# Hesitant Fuzzy Linguistic Hamy Mean Aggregation Operators and Their Application to Linguistic Multiple Attribute Decision-Making

Authors: Yuan Rong; Zheng Pei; Yi Liu
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

Linguistic aggregation operator is a paramount appliance to fix linguistic multiple attribute decision-making (MADM) issues. In the article, the Hamy mean (HM) operator is utilized to fuse hesitant fuzzy linguistic (HFL) information and several novel HFL aggregation operators including the hesitant fuzzy linguistic Hamy mean (HFLHM) operator, weighted hesitant fuzzy linguistic Hamy mean (WHFLHM) operator, hesitant fuzzy linguistic dual Hamy mean (HFLDHM) operator, and weighted hesitant fuzzy linguistic dual Hamy mean (WHFLDHM) operator are proposed. Besides, several paramount theorems and particular cases of these aggregation operators are investigated in detail, and then a novel MADM approach is presented by using the proposed aggregation operators. Ultimately, a practical example is utilized to manifest the effectiveness and practicability of the propounded method.


---

## Body

## 1. Introduction

The essence of MADM is the process of sorting a limited number of alternatives under given attributes and selecting the optimal one through the evaluation information provided by experts. It has been widely applied in multitudinous aspects [1–9]. Owing to the complicacy and nondeterminacy of the decision-making (DM) context, one of the most arduous challenges in course of MADM is how to represent the evaluation information of attributes in fuzzy and uncertain circumstances accurately. For handling this limitation, Zadeh [10] originally brought forward the notion of fuzzy set (FS), which can effectively depict vague and uncertain information through a membership degree. Thereafter, FS has been further investigated and acquired a large number of research achievements by multitude scholars in diverse aspects. Among them, the most important aspect is the extension of FS, for instance, intuitionistic FS [11], interval-valued intuitionistic FS [12], pythagorean FS [13], q-rung orthopair FS [14], and neutrosophic set [15] . Since then, a multitude of affluent research achievements were acquired on the basis of these extended FSs [16–22]. However, the aforementioned FSs have a common limitation which they only have a membership degree or no-membership degree to express preference information of decision makers (DMs). It cannot deal with several situations which DMs cannot ascertain a satisfied membership degree from a collection of membership degrees. Accordingly, for coping with this circumstance, Torra and Narukawa [23, 24] put forward the notion of hesitant FS (HFS), which enables the membership degree of an element belonging to a fixed objective to have multiple feasible values between 0 and 1. Subsequently, a lot of plentiful research results including theory and methodology on HFS have been achieved. These achievements are summarized as the following aspects: (1) the foundation theory aspect: operations and algebraic structures [25, 26], information entropy measure [27, 28] distance measure, and similarly measure [29–31]; (2) the MADM approaches on HFS: Zhang and Wei [32] proposed the extension VIKOR approach to resolve the DM problem with hesitant fuzzy information. Zhang and Xu [33] propounded TODIM approach on the basis of a new measure function to deal with DM issues. Chen and Xu [34] extended the traditional ELECTRE II approach to handle hesitant fuzzy MADM problems; (3) the aggregation operator aspect: Xia and Xu [35] presented several aggregation operators for aggregating hesitant fuzzy information. He et al. [36] combined the power operator and Bonferroni mean (BM) operator to fuse hesitant fuzzy information, which can take into account the importance and inter-relationship of the aggregated data. Hong et al. [37] propounded the hesitant fuzzy dual Muirhead mean operators whose characteristic is that the correlation of attributes is taken into consideration.Influenced by the complicacy and vagueness of information, DMs cannot provide their preference information in the form of a precise number in actual world. Especially, the common evaluation langue terms are “very good,” “good,” “bad,” and “very bad.” These linguistic terms are in line with DMs’ assessment thinking for providing evaluation information to a particular project. Accordingly, the linguistic term set (LTS) is more convenient to express assessment information of alternatives than numerical number. Inspired by HFS, Rodriguez et al. [38] developed the hesitant fuzzy linguistic term set by combining the HFS and the LTS where DMs provide their evaluation information for a linguistic variable through some orderly and continuous linguistic terms. Although HFLTS can express assessment information qualitative, it cannot take into account the situation that the membership degree of a linguistic term belongs to a selected objective. To surmount this limitation, Lin et al. [39] introduced the notion of hesitant fuzzy linguistic set (HFLS), which allows several possible membership degrees to belong to a selected linguistic term, and the foundation element of HFLS is named HFLE. At the same time, several aggregation operators including the HFL geometric operator, HFL correlated operator, induced HFL operator, HFL prioritized operator, and HFL power operator are investigated. From the definition of HFLS, we can know that a HFLE includes two aspect information; on the one hand, the nature langue is used to express evaluation information through the linguistic term, and on the other hand, the hesitancy of DMs to a linguistic term is taken into consideration. Thereafter, Wang et al. [40] proposed the Hausdorff distance of HFL numbers and established novel HFL-TOPSIS and TODIM approach. Gou et al. [41] presented the HFL Bonferroni mean (HFLBM) operator and weighted HFLBM (WHFLBM) operator. It is obvious that the HFLS has stronger capacity to portray no-indeterminacy and vague information than HFS or LTS.Aggregation function [42, 43] is one of the hot spots in the domain of the information process system, such as information fusion, pattern recognition, and decision support system. It is widely utilized to reduce the dimensions of assessment information during information aggregation procedure, such as triangular norms [44, 45], copulas [46] and arithmetic mean. It is of great importance in the process for collecting related assessment information from multiple evaluators in MADM issues. The main thought of the aggregation operator is to aggregate the evaluation information of alternatives under different attributes to get the total comprehensive evaluation information of alternatives. Traditional aggregation operators like weighted average, weighted geometric, and ordered weighted average (OWA) operator have been widely applied in diverse kinds of FSs, such as IFS [47], HFS [48], and NS [49]. Furthermore, Li [50] and Li et al. [51] propounded the generalized OWA operator under IFS context and established DM methodology based upon these operators. However, these operators suppose that the attributes are independent, which can lead to information loss and produce inconsistent phenomena in actual problems. Hence, scholars have concerned several special aggregation functions which can take into account the inter-relationship among diverse input arguments. The BM operator is a common operator to fuse vague information between any two variables. Wei et al. [52] presented the generalized interval neutrosophic number BM operator to construct DM approach for evaluating technological innovation capability of companies. Liu and Zhang [53] proposed several intuitionistic uncertain linguistic BM operators and utilized them to solve group decision problems. As the complexity of the assessment information and DM setting increases, the inter-relationship of all input arguments should be taken into consideration in actual applications. In light of this situation, Hara et al. [54] initially developed the Hamy mean operator, which can capture the correlation of multiarguments and have an alterable parameter that enables the decision procedure more flexible. Recently, the HM operator has been investigated to fuse fuzzy information under various fuzzy settings, such as Interval type-2 context [55], 2-tuple linguistic neutrosophic [56], linguistic intuitionistic fuzzy number [57], and q-rung orthopair fuzzy environment [58]. Up to now, to the best of our knowledge, no study has been investigated on the HM operator under HFLS setting.Based on the aforementioned analysis and literature review, the motivations of this article are displayed as follows: (1) the HFLS is much better than HFS and LTS to express fuzzy and indeterminate information by synthesizing the advantages of HFS and LTS; (2) the HM operator has a strength to can take into account the inter-relationships of fused arguments, and it also can enable the DM model more reasonable for addressing MADM issues; and (3) the adjustable parameter of the HM operator can make the DM procedure more flexible, and the evaluator can adjust preference attitude according to it.In view of the aforementioned comprehensive analysis, the main research objective of this article is to extend the HM operator to HFLS context and to propound several novel operators to solve MADM issues with HFL information. The contributions and goals of this study are as follows:(1)
To extend the HM operator to HFL setting(2)
To propound the HFLHM operator, WHFLHM operator, HFLDHM operator, and WHFLDHM operator(3)
To explore the desirable properties and several particular cases of the presented operators(4)
To propose a novel MADM approach on the basis of the proposed operators(5)
To elaborate the validity and merits of the proposed approach by numerical example and comparison analysisFor the sake of the above goals, the organizational structure of this study is displayed as follows. In Section2, we retrospect several fundamental notions including the HFS, LTS, HFLS, and HM operator. Section 3 puts forward the HFLHM and WHFLHM operator and investigates the desirable properties of them in detail. Section 4 puts forward the HFLDHM operator and WHFLDHM operator and discusses related theorems. In Section 5, a novel MADM approach is developed to tackle MADM problems based on the WHFLHM operator and WHFLDHM operator. In Section 6, an illustration example is provided to verify the validity of the proposed approach in this article, and a comparative analysis is given to manifest the advantages. The conclusion remarks are drawn in the end.


## 2. Preliminaries

In this part, we will retrospect several related fundamental definitions including HFS, LTS, HFLS, HM operator, and DHM operator.


### 2.1. HFS

Definition 1 (see [24]).
Supposing thatX is a fixed set, the mathematical expression of a HFS on X is depicted as follows:(1)U=x,hUxx∈X,in which hUx is a collection of a slice of values in 0,1 which indicate the possible membership degrees. For any x, h=hUx is called the hesitant fuzzy element.Definition 2 (see [35]).
Assuming that ς be an HFE, thenℓς=1/♯ς∑γ∈ςγ is called the score function of ς, where ♯ς denotes the number of all elements in ς. For any two HFEs ς1 and ς2, if ℓς1>ℓς2, then ς1>ς2; if ℓς1<ℓς2, then ς1<ς2.


### 2.2. LTS

Definition 3 (see [59]).
LetS=sαα=0,1,…,t be a LTS with odd cardinality, and sα is indicative of a possible value for a linguistic variable and meets the following characteristics:(1)
Ifα>β, then sα>sβ(2)
Existing negative operators:Negsα=sβ, where α+β=t(3)
Maximum operator: ifsα≥sβ, then Maxsα,sβ=sα(4)
Minimum operator: ifsα≤sβ, then Minsα,sβ=sα


### 2.3. HFLS

Lin et al. [39] introduced the notation of HFLSs by combining the HFSs and LTSs to express fuzzy and indeterminate information more effectively.Definition 4.
Assume thatX=x1,x2,…,xn be a referenced set. A HFLS in the mathematical form is described as(2)A=x,sθx,hAxx∈X,where hAx is a collection of several possible values in 0,1 which indicate the possible membership degree that x∈X belongs to the linguistic term set sθx, and θx∈0,t. For simplicity, a˜=sθx,hAx is represented as a HFLE.Example 1.
Assume thatX=x1,x2,x3 be a regular set. A HFLS on X is expressed as A=x1,s4,0.3,x2,s5,0.5,0.8,x1,s7,0.2,0.6,0.9. HFLS A can be divided into three subsets which only have one goal, i.e., A has three HFLEs, s4,0.3, s5,0.5,0.8, and s7,0.2,0.6,0.9. Take s7,0.2,0.6,0.9 as an example; s7 denotes a linguistic term which can qualitatively evaluate the goal, and 0.2,0.6,0.9 indicates some possible membership degrees that x3 belongs to s7. Specially, s4,0.3 denotes that the degree to which x1 belongs to s4 which is 0.3; it is a particular case of HFLE, namely, fuzzy linguistic number.
As a tool that can combine the characteristic of quantitative and qualitative to express vague and uncertain information validly. HFLS synthesizes the merits of LTS and HFS, which makes the views expressed by decision makers belong to the corresponding linguistic term. LTS is more in line with the evaluation thinking of individuals, and HFE can allow multiple decision makers to give their own possible value to an alternative. Considering the complexity of evaluation information and human cognitive thinking, it is formidable for DMs to express the viewpoint with a precise numerical value. However, the HFLE can express evaluation information by a linguistic term. For instance, a linguistic terms5 “good” can be utilized to evaluate performance or quality of a car. Four decision makers give their evaluation value based on the linguistic term s5 “good” {0.2, 0.3, 0.4, 0.6}, and then the evaluation information by the form of HFLE can be indicated as s5,0.2,0.3,0.4,0.6. Accordingly, HFLS is a powerful and valid tool to cope with decision-making problems.Definition 5 (see [39]).
Leta˜=sθa˜,ha˜, a˜1=sθa˜1,hAa˜1, and a˜2=sθa˜2,hAa˜2 be three HFLEs, and κ be a real number, then(3)a˜1⊕a˜2=sθa˜1+sθa˜2,∪γa˜1∈ha˜1,γa˜2∈ha˜2γa˜1+γa˜2−γa˜1γa˜2;a˜1⊗a˜2=sθa˜1×sθa˜2,∪γa˜1∈ha1,γa˜2∈ha˜2γa˜1γa˜2;κa˜=sκsθa˜,∪γa˜∈ha˜1−1−γa˜κ;a˜κ=sθa˜κ,∪γa˜∈ha˜γa˜κ.
In order to compare HFLEs, Lin et al. [39] gave the ranking method as follows.Definition 6 (see [39]).
Supposing thata˜=sθx,hAx be a HFLE, and Ja˜=1/♯h∑γ∈hγsθx is called the score function of a˜, where ♯h is denoted by the number of all elements in h. For any two HFLEs, a˜1=sθa˜1,hAa˜1 and a˜2=sθa˜2,hAa˜2; if Ja˜1>Ja˜2, then a˜1>a˜2, and if Ja˜1=Ja˜2, then a˜1=a˜2.


### 2.4. Hamy Mean Operator

Definition 7 (see [54]).
LetA=a˜ii=1,2,…,m be a group of positive real numbers. The definition of the HMξ operator is depicted as(4)HMξa˜1,a˜2,…,a˜m=∑1≤i1<⋯<iξ≤m∏j=1ξa˜ij1/ξCmξ,in which ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the collection 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.
The HM operator has the following properties:(1)
HMξ0,0,…,0=0,ifa˜i=0i=1,2,…,m(2)
HMξa˜,a˜,…,a˜=a,ifa˜i=a˜i=1,2,…,m(3)
HMξa˜1,a˜2,…,a˜m≤HMξb˜1,b˜2,…,b˜m,ifa˜m≤b˜mi=1,2,…,m;(4)
mina˜i≤HMξa˜1,a˜2,…,a˜m≤maxa˜i
There are two particular cases of the HM operator; numbered lists can be added as follows:(1)
Whenξ=1, HMξa˜1,a˜2,…,a˜m=1/m∑i=1ma˜i, and it degenerates to the arithmetic mean operator [60](2)
Whenξ=m, HMξa˜1,a˜2,…,a˜m=∑i=1ma˜i1/m, and it degenerates to the geometric mean operator [61]Definition 8 (see [56]).
LetA=a˜ii=1,2,…,m be a set of positive real numbers. DHMx is called the dual Hamy mean operator, which is depicted as follows:(5)DHMξa˜1,a˜2,…,a˜m=∏1≤i1<⋯<iξ≤m∑j=1ξaijξ1/Cmξ,in which ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.


## 2.1. HFS

Definition 1 (see [24]).
Supposing thatX is a fixed set, the mathematical expression of a HFS on X is depicted as follows:(1)U=x,hUxx∈X,in which hUx is a collection of a slice of values in 0,1 which indicate the possible membership degrees. For any x, h=hUx is called the hesitant fuzzy element.Definition 2 (see [35]).
Assuming that ς be an HFE, thenℓς=1/♯ς∑γ∈ςγ is called the score function of ς, where ♯ς denotes the number of all elements in ς. For any two HFEs ς1 and ς2, if ℓς1>ℓς2, then ς1>ς2; if ℓς1<ℓς2, then ς1<ς2.


## 2.2. LTS

Definition 3 (see [59]).
LetS=sαα=0,1,…,t be a LTS with odd cardinality, and sα is indicative of a possible value for a linguistic variable and meets the following characteristics:(1)
Ifα>β, then sα>sβ(2)
Existing negative operators:Negsα=sβ, where α+β=t(3)
Maximum operator: ifsα≥sβ, then Maxsα,sβ=sα(4)
Minimum operator: ifsα≤sβ, then Minsα,sβ=sα


## 2.3. HFLS

Lin et al. [39] introduced the notation of HFLSs by combining the HFSs and LTSs to express fuzzy and indeterminate information more effectively.Definition 4.
Assume thatX=x1,x2,…,xn be a referenced set. A HFLS in the mathematical form is described as(2)A=x,sθx,hAxx∈X,where hAx is a collection of several possible values in 0,1 which indicate the possible membership degree that x∈X belongs to the linguistic term set sθx, and θx∈0,t. For simplicity, a˜=sθx,hAx is represented as a HFLE.Example 1.
Assume thatX=x1,x2,x3 be a regular set. A HFLS on X is expressed as A=x1,s4,0.3,x2,s5,0.5,0.8,x1,s7,0.2,0.6,0.9. HFLS A can be divided into three subsets which only have one goal, i.e., A has three HFLEs, s4,0.3, s5,0.5,0.8, and s7,0.2,0.6,0.9. Take s7,0.2,0.6,0.9 as an example; s7 denotes a linguistic term which can qualitatively evaluate the goal, and 0.2,0.6,0.9 indicates some possible membership degrees that x3 belongs to s7. Specially, s4,0.3 denotes that the degree to which x1 belongs to s4 which is 0.3; it is a particular case of HFLE, namely, fuzzy linguistic number.
As a tool that can combine the characteristic of quantitative and qualitative to express vague and uncertain information validly. HFLS synthesizes the merits of LTS and HFS, which makes the views expressed by decision makers belong to the corresponding linguistic term. LTS is more in line with the evaluation thinking of individuals, and HFE can allow multiple decision makers to give their own possible value to an alternative. Considering the complexity of evaluation information and human cognitive thinking, it is formidable for DMs to express the viewpoint with a precise numerical value. However, the HFLE can express evaluation information by a linguistic term. For instance, a linguistic terms5 “good” can be utilized to evaluate performance or quality of a car. Four decision makers give their evaluation value based on the linguistic term s5 “good” {0.2, 0.3, 0.4, 0.6}, and then the evaluation information by the form of HFLE can be indicated as s5,0.2,0.3,0.4,0.6. Accordingly, HFLS is a powerful and valid tool to cope with decision-making problems.Definition 5 (see [39]).
Leta˜=sθa˜,ha˜, a˜1=sθa˜1,hAa˜1, and a˜2=sθa˜2,hAa˜2 be three HFLEs, and κ be a real number, then(3)a˜1⊕a˜2=sθa˜1+sθa˜2,∪γa˜1∈ha˜1,γa˜2∈ha˜2γa˜1+γa˜2−γa˜1γa˜2;a˜1⊗a˜2=sθa˜1×sθa˜2,∪γa˜1∈ha1,γa˜2∈ha˜2γa˜1γa˜2;κa˜=sκsθa˜,∪γa˜∈ha˜1−1−γa˜κ;a˜κ=sθa˜κ,∪γa˜∈ha˜γa˜κ.
In order to compare HFLEs, Lin et al. [39] gave the ranking method as follows.Definition 6 (see [39]).
Supposing thata˜=sθx,hAx be a HFLE, and Ja˜=1/♯h∑γ∈hγsθx is called the score function of a˜, where ♯h is denoted by the number of all elements in h. For any two HFLEs, a˜1=sθa˜1,hAa˜1 and a˜2=sθa˜2,hAa˜2; if Ja˜1>Ja˜2, then a˜1>a˜2, and if Ja˜1=Ja˜2, then a˜1=a˜2.


## 2.4. Hamy Mean Operator

Definition 7 (see [54]).
LetA=a˜ii=1,2,…,m be a group of positive real numbers. The definition of the HMξ operator is depicted as(4)HMξa˜1,a˜2,…,a˜m=∑1≤i1<⋯<iξ≤m∏j=1ξa˜ij1/ξCmξ,in which ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the collection 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.
The HM operator has the following properties:(1)
HMξ0,0,…,0=0,ifa˜i=0i=1,2,…,m(2)
HMξa˜,a˜,…,a˜=a,ifa˜i=a˜i=1,2,…,m(3)
HMξa˜1,a˜2,…,a˜m≤HMξb˜1,b˜2,…,b˜m,ifa˜m≤b˜mi=1,2,…,m;(4)
mina˜i≤HMξa˜1,a˜2,…,a˜m≤maxa˜i
There are two particular cases of the HM operator; numbered lists can be added as follows:(1)
Whenξ=1, HMξa˜1,a˜2,…,a˜m=1/m∑i=1ma˜i, and it degenerates to the arithmetic mean operator [60](2)
Whenξ=m, HMξa˜1,a˜2,…,a˜m=∑i=1ma˜i1/m, and it degenerates to the geometric mean operator [61]Definition 8 (see [56]).
LetA=a˜ii=1,2,…,m be a set of positive real numbers. DHMx is called the dual Hamy mean operator, which is depicted as follows:(5)DHMξa˜1,a˜2,…,a˜m=∏1≤i1<⋯<iξ≤m∑j=1ξaijξ1/Cmξ,in which ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.


## 3. Hesitant Fuzzy Linguistic Hamy Mean Aggregation Operators

In this part, we will extend the HM operator and the DHM operator to HFL setting to present the HFLHM operator and the HFLDHM operator for aggregating HFLEs, and we further discuss particular cases and relevant properties of the proposed operators.


### 3.1. HFLHM Operator

Definition 9.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, then the HFLHM operator is shown as(6)HFLHMξa˜1,a˜2,…,a˜m=1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ,where ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.
According to the operational laws of HFLE, we can obtain the following theorems of the HFLHM operator.Theorem 1.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs; then, the fused value obtained by the HFLHM operator is also a HFLE and it can be expressed as(7)HFLHMξa˜1,a˜2,…,a˜m=1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ,s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ.Proof.
(1)
Firstly, we prove equation (7) holds, and according to the operational laws of HFLEs, we have(8)⊗j=1ξa˜ij=s∏j=1ξθa˜ij,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij.Therefore,(9)⊗j=1ξa˜ij1/ξ=s∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij1/ξ.Furthermore,(10)⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij=s∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ.Then,(11)1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ=s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ.Hence,(12)HFLHMξa˜1,a˜2,…,a˜m=s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ.(2)
Secondly, we prove equation (7) is a HFLE. It is known that a˜ij∈0,1, and we can acquire(13)∏j=1ξγa˜ij∈0,1⟹∏j=1ξγa˜ij1/ξ∈0,1⟹1−∏j=1ξγa˜ij1/ξ∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ⟹1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ∈0,1.Example 2.
Leta^1=s3,0.2,0.3, a^2=s4,0.4,0.5, and a^3=s6,0.6,0.7 be three HFLEs, then the aggregation value of them can be obtained by the HFLHM operator, which is shown as follows (suppose ξ=2):(14)HFLHMξa˜1,a˜2,a˜3=s1/Ckξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ=s4.2019,0.3793,0.4044,0.4131,0.4409,0.4232,0.4496,0.4576,0.4863.
Hereinbelow, we will talk about several desired theorems of the HFLHM operator.Theorem 2 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs; if a˜i=a˜=sθa˜,hAa˜i=1,2,…,m, then(15)HFLHMξa˜1,a˜2,…,a˜m=a˜.Proof.
Sincea˜=sθa˜,hAa˜, on the basis of Theorem 1, we have(16)HFLHMξa˜1,a˜2,…,a˜m=s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ=s1/Cmξ∑1≤i1<⋯<iξ≤mθa˜ij1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−1−γa˜ξ1/ξ1/Cmξ=s1/CmξCmξθa˜,∪γa˜∈ha˜1−1−γa˜Cmξ1/Cmξ=sθa˜,∪γa˜∈ha˜1−1−γa˜=sθa˜,∪γa˜∈ha˜γa˜=sθa˜,ha˜=a˜.Theorem 3 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and ℬ=b˜i=sθb˜i,hℬb˜ii=1,2,…,m be two set of HFLEs; if θa˜i≥θb˜i and hAa˜i≥hℬb˜i for each ii=1,2,…,m, then(17)HFLHMξa˜1,a˜2,…,a˜m≥HFLHMξb˜1,b˜2,…,b˜m.Proof.
Sinceθa˜i≥θb˜i, we have(18)∏j=1ξθa˜ij≥∏j=1ξθb˜ij,∏j=1ξθa˜ij1/ξ≥∏j=1ξθb˜ij1/ξ.
Then,(19)∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ≥∑1≤i1<⋯<iξ≤m∏j=1ξθb˜ij1/ξ.
Accordingly,(20)1Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ≥1Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθb˜ij1/ξ,that is, sθa˜i≥sθb˜i.
Sinceha˜i≥hb˜i and γa˜ij≥γb˜ij, we have(21)∏j=1ξγa˜ij≥∏j=1ξγb˜ij,1−∏j=1ξγa˜ij1/ξ≤1−∏j=1ξγb˜ij1/ξ.
Then,(22)∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ≤∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ.
Therefore,(23)1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ≥1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ1/Cmξ.
Besides, letJa˜=1/♯h∑γ∈ha˜γJθa˜ and Jb˜=1/♯h∑γ∈hb˜γJθb˜ be the score function of a˜ and b˜, respectively. Based on the above analysis, we have Ja˜≥Jb˜, that is, HFLHMξa˜1,a˜2,…,a˜m≥HFLHMξb˜1,b˜2,…,b˜m.Theorem 4 (boundedness).
Assuming thatA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLE. Assume that a˜−=mina˜i,a˜2,…,a˜m and a˜+=maxa˜i,a˜2,…,a˜m, then(24)a˜−≤HFLHMξa˜1,a˜2,…,a˜m≤a˜+.Proof.
According to Theorem3, we have(25)HFLHMξa˜−,a˜−,…,a˜−≤HFLHMξa˜1,a˜2,…,a˜m≤HFLHMξa˜+,a˜+,…,a˜+.
According to Theorem2, we have(26)HFLHMξa˜−,a˜−,…,a˜−=a˜−,HFLHMξa˜+,a˜+,…,a˜+=a˜+.
Hence, we obtain(27)a˜−≤HFLHMξa˜1,a˜2,…,a˜m≤a˜+.Theorem5 (commutativity).
Assumea˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθb˜i,hb˜ii=1,2,…,m be a collection of HFLEs. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(28)HFLHMξa˜1,a˜2,…,a˜m=HFLHMξb˜1,b˜2,…,b˜m.Proof.
Sinceb˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(29)1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ=1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξb˜ij1/ξ.
Accordingly,HFLHMξa˜1,a˜2,…,a˜m=HFLHMξb˜1,b˜2,…,b˜m.
In what follows, we shall explore some particular cases of the HFLHM operator obtained by assigning different values of parameterξ.Case 1.
Whenξ=1, the HFLHM operator shall degenerate to the hesitant fuzzy linguistic arithmetic average operator.(30)HFLHM1a˜1,a˜2,…,a˜m=1m∑i=1ma˜i=s1/m∑i=1mθa˜i,∪γa˜i∈ha˜ii=1,2,…,m1−∏i=1m1−γa˜ij1/m.Case 2.
Whenξ=m, the HFLHM operator shall degenerate to the hesitant fuzzy linguistic geometric average operator.(31)HFLHMma˜1,a˜2,…,a˜m=∏i=1ma˜i1/m=s∏i=1mθa˜i1/m,∪γa˜i∈ha˜ii=1,2,…,m∏i=1mγa˜ij1/m.


### 3.2. Hesitant Fuzzy Linguistic Weighted Hamy Mean Operator

The attribute weights play an essential role in actual decision-making issues, and they shall affect the ultimate decision results. Accordingly, it is indispensable to take into consideration the weights of attribute during the information fusion course. So, we will present the WHFLHM operator which is as follows.Definition 10.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. Then, the WHFLHM operator is described as follows:(32)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ,1≤ξ≤m,⊗i=1ξa˜i1−wi/m−1,ξ=m,where ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.Theorem 6.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and the associated weight vector wi∈0,1 and ∑i=1mwi=1. Then, the aggregation result obtained by Definition 10 is also a HFLE, and it can be expressed as(33)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ1≤ξ<m,orWHFLHMwξ=a˜1,a˜2,…,a˜m=⊗j=1ξa˜i1−wi/m−1=s∏j=1ξθa˜i1−wi/m−1,∪γa˜i∈ha˜ii=1,2,…,m∏i=1ξγa˜i1−wi/m−1ξ=m.Proof.
(1)
Firstly, we prove equation (33) holds. For the first situation, when 1≤ξ≤m, according to the operational laws of HFLEs, we have(34)⊗j=1ξa˜ij=s∏j=1ξθa˜ij,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij.
Therefore,(35)⊗j=1ξa˜ij1/ξ=s∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij1/ξ.
Furthermore,(36)1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij.
Moreover,(37)⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ.
Accordingly,(38)1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ.
For the second situation, whenξ=m, we have
(39)
a
˜
i
1
−
w
i
/
m
−
1
=
s
θ
a
˜
i
1
−
w
i
/
m
−
1
,
∪
γ
a
˜
i
∈
h
a
˜
i
i
=
1,2
,
…
,
m
γ
a
˜
i
1
−
w
i
/
m
−
1
.
Hence,(40)⊗i=1ξa˜i1−wi/m−1=s∏i=1ξθa˜i1−wi/m−1,∪γa˜i∈ha˜ii=1,2,…,m∏i=1ξγa˜i1−wi/m−1.
(2)
Secondly, we prove equation (33) is a HFLE. It is known that γa˜ij∈0,1; when 1≤ξ≤m, we can acquire(41)∏j=1ξγa˜ij∈0,1⟹∏j=1ξγa˜ij1/ξ∈0,1⟹1−∏j=1ξγa˜ij1/ξ∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ∈0,1⟹1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ∈0,1.
For the other situation, whenξ=m, it is easy to prove, so we omit the proof process here.
In the following, we will investigate some desirable properties of the WHFLHM operator.Theorem 7 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If a˜i=a˜=sθa˜,hAa˜i=1,2,…,m, then(42)WHFLHMwξa˜1,a˜2,…,a˜m=a˜.Proof.
Sincea˜=sθa˜,hAa˜, based on Theorem 6, we have
(1) For the first situation, when1≤ξ≤m,(43)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜1/ξ1−∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−γa˜ξ1/ξ1−∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−γa˜1−∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−1−γa˜Cmξ−∏1≤i1<⋯<iξ≤m∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξCmξ−∑i=1mCm−1ξ−1wiθa˜,∪γa˜∈ha˜1−1−γa˜Cmξ−∑i=1mCm−1ξ−1wi1/Cm−1ξ=s1/Cm−1ξCmξ−Cm−1ξ−1∑i=1mwiθa˜,∪γa˜∈ha˜1−1−γa˜Cmξ−Cm−1ξ−1∑i=1mwi1/Cm−1ξ.
Due to∑i=1mwi=1, we can obtain(44)WHFLHMwξ=a˜1,a˜2,…,a˜m=s1/Cm−1ξCmξ−Cm−1ξ−1θa˜,∪γa˜∈ha˜1−1−γa˜Cmξ−Cm−1ξ−11/Cm−1ξ=sξ!m−1−ξ!/m−1!m−1!/ξ!m−1−ξ!θa˜,∪γa˜∈ha˜1−1−γa˜m−1!/ξ!m−1−ξ!ξ!m−1−ξ!/m−1!=sθa˜,ha˜=a˜.
(2) For the second situation, whenξ=m, we have(45)WHFLHMwξa˜1,a˜2,…,a˜m=s∏j=1ξθa˜i1−wi/m−1,∪γa˜∈ha˜∏j=1ξγa˜i1−wi/m−1=sθa˜im−∑j=1mwi/m−1,∂∪γa˜∈ha˜γa˜m−∑j=1mwi/m−1.
Due to∑i=1mwi=1, we can obtain(46)WHFLHMwξ=a˜1,a˜2,…,a˜m=sθa˜im−1/m−1,∪γa˜∈ha˜γa˜m−1/m−1=sθa˜i,∪γa˜∈ha˜γa˜=a˜.Theorem 8 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and ℬ=b˜i=sθb˜i,hℬa˜ii=1,2,…,m be two sets of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If θa˜i≥θb˜i and hAa˜i≥hℬb˜i for each i, then(47)WHFLHMwξa˜1,a˜2,…,a˜m≥WHFLHMwξb˜1,b˜2,…,b˜m.Proof.
Sinceξ≥1,θa˜ij≥θb˜ij≥0, we have(48)∏j=1ξθa˜ij≥∏j=1ξθb˜ij,∏j=1ξθa˜ij1/ξ≥∏j=1ξθb˜ij1/ξ.
Then,(49)⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ≥⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθb˜ij1/ξ.
Accordingly,(50)1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ≥1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθb˜ij1/ξ.
That is,sθa˜i≥sθb˜i.
Sinceha˜ij≥hb˜ij and γa˜ij≥γb˜ij, we have(51)∏j=1ξγa˜ij≥∏j=1ξγb˜ij,1−∏j=1ξγa˜ij1/ξ≤1−∏j=1ξγb˜ij1/ξ.
Then,(52)∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij≤∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ1−∑j=1ξwij.
Therefore,(53)1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ≥1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ.
Besides, letsa˜=1/♯h∑γ∈ha˜γsθa˜ and sb˜=1/♯h∑γ∈ha˜γsθb˜ be the score function of a˜ and b˜, respectively. Based on the above analysis, we have sa˜≥sb˜. That is, WHFLHMξa˜1,a˜2,…,a˜m≥WHFLHMξb˜1,b˜2,…,b˜m.Theorem 9 (boundedness).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and weight vector wi∈0,1,∑i=1mwi=1. Assume that a˜−=mina˜i,a˜2,…,a˜m and a˜+=maxa˜i,a˜2,…,a˜m, then(54)a˜−≤WHFLHMwξa˜1,a˜2,…,a˜m≤a˜+.Proof.
According to Theorem8, we have(55)WHFLHMwξa˜−,a˜−,…,a˜−≤WHFLHMξa˜1,a˜2,…,a˜m≤HFLHMξa˜+,a˜+,…,a˜+.
According to Theorem7, we have(56)WHFLHMwξa˜−,a˜−,…,a˜−=a˜−,WHFLHMξa˜+,a˜+,…,a˜+=a˜+.
Hence, we obtain(57)a˜−≤WHFLHMwξa˜1,a˜2,…,a˜m≤a˜+.Theorem 10 (commutativity).
Leta˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθa˜i,hb˜ii=1,2,…,m be a collection of HFLEs, and wi∈0,1,∑i=1mwi=1. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(58)WHFLHMwξa˜1,a˜2,…,a˜m=WHFLHMwξb˜1,b˜2,…,b˜m.Proof.
Sinceb˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(59)1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξb˜ij1/ξ1≤ξ≤m,⊗i=1ξa˜i1−wi/m−1=⊗i=1ξb˜i1−wi/m−1ξ=m.
Accordingly,WHFLHMwξa˜1,a˜2,…,a˜m=WHFLHMwξb˜1,b˜2,…,b˜m.
In the following, we shall discuss several particular cases of the HFLHM operator obtained by assigning different values of parameterξ.Case 3.
Whenξ=1, then the WHFLHMwξ operator will degenerate as(60)WHFLHMw1=a˜1,a˜2,…,a˜m=1Cm−11⊕1≤i1<⋯<iξ≤m1−∑j=11wij⊗j=11a˜ij1/1=s1/m−1∑1≤i1≤m1−wiθai˜,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1≤m1−γai˜1−wi1/m−1.Case 4.
Whenξ=m, then the WHFLHMwξ operator will degenerate as(61)WHFLHMwma˜1,a˜2,…,a˜m=⊗i=1ξa˜i1−wi/m−1=s∏i=1ξθa˜i1−wi/m−1,∪γa˜i∈ha˜ii=1,2,…,m∏i=1ξγa˜i1−wi/m−1.Example 3.
Leta^1=s2,0.3,0.4, a^2=s5,0.6,0.7, and a^3=s6,0.8,0.9 be three HFLEs, and the weighting vector be w=0.2,0.3,0.5T, then the aggregation value of them can be obtained by the WHFLHM operator, which is shown as follows (suppose ξ=2):(62)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ=s1.2386,0.5103,0.5330,0.5436,0.5692,0.5608,0.5839,0.5945,0.6199.


## 3.1. HFLHM Operator

Definition 9.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, then the HFLHM operator is shown as(6)HFLHMξa˜1,a˜2,…,a˜m=1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ,where ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.
According to the operational laws of HFLE, we can obtain the following theorems of the HFLHM operator.Theorem 1.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs; then, the fused value obtained by the HFLHM operator is also a HFLE and it can be expressed as(7)HFLHMξa˜1,a˜2,…,a˜m=1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ,s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ.Proof.
(1)
Firstly, we prove equation (7) holds, and according to the operational laws of HFLEs, we have(8)⊗j=1ξa˜ij=s∏j=1ξθa˜ij,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij.Therefore,(9)⊗j=1ξa˜ij1/ξ=s∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij1/ξ.Furthermore,(10)⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij=s∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ.Then,(11)1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ=s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ.Hence,(12)HFLHMξa˜1,a˜2,…,a˜m=s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ.(2)
Secondly, we prove equation (7) is a HFLE. It is known that a˜ij∈0,1, and we can acquire(13)∏j=1ξγa˜ij∈0,1⟹∏j=1ξγa˜ij1/ξ∈0,1⟹1−∏j=1ξγa˜ij1/ξ∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ⟹1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ∈0,1.Example 2.
Leta^1=s3,0.2,0.3, a^2=s4,0.4,0.5, and a^3=s6,0.6,0.7 be three HFLEs, then the aggregation value of them can be obtained by the HFLHM operator, which is shown as follows (suppose ξ=2):(14)HFLHMξa˜1,a˜2,a˜3=s1/Ckξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ=s4.2019,0.3793,0.4044,0.4131,0.4409,0.4232,0.4496,0.4576,0.4863.
Hereinbelow, we will talk about several desired theorems of the HFLHM operator.Theorem 2 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs; if a˜i=a˜=sθa˜,hAa˜i=1,2,…,m, then(15)HFLHMξa˜1,a˜2,…,a˜m=a˜.Proof.
Sincea˜=sθa˜,hAa˜, on the basis of Theorem 1, we have(16)HFLHMξa˜1,a˜2,…,a˜m=s1/Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ=s1/Cmξ∑1≤i1<⋯<iξ≤mθa˜ij1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−1−γa˜ξ1/ξ1/Cmξ=s1/CmξCmξθa˜,∪γa˜∈ha˜1−1−γa˜Cmξ1/Cmξ=sθa˜,∪γa˜∈ha˜1−1−γa˜=sθa˜,∪γa˜∈ha˜γa˜=sθa˜,ha˜=a˜.Theorem 3 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and ℬ=b˜i=sθb˜i,hℬb˜ii=1,2,…,m be two set of HFLEs; if θa˜i≥θb˜i and hAa˜i≥hℬb˜i for each ii=1,2,…,m, then(17)HFLHMξa˜1,a˜2,…,a˜m≥HFLHMξb˜1,b˜2,…,b˜m.Proof.
Sinceθa˜i≥θb˜i, we have(18)∏j=1ξθa˜ij≥∏j=1ξθb˜ij,∏j=1ξθa˜ij1/ξ≥∏j=1ξθb˜ij1/ξ.
Then,(19)∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ≥∑1≤i1<⋯<iξ≤m∏j=1ξθb˜ij1/ξ.
Accordingly,(20)1Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθa˜ij1/ξ≥1Cmξ∑1≤i1<⋯<iξ≤m∏j=1ξθb˜ij1/ξ,that is, sθa˜i≥sθb˜i.
Sinceha˜i≥hb˜i and γa˜ij≥γb˜ij, we have(21)∏j=1ξγa˜ij≥∏j=1ξγb˜ij,1−∏j=1ξγa˜ij1/ξ≤1−∏j=1ξγb˜ij1/ξ.
Then,(22)∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ≤∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ.
Therefore,(23)1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1/Cmξ≥1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ1/Cmξ.
Besides, letJa˜=1/♯h∑γ∈ha˜γJθa˜ and Jb˜=1/♯h∑γ∈hb˜γJθb˜ be the score function of a˜ and b˜, respectively. Based on the above analysis, we have Ja˜≥Jb˜, that is, HFLHMξa˜1,a˜2,…,a˜m≥HFLHMξb˜1,b˜2,…,b˜m.Theorem 4 (boundedness).
Assuming thatA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLE. Assume that a˜−=mina˜i,a˜2,…,a˜m and a˜+=maxa˜i,a˜2,…,a˜m, then(24)a˜−≤HFLHMξa˜1,a˜2,…,a˜m≤a˜+.Proof.
According to Theorem3, we have(25)HFLHMξa˜−,a˜−,…,a˜−≤HFLHMξa˜1,a˜2,…,a˜m≤HFLHMξa˜+,a˜+,…,a˜+.
According to Theorem2, we have(26)HFLHMξa˜−,a˜−,…,a˜−=a˜−,HFLHMξa˜+,a˜+,…,a˜+=a˜+.
Hence, we obtain(27)a˜−≤HFLHMξa˜1,a˜2,…,a˜m≤a˜+.Theorem5 (commutativity).
Assumea˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθb˜i,hb˜ii=1,2,…,m be a collection of HFLEs. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(28)HFLHMξa˜1,a˜2,…,a˜m=HFLHMξb˜1,b˜2,…,b˜m.Proof.
Sinceb˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(29)1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξa˜ij1/ξ=1Cmξ⊕1≤i1<⋯<iξ≤m⊗j=1ξb˜ij1/ξ.
Accordingly,HFLHMξa˜1,a˜2,…,a˜m=HFLHMξb˜1,b˜2,…,b˜m.
In what follows, we shall explore some particular cases of the HFLHM operator obtained by assigning different values of parameterξ.Case 1.
Whenξ=1, the HFLHM operator shall degenerate to the hesitant fuzzy linguistic arithmetic average operator.(30)HFLHM1a˜1,a˜2,…,a˜m=1m∑i=1ma˜i=s1/m∑i=1mθa˜i,∪γa˜i∈ha˜ii=1,2,…,m1−∏i=1m1−γa˜ij1/m.Case 2.
Whenξ=m, the HFLHM operator shall degenerate to the hesitant fuzzy linguistic geometric average operator.(31)HFLHMma˜1,a˜2,…,a˜m=∏i=1ma˜i1/m=s∏i=1mθa˜i1/m,∪γa˜i∈ha˜ii=1,2,…,m∏i=1mγa˜ij1/m.


## 3.2. Hesitant Fuzzy Linguistic Weighted Hamy Mean Operator

The attribute weights play an essential role in actual decision-making issues, and they shall affect the ultimate decision results. Accordingly, it is indispensable to take into consideration the weights of attribute during the information fusion course. So, we will present the WHFLHM operator which is as follows.Definition 10.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. Then, the WHFLHM operator is described as follows:(32)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ,1≤ξ≤m,⊗i=1ξa˜i1−wi/m−1,ξ=m,where ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.Theorem 6.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and the associated weight vector wi∈0,1 and ∑i=1mwi=1. Then, the aggregation result obtained by Definition 10 is also a HFLE, and it can be expressed as(33)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ1≤ξ<m,orWHFLHMwξ=a˜1,a˜2,…,a˜m=⊗j=1ξa˜i1−wi/m−1=s∏j=1ξθa˜i1−wi/m−1,∪γa˜i∈ha˜ii=1,2,…,m∏i=1ξγa˜i1−wi/m−1ξ=m.Proof.
(1)
Firstly, we prove equation (33) holds. For the first situation, when 1≤ξ≤m, according to the operational laws of HFLEs, we have(34)⊗j=1ξa˜ij=s∏j=1ξθa˜ij,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij.
Therefore,(35)⊗j=1ξa˜ij1/ξ=s∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m∏j=1ξγa˜ij1/ξ.
Furthermore,(36)1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij.
Moreover,(37)⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ.
Accordingly,(38)1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ.
For the second situation, whenξ=m, we have
(39)
a
˜
i
1
−
w
i
/
m
−
1
=
s
θ
a
˜
i
1
−
w
i
/
m
−
1
,
∪
γ
a
˜
i
∈
h
a
˜
i
i
=
1,2
,
…
,
m
γ
a
˜
i
1
−
w
i
/
m
−
1
.
Hence,(40)⊗i=1ξa˜i1−wi/m−1=s∏i=1ξθa˜i1−wi/m−1,∪γa˜i∈ha˜ii=1,2,…,m∏i=1ξγa˜i1−wi/m−1.
(2)
Secondly, we prove equation (33) is a HFLE. It is known that γa˜ij∈0,1; when 1≤ξ≤m, we can acquire(41)∏j=1ξγa˜ij∈0,1⟹∏j=1ξγa˜ij1/ξ∈0,1⟹1−∏j=1ξγa˜ij1/ξ∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij∈0,1⟹∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ∈0,1⟹1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ∈0,1.
For the other situation, whenξ=m, it is easy to prove, so we omit the proof process here.
In the following, we will investigate some desirable properties of the WHFLHM operator.Theorem 7 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If a˜i=a˜=sθa˜,hAa˜i=1,2,…,m, then(42)WHFLHMwξa˜1,a˜2,…,a˜m=a˜.Proof.
Sincea˜=sθa˜,hAa˜, based on Theorem 6, we have
(1) For the first situation, when1≤ξ≤m,(43)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜1/ξ1−∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−γa˜ξ1/ξ1−∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−∏1≤i1<⋯<iξ≤m1−γa˜1−∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜1/ξ,∪γa˜∈ha˜1−1−γa˜Cmξ−∏1≤i1<⋯<iξ≤m∑j=1ξwij1/Cm−1ξ=s1/Cm−1ξCmξ−∑i=1mCm−1ξ−1wiθa˜,∪γa˜∈ha˜1−1−γa˜Cmξ−∑i=1mCm−1ξ−1wi1/Cm−1ξ=s1/Cm−1ξCmξ−Cm−1ξ−1∑i=1mwiθa˜,∪γa˜∈ha˜1−1−γa˜Cmξ−Cm−1ξ−1∑i=1mwi1/Cm−1ξ.
Due to∑i=1mwi=1, we can obtain(44)WHFLHMwξ=a˜1,a˜2,…,a˜m=s1/Cm−1ξCmξ−Cm−1ξ−1θa˜,∪γa˜∈ha˜1−1−γa˜Cmξ−Cm−1ξ−11/Cm−1ξ=sξ!m−1−ξ!/m−1!m−1!/ξ!m−1−ξ!θa˜,∪γa˜∈ha˜1−1−γa˜m−1!/ξ!m−1−ξ!ξ!m−1−ξ!/m−1!=sθa˜,ha˜=a˜.
(2) For the second situation, whenξ=m, we have(45)WHFLHMwξa˜1,a˜2,…,a˜m=s∏j=1ξθa˜i1−wi/m−1,∪γa˜∈ha˜∏j=1ξγa˜i1−wi/m−1=sθa˜im−∑j=1mwi/m−1,∂∪γa˜∈ha˜γa˜m−∑j=1mwi/m−1.
Due to∑i=1mwi=1, we can obtain(46)WHFLHMwξ=a˜1,a˜2,…,a˜m=sθa˜im−1/m−1,∪γa˜∈ha˜γa˜m−1/m−1=sθa˜i,∪γa˜∈ha˜γa˜=a˜.Theorem 8 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and ℬ=b˜i=sθb˜i,hℬa˜ii=1,2,…,m be two sets of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If θa˜i≥θb˜i and hAa˜i≥hℬb˜i for each i, then(47)WHFLHMwξa˜1,a˜2,…,a˜m≥WHFLHMwξb˜1,b˜2,…,b˜m.Proof.
Sinceξ≥1,θa˜ij≥θb˜ij≥0, we have(48)∏j=1ξθa˜ij≥∏j=1ξθb˜ij,∏j=1ξθa˜ij1/ξ≥∏j=1ξθb˜ij1/ξ.
Then,(49)⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ≥⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθb˜ij1/ξ.
Accordingly,(50)1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ≥1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθb˜ij1/ξ.
That is,sθa˜i≥sθb˜i.
Sinceha˜ij≥hb˜ij and γa˜ij≥γb˜ij, we have(51)∏j=1ξγa˜ij≥∏j=1ξγb˜ij,1−∏j=1ξγa˜ij1/ξ≤1−∏j=1ξγb˜ij1/ξ.
Then,(52)∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij≤∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ1−∑j=1ξwij.
Therefore,(53)1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ≥1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγb˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ.
Besides, letsa˜=1/♯h∑γ∈ha˜γsθa˜ and sb˜=1/♯h∑γ∈ha˜γsθb˜ be the score function of a˜ and b˜, respectively. Based on the above analysis, we have sa˜≥sb˜. That is, WHFLHMξa˜1,a˜2,…,a˜m≥WHFLHMξb˜1,b˜2,…,b˜m.Theorem 9 (boundedness).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and weight vector wi∈0,1,∑i=1mwi=1. Assume that a˜−=mina˜i,a˜2,…,a˜m and a˜+=maxa˜i,a˜2,…,a˜m, then(54)a˜−≤WHFLHMwξa˜1,a˜2,…,a˜m≤a˜+.Proof.
According to Theorem8, we have(55)WHFLHMwξa˜−,a˜−,…,a˜−≤WHFLHMξa˜1,a˜2,…,a˜m≤HFLHMξa˜+,a˜+,…,a˜+.
According to Theorem7, we have(56)WHFLHMwξa˜−,a˜−,…,a˜−=a˜−,WHFLHMξa˜+,a˜+,…,a˜+=a˜+.
Hence, we obtain(57)a˜−≤WHFLHMwξa˜1,a˜2,…,a˜m≤a˜+.Theorem 10 (commutativity).
Leta˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθa˜i,hb˜ii=1,2,…,m be a collection of HFLEs, and wi∈0,1,∑i=1mwi=1. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(58)WHFLHMwξa˜1,a˜2,…,a˜m=WHFLHMwξb˜1,b˜2,…,b˜m.Proof.
Sinceb˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(59)1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξb˜ij1/ξ1≤ξ≤m,⊗i=1ξa˜i1−wi/m−1=⊗i=1ξb˜i1−wi/m−1ξ=m.
Accordingly,WHFLHMwξa˜1,a˜2,…,a˜m=WHFLHMwξb˜1,b˜2,…,b˜m.
In the following, we shall discuss several particular cases of the HFLHM operator obtained by assigning different values of parameterξ.Case 3.
Whenξ=1, then the WHFLHMwξ operator will degenerate as(60)WHFLHMw1=a˜1,a˜2,…,a˜m=1Cm−11⊕1≤i1<⋯<iξ≤m1−∑j=11wij⊗j=11a˜ij1/1=s1/m−1∑1≤i1≤m1−wiθai˜,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1≤m1−γai˜1−wi1/m−1.Case 4.
Whenξ=m, then the WHFLHMwξ operator will degenerate as(61)WHFLHMwma˜1,a˜2,…,a˜m=⊗i=1ξa˜i1−wi/m−1=s∏i=1ξθa˜i1−wi/m−1,∪γa˜i∈ha˜ii=1,2,…,m∏i=1ξγa˜i1−wi/m−1.Example 3.
Leta^1=s2,0.3,0.4, a^2=s5,0.6,0.7, and a^3=s6,0.8,0.9 be three HFLEs, and the weighting vector be w=0.2,0.3,0.5T, then the aggregation value of them can be obtained by the WHFLHM operator, which is shown as follows (suppose ξ=2):(62)WHFLHMwξ=a˜1,a˜2,…,a˜m=1Cm−1ξ⊕1≤i1<⋯<iξ≤m1−∑j=1ξwij⊗j=1ξa˜ij1/ξ=s1/Cm−1ξ∑1≤i1<⋯<iξ≤m1−∑j=1ξwij∏j=1ξθa˜ij1/ξ,∪γa˜i∈ha˜ii=1,2,…,m1−∏1≤i1<⋯<iξ≤m1−∏j=1ξγa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ=s1.2386,0.5103,0.5330,0.5436,0.5692,0.5608,0.5839,0.5945,0.6199.


## 4. Hesitant Fuzzy Linguistic Dual Hamy Mean Operators

In the theory research of aggregation operators, there are two kinds of operator including the original operator and its dual operator, for instance, the arithmetic average operator and the geometric operator. So, we will investigate the DHM operator under HFLS setting.


### 4.1. HFLDHM Operator

Definition 11 (see [56]).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a set of HFLEs, then the HFLDHM operator is depicted as(63)HFLDHMξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m⊕j=1ξa˜ijξ1/Cmξ,where ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.Theorem 11.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, then the fused value obtained by the HFLHM operator is also a HFLE, and it can be expressed as(64)HFLDHMξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m⊕j=1ξa˜ijξ1/Cmξ=s∏1≤i1<⋯<iξ≤m∑j=1ξa˜ij/ξ1/Cmξ,∪γa˜i∈ha˜ii=1,2,…,m∏1≤i1<⋯<iξ≤m1−∏j=1ξ1−a˜ij1/ξ1/Cmξ.The proof of Theorem11 is analogue to Theorem 1, so we omit it here.Similar to the HFLHM operator, the following properties of the HFLDHM operator are investigated.Theorem 12 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs; if a˜=a˜=sθa˜i,hAa˜ii=1,2,…,m, then(65)HFLDHMξa˜1,a˜2,…,a˜m=a˜.Theorem 13 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and A=a˜i=sθa˜i,hAb˜ii=1,2,…,m be two sets of HFLEs; if θa˜i≥θb˜i and hAa˜i≥hℬb˜i for each ii=1,2,…,m, then(66)HFLDHMξa˜1,a˜2,…,a˜m≥HFLDHMξb˜1,b˜2,…,b˜m.Theorem 14 (boundedness).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs. Assume that a˜−=mina˜i,a˜2,…,a˜m and a˜+=maxa˜i,a˜2,…,a˜m, then(67)a˜−≤HFLDHMξa˜1,a˜2,…,a˜m≤a˜+.Theorem 15 (commutativity).
Leta˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθa˜i,hb˜ii=1,2,…,m be two collections of HFLEs. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(68)HFLDHMξa˜1,a˜2,…,a˜m=HFLDHMξb˜1,b˜2,…,b˜m.


### 4.2. Hesitant Fuzzy Linguistic Weighted Dual Hamy Mean Operator

Definition 12.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wnT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1nwi=1. Then, the WHFLDHM operator is described as follows:(69)WHFLDHMwξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m1−∑j=1ξwij⊕j=1ξa˜ijξ1/Cm−1ξ,1≤ξ<m,⊕i=1ξa˜i1−wi/m−1,ξ=m,whereξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.Theorem 16.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and the associated weight vector wi∈0,1 and ∑i=1nwi=1. Then, the aggregation result obtained by Definition 12 is also a HFLE, and it can be expressed as(70)WHFLDHMwξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m1−∑j=1ξwij⊕j=1ξa˜ijξ1/Cm−1ξ=s∏1≤i1<⋯<iξ≤m1−∑j=1ξwij∑j=1ξa˜ij/ξ1/Cm−1ξ,∪γa˜i∈ha˜ii=1,2,…,m∏1≤i1<⋯<iξ≤m1−∏j=1ξ1−γa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ,or(71)WHFLDHMwξa˜1,a˜2,…,a˜m=⊕i=1ξa˜i1−wi/m−1=s∑i=1mθa˜i1−wi/m−1,∪​γa˜i∈ha˜ii=1,2,…,m1−∏i=1m1−γa˜i1−wi/m−1ξ=m.The proof of Theorem16 is analogue to Theorem 6, so we omit it here.Similar to the HFLHM operator, the following properties of the WHFLDHM operator are investigated.Theorem 17 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If a˜i=a˜=sθa˜,hAa˜i=1,2,…,m, then(72)WHFLDHMwξa˜1,a˜2,…,a˜m=a˜.Theorem 18 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and ℬ=a˜i=sθa˜i,hAb˜ii=1,2,…,m be two sets of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If θa˜i≥θb˜i and ha˜i≥hb˜i for each ii=1,2,…,m, then(73)WHFLDHMwξa˜1,a˜2,…,a˜m≥WHFLDHMwξb˜1,b˜2,…,b˜m.Theorem 19 (boundedness).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and weight vector wi∈0,1,∑i=1mwi=1. Assume that a˜−=mina˜i,a˜2,…,a˜m and then,(74)a˜−≤WHFLDHMwξa˜1,a˜2,…,a˜m≤a˜+.Theorem 20 (commutativity).
Leta˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθb˜i,hb˜ii=1,2,…,m be two collections of HFLEs. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(75)WHFLDHMwξa˜1,a˜2,…,a˜m=WHFLDHMwξb˜1,b˜2,…,b˜m.


## 4.1. HFLDHM Operator

Definition 11 (see [56]).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a set of HFLEs, then the HFLDHM operator is depicted as(63)HFLDHMξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m⊕j=1ξa˜ijξ1/Cmξ,where ξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.Theorem 11.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, then the fused value obtained by the HFLHM operator is also a HFLE, and it can be expressed as(64)HFLDHMξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m⊕j=1ξa˜ijξ1/Cmξ=s∏1≤i1<⋯<iξ≤m∑j=1ξa˜ij/ξ1/Cmξ,∪γa˜i∈ha˜ii=1,2,…,m∏1≤i1<⋯<iξ≤m1−∏j=1ξ1−a˜ij1/ξ1/Cmξ.The proof of Theorem11 is analogue to Theorem 1, so we omit it here.Similar to the HFLHM operator, the following properties of the HFLDHM operator are investigated.Theorem 12 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs; if a˜=a˜=sθa˜i,hAa˜ii=1,2,…,m, then(65)HFLDHMξa˜1,a˜2,…,a˜m=a˜.Theorem 13 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and A=a˜i=sθa˜i,hAb˜ii=1,2,…,m be two sets of HFLEs; if θa˜i≥θb˜i and hAa˜i≥hℬb˜i for each ii=1,2,…,m, then(66)HFLDHMξa˜1,a˜2,…,a˜m≥HFLDHMξb˜1,b˜2,…,b˜m.Theorem 14 (boundedness).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs. Assume that a˜−=mina˜i,a˜2,…,a˜m and a˜+=maxa˜i,a˜2,…,a˜m, then(67)a˜−≤HFLDHMξa˜1,a˜2,…,a˜m≤a˜+.Theorem 15 (commutativity).
Leta˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθa˜i,hb˜ii=1,2,…,m be two collections of HFLEs. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(68)HFLDHMξa˜1,a˜2,…,a˜m=HFLDHMξb˜1,b˜2,…,b˜m.


## 4.2. Hesitant Fuzzy Linguistic Weighted Dual Hamy Mean Operator

Definition 12.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wnT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1nwi=1. Then, the WHFLDHM operator is described as follows:(69)WHFLDHMwξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m1−∑j=1ξwij⊕j=1ξa˜ijξ1/Cm−1ξ,1≤ξ<m,⊕i=1ξa˜i1−wi/m−1,ξ=m,whereξ is a parameter, ξ=1,2,…,m, i1,i2,…,iξ are ξ integer values taken from the set 1,2,…,m of m integer values, Cmξ is the binomial coefficient, and Cmξ=m!/ξ!m−ξ!.Theorem 16.
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and the associated weight vector wi∈0,1 and ∑i=1nwi=1. Then, the aggregation result obtained by Definition 12 is also a HFLE, and it can be expressed as(70)WHFLDHMwξa˜1,a˜2,…,a˜m=⊗1≤i1<⋯<iξ≤m1−∑j=1ξwij⊕j=1ξa˜ijξ1/Cm−1ξ=s∏1≤i1<⋯<iξ≤m1−∑j=1ξwij∑j=1ξa˜ij/ξ1/Cm−1ξ,∪γa˜i∈ha˜ii=1,2,…,m∏1≤i1<⋯<iξ≤m1−∏j=1ξ1−γa˜ij1/ξ1−∑j=1ξwij1/Cm−1ξ,or(71)WHFLDHMwξa˜1,a˜2,…,a˜m=⊕i=1ξa˜i1−wi/m−1=s∑i=1mθa˜i1−wi/m−1,∪​γa˜i∈ha˜ii=1,2,…,m1−∏i=1m1−γa˜i1−wi/m−1ξ=m.The proof of Theorem16 is analogue to Theorem 6, so we omit it here.Similar to the HFLHM operator, the following properties of the WHFLDHM operator are investigated.Theorem 17 (idempotency).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If a˜i=a˜=sθa˜,hAa˜i=1,2,…,m, then(72)WHFLDHMwξa˜1,a˜2,…,a˜m=a˜.Theorem 18 (monotonicity).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m and ℬ=a˜i=sθa˜i,hAb˜ii=1,2,…,m be two sets of HFLEs, and w=w1,w2,…,wmT is the associated weight vector of a˜ii=1,2,…,m such that wi∈0,1 and ∑i=1mwi=1. If θa˜i≥θb˜i and ha˜i≥hb˜i for each ii=1,2,…,m, then(73)WHFLDHMwξa˜1,a˜2,…,a˜m≥WHFLDHMwξb˜1,b˜2,…,b˜m.Theorem 19 (boundedness).
LetA=a˜i=sθa˜i,hAa˜ii=1,2,…,m be a collection of HFLEs, and weight vector wi∈0,1,∑i=1mwi=1. Assume that a˜−=mina˜i,a˜2,…,a˜m and then,(74)a˜−≤WHFLDHMwξa˜1,a˜2,…,a˜m≤a˜+.Theorem 20 (commutativity).
Leta˜i=sθa˜i,ha˜ii=1,2,…,m and b˜i=sθb˜i,hb˜ii=1,2,…,m be two collections of HFLEs. Assume that b˜1,b˜2,…,b˜m is any permutation of a˜1,a˜2,…,a˜m, then(75)WHFLDHMwξa˜1,a˜2,…,a˜m=WHFLDHMwξb˜1,b˜2,…,b˜m.


## 5. A New Approach to MADM Based on the WHFLHM Operator and the WHFLDHM Operator

In this part, we shall present a novel MADM approach based on the WHFLHM operator and the WHFLDHM operator to solve the decision problem with HFL information.A classical MADM problem for selecting a best alternative with HFL information is stated as follows. LetX=xkk=1,2,…,p be a group of alternatives. A=all=1,2,…,q is a collection of attributes with the weight vector w=w1,w2,…,wqT, wl∈0,1l=1,2,…,q, and ∑l=1qwl=1. The HFL decision matrix is represented as H˜=a˜klp×q, in which a˜kl=sθakl,ha˜kl is a HFLE, which denotes the assessment value of xkk=1,2,…,p on all=1,2,…,q. In the following, we will establish a novel algorithm (see algorithm 1) to carry out the HFLMADM problem on the basis of the presented aggregation operators.Algorithm 1: For HFLMADM with HFL information.
Input: the HFL decision matrix H˜=a˜klp×q and the attribute weight vector w=w1,w2,…,wqT.
Begin
Step 1: normalize the HFLDM H˜=a˜klp×q to H=aklp×q by the following method.
akl=sθakl,hakl=a˜kl=sθakl,ha˜kl,alis benefit−type,a˜klc=sθakl,1−ha˜kl,alis cost−type.
Step 2: compute the comprehensive assessment value ak for alternative xk by
ak=WHFLHMwxak1,ak2,…,akq
or
ak=WHFLDHMwxak1,ak2,…,akq.
to aggregate the HFL assessment information.
Step 3: compute the score of the comprehensive assessment value ak by Definition 6.
Step 4: ranking the alternatives and obtaining the optimal one(s).
End.


## 6. Illustration Example

In this part, we provide an actual application example which is cited from [40] to prove the validity and applicability of the new approach.A state-owned enterprise engaged in the sale and production of nonferrous metals has set up an overseas affairs investment department composed of investment experts to invest in countries with great potential worldwide to provide sales channels for nonferrous metals in order to expand its business scope. Recently, the investment sector will choose a country with the greatest potential for industrial investment. Through investigation and research, the expert group takes into account five countries as alternative targets, i.e.,x1,x2,x3,x4,x5. Because there are a host of influence factors in the investment problem, after the analysis and discussion of the expert group, the four most influential factors are identified, including c1: resource reserve (for example, mineral reserve and exploration); c2: political policy (for example, policy support); c3: economic (for example, capital reserve strength and economic stability); and c4: infrastructure (for example, transportation and machinery facilities). The homologous attribute weight vector is w=0.2,0.4,0.3,0.1T.Considering the above factors, the expert group gave the evaluation information of each alternative. The LTSS=s0,s1,s2,s3,s4,s5,s6 = {very poor, slightly poor, fair, slightly good, good, very good} is utilized to evaluate alternatives. The assessment information is provided by experts with HFLE, in which ajxi represents the assessment value of xi on the attribute cj. Considering the assessment value ajxi, experts reach a consensus based on the selected linguistic term and give their opinion by a certainty value in 0,1 based on their knowledge and cognition, which represents the membership degree of xi to the selected linguistic term under the attribute aj. After that, all the possible values based on each selected linguistic term are collected together, and the same MD in a linguistic term only emerge once. Finally, experts reach consensus for the assessment information of all alternatives under considered attributes after discussion and investigation, and the ultimate assessment information matrix expressed by HFLE is determined as shown in Table 1.Table 1
The hesitant fuzzy linguistic decision matrix.
Alternative
a
1
a
2
a
3
a
4
x
1
s
4
,
0.3
,
0.4
s
5
,
0.2
,
0.4
s
4
,
0.5
,
0.7
s
3
,
0.4
x
2
s
2
,
0.5
,
0.6
s
4
,
0.3
,
0.5
s
3
,
0.7
s
4
,
0.4
,
0.6
x
3
s
3
,
0.4
,
0.6
s
4
,
0.4
s
6
,
0.7
,
0.8
s
2
,
0.6
x
4
s
2
,
0.3
,
0.4
,
0.5
s
3
,
0.6
s
2
,
0.4
,
0.7
s
2
,
0.8
x
5
s
5
,
0.5
,
0.7
s
5
,
0.5
,
0.6
s
3
,
0.6
,
0.8
s
4
,
0.7


### 6.1. Decision-Making Steps

Step 1.
Standardize the HFLDMH˜=a˜ijm×n.
Since all considered attributes are benefit-type, we need not to standardize the attribute value.Step 2.
Compute comprehensive assessment valueai for alternative xi by the equation in Algorithm 1 and suppose ξ=2.(76)a˜1=s3.5986,0.3497,0.3885,0.3938,0.4340,0.3777,0.4184,0.4225,0.4642;a˜2=s2.8846,0.4704,0.5332,0.5128,0.5758,0.4972,0.5612,0.5397,0.6032,;a˜3=s3.2061,0.5263,0.5488,0.5827,0.6062;a˜4=s2.0165,0.5192,0.4695,0.5508,0.6273,0.5806,0.6555;a˜5=s3.8830,0.5815,0.6297,0.6021,0.6499,0.6365,0.6851,0.6566,0.7042.Step 3.
Calculate the score ofai by Definition 6.(77)Jx1=1.4614,Jx2=1.5481,Jx3=1.8146,Jx4=1.1437,Jx5=2.4975.Step 4.
Rank the alternatives in the descending order, and obtain the best one.(78)x5≻x3≻x2≻x1≻x4.
Accordingly, the best alternative isx5.
In Step 2, if we utilize the equation in Algorithm1 to fuse the attribute value, then we obtain(79)a˜1=s3.6843,0.3562,0.4048,0.3974,0.4478,0.3833,0.4327,0.4236,0.4741;a˜2=s2.4039,0.4810,0.5427,0.5216,0.5797,0.5096,0.5698,0.5496,0.6053;a˜3=s3.0992,0.5360,0.5648,0.5873,0.6153;a˜4=s1.1635,0.5534,0.6275,0.5773,0.6472,0.5773,0.6472;a˜5=s4.1494,0.5877,0.6070,0.6402,0.6590,0.6406,0.6585,0.6899,0.7068.Step 5.
Calculate the score of the comprehensive assessment valueai by Definition 6.(80)Jx1=1.5289,Jx2=1.3099,Jx3=1.7846,Jx4=0.7039,Jx5=2.6918.Step 6.
Rank the alternatives in the descending order, and obtain the best one.(81)x5≻x3≻x1≻x2≻x4.
Accordingly, the best alternative isx5.


### 6.2. Sensitivity Analysis

The parameter plays a vital role in the process of information integration for ultimate ranking results. Because the WHFLHM operator and the WHFLDHM operator have a changeable parameter, different parameter values reflect different risk preference attitudes of decision makers in solving MADM problems. So, it is indispensable to discuss the influence of different parameters for ranking result. In the next, we take diverse parameter value ofξ in the WHFLHM operator and the WHFLDHM operator to obtain the corresponding ranking results, which are displayed in Tables 2 and 3.Table 2
Score value and order relation of alternatives by the WHFLHM operator.
Parameter
Score values ofxii=1,2,3,4,5
Ranking order
ξ
=
1
J
x
1
=
1.6598
,
J
x
2
=
1.7974
,
J
x
3
=
2.0881
,
J
x
4
=
1.3914
,
J
x
5
=
2.7724
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
ξ
=
2
J
x
1
=
1.4614
,
J
x
2
=
1.5481
,
J
x
3
=
1.8146
,
J
x
4
=
1.1437
,
J
x
5
=
2.4975
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
ξ
=
3
J
x
1
=
1.5354
,
J
x
2
=
1.6544
,
J
x
3
=
1.8577
,
J
x
4
=
1.2438
,
J
x
5
=
2.6605
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
ξ
=
4
J
x
1
=
1.5157
,
J
x
2
=
1.6327
,
J
x
3
=
1.8283
,
J
x
4
=
1.2339
,
J
x
5
=
2.6433
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4Table 3
Score value and order relation of alternatives by the WHFLDHM operator.
Parameter
Score values ofxii=1,2,3,4,5
Ranking order
ξ
=
1
J
x
1
=
1.1308
,
J
x
2
=
1.1281
,
J
x
3
=
1.5198
,
J
x
4
=
1.0257
,
J
x
5
=
1.8264
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
ξ
=
2
J
x
1
=
1.5289
,
J
x
2
=
1.3099
,
J
x
3
=
1.7846
,
J
x
4
=
0.7039
,
J
x
5
=
2.6918
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
ξ
=
3
J
x
1
=
0.2547
,
J
x
2
=
0.1456
,
J
x
3
=
0.2673
,
J
x
4
=
0.0383
,
J
x
5
=
0.5068
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
ξ
=
4
J
x
1
=
2.9632
,
J
x
2
=
2.4049
,
J
x
3
=
3.1950
,
J
x
4
=
2.3032
,
J
x
5
=
3.7484
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4From Tables2 and 3, the following conclusions for parameter ξ on the decision-making results are summarized:(1)
We can obtain the different score values and decision results by assigning different parameter values ofξ. Although the final ordering results of alternatives are slightly different, the best alternative is x5. This is because the parameter is taken as different values, and the different correlation of attributes is taken into account. For example, if ξ=2, the correlation of two attributes is considered; if ξ=3, the inter-relationship of three attributes is considered, which shows that the HM operator has excellent robustness in the issues of information fusion.(2)
The HM operator can reduce to the arithmetic average operator and geometric average operator by taking different parameter valueξ. In practice situation, decision makers can analyse the generalization results by some special cases.(3)
When parameters are taken different values, we can get different score values and decision results; it means that the parameterξ has a good control capability, and different parameters can be regarded as risk preferences of evaluators. Evaluators can choose favourable parameter ξ according to their preferences attitude.


### 6.3. Comparison Analysis

In order to verify the validity and the remarkable merits of the propounded method, we will address the same demonstrative example by utilizing other approaches including the HFL-TOPSIS method [40], HFL weight average (HFLWA) operator [39], HFL weight geometric (HFLWG) operator, and HFLBM operator. The ranking results and the characteristic comparative with other approaches are manifested in Tables 4 and 5, respectively.Table 4
Decision results by utilizing different approaches.
Approaches
Parameter value
Order relation
HFLWA operator [39]
NO
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
HFLWG operator [39]
NO
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
HFLWPA operator [39]
NO
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
HFL-TOPSIS [40]
NO
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
HFLWBM operator [62]
p
=
q
=
1
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
WHFLHM operator
ξ
=
2
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
WHFLDHM operator
ξ
=
2
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4Table 5
Comparison of characteristics with different MADM approaches.
Approaches
Whether to capture the inter-relationship between two attributes
Whether to capture the inter-relationship between multiple attributes
Whether has generalized characteristics by the parameter
HFLWA operator [39]
No
No
No
HFLWG operator [39]
No
No
No
HFL-TOPSIS [40]
No
No
No
HFLWBM operator [62]
Yes
No
Yes
HFLWHM operator
Yes
Yes
Yes
HFLWDHM operator
Yes
Yes
YesIn the next, we will perform a detailed comparative analysis on the basis of the above approaches with our proposed approach. According to our further analysis and the character comparison in Table5, we obtain the following summarization:(1)
Compared with the approach based on the HFLWA operator proposed by Lin et al. [39], although the HFLWA operator can be utilized to aggregate HFL information, it usually assumes that attributes are independent of each other, and it lacks flexibility and robustness in the course information fusion. In comparison, the proposed approach based on the WHFLHM operator can not only capture the correlation of overall attributes but also can enable the decision process dynamic through a flexible parameter. Moreover, the WHFLHM operator is more universal than the HFLWA operator because it is a particular case of the WHFLHM operator. Accordingly, our developed approach is more appropriate to cope with practical MADM problems.(2)
Compared with HFL-TOPSIS approach in [40], the TOPSIS approach only provides a ranking result of alternatives, and the distance computation of ideal points will lead to information loss phenomenon happening. However, the presented approach on the basis of the WHFLHM operator can directly obtain the comprehensive evaluation value as well as the sorting of alternatives, and the inter-relationship among any number of arbitrary attributes is taken into consideration, i.e., it can reduce information loss to some extent in information integration procedure. At the same time, the adjustable parameter ξ can be assigned different values through evaluator’s risk preference. In summary, our developed method is more general and flexible than the HFL-TOPSIS method.(3)
Compared with the HFLWBM operator in [62], the HFLWBM operator can only capture the inter-relation between any two input arguments. However, in the practical application issues, we often need to take into account all arguments provided by the evaluator. The developed HFLWHM and HFLWDHM operator has an excellent capability to model the correlation among multi-input parameters, and it can be applied more flexibly during the process of information fusion.As analysed and compared the abovementioned summarization points, the superiorities of the propounded approach are summarized as follows:Firstly, the HFLS can express the assessment information more comprehensive because it combines the excellent feature of the linguistic term and the hesitant fuzzy set. The HFLS can cope with practical problems from both quantitative and qualitative perspectives.Secondly, the comparative analysis show that the novel approach based on the WHFLHM operator thinks over the inter-relationship among multi-input arguments, which can further reduce the loss of evaluation information and enable the decision process more flexible and valid.Thirdly, the HFLWHM operator can be applied in different situations by assigning preference values with different parametersξ, which can be seen as different risk preference by decision makers based on processing practical problems.Because of the complicacy of the decision environment, decision makers tend to estimate the alternatives according to their preference in linguistic terms and may hesitate to determine the degree of membership. At the same time, in order to conduct a more reasonable decision in actual problems, it is indispensable for decision makers to concern the inter-relation of the multiple input argument. Accordingly, the propounded approach can be regarded as an effective technique for addressing practical issues.


## 6.1. Decision-Making Steps

Step 1.
Standardize the HFLDMH˜=a˜ijm×n.
Since all considered attributes are benefit-type, we need not to standardize the attribute value.Step 2.
Compute comprehensive assessment valueai for alternative xi by the equation in Algorithm 1 and suppose ξ=2.(76)a˜1=s3.5986,0.3497,0.3885,0.3938,0.4340,0.3777,0.4184,0.4225,0.4642;a˜2=s2.8846,0.4704,0.5332,0.5128,0.5758,0.4972,0.5612,0.5397,0.6032,;a˜3=s3.2061,0.5263,0.5488,0.5827,0.6062;a˜4=s2.0165,0.5192,0.4695,0.5508,0.6273,0.5806,0.6555;a˜5=s3.8830,0.5815,0.6297,0.6021,0.6499,0.6365,0.6851,0.6566,0.7042.Step 3.
Calculate the score ofai by Definition 6.(77)Jx1=1.4614,Jx2=1.5481,Jx3=1.8146,Jx4=1.1437,Jx5=2.4975.Step 4.
Rank the alternatives in the descending order, and obtain the best one.(78)x5≻x3≻x2≻x1≻x4.
Accordingly, the best alternative isx5.
In Step 2, if we utilize the equation in Algorithm1 to fuse the attribute value, then we obtain(79)a˜1=s3.6843,0.3562,0.4048,0.3974,0.4478,0.3833,0.4327,0.4236,0.4741;a˜2=s2.4039,0.4810,0.5427,0.5216,0.5797,0.5096,0.5698,0.5496,0.6053;a˜3=s3.0992,0.5360,0.5648,0.5873,0.6153;a˜4=s1.1635,0.5534,0.6275,0.5773,0.6472,0.5773,0.6472;a˜5=s4.1494,0.5877,0.6070,0.6402,0.6590,0.6406,0.6585,0.6899,0.7068.Step 5.
Calculate the score of the comprehensive assessment valueai by Definition 6.(80)Jx1=1.5289,Jx2=1.3099,Jx3=1.7846,Jx4=0.7039,Jx5=2.6918.Step 6.
Rank the alternatives in the descending order, and obtain the best one.(81)x5≻x3≻x1≻x2≻x4.
Accordingly, the best alternative isx5.


## 6.2. Sensitivity Analysis

The parameter plays a vital role in the process of information integration for ultimate ranking results. Because the WHFLHM operator and the WHFLDHM operator have a changeable parameter, different parameter values reflect different risk preference attitudes of decision makers in solving MADM problems. So, it is indispensable to discuss the influence of different parameters for ranking result. In the next, we take diverse parameter value ofξ in the WHFLHM operator and the WHFLDHM operator to obtain the corresponding ranking results, which are displayed in Tables 2 and 3.Table 2
Score value and order relation of alternatives by the WHFLHM operator.
Parameter
Score values ofxii=1,2,3,4,5
Ranking order
ξ
=
1
J
x
1
=
1.6598
,
J
x
2
=
1.7974
,
J
x
3
=
2.0881
,
J
x
4
=
1.3914
,
J
x
5
=
2.7724
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
ξ
=
2
J
x
1
=
1.4614
,
J
x
2
=
1.5481
,
J
x
3
=
1.8146
,
J
x
4
=
1.1437
,
J
x
5
=
2.4975
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
ξ
=
3
J
x
1
=
1.5354
,
J
x
2
=
1.6544
,
J
x
3
=
1.8577
,
J
x
4
=
1.2438
,
J
x
5
=
2.6605
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
ξ
=
4
J
x
1
=
1.5157
,
J
x
2
=
1.6327
,
J
x
3
=
1.8283
,
J
x
4
=
1.2339
,
J
x
5
=
2.6433
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4Table 3
Score value and order relation of alternatives by the WHFLDHM operator.
Parameter
Score values ofxii=1,2,3,4,5
Ranking order
ξ
=
1
J
x
1
=
1.1308
,
J
x
2
=
1.1281
,
J
x
3
=
1.5198
,
J
x
4
=
1.0257
,
J
x
5
=
1.8264
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
ξ
=
2
J
x
1
=
1.5289
,
J
x
2
=
1.3099
,
J
x
3
=
1.7846
,
J
x
4
=
0.7039
,
J
x
5
=
2.6918
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
ξ
=
3
J
x
1
=
0.2547
,
J
x
2
=
0.1456
,
J
x
3
=
0.2673
,
J
x
4
=
0.0383
,
J
x
5
=
0.5068
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
ξ
=
4
J
x
1
=
2.9632
,
J
x
2
=
2.4049
,
J
x
3
=
3.1950
,
J
x
4
=
2.3032
,
J
x
5
=
3.7484
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4From Tables2 and 3, the following conclusions for parameter ξ on the decision-making results are summarized:(1)
We can obtain the different score values and decision results by assigning different parameter values ofξ. Although the final ordering results of alternatives are slightly different, the best alternative is x5. This is because the parameter is taken as different values, and the different correlation of attributes is taken into account. For example, if ξ=2, the correlation of two attributes is considered; if ξ=3, the inter-relationship of three attributes is considered, which shows that the HM operator has excellent robustness in the issues of information fusion.(2)
The HM operator can reduce to the arithmetic average operator and geometric average operator by taking different parameter valueξ. In practice situation, decision makers can analyse the generalization results by some special cases.(3)
When parameters are taken different values, we can get different score values and decision results; it means that the parameterξ has a good control capability, and different parameters can be regarded as risk preferences of evaluators. Evaluators can choose favourable parameter ξ according to their preferences attitude.


## 6.3. Comparison Analysis

In order to verify the validity and the remarkable merits of the propounded method, we will address the same demonstrative example by utilizing other approaches including the HFL-TOPSIS method [40], HFL weight average (HFLWA) operator [39], HFL weight geometric (HFLWG) operator, and HFLBM operator. The ranking results and the characteristic comparative with other approaches are manifested in Tables 4 and 5, respectively.Table 4
Decision results by utilizing different approaches.
Approaches
Parameter value
Order relation
HFLWA operator [39]
NO
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
HFLWG operator [39]
NO
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
HFLWPA operator [39]
NO
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
HFL-TOPSIS [40]
NO
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
HFLWBM operator [62]
p
=
q
=
1
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4
WHFLHM operator
ξ
=
2
x
5
≻
x
3
≻
x
2
≻
x
1
≻
x
4
WHFLDHM operator
ξ
=
2
x
5
≻
x
3
≻
x
1
≻
x
2
≻
x
4Table 5
Comparison of characteristics with different MADM approaches.
Approaches
Whether to capture the inter-relationship between two attributes
Whether to capture the inter-relationship between multiple attributes
Whether has generalized characteristics by the parameter
HFLWA operator [39]
No
No
No
HFLWG operator [39]
No
No
No
HFL-TOPSIS [40]
No
No
No
HFLWBM operator [62]
Yes
No
Yes
HFLWHM operator
Yes
Yes
Yes
HFLWDHM operator
Yes
Yes
YesIn the next, we will perform a detailed comparative analysis on the basis of the above approaches with our proposed approach. According to our further analysis and the character comparison in Table5, we obtain the following summarization:(1)
Compared with the approach based on the HFLWA operator proposed by Lin et al. [39], although the HFLWA operator can be utilized to aggregate HFL information, it usually assumes that attributes are independent of each other, and it lacks flexibility and robustness in the course information fusion. In comparison, the proposed approach based on the WHFLHM operator can not only capture the correlation of overall attributes but also can enable the decision process dynamic through a flexible parameter. Moreover, the WHFLHM operator is more universal than the HFLWA operator because it is a particular case of the WHFLHM operator. Accordingly, our developed approach is more appropriate to cope with practical MADM problems.(2)
Compared with HFL-TOPSIS approach in [40], the TOPSIS approach only provides a ranking result of alternatives, and the distance computation of ideal points will lead to information loss phenomenon happening. However, the presented approach on the basis of the WHFLHM operator can directly obtain the comprehensive evaluation value as well as the sorting of alternatives, and the inter-relationship among any number of arbitrary attributes is taken into consideration, i.e., it can reduce information loss to some extent in information integration procedure. At the same time, the adjustable parameter ξ can be assigned different values through evaluator’s risk preference. In summary, our developed method is more general and flexible than the HFL-TOPSIS method.(3)
Compared with the HFLWBM operator in [62], the HFLWBM operator can only capture the inter-relation between any two input arguments. However, in the practical application issues, we often need to take into account all arguments provided by the evaluator. The developed HFLWHM and HFLWDHM operator has an excellent capability to model the correlation among multi-input parameters, and it can be applied more flexibly during the process of information fusion.As analysed and compared the abovementioned summarization points, the superiorities of the propounded approach are summarized as follows:Firstly, the HFLS can express the assessment information more comprehensive because it combines the excellent feature of the linguistic term and the hesitant fuzzy set. The HFLS can cope with practical problems from both quantitative and qualitative perspectives.Secondly, the comparative analysis show that the novel approach based on the WHFLHM operator thinks over the inter-relationship among multi-input arguments, which can further reduce the loss of evaluation information and enable the decision process more flexible and valid.Thirdly, the HFLWHM operator can be applied in different situations by assigning preference values with different parametersξ, which can be seen as different risk preference by decision makers based on processing practical problems.Because of the complicacy of the decision environment, decision makers tend to estimate the alternatives according to their preference in linguistic terms and may hesitate to determine the degree of membership. At the same time, in order to conduct a more reasonable decision in actual problems, it is indispensable for decision makers to concern the inter-relation of the multiple input argument. Accordingly, the propounded approach can be regarded as an effective technique for addressing practical issues.


## 7. Conclusion

In this article, by taking into account the merits of the linguistic term set and the HM operator, we extend the HM operator to HFL setting to present several HFL aggregation operators for dealing with MADM problems effectively. To begin with, we present the HFLHM operator, HFLWHM operator, HFLDHM operator, and HFLWDHM operator. Thereafter, we develop a novel MADM algorithm based on the presented operators. Ultimately, the proposed algorithm is applied to solve the practical problem, and a comparative analysis is preformed through comparing it with other MADM approaches. The most remarkable feature of the novel method is that it fully considers the correlation among multiple attributes. In addition, the proposed method has flexibility because of the adjustable parameter in the HM operator. As the parameter value changes, different ranking results of alternatives will be obtained in tackling MADM problems, which can be viewed as the risk preference of the decision makers. Furthermore, the hesitant fuzzy linguistic set can portray the incomplete, vague, and uncertain information more valid through combining the linguistic term set and the hesitant fuzzy set. In future research works, the developed aggregation operators can be utilized for other actual applications. What is more, considering the merits of the HM operator, we can generalize it to other fuzzy contexts, for instance, unbalanced linguistic term sets [59], hesitant pythagorean fuzzy set [63], and q-rung orthopair fuzzy linguistic set [64].



---

*Source: 3262618-2020-02-19.xml*