⋯≥αn, and W=w1,w2,…,wnT is the associated weight vector with 0≤wj≤1,j=1,2,…,n and ∑j=1nwj=1.Remark 18.
Ifξ=1, the I-IVHFHOWA operator is reduced to the following one:(34)I-IVHFOWAα1,α2,…,αn=⋃t=1l1-∏j=1n1-γjtLwj,1-∏j=1n1-γjtUwj,which is called the improved interval-valued hesitant fuzzy ordered weighted averaging (I-IVHFOWA) operator.
Ifξ=2, the I-IVHFHOWA operator is reduced to the following one:(35)I-IVHFEOWAα1,α2,…,αn=⋃t=1l∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj,∏j=1n1+γjtUwj-∏j=1n1-γjtUwj∏j=1n1+γjtUwj+∏j=1n1-γjtUwj,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.Definition 19.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs; the improved IVHFHOWG (I-IVHFHOWG) operator is a mapping I-IVHFHOWG:Hn→H, such that(36)I-IVHFHOWGα1,α2,…,αn=⨂˙j=1nαjwj=⋃t=1lξ∏j=1nγjtLwj∏j=1n1+ξ-11-γjtLwj+ξ-1∏j=1nγjtLwj,ξ∏j=1nγjtUwj∏j=1n1+ξ-11-γjtUwj+ξ-1∏j=1nγjtUwj,where j:1,2,…,n→1,2,…,n is a permutation such that α1≥α2≥⋯≥αn, and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n and ∑j=1nwj=1.Remark 20.
Ifξ=1, the I-IVHFHOWG operator is degenerated to the following:(37)I-IVHFOWGα1,α2,…,αn=⋃t=1l∏j=1nγjtLwj,∏j=1nγjtUwj,which is called the improved interval-valued hesitant fuzzy ordered weighted geometric (I-IVHFOWG) operator.Ifξ=2, the I-IVHFHOWG operator is degenerated to the following operator:(38)I-IVHFEOWGα1,α2,…,αn=⋃t=1l2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj,2∏j=1nγjtUwj∏j=1n2-γjtUwj+∏j=1nγjtUwj,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.Theorem 21 (idempotency).
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs; if all αj,j=1,2,…,n, are equal, that is, αj=α for all j, then(39)I-IVHFHOWAα1,α2,…,αn=α,I-IVHFHOWGα1,α2,…,αn=α.Proof.
Sinceαj=α for all j, that is, α1=α2=⋯=αn=α=γ1L,γ1U,γ2L,γ2U,…,γlL,γlU, then we have(40)I-IVHFHOWAα1,α2,…,αn=⨁˙j=1nwjαj=⋃t=1l∏j=1n1+ξ-1γjtLwj-∏j=1n1-γjtLwj∏j=1n1+ξ-1γjtLwj+ξ-1∏j=1n1-γjtLwj,∏j=1n1+ξ-1γjtUwj-∏j=1n1-γjtUwj∏j=1n1+ξ-1γjtUwj+ξ-1∏j=1n1-γjtUwj=⋃t=1l∏j=1n1+ξ-1γtLwj-∏j=1n1-γtLwj∏j=1n1+ξ-1γtLwj+ξ-1∏j=1n1-γtLwj,∏j=1n1+ξ-1γtUwj-∏j=1n1-γtUwj∏j=1n1+ξ-1γtUwj+ξ-1∏j=1n1-γtUwj=⋃t=1l1+ξ-1γtL∑j=1nwj-1-γtL∑j=1nwj1+ξ-1γtL∑j=1nwj+ξ-1·1-γtL∑j=1nwj,1+ξ-1γtU∑j=1nwj-1-γtU∑j=1nwj1+ξ-1γtU∑j=1nwj+ξ-1·1-γtU∑j=1nwj=⋃t=1l1+ξ-1γtL-1-γtL1+ξ-1γtL+ξ-1·1-γtL,1+ξ-1γtU-1-γtU1+ξ-1γtU+ξ-1·1-γtU=⋃t=1lγtL,γtU=γ1L,γ1U,γ2L,γ2U,…,γlL,γlU=α.
Similarly, we haveI-IVHFHOWGα1,α2,…,αn=α, which completes the proof of Theorem  3.It can be easily proved that the I-IVHFHOWA and I-IVHFHOWG operators have the following property.Theorem 22 (boundedness).
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs, αL=⋃t=1lmin1≤j≤n⁡γjtL,min1≤j≤n⁡γjtU,  αU=⋃t=1lmax1≤j≤n⁡γjtL,max1≤j≤n⁡γjtU; then(41)αL≤I-IVHFHOWAα1,α2,…,αn≤αU,αL≤I-IVHFHOWGα1,α2,…,αn≤αU.


### 4.2. Relationship among the Improved Interval-Valued Hesitant Fuzzy Prioritized Aggregation Operators

In what follows, we analyze the relationship among the I-IVHFOWA operator, I-IVHFOWG operator, I-IVHFEOWA operator, and I-IVHFEOWG operator.In order to study the relationship among these interval-valued hesitant fuzzy information aggregation operators, we first introduce the following lemmas.Lemma 23 (see [45]).
Letxj>0,wj>0,j=1,2,…,n, and ∑j=1nwj=1; then ∏j=1nxjwj≤∑j=1nwjxj, with equality if and only if x1=x2=⋯=xn.Lemma 24 (see [46]).
Letaj,bj,cj,djj=1,2,…,n be four collections of nonnegative numbers; if aj-bj-cj-dj=0,j=1,2,…,n, then(42)∏j=1naj-∏j=1nbj-∏j=1ncj-∏j=1ndj≥0.Theorem 25.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(43)I-IVHFEOWAα1,α2,…,αn≤I-IVHFOWAα1,α2,…,αn.Proof.
Since∑j=1nwj=1 and wj≥0,j=1,2,…,n, and according to Lemma 23, for any γ1∈α1,γ2∈α2,…,γn∈αn, we have(44)∏j=1n1+γjtLwj+∏j=1n1-γjtLwj≤∑j=1nwj1+γjtL+∑j=1nwj1-γjtL=∑j=1nwj1+γjtL+1-γjtL=2.Then(45)∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj=1-2∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj≤1-2∏j=1n1-γjtLwj2=1-∏j=1n1-γjtLwj.
Similarly, we also get that(46)∏j=1n1+γjtUwj-∏j=1n1-γjtUwj∏j=1n1+γjtUwj+∏j=1n1-γjtUwj≤1-∏j=1n1-γjtUwj.
Then according to Definitions4 and 5, we know that the score function of I-IVHFEOWAα1,α2,…,αn is less than the score function of I-IVHFOWAα1,α2,…,αn; that is, (47)sI-IVHFEOWAα1,α2,…,αn≤sI-IVHFOWAα1,α2,…,αn.It follows that(48)I-IVHFEOWAα1,α2,…,αn≤I-IVHFOWAα1,α2,…,αn.Theorem 26.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs, let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(49)I-IVHFOWGα1,α2,…,αn≤I-IVHFEOWGα1,α2,…,αn.Proof.
Since∑j=1nwj=1 and wj≥0,j=1,2,…,n, and according to Lemma 23, for any γ1∈α1,γ2∈α2,…,γn∈αn, we have(50)∏j=1n2-γjtLwj+∏j=1nγjtLwj≤∑j=1nwj2-γjtL+∑j=1nwjγjtL=∑j=1nwj2-γjtL+wjγjtL=2.Then(51)2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj≥2∏j=1nγjtLwj2=∏j=1nγjtLwj.
Similarly, we have(52)2∏j=1nγjtUwj∏j=1n2-γjtUwj+∏j=1nγjtUwj≥∏j=1nγjtUwj.
According to Definitions4 and 5, we obtain that the score function of I-IVHFOWGα1,α2,…,αn is less than the score function of I-IVHFEOWGα1,α2,…,αn; that is, (53)sI-IVHFOWGα1,α2,…,αn≤sI-IVHFEOWGα1,α2,…,αn.It follows that(54)I-IVHFOWGα1,α2,…,αn≤I-IVHFEOWGα1,α2,…,αn.Theorem 27.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(55)I-IVHFEOWAα1,α2,…,αn≥I-IVHFEOWGα1,α2,…,αn.Proof.
Since∑j=1nwj=1 and wj≥0,j=1,2,…,n, and for any γ1∈α1,γ2∈α2,…,γn∈αn, we have (56)∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj=∏j=1n2+γjtL-γjtL2wj-∏j=1n2-3γjtL+γjtL2wj-∏j=1nγjtL+γjtL2wj-3∏j=1nγjtL-γjtL2wj×∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-1×∏j=1n2-γjtLwj+∏j=1nγjtLwj-1=∏j=1n2+γjtL-γjtL2wj-∏j=1n2-3γjtL+γjtL2wj-∏j=1nγjtL+γjtL2wj-∏j=1n3γjtL-3γjtL2wj×∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-1×∏j=1n2-γjtLwj+∏j=1nγjtLwj-1.As 0≤γjtL≤γjtU≤1,j=1,2,…,n,t=1,2,…,l, then(57)2+γjtL-γjtL2≥0,2-3γjtL+γjtL2≥0,γjtL+γjtL2≥0,3γjtL-3γjtL2≥0,j=1,2,…,n,t=1,2,…,l.
Note that(58)2+γjtL-γjtL2-2-3γjtL+γjtL2-γjtL+γjtL2-3γjtL-3γjtL2=2+γjtL-γjtL2-2+3γjtL-γjtL2-γjtL-γjtL2-3γjtL+3γjtL2=0,j=1,2,…,n,t=1,2,…,l.
Using Lemma24, for t=1,2,…,l, we have(59)∏j=1n2+γjtL-γjtL2wj-∏j=1n2-3γjtL+γjtL2wj-∏j=1nγjtL+γjtL2wj-∏j=1n3γjtL-3γjtL2wj≥0.It follows that(60)∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj≥0.
Similarly, we have(61)∏j=1n1+γjtUwj-∏j=1n1-γjtUwj∏j=1n1+γjtUwj+∏j=1n1-γjtUwj-2∏j=1nγjtUwj∏j=1n2-γjtUwj+∏j=1nγjtUwj≥0.According to Definitions4 and 5, the score function of I-IVHFEOWGα1,α2,…,αn is less than the score function of I-IVHFEOWAα1,α2,…,αn; that is,(62)sI-IVHFEOWAα1,α2,…,αn≥sI-IVHFEOWGα1,α2,…,αn.It follows that(63)I-IVHFEOWAα1,α2,…,αn≥I-IVHFEOWGα1,α2,…,αn.According to Theorems21–27, we haveTheorem 28.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(64)I-IVHFOWGα1,α2,…,αn≤I-IVHFEOWGα1,α2,…,αn≤I-IVHFEOWAα1,α2,…,αn≤I-IVHFOWAα1,α2,…,αn.


## 4.1. Improved Interval-Valued Hesitant Fuzzy Hamacher Information Aggregation Operators

Example 15.
Suppose thatα1 and α2 are two IVHFEs, and α1=α2=α=0.1,0.2,0.3,0.5,W=0.5,0.5T. Without loss of generality, take ξ=1. According to Definition 8, it follows that(26)IVHFHOWAα,α=IVHFHOWAα1,α2=1-∏j=121-γjLwj,1-∏j=121-γjUwj∣γ1∈α1,γ2∈α2=1-1-0.10.3×1-0.10.7,1-1-0.20.3×1-0.20.7,1-1-0.10.3×1-0.30.7,1-1-0.20.3×1-0.50.7,1-1-0.30.3×1-0.10.7,1-1-0.50.3×1-0.20.7,1-1-0.30.3×1-0.50.7,1-1-0.30.3×1-0.50.7=0.1000,0.2000,0.2452,0.4243,0.1654,0.3052,0.3000,0.5000.Let β=IVHFHOWAα,α; then the score values of α and β can be calculated as sα=0.2,0.35, sβ=0.2027,0.3574. Therefore, the degree of possibility of sα≥sβ is(27)psα≥sβ=max⁡1-max⁡0.3574-0.20.15+0.1547,0,0=0.4834<0.5.Then we have sβ>sα; thus β>α; that is,(28)IVHFHOWAα,α>α.
On the other hand, by the IVHFHOWG operator in Definition9, it is obtained that(29)IVHFHOWGα,α=IVHFHOWGα1,α2=∏j=12γjLwj,∏j=12γjUwj∣γ1∈α1,γ2∈α2=0.10.3×0.10.7,0.20.3×0.20.7,0.10.3×0.30.7,0.20.3×0.50.7,0.30.3×0.10.7,0.50.3×0.20.7,0.30.3×0.50.7,0.30.3×0.50.7=0.1000,0.2000,0.2158,0.3798,0.1390,0.2633,0.3000,0.5000.Let χ=IVHFHOWGα,α; then the score value of χ is sχ=0.1887,0.3358. Therefore, the degree of possibility of sα≥sχ is(30)psα≥sχ=max⁡1-max⁡0.3358-0.20.15+0.1471,0,0=0.5429>0.5.Then we have sα>sχ; thus, χ<α; that is,(31)IVHFHOWGα,α<α.Example15 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.
Letα, α1, and α2 be three IVHFEs; then(32)1α1⊕˙α2=⋃t=1lγ1tL+γ2tL-γ1tLγ2tL-1-ξγ1tLγ2tL1-1-ξγ1tLγ2tL,γ1tU+γ2tU-γ1tUγ2tU-1-ξγ1tUγ2tU1-1-ξγ1tUγ2tU;2α1⊗˙α2=⋃t=1lγ1tLγ2tLξ+1-ξγ1tL+γ2tL-γ1tLγ2tL,γ1tUγ2tUξ+1-ξγ1tU+γ2tU-γ1tUγ2tU;3ρα=⋃t=1l1+ξ-1γtLρ-1-γtLρ1+ξ-1γtLρ+ξ-11-γtLρ,1+ξ-1γtUρ-1-γtUρ1+ξ-1γtUρ+ξ-11-γtUρ,ρ>0;4αρ=⋃t=1lξγtLρ1+ξ-11-γtLρ+ξ-1γtLρ,ξγtUρ1+ξ-11-γtUρ+ξ-1γtUρ,ρ>0.
According to Definition16, we obtain that α1,α2,α1⊕˙α2, and α1⊗˙α2 have the same length l. Based on the adjusted operational principle for IVHFEs, we develop the improved interval-valued hesitant fuzzy Hamacher aggregation operators as follows.Definition 17.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs; the improved IVHFHOWA (I-IVHFHOWA) operator is a mapping I-IVHFHOWA:Hn→H, such that(33)I-IVHFHOWAα1,α2,…,αn=⨁˙j=1nwjαj=⋃t=1l∏j=1n1+ξ-1γjtLwj-∏j=1n1-γjtLwj∏j=1n1+ξ-1γjtLwj+ξ-1∏j=1n1-γjtLwj,∏j=1n1+ξ-1γjtUwj-∏j=1n1-γjtUwj∏j=1n1+ξ-1γjtUwj+ξ-1∏j=1n1-γjtUwj,where j:1,2,…,n→1,2,…,n is a permutation such that α1≥α2≥⋯≥αn, and W=w1,w2,…,wnT is the associated weight vector with 0≤wj≤1,j=1,2,…,n and ∑j=1nwj=1.Remark 18.
Ifξ=1, the I-IVHFHOWA operator is reduced to the following one:(34)I-IVHFOWAα1,α2,…,αn=⋃t=1l1-∏j=1n1-γjtLwj,1-∏j=1n1-γjtUwj,which is called the improved interval-valued hesitant fuzzy ordered weighted averaging (I-IVHFOWA) operator.
Ifξ=2, the I-IVHFHOWA operator is reduced to the following one:(35)I-IVHFEOWAα1,α2,…,αn=⋃t=1l∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj,∏j=1n1+γjtUwj-∏j=1n1-γjtUwj∏j=1n1+γjtUwj+∏j=1n1-γjtUwj,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.Definition 19.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs; the improved IVHFHOWG (I-IVHFHOWG) operator is a mapping I-IVHFHOWG:Hn→H, such that(36)I-IVHFHOWGα1,α2,…,αn=⨂˙j=1nαjwj=⋃t=1lξ∏j=1nγjtLwj∏j=1n1+ξ-11-γjtLwj+ξ-1∏j=1nγjtLwj,ξ∏j=1nγjtUwj∏j=1n1+ξ-11-γjtUwj+ξ-1∏j=1nγjtUwj,where j:1,2,…,n→1,2,…,n is a permutation such that α1≥α2≥⋯≥αn, and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n and ∑j=1nwj=1.Remark 20.
Ifξ=1, the I-IVHFHOWG operator is degenerated to the following:(37)I-IVHFOWGα1,α2,…,αn=⋃t=1l∏j=1nγjtLwj,∏j=1nγjtUwj,which is called the improved interval-valued hesitant fuzzy ordered weighted geometric (I-IVHFOWG) operator.Ifξ=2, the I-IVHFHOWG operator is degenerated to the following operator:(38)I-IVHFEOWGα1,α2,…,αn=⋃t=1l2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj,2∏j=1nγjtUwj∏j=1n2-γjtUwj+∏j=1nγjtUwj,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.Theorem 21 (idempotency).
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs; if all αj,j=1,2,…,n, are equal, that is, αj=α for all j, then(39)I-IVHFHOWAα1,α2,…,αn=α,I-IVHFHOWGα1,α2,…,αn=α.Proof.
Sinceαj=α for all j, that is, α1=α2=⋯=αn=α=γ1L,γ1U,γ2L,γ2U,…,γlL,γlU, then we have(40)I-IVHFHOWAα1,α2,…,αn=⨁˙j=1nwjαj=⋃t=1l∏j=1n1+ξ-1γjtLwj-∏j=1n1-γjtLwj∏j=1n1+ξ-1γjtLwj+ξ-1∏j=1n1-γjtLwj,∏j=1n1+ξ-1γjtUwj-∏j=1n1-γjtUwj∏j=1n1+ξ-1γjtUwj+ξ-1∏j=1n1-γjtUwj=⋃t=1l∏j=1n1+ξ-1γtLwj-∏j=1n1-γtLwj∏j=1n1+ξ-1γtLwj+ξ-1∏j=1n1-γtLwj,∏j=1n1+ξ-1γtUwj-∏j=1n1-γtUwj∏j=1n1+ξ-1γtUwj+ξ-1∏j=1n1-γtUwj=⋃t=1l1+ξ-1γtL∑j=1nwj-1-γtL∑j=1nwj1+ξ-1γtL∑j=1nwj+ξ-1·1-γtL∑j=1nwj,1+ξ-1γtU∑j=1nwj-1-γtU∑j=1nwj1+ξ-1γtU∑j=1nwj+ξ-1·1-γtU∑j=1nwj=⋃t=1l1+ξ-1γtL-1-γtL1+ξ-1γtL+ξ-1·1-γtL,1+ξ-1γtU-1-γtU1+ξ-1γtU+ξ-1·1-γtU=⋃t=1lγtL,γtU=γ1L,γ1U,γ2L,γ2U,…,γlL,γlU=α.
Similarly, we haveI-IVHFHOWGα1,α2,…,αn=α, which completes the proof of Theorem  3.It can be easily proved that the I-IVHFHOWA and I-IVHFHOWG operators have the following property.Theorem 22 (boundedness).
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs, αL=⋃t=1lmin1≤j≤n⁡γjtL,min1≤j≤n⁡γjtU,  αU=⋃t=1lmax1≤j≤n⁡γjtL,max1≤j≤n⁡γjtU; then(41)αL≤I-IVHFHOWAα1,α2,…,αn≤αU,αL≤I-IVHFHOWGα1,α2,…,αn≤αU.


## 4.2. Relationship among the Improved Interval-Valued Hesitant Fuzzy Prioritized Aggregation Operators

In what follows, we analyze the relationship among the I-IVHFOWA operator, I-IVHFOWG operator, I-IVHFEOWA operator, and I-IVHFEOWG operator.In order to study the relationship among these interval-valued hesitant fuzzy information aggregation operators, we first introduce the following lemmas.Lemma 23 (see [45]).
Letxj>0,wj>0,j=1,2,…,n, and ∑j=1nwj=1; then ∏j=1nxjwj≤∑j=1nwjxj, with equality if and only if x1=x2=⋯=xn.Lemma 24 (see [46]).
Letaj,bj,cj,djj=1,2,…,n be four collections of nonnegative numbers; if aj-bj-cj-dj=0,j=1,2,…,n, then(42)∏j=1naj-∏j=1nbj-∏j=1ncj-∏j=1ndj≥0.Theorem 25.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(43)I-IVHFEOWAα1,α2,…,αn≤I-IVHFOWAα1,α2,…,αn.Proof.
Since∑j=1nwj=1 and wj≥0,j=1,2,…,n, and according to Lemma 23, for any γ1∈α1,γ2∈α2,…,γn∈αn, we have(44)∏j=1n1+γjtLwj+∏j=1n1-γjtLwj≤∑j=1nwj1+γjtL+∑j=1nwj1-γjtL=∑j=1nwj1+γjtL+1-γjtL=2.Then(45)∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj=1-2∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj≤1-2∏j=1n1-γjtLwj2=1-∏j=1n1-γjtLwj.
Similarly, we also get that(46)∏j=1n1+γjtUwj-∏j=1n1-γjtUwj∏j=1n1+γjtUwj+∏j=1n1-γjtUwj≤1-∏j=1n1-γjtUwj.
Then according to Definitions4 and 5, we know that the score function of I-IVHFEOWAα1,α2,…,αn is less than the score function of I-IVHFOWAα1,α2,…,αn; that is, (47)sI-IVHFEOWAα1,α2,…,αn≤sI-IVHFOWAα1,α2,…,αn.It follows that(48)I-IVHFEOWAα1,α2,…,αn≤I-IVHFOWAα1,α2,…,αn.Theorem 26.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs, let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(49)I-IVHFOWGα1,α2,…,αn≤I-IVHFEOWGα1,α2,…,αn.Proof.
Since∑j=1nwj=1 and wj≥0,j=1,2,…,n, and according to Lemma 23, for any γ1∈α1,γ2∈α2,…,γn∈αn, we have(50)∏j=1n2-γjtLwj+∏j=1nγjtLwj≤∑j=1nwj2-γjtL+∑j=1nwjγjtL=∑j=1nwj2-γjtL+wjγjtL=2.Then(51)2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj≥2∏j=1nγjtLwj2=∏j=1nγjtLwj.
Similarly, we have(52)2∏j=1nγjtUwj∏j=1n2-γjtUwj+∏j=1nγjtUwj≥∏j=1nγjtUwj.
According to Definitions4 and 5, we obtain that the score function of I-IVHFOWGα1,α2,…,αn is less than the score function of I-IVHFEOWGα1,α2,…,αn; that is, (53)sI-IVHFOWGα1,α2,…,αn≤sI-IVHFEOWGα1,α2,…,αn.It follows that(54)I-IVHFOWGα1,α2,…,αn≤I-IVHFEOWGα1,α2,…,αn.Theorem 27.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(55)I-IVHFEOWAα1,α2,…,αn≥I-IVHFEOWGα1,α2,…,αn.Proof.
Since∑j=1nwj=1 and wj≥0,j=1,2,…,n, and for any γ1∈α1,γ2∈α2,…,γn∈αn, we have (56)∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj=∏j=1n2+γjtL-γjtL2wj-∏j=1n2-3γjtL+γjtL2wj-∏j=1nγjtL+γjtL2wj-3∏j=1nγjtL-γjtL2wj×∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-1×∏j=1n2-γjtLwj+∏j=1nγjtLwj-1=∏j=1n2+γjtL-γjtL2wj-∏j=1n2-3γjtL+γjtL2wj-∏j=1nγjtL+γjtL2wj-∏j=1n3γjtL-3γjtL2wj×∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-1×∏j=1n2-γjtLwj+∏j=1nγjtLwj-1.As 0≤γjtL≤γjtU≤1,j=1,2,…,n,t=1,2,…,l, then(57)2+γjtL-γjtL2≥0,2-3γjtL+γjtL2≥0,γjtL+γjtL2≥0,3γjtL-3γjtL2≥0,j=1,2,…,n,t=1,2,…,l.
Note that(58)2+γjtL-γjtL2-2-3γjtL+γjtL2-γjtL+γjtL2-3γjtL-3γjtL2=2+γjtL-γjtL2-2+3γjtL-γjtL2-γjtL-γjtL2-3γjtL+3γjtL2=0,j=1,2,…,n,t=1,2,…,l.
Using Lemma24, for t=1,2,…,l, we have(59)∏j=1n2+γjtL-γjtL2wj-∏j=1n2-3γjtL+γjtL2wj-∏j=1nγjtL+γjtL2wj-∏j=1n3γjtL-3γjtL2wj≥0.It follows that(60)∏j=1n1+γjtLwj-∏j=1n1-γjtLwj∏j=1n1+γjtLwj+∏j=1n1-γjtLwj-2∏j=1nγjtLwj∏j=1n2-γjtLwj+∏j=1nγjtLwj≥0.
Similarly, we have(61)∏j=1n1+γjtUwj-∏j=1n1-γjtUwj∏j=1n1+γjtUwj+∏j=1n1-γjtUwj-2∏j=1nγjtUwj∏j=1n2-γjtUwj+∏j=1nγjtUwj≥0.According to Definitions4 and 5, the score function of I-IVHFEOWGα1,α2,…,αn is less than the score function of I-IVHFEOWAα1,α2,…,αn; that is,(62)sI-IVHFEOWAα1,α2,…,αn≥sI-IVHFEOWGα1,α2,…,αn.It follows that(63)I-IVHFEOWAα1,α2,…,αn≥I-IVHFEOWGα1,α2,…,αn.According to Theorems21–27, we haveTheorem 28.
Letαj=γjL,γjU∣γj∈αjj=1,2,…,n be a collection of IVHFEs and let W=w1,w2,…,wnT be the associated weight vector with 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1; then(64)I-IVHFOWGα1,α2,…,αn≤I-IVHFEOWGα1,α2,…,αn≤I-IVHFEOWAα1,α2,…,αn≤I-IVHFOWAα1,α2,…,αn.


## 5. MAGDM Method under Interval-Valued Hesitant Fuzzy Environment

LetX=X1,X2,…,Xm be a set of alternatives and let C=C1,C2,…,Cn be a collection of attributes. Suppose that the decision makers are required to provide not only the information that the alternative Xi satisfies the attribute Cj with anonymity, but also the information expressed by interval-valued values; then these values can be considered as an IVHFE αij. When all the performances of the alternatives are provided, the interval-valued hesitant fuzzy decision matrix D=αijm×n can be constructed, where αiji=1,2,…,m,j=1,2,…,n are in the form of IVHFEs. Assume that the weight vector of attribute is W=w1,w2,…,wnΤ, where 0≤wj≤1,j=1,2,…,n, and ∑j=1nwj=1.


### 5.1. 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 attributeCj, the averaging entropy ECj of the attribute Cj is given as(65)ECj=1m∑i=1mEαij,and each Eαij 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 weightwj of the attribute Cj,j=1,2,…,n is completely unknown, we can establish the following equations for determining attribute weights:(66)wj=1-ECj∑j=1n1-ECj,j=1,2,…,n.If the information about weightwj of the attribute Cj,j=1,2,…,n 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:(67)MODmaxEW=∑j=1nwj1-ECjs.t.W∈Ω,∑j=1nwj=1,wj≥0,j=1,2,…,n.


### 5.2. 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 1.
If all the attributesCj,j=1,2,…,n are of the benefit type, then the attribute values do not be normalized. Otherwise, we transform D=αijm×n into the normalized interval-valued hesitant fuzzy decision matrix B=βijm×n, where(68)βij=αij,for  benefit  attributeCjαijc,forcostattributeCj,i=1,2,…,m,j=1,2,…,n.Step 2.
Utilize (66) or model (67) to determine the weight vector of attribute: W=w1,w2,…,wnT.Step 3.
Based on the interval-valued hesitant fuzzy decision matrixB=βijm×n, utilize the I-IVHFHOWA operator:(69)βi=I-IVHFHOWAβi1,βi2,…,βin=⨁˙j=1nwjβij=⋃t=1l∏j=1n1+ξ-1ηijtLwj-∏j=1n1-ηijtLwj∏j=1n1+ξ-1ηijtLwj-ξ-1∏j=1n1-ηijtLwj,∏j=1n1+ξ-1ηijtUwj-∏j=1n1-ηijtUwj∏j=1n1+ξ-1ηijtUwj-ξ-1∏j=1n1-ηijtUwj,or the I-IVHFHOWG operator:(70)βi=I-IVHFHOWGβi1,βi2,…,βin=⨂˙j=1nβijwj=⋃t=1lξ∏j=1nηijtLwj∏j=1n1+ξ-11-ηijtLwj+ξ-1∏j=1nηijtLwj,ξ∏j=1nηijtUwj∏j=1n1+ξ-11-ηijtUwj-ξ-1∏j=1nηijtUwj,to aggregate the whole individual IVHFEs βiji=1,2,…,m,j=1,2,…,n into the overall IVHFEs βii=1,2,…,m of the alternatives Xi.Step 4.
Calculate the score functionssβii=1,2,…,m of the overall IVHFEs βii=1,2,…,m by Definition 5.Step 5.
Rank the overall IVHFEsβii=1,2,…,m in accordance with Definition 5.Step 6.
Select the priority of the alternatives according to the ranking ofβii=1,2,…,m. Note that the bigger the βi, the better the alternative Xi.Step 7.
End.


## 5.1. 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 attributeCj, the averaging entropy ECj of the attribute Cj is given as(65)ECj=1m∑i=1mEαij,and each Eαij 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 weightwj of the attribute Cj,j=1,2,…,n is completely unknown, we can establish the following equations for determining attribute weights:(66)wj=1-ECj∑j=1n1-ECj,j=1,2,…,n.If the information about weightwj of the attribute Cj,j=1,2,…,n 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:(67)MODmaxEW=∑j=1nwj1-ECjs.t.W∈Ω,∑j=1nwj=1,wj≥0,j=1,2,…,n.


## 5.2. 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 1.
If all the attributesCj,j=1,2,…,n are of the benefit type, then the attribute values do not be normalized. Otherwise, we transform D=αijm×n into the normalized interval-valued hesitant fuzzy decision matrix B=βijm×n, where(68)βij=αij,for  benefit  attributeCjαijc,forcostattributeCj,i=1,2,…,m,j=1,2,…,n.Step 2.
Utilize (66) or model (67) to determine the weight vector of attribute: W=w1,w2,…,wnT.Step 3.
Based on the interval-valued hesitant fuzzy decision matrixB=βijm×n, utilize the I-IVHFHOWA operator:(69)βi=I-IVHFHOWAβi1,βi2,…,βin=⨁˙j=1nwjβij=⋃t=1l∏j=1n1+ξ-1ηijtLwj-∏j=1n1-ηijtLwj∏j=1n1+ξ-1ηijtLwj-ξ-1∏j=1n1-ηijtLwj,∏j=1n1+ξ-1ηijtUwj-∏j=1n1-ηijtUwj∏j=1n1+ξ-1ηijtUwj-ξ-1∏j=1n1-ηijtUwj,or the I-IVHFHOWG operator:(70)βi=I-IVHFHOWGβi1,βi2,…,βin=⨂˙j=1nβijwj=⋃t=1lξ∏j=1nηijtLwj∏j=1n1+ξ-11-ηijtLwj+ξ-1∏j=1nηijtLwj,ξ∏j=1nηijtUwj∏j=1n1+ξ-11-ηijtUwj-ξ-1∏j=1nηijtUwj,to aggregate the whole individual IVHFEs βiji=1,2,…,m,j=1,2,…,n into the overall IVHFEs βii=1,2,…,m of the alternatives Xi.Step 4.
Calculate the score functionssβii=1,2,…,m of the overall IVHFEs βii=1,2,…,m by Definition 5.Step 5.
Rank the overall IVHFEsβii=1,2,…,m in accordance with Definition 5.Step 6.
Select the priority of the alternatives according to the ranking ofβii=1,2,…,m. Note that the bigger the βi, the better the alternative Xi.Step 7.
End.


## 6. 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 EOCsX=X1,X2,X3,X4 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, includingC1: energy, C2: health and medical services, C3: communication equipment, C4: emergency medical personnel, and C5: human resource.(2)
The attribute weight vectorW=w1,w2,w3,w4,w5T is completely unknown.(3)
The fuzzy decision matrixD=αij4×5 is provided by evaluators, and the evaluated values are expressed by interval-valued hesitant fuzzy information, where αiji=1,2,3,4,j=1,2,…,5 denotes the evaluated value of EOC Xi under attribute Cj. The interval-valued hesitant fuzzy decision matrix D=αij4×5 is shown in Table 1.Table 1
Interval-valued hesitant fuzzy decision matrix.
C
1
C
2
C
3
C
4
C
5
X
1
0.4,0.5
,
0.3,0.4
0.4,0.5
,
0.2,0.3
0.7,0.8
,
0.5,0.6
0.4,0.5
,
0.4,0.4
0.4,0.6
,
0.4,0.5
X
2
0.5,0.6
,
0.4,0.5
0.5,0.7
,
0.4,0.7
0.6,0.8
,
0.3,0.5
0.8,0.9
,
0.7,0.8
0.5,0.5
,
0.3,0.5
X
3
0.5,0.7
,
0.3,0.4
0.5,0.5
,
0.3,0.4
0.8,0.9
,
0.7,0.8
0.6,0.7
,
0.5,0.7
0.5,0.8
,
0.5,0.7
X
4
0.5,0.8
,
0.2,0.3
0.4,0.7
,
0.3,0.4
0.6,0.8
,
0.6,0.7
0.5,0.6
,
0.3,0.4
0.5,0.6
,
0.4,0.6The proposed MAGDM method is used to rank the EOCs and get the most desirable EOC. The following steps are involved.Step 1.
Considering that all the attributesCjj=1,2,…,5 are the benefit attributes, then the attribute values are not normalized.Step 2.
Without loss of generality, take the BUM functionQy=y, and then λ=0.5 (medium value). For each attribute weight is completely unknown, we calculate the weight vector of attribute by the following equations:(71)wj=1-ECj∑j=151-ECj,j=1,2,…,5.By (10) and (65), we obtain the attribute weights as follows:(72)w1=0.0191,w2=0.0424,w3=0.1416,w4=0.3774,w5=0.4196.Step 3.
Letξ=1 or 2. Utilize the I-IVHFHOWA operator and the I-IVHFHOWG operator expressed by (69) and (70), respectively, to aggregate the all individual IVHFEs αijj=1,2,…,5 into the overall IVHFEs αii=1,2,3,4 of the EOCs Xi, which are shown in Table 2.Table 2
The overall IVHFEs of the candidates by utilizing different operators.
ξ=1
ξ=2
I-IVHFHOWA
I-IVHFHOWG
I-IVHFHOWA
I-IVHFHOWG
α1
0.4561,0.6001
,
0.4064,0.4718
{
[
0.4329,0.5769
]
,
[
0.3987,0.4595
]
}
0.4514,0.5964
,
0.4053,0.4697
0.4359,0.5801
,
0.3999,0.4615
α2
0.6572,0.7669
,
0.4964,0.6538
0.6126,0.6790
,
0.4204,0.6056
0.6510,0.7581
,
0.4831,0.6470
0.6194,0.6921
,
0.4309,0.6129
α3
0.5963,0.7787
,
0.5252,0.7044
0.5724,0.7563
,
0.5081,0.6892
0.5924,0.7765
,
0.5224,0.7028
0.5758,0.7595
,
0.5108,0.6923
α4
0.5118,0.6465
,
0.3923,0.5399
0.5082,0.6325
,
0.3705,0.5104
0.5112,0.6445
,
0.3882,0.5352
0.5088,0.6344
,
0.3733,0.5153Step 4.
Calculate the score functionssαii=1,2,3,4 of the overall IVHFEs αii=1,2,3,4 by Definition 5, as shown in Table 3.Table 3
The score functions obtained by the I-IVHFHOWA operator and I-IVHFHOWG operator.
s
(
α
1
)
s
(
α
2
)
s
(
α
3
)
s
(
α
4
)
ξ
=
1
I-IVHFHOWA
0.4313,0.5760
0.5768,0.7103
0.5608,0.7416
0.4521,0.5932
I-IVHFHOWG
0.4158,0.5182
0.5165,0.6423
0.5403,0.7227
0.4393,0.5715
ξ
=
2
I-IVHFHOWA
0.4284,0.5331
0.5671,0.7025
0.5574,0.7397
0.4497,0.5898
I-IVHFHOWG
0.4179,0.5208
0.5252,0.6525
0.5433,0.7259
0.4411,0.5748Step 5.
Rank the overall IVHFEsαii=1,2,3,4, which is shown in Table 4.Table 4
Ordering of the overall IVHFEs.
Ordering
ξ
=
1
I-IVHFHOWA
α
3
>
α
2
>
α
4
>
α
1
I-IVHFHOWG
α
3
>
α
2
>
α
4
>
α
1
ξ
=
2
I-IVHFHOWA
α
3
>
α
2
>
α
4
>
α
1
I-IVHFHOWG
α
3
>
α
2
>
α
4
>
α
1Step 6.
Rank the EOCs.According to the ordering of the overall IVHFEsαii=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 X3.Table 5
Ordering of the EOCs.
Ordering
ξ
=
1
I-IVHFHOWA
X
3
≻
X
2
≻
X
4
≻
X
1
I-IVHFHOWA
X
3
≻
X
2
≻
X
4
≻
X
1
ξ
=
2
I-IVHFHOWA
X
3
≻
X
2
≻
X
4
≻
X
1
I-IVHFHOWA
X
3
≻
X
2
≻
X
4
≻
X
1Moreover, we can analyze how the different parameter valueξ affects the rankings of the EOCs; in this case, we consider different values of ξ, 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 ξ 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 ξ 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 ξ increases, we can find that the deviation decreases.Figure 1
Lower limits of scores for EOCs obtained by I-IVHFHOWA operator.Figure 2
Upper limits of scores for EOCs obtained by I-IVHFHOWA operator.Figure 3
Lower limits of scores for EOCs obtained by I-IVHFHOWG operator.Figure 4
Upper limits of scores for EOCs obtained by I-IVHFHOWG operator.Figure 5
Deviation values of lower limits of scores for EOCs obtained by I-IVHFHOWA operator and I-IVHFHOWG operator.Figure 6
Deviation values of upper limits of scores for EOCs obtained by I-IVHFHOWA operator and I-IVHFHOWG operator.As can be seen, depending on the different parameter valuesξ used, the ranking of the EOCs does not change, and Y5 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.Steps
1′and 2′. See Steps 1 and 2.Step
3′. Let ξ=1. By using the IVHFHOWG operator proposed by Li and Peng [43], we derive the overall IVHFEs αii=1,2,3,4 of the EOCs Xi as follows:(73)α1=0.6126,0.6790,0.4204,0.6056,0.4338,0.5987,0.4884,0.6125,0.6412,0.7555,0.6341,0.7418,0.4011,0.5424,0.5145,0.6337,0.4155,0.6408,0.3849,0.5224,0.4250,0.5967,0.6874,0.7705,0.6215,0.7342,0.5124,0.6845,0.5420,0.6406,0.4054,0.6255,0.6121,0.6748,0.5545,0.5987,0.4620,0.5807,0.5622,0.6500,0.5008,0.6132,0.6414,0.7120,0.4111,0.5865,0.5215,0.6222,0.4335,0.5619,0.5656,0.6225,0.5525,0.6254,0.3994,0.5488,0.6215,0.6782,0.5303,0.6245,0.4664,0.5440,0.5352,0.6357,α2=0.4329,0.5769,0.3987,0.4595,0.4227,0.5454,0.4007,0.4668,0.3654,0.4333,0.3848,0.4411,0.3999,0.4765,0.5145,0.6337,0.4155,0.6408,0.3849,0.5224,0.4250,0.5967,0.6874,0.7705,0.6215,0.7342,0.5124,0.6845,0.3615,0.4512,0.3845,0.4778,0.4213,0.5557,0.3390,0.4027,0.3668,0.4750,0.4343,0.5602,0.4517,0.5710,0.4225,0.5419,0.3688,0.4627,0.4115,0.5472,0.3612,0.4470,0.4519,0.5921,0.4697,0.6002,0.4411,0.5438,0.4217,0.5458,0.3789,0.4667,0.4125,0.5448,0.3860,0.4411,α3=0.5724,0.7563,0.5081,0.6892,0.4343,0.5663,0.4333,0.5779,0.4003,0.5217,0.3567,0.4764,0.4102,0.4965,0.5002,0.5844,0.4121,0.5117,0.4090,0.5214,0.4555,0.5658,0.4452,0.5543,0.4300,0.5617,0.4221,0.5388,0.4076,0.4648,0.4562,0.4948,0.4150,0.5641,0.4127,0.5324,0.4119,0.5600,0.4553,0.5225,0.4411,0.5455,0.6222,0.7017,0.4518,0.5222,0.4333,0.5757,0.4313,0.4924,0.5124,0.6007,0.5399,0.6417,0.5544,0.6488,0.4335,0.5267,0.4149,0.5990,0.5100,0.6254,0.4322,0.5458,α4=0.5082,0.6325,0.3705,0.5104,0.3661,0.4949,0.5121,0.6242,0.4313,0.4966,0.3774,0.4361,0.4216,0.5100,0.3942,0.5233,0.3887,0.4685,0.4175,0.5120,0.3770,0.4500,0.3822,0.5013,0.3923,0.4578,0.3666,0.4557,0.3505,0.4394,0.3770,0.4625,0.4111,0.5239,0.3405,0.4299,0.3588,0.4607,0.3958,0.5110,0.4332,0.51020.3925,0.4698,0.3668,0.4445,0.3917,0.5043,0.3778,0.5520,0.4219,0.5476,0.4191,0.6485,0.5334,0.6525,0.3971,0.5063,0.3862,0.4557,0.3993,0.5128,0.3430,0.4447.Step
4′. Based on the above overall IVHFEs αii=1,2,3,4 and Definition 5, we can calculate the score functions sαii=1,2,3,4 as follows:(74)sα1=0.4227,0.5198,sα2=0.5089,0.6780,sα3=0.5641,0.6973,sα4=0.4648,0.5466.Step
5′. According to Definition 4, we have sα3>sα2>sα4>sα1; then the ranking order of the EOCs can be determined: that is, α3>α2>α4>α1, and the most desirable EOC is X3.By using the proposed interval-valued hesitant fuzzy entropy formula, Hu and Zhou [49] developed a new approach for interval-valued hesitant fuzzy MAGDM problems. If we utilize the approach in [49] to cope with the aforementioned problem, then the main steps are follows.Step
1′′. See Step 1.Step
2′′. Because the attribute weight vector W is completely unknown, by the equations(75)wj′=1-Ej∑j=151-Ej,j=1,2,…,5,where Ej=1/m∑i=1mE~αij and E~αij=1/l∑j=1l1/21-s-1γtL+γl-t+1U/2s+1-γtL+γl-t+1U/2s-1 (without loss of generality, let s=2), then we can calculate the attribute weights as follows:(76)w1′=0.1147,w2′=0.1343,w3′=0.1806,w4′=0.3449,w5′=0.2255.Step
3′′. Utilizing the interval-valued hesitant fuzzy cross-entropy equations (7) and (8) in [49], we obtain the positive cross-entropy Ci+i=1,2,3,4 and negative cross-entropy Ci-i=1,2,3,4 of each EOC: C1+=0.5425,C1-=0.5377,C2+=0.4201,C2-=0.4526,C3+=0.5119,C3-=0.5833,C4+=0.6434,C24-=0.5770.Step
4′′. By (9) in [49], we determine the cross closeness degree Tii=1,2,3,4 of the EOC Xi:(77)T1=0.4978,T2=0.5186,T3=0.5326,T4=0.4728.Step
5′′. Since T3>T2>T1>T4, then the ranking of all the EOCs Xii=1,2,3,4 is X3≻X2≻X1≻X4, and the most desirable EOC is X3.Through the above example, we find that compared with the methods developed by Li and Peng [43] and Hu and Zhou [49] to rank the EOCs and determine the most desirable EOC, our proposed method has some advantages.(
1
) Compared with the method developed by Li and Peng [43], although the developed method in this paper and that of Li and Peng [43] produce the same ranking of the four EOCs, that is, X3≻X2≻X4≻X1, our proposed operators meet idempotency, and the decision making process of the information aggregation proposed by our method is more simplified than that of Li and Peng [43]. Therefore, the proposed MAGDM approach is both valid and practical to deal with group decision making problems.(
2
) Comparing the four EOCs ranking derived by our approach with that of Hu and Zhou [49], we find that the method of Hu and Zhou [49] derives a little different ranking of these EOCs. However, from the original interval-valued hesitant fuzzy decision matrix D=αij4×5, we find that most of the attribute values of EOC X4 are greater than X1, which means that X4 is preferred to X1. Therefore, our MAGDM approach is more rational than that of Hu and Zhou [49] in this case.(
3
) In addition, we can consider a wide range of particular operators, including I-IVHFOWA operator, I-IVHFOWG operator, I-IVHFEOWA operator, and I-IVHFEOWG operator. The parameter ξ can be viewed as the measure of the decision maker’s attitudinal character; therefore, when people make a decision, they can choose the different parameter ξ according to their own risk attitudes.


## 7. Conclusions

In FSs theory and modern decision making science in the uncertain environment, interval-valued hesitant fuzzy MAGDM is an important research field. In this paper, we first presented the axiomatic definition of entropy for IVHFEs and constructed a continuous entropy formula for IVHFE to determine the entropy weights with respect to the decision matrix. Then, based on the Hamachert-conorm andt-norm, we introduced some new adjusted operations on IVHFEs, and some improved interval-valued hesitant fuzzy Hamacher information aggregation operators have been investigated, such as I-IVHFHOWA operator and I-IVHFHOWG operator. Some of their desirable properties have been discussed. In addition, the relationship among the I-IVHFOWA operator, I-IVHFOWG operator, I-IVHFEOWA operator, and I-IVHFEOWG operator has also been analyzed. We also establish the programming model to determine optimal weight of attribute with the principle of minimum entropy. Finally, an approach to MAGDM has been developed, which is based on interval-valued hesitant fuzzy continuous entropy and the proposed operators. A numerical example for EOC selection is given to illustrate that the proposed aggregation methods provide a useful way to deal with MAGDM under interval-valued hesitant fuzzy environment.However, there are still lots of work to be done in the future, including how to investigate sensitivity analysis and to give some practical applications of our method to the fields of decision making and supply chain management and medical diagnosis.



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*Source: 2931482-2017-09-26.xml*