# Extended Heronian Mean Based on Hesitant Fuzzy Linguistic Information for Multiple Attribute Group Decision-Making

Authors: Minghua Shi; Yuewen Xiao; Qing Wan
Journal: Complexity
 (2019)

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## Abstract

This paper investigates the hesitant fuzzy linguistic multiple attribute group decision-making (MAGDM) problem with the heterogeneous relationship among the attribute variables that cannot be solved by most existing decision-making methods. To address this problem, a new operator is introduced based on partitioning attribute variables into different sets according to their interrelationship. This operator is called the extended Heronian mean (EHM) operator. To obtain each expert’s comprehensive values of the alternatives in the hesitant fuzzy linguistic MAGDM problem, we investigate the EHM operator under a hesitant fuzzy linguistic environment and propose the hesitant fuzzy linguistic EHM operator and the hesitant fuzzy linguistic linear support degree weighted EHM operator. In addition, the axiom definition of a linguistic type similarity measure of hesitant fuzzy linguistic term sets is proposed. The weight of the experts can be determined based on this type similarity measure. Finally, a practical case is presented to demonstrate the steps of our method, and a comparison analysis illustrates its feasibility and effectiveness.


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## Body

## 1. Introduction

In such a complicated and changeable economic and social environment, it is difficult for a single expert to fully understand and master the decision-making problem. Therefore, it is necessary that experts from different disciplines take part in the decision-making process, which evolves the process into multiple attribute group decision-making (MAGDM). MAGDM has been applied in many fields, such as academic assessments of higher education institutions [1], financial risk evaluation [2, 3], investment objective selection [4], and green supply chain management [5]. The decision-maker may use qualitative values instead of quantitative values to express his/her assessment information due to characteristics of decision-making objects, information integrality cannot be guaranteed, and human knowledge is limited. For example, when evaluating the degree of economic activity in one location, people may use linguistic terms such as “low,” “medium,” and “high.”A reasonable selection of the means of information representation is essential to solving linguistic MAGDM problems. In early research, fuzzy linguistic term sets (FLTS) were commonly used to represent decision-making information and were described and analyzed by Zadeh [6]. Based on FLTS, many other types of information representation models have been proposed in recent decades, such as the 2-tuple fuzzy linguistic representation model [7], the linguistic intuitionistic fuzzy set [8, 9], the uncertain linguistic fuzzy soft set [10], the interval neutrosophic uncertain linguistic model [11], the probabilistic linguistic term set [12], and picture 2-tuple linguistic set [13]. However, when experts vacillate among several linguistic terms, it is quite difficult for them to model such situations by the abovementioned representation models. Therefore, Rodriguez et al. [14] presented hesitant fuzzy linguistic term set (HFLTSs) which can reflect the hesitant psychology feature of the decision-maker. Hesitant fuzzy linguistic decision-making has received much attention by many scholars and has produced a large amount of research results [15–17].Similarity measure is a very important research topic in the field of linguistic decision-making. It has been widely used in many fields, such as semantic translation, cluster analysis, and machine learning. For example, Chen et al. [18] introduced a new similarity measure formula for interval linguistic terms based on the likelihood of the comparison between two intervals. For the trapezoid fuzzy linguistic variables, Xu [19] put forward the similarity measure calculation method. Based on Xu’s work, Liao et al. [20] further proposed the axioms of similarity measures for HFLTSs and then used them to sort alternatives in multiple attribute decision-making (MADM). To enrich the hesitant fuzzy linguistic similarity measure calculation, Hesamian et al. [21] proposed Gower-Legendre and Tversky similarity measures, Liao et al. [22] introduced cosine similarity measures, Gou et al. [23] developed some cross-entropy measures, Farhadinia et al. [24] defined some entropy measures, and Song et al. [25] proposed vector similarity measures.Information aggregation is also an important component of linguistic decision-making. An operator is a primary tool for linguistic information aggregation, which is widely accepted and used in practical decision-making. In earlier research, linguistic aggregation operators were made based on the assumption that the input arguments are independent of each other, such as the linguistic geometric averaging operator [26], the linguistic hybrid geometric operator [27], and the uncertain linguistic ordered weighted averaging operator [28]. However, in recent years, scholars have gradually realized the significance of considering the associated relationship in information aggregation [29, 30]. The Heronian mean (HM) operator and Bonferroni mean (BM) operator can capture the interrelationship of the aggregation variables through the crossover operation of aggregation variables. Wei et al. and Tian et al. investigated a weighted BM operator under uncertain linguistic fuzzy environment [31] and simplified neutrosophic linguistic environment [32], respectively. Nie et al. proposed a Pythagorean fuzzy partitioned normalized weighted Bonferroni mean [33]. Liu et al. [34] analyzed the structure principle of the BM operator and HM operator and then pointed out that the HM operator can avoid the disadvantage of redundant information and make the information fusion more efficient. Li et al. [35] developed some 2-tuple linguistic HM aggregation operators and studied their properties. Liu et al. [36] proposed some new HM operators, based on the HM operator and the geometric HM operator, to solve intuitionistic uncertain linguistic MAGDM problems. Yu et al. [37] defined some of the reducible weighted linguistic hesitant fuzzy HM operators and then discussed their special cases and desirable properties.Analyzing the existing research literature on linguistic similarity measures and linguistic aggregation operators shows that the following problems can be found:(1) Most of the existing studies on the similarity measures of linguistic fuzzy sets use a crisp number to measure the degree of similarity. Whether this numerical value measurement is appropriate for linguistic input arguments is worth discussing because it conflicts with the motivation for using linguistic fuzzy set in decision-making that is close to the cognitive and understanding structure of people [38]. For example, consider a medical and health institution that wants to develop an online medical diagnosis system (OMDS) to provide the results of identified illnesses in an easily understandable way. The OMDS may suggest “similar,” “moderately similar,” and “high degree of similarity” as description forms. Thus, it is necessary to carry out research on the linguistic similarity measures of HFLTS to enrich and develop the theories and means of hesitant fuzzy linguistic decision-making.(2) Although existing HM operators consider the interrelationship of the argument variables, they are built on the assumption that a relationship exists between any two input arguments. However, in some real-world problems, this assumption is not always true. For example, consider an international company selecting a suitable department manager according to the following attributes:C1: Educational background; C2: Work experience; C3: Professional knowledge; C4: Working skill; C5: Age; and C6: Health. Based on their interrelationship, the attributes can be divided into two classes: S1={C1,C2,C3,C4} and S2={C5,C6}. It is easy to see that the elements in the same set are connected, but there is no relationship between the elements from different sets. Therefore, if we use the existing HM operators, they would generate disturbing information in the process of aggregation which would affect the reliability of decision-making.In accordance with the above analysis, this paper begins by providing the axiom definition of linguistic similarity measures for HFLTS and then proposes a new operator under the inspiration of previous studies. This new operator is called extended Heronian mean (EHM). Next, we investigate the EHM operator in a hesitant fuzzy environment and propose some operators to infuse HFLTSs. Based on these operators and linguistic similarity measures, a new method for solving the hesitant fuzzy linguistic MAGDM problem is designed. Weights that are completely unknown or represented by a crisp numbers (linguistic terms) situation are taken into consideration in the decision-making process. This makes the new method more practical. The framework of this article is as follows:Section2 reviews some concepts of HFLTSs and proposes linguistic similarity measures for HFLTS and EHM operators. Section 3 introduces the HFLEHM and HFLLSDWEHM operators and investigates their properties. Section 4 designs a group decision-making method under hesitant fuzzy linguistic environments. Section 5 provides a practical example to illustrate the method and discuss the effect of parameters on decision-making results. Section 6 illustrates the feasibility and effectiveness of the proposed operators and method through comparison and analysis. Lastly, Section 7 summarizes this study.


## 2. Preliminaries

### 2.1. Hesitant Fuzzy Linguistic Term Set

To build a linguistic decision model, a set of evaluation scales can be built asS=sα∣α=0,1,…,t, where sα denotes a linguistic measurement level of evaluation objectives, and t is an even number which always takes the values of 4, 6, and 8 [26]. This requires the following properties to be satisfied: (1) sα<sβiffα<β and (2) N(sα)=st-α. To make the linguistic decision model have an obvious logicality and system, Xu extended the above discrete evaluation scales set to a continuous set S¯=sα∣α∈[0,q],q>t and defined the following operational laws [39]:Definition 1.
Letsα,sβ∈S¯,λ1,λ2,λ>0; then, the following are given.
(1)sα⊕sβ=sβ⊕sα=sα+β; (2) λsα=sλα
(3)(λ1+λ2)sα=λ1sα⊕λ2sα; (4) λ(sα⊕sβ)=λsα⊕λsβ
With the decision environment becoming more and more complex and uncertain, this kind of linguistic term sets is difficult to satisfy with the request of decision modeling. Because of this, Rodriguez et al. [14] proposed the HFLTS, and then Liao et al. [20] further put forward its mathematical form.Definition 2 (see [20]).
LetX=x1,x2,…,xn be a universe of discourse, and let S=sα∣α=0,1,…,t be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS) on X is characterized by the following:(1)HS=xi,hSxi∣xi∈X,i=1,2,…,nwhere hS(xi)=sηl(xi)∣sηl(xi)∈S,l=1,2,…,L is a subset of S, and L denotes the number of linguistic terms in hS(xi). For convenience, we call hS(xi) a hesitant fuzzy linguistic element (HFLE), denoted by hS.Definition 3 (see [20]).
LetS=sα∣α=0,1,…,t be a linguistic term set, and let hS=sηl∣sηl∈S,l=1,2,…,L be a HFLE. μ(hS)=(1/L)∑i=1Lηi is called the score of hS, and υ(hS)=(1/L)∑i≠kLηi-ηk2 is called the deviation degree of hS. Then, the comparison law between two HFLEs (hS1=sηl1∣sηl1∈S,l=1,2,…,L and hS2=sηl2∣sηl2∈S,l=1,2,…,L) can be defined as follows:(a)
Ifμ(hS1)>μ(hS2), then hS1>hS2(b)
Ifμ(hS1)=μ(hS2), then
(i)
ifυ(hS1)=υ(hS2), then hS1=hS2(ii)
ifυ(hS1)>υ(hS2), then hS1<hS2There are also other comparison rules of HFLEs in the literature. For example, Liu et al. proposed a comparison rule between HFLEs based on a linguistic scale function [40]. Wei et al. introduced a novel comparison rule between HFLEs, which takes into account the average linguistic term and the hesitant degree [41]. Motivated by the possibility degree of intervals, Lee et al. introduced a likelihood based comparison rule between HFLEs [42]. In consideration of both the hesitant degrees and the unbalanced linguistic terms in evaluations, Liao et al. introduced a new comparison rule between HFLEs based on the psychological characteristics of experts [43]. On the basis of pairwise comparison of each linguistic term in the two HELEs, Huang et al. offered a comparison rule between HELEs [44]. For details, please refer to the survey papers [45]. However, the comparison rule proposed by Liao et al. [20], as shown in Definition 3, is simple and convenient and is thus widely used in decision-making.


### 2.2. A New Similarity Measure for HFLTSs

The similarity measure is an important tool for studying and applying linguistic fuzzy sets. Liao et al. proposed the numeric axiom definition of similarity measures of HFLTSs [22]. Based on this, we propose a linguistic axiom definition as follows:Definition 4.
LetS=sα∣α=0,1,…,t and S∗=sα∗∣α=0,1,…,t be two linguistic term sets, and HS1 and HS2 are two HFLTSs on S; then the linguistic similarity measure between HS1 and HS2 is defined as ρl(HS1,HS2), which satisfies the following:
(1)s0∗≤ρl(HS1,HS2)≤st∗; (2) ρl(HS1,HS2)=st∗iffHS1=HS2; (3) ρl(HS1,HS2)=ρl(HS2,HS1).Remark 5.
s α ∗ denotes a linguistic similarity measure level and st∗ and s0∗ represent the maximum and minimum linguistic similarity measure levels, respectively. Moreover, S∗ must have the same number of elements as S.Definition 6.
LetS=sα∣α=0,1,…,t and S∗=sα∗∣α=0,1,…,t be two linguistic term sets, HS1=hS11,hS12,…hS1k and HS2=hS21,hS22,…hS2k are two HFLTSs on S, hS1i=sα11i,sα21i,…,sαl1i, and hS2i=sβ12i,sβ22i,…,sβl2i,i=0,1,…,k. Then ρl(HS1,HS2)=S(1/lk)∑i=1k∑j=1lt-αj1i-βj2i∗ is called the linguistic similarity measure between HS1 and HS2.Remark 7.
Ifk=1, then the linguistic similarity measure between two HFLEs, hS11 and hS21, is defined as ρl(hS11,hS21)=S(1/l)∑j=1lt-αj11-βj21∗.Example 8.
LethS1=s2,s3,s5, hS2=s2,s4,s6, and hS3=s3,s4,s6 be three HFLEs on S=sα∣α=0,1,…,6. S∗ can be built as s0∗: extremely far, s1∗: far, s2∗: slightly far, s3∗: medium, s4∗: slightly close, s5∗: close, and s6∗: extremely close. According to the definition of linguistic similarity measure of HFLTSs, we obtain the following:(2)ρlhS1,hS2=S1/36-2-2+6-3-4+6-5-6∗=S16/3∗,ρlhS1,hS3=S1/36-2-3+6-3-4+6-5-6∗=S5∗.This means that the similarity degree between hS1 and hS2 is higher than close, and the similarity degree between hS1 and hS3 is close. Therefore, ρl(hS1,hS3)<ρl(hS1,hS2).Utilizing the similarity measure defined by Hesamian et al. [21], we can then obtain the following: (3)ρ21hS1,hS2=1-hS1△hS27=1-47=37,ρ21hS1,hS3=1-hS1△hS37=1-47=37,which means that ρ21(hS1,hS2)=ρ21(hS1,hS3).Utilizing the similarity measure defined by Liao et al. based on the Hamming distance [20], we can then obtain the following:(4)ShS1,hS2=1-132-27+3-47+5-67=1921,ShS1,hS3=1-132-37+3-47+5-67=1821,which means that ρl(hS1,hS3)<ρl(hS1,hS2).From the above calculation results, we can see that the method proposed in this paper and the method given by Liao et al. have a high degree of discrimination. However, the representation of similarity degree of the two methods is quite different. One is a quantitative representation and the other is a qualitative representation. Users should reasonably choose according to the application environment.


### 2.3. Extended Heronian Mean Operator

Definition 9 (see [34]).
Letp≥0, q≥0, p+q≠0 and ai(i=1,2,…,n) be nonnegative real numbers. Then (5)HMp,qa1,a2,…,an=2nn+1∑i=1,j=inaipajq1/p+qis called Heronian mean (HM).The HM operator is based on the assumption that a correlation exists between any two input arguments. However, in some real-world applications the above assumption does not hold because a partial correlation may exist between input arguments. Hence, an EHM is introduced to provide more precise aggregate information.Definition 10.
Letp≥0, q≥0, p+q≠0 and ai(i=1,2,…,n) be nonnegative real numbers. Then, the extended Heronian mean (EHM) operator is defined as follows:(6)EHMp,qa1,a2,…,an=1∑i=1nIAi∑i=1naip∑aij∈Aiaijq1/p+qwhere set Ai consists of some elements of ai,ai+1,…,an which have a correlation with ai, and I(Ai) represents the cardinality of Ai.Remark 11.
IfI(Ai)=1(i=1,2,…,n), that is to say, all input arguments are independent, then the EHM operator reduces to the following:(7)EHMp,qa1,a2,…,an=1∑i=1nIAi∑i=1naip∑aij∈Aiaijq1/p+q=1n∑i=1naipaiq1/p+q=1n∑i=1nair1/r,r=p+qwhich is called generalized arithmetic averaging (GAA) operator.Remark 12.
IfI(Ai)=n-i+1(i=1,2,…,n), that is to say, all input arguments are related to each other, then EHM operator reduces to the HM operator.(8)EHMp,qa1,a2,…,an=1∑i=1nIAi∑i=1naip∑aij∈Aiaijq1/p+q=2nn+1∑i=1,j=inaipajq1/p+qObviously, the EHM operator has the following properties:(1) Ifai=a(i=1,2,…,n), then EHMp,q(a,a,…,a)=a(2) Ifai≥di(i=1,2,…,n), then EHMp,q(a1,a2,…,an)≤EHMp,q(d1,d2,…,dn)Remark 13.
From properties (1) and (2), we can get(9)min1≤i≤n⁡ai≤EHMp,qa1,a2,…,an≤max1≤i≤n⁡ai.In the following section, we further extend the EHM operator to the hesitant fuzzy linguistic environment and discuss its relevant properties.


## 2.1. Hesitant Fuzzy Linguistic Term Set

To build a linguistic decision model, a set of evaluation scales can be built asS=sα∣α=0,1,…,t, where sα denotes a linguistic measurement level of evaluation objectives, and t is an even number which always takes the values of 4, 6, and 8 [26]. This requires the following properties to be satisfied: (1) sα<sβiffα<β and (2) N(sα)=st-α. To make the linguistic decision model have an obvious logicality and system, Xu extended the above discrete evaluation scales set to a continuous set S¯=sα∣α∈[0,q],q>t and defined the following operational laws [39]:Definition 1.
Letsα,sβ∈S¯,λ1,λ2,λ>0; then, the following are given.
(1)sα⊕sβ=sβ⊕sα=sα+β; (2) λsα=sλα
(3)(λ1+λ2)sα=λ1sα⊕λ2sα; (4) λ(sα⊕sβ)=λsα⊕λsβ
With the decision environment becoming more and more complex and uncertain, this kind of linguistic term sets is difficult to satisfy with the request of decision modeling. Because of this, Rodriguez et al. [14] proposed the HFLTS, and then Liao et al. [20] further put forward its mathematical form.Definition 2 (see [20]).
LetX=x1,x2,…,xn be a universe of discourse, and let S=sα∣α=0,1,…,t be a linguistic term set. A hesitant fuzzy linguistic term set (HFLTS) on X is characterized by the following:(1)HS=xi,hSxi∣xi∈X,i=1,2,…,nwhere hS(xi)=sηl(xi)∣sηl(xi)∈S,l=1,2,…,L is a subset of S, and L denotes the number of linguistic terms in hS(xi). For convenience, we call hS(xi) a hesitant fuzzy linguistic element (HFLE), denoted by hS.Definition 3 (see [20]).
LetS=sα∣α=0,1,…,t be a linguistic term set, and let hS=sηl∣sηl∈S,l=1,2,…,L be a HFLE. μ(hS)=(1/L)∑i=1Lηi is called the score of hS, and υ(hS)=(1/L)∑i≠kLηi-ηk2 is called the deviation degree of hS. Then, the comparison law between two HFLEs (hS1=sηl1∣sηl1∈S,l=1,2,…,L and hS2=sηl2∣sηl2∈S,l=1,2,…,L) can be defined as follows:(a)
Ifμ(hS1)>μ(hS2), then hS1>hS2(b)
Ifμ(hS1)=μ(hS2), then
(i)
ifυ(hS1)=υ(hS2), then hS1=hS2(ii)
ifυ(hS1)>υ(hS2), then hS1<hS2There are also other comparison rules of HFLEs in the literature. For example, Liu et al. proposed a comparison rule between HFLEs based on a linguistic scale function [40]. Wei et al. introduced a novel comparison rule between HFLEs, which takes into account the average linguistic term and the hesitant degree [41]. Motivated by the possibility degree of intervals, Lee et al. introduced a likelihood based comparison rule between HFLEs [42]. In consideration of both the hesitant degrees and the unbalanced linguistic terms in evaluations, Liao et al. introduced a new comparison rule between HFLEs based on the psychological characteristics of experts [43]. On the basis of pairwise comparison of each linguistic term in the two HELEs, Huang et al. offered a comparison rule between HELEs [44]. For details, please refer to the survey papers [45]. However, the comparison rule proposed by Liao et al. [20], as shown in Definition 3, is simple and convenient and is thus widely used in decision-making.


## 2.2. A New Similarity Measure for HFLTSs

The similarity measure is an important tool for studying and applying linguistic fuzzy sets. Liao et al. proposed the numeric axiom definition of similarity measures of HFLTSs [22]. Based on this, we propose a linguistic axiom definition as follows:Definition 4.
LetS=sα∣α=0,1,…,t and S∗=sα∗∣α=0,1,…,t be two linguistic term sets, and HS1 and HS2 are two HFLTSs on S; then the linguistic similarity measure between HS1 and HS2 is defined as ρl(HS1,HS2), which satisfies the following:
(1)s0∗≤ρl(HS1,HS2)≤st∗; (2) ρl(HS1,HS2)=st∗iffHS1=HS2; (3) ρl(HS1,HS2)=ρl(HS2,HS1).Remark 5.
s α ∗ denotes a linguistic similarity measure level and st∗ and s0∗ represent the maximum and minimum linguistic similarity measure levels, respectively. Moreover, S∗ must have the same number of elements as S.Definition 6.
LetS=sα∣α=0,1,…,t and S∗=sα∗∣α=0,1,…,t be two linguistic term sets, HS1=hS11,hS12,…hS1k and HS2=hS21,hS22,…hS2k are two HFLTSs on S, hS1i=sα11i,sα21i,…,sαl1i, and hS2i=sβ12i,sβ22i,…,sβl2i,i=0,1,…,k. Then ρl(HS1,HS2)=S(1/lk)∑i=1k∑j=1lt-αj1i-βj2i∗ is called the linguistic similarity measure between HS1 and HS2.Remark 7.
Ifk=1, then the linguistic similarity measure between two HFLEs, hS11 and hS21, is defined as ρl(hS11,hS21)=S(1/l)∑j=1lt-αj11-βj21∗.Example 8.
LethS1=s2,s3,s5, hS2=s2,s4,s6, and hS3=s3,s4,s6 be three HFLEs on S=sα∣α=0,1,…,6. S∗ can be built as s0∗: extremely far, s1∗: far, s2∗: slightly far, s3∗: medium, s4∗: slightly close, s5∗: close, and s6∗: extremely close. According to the definition of linguistic similarity measure of HFLTSs, we obtain the following:(2)ρlhS1,hS2=S1/36-2-2+6-3-4+6-5-6∗=S16/3∗,ρlhS1,hS3=S1/36-2-3+6-3-4+6-5-6∗=S5∗.This means that the similarity degree between hS1 and hS2 is higher than close, and the similarity degree between hS1 and hS3 is close. Therefore, ρl(hS1,hS3)<ρl(hS1,hS2).Utilizing the similarity measure defined by Hesamian et al. [21], we can then obtain the following: (3)ρ21hS1,hS2=1-hS1△hS27=1-47=37,ρ21hS1,hS3=1-hS1△hS37=1-47=37,which means that ρ21(hS1,hS2)=ρ21(hS1,hS3).Utilizing the similarity measure defined by Liao et al. based on the Hamming distance [20], we can then obtain the following:(4)ShS1,hS2=1-132-27+3-47+5-67=1921,ShS1,hS3=1-132-37+3-47+5-67=1821,which means that ρl(hS1,hS3)<ρl(hS1,hS2).From the above calculation results, we can see that the method proposed in this paper and the method given by Liao et al. have a high degree of discrimination. However, the representation of similarity degree of the two methods is quite different. One is a quantitative representation and the other is a qualitative representation. Users should reasonably choose according to the application environment.


## 2.3. Extended Heronian Mean Operator

Definition 9 (see [34]).
Letp≥0, q≥0, p+q≠0 and ai(i=1,2,…,n) be nonnegative real numbers. Then (5)HMp,qa1,a2,…,an=2nn+1∑i=1,j=inaipajq1/p+qis called Heronian mean (HM).The HM operator is based on the assumption that a correlation exists between any two input arguments. However, in some real-world applications the above assumption does not hold because a partial correlation may exist between input arguments. Hence, an EHM is introduced to provide more precise aggregate information.Definition 10.
Letp≥0, q≥0, p+q≠0 and ai(i=1,2,…,n) be nonnegative real numbers. Then, the extended Heronian mean (EHM) operator is defined as follows:(6)EHMp,qa1,a2,…,an=1∑i=1nIAi∑i=1naip∑aij∈Aiaijq1/p+qwhere set Ai consists of some elements of ai,ai+1,…,an which have a correlation with ai, and I(Ai) represents the cardinality of Ai.Remark 11.
IfI(Ai)=1(i=1,2,…,n), that is to say, all input arguments are independent, then the EHM operator reduces to the following:(7)EHMp,qa1,a2,…,an=1∑i=1nIAi∑i=1naip∑aij∈Aiaijq1/p+q=1n∑i=1naipaiq1/p+q=1n∑i=1nair1/r,r=p+qwhich is called generalized arithmetic averaging (GAA) operator.Remark 12.
IfI(Ai)=n-i+1(i=1,2,…,n), that is to say, all input arguments are related to each other, then EHM operator reduces to the HM operator.(8)EHMp,qa1,a2,…,an=1∑i=1nIAi∑i=1naip∑aij∈Aiaijq1/p+q=2nn+1∑i=1,j=inaipajq1/p+qObviously, the EHM operator has the following properties:(1) Ifai=a(i=1,2,…,n), then EHMp,q(a,a,…,a)=a(2) Ifai≥di(i=1,2,…,n), then EHMp,q(a1,a2,…,an)≤EHMp,q(d1,d2,…,dn)Remark 13.
From properties (1) and (2), we can get(9)min1≤i≤n⁡ai≤EHMp,qa1,a2,…,an≤max1≤i≤n⁡ai.In the following section, we further extend the EHM operator to the hesitant fuzzy linguistic environment and discuss its relevant properties.


## 3. Hesitant Fuzzy Linguistic Extended Heronian Means

### 3.1. HFLEHM Operator

Definition 14.
Lethi=sα1i,sα2i,…,sαmi(i=1,2,…,n) be a collection of HFLEs, for any p,q≥0,(10)HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+qWe call HFLEHMp,q a hesitant fuzzy linguistic extended Heronian mean (HFLEHM) operator, where set Bki consists of some elements of αki,αki+1,…,αkn which have correlation with αk(i), and I(Bki) represents the cardinality of Bki.Remark 15.
IfI(Bki)=1(i=1,2,…,n;k=1,2,…,m), that is to say, all hi(i=1,2,…,n) are independent, then HFLEHM operator reduces to(11)HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1n⨁i=1nsαkip⊗sαkiq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1n⨁i=1nsαkip+q1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1n⨁i=1nsαkir1/r,r=p+q,which is called hesitant fuzzy linguistic generalized arithmetic averaging (HFLGAA) operator.Remark 16.
IfI(Bki)=n-i+1(i=1,2,…,n), that is to say, all hi(i=1,2,…,n) are related to each other, then HFLEHM operator reduces to HFLHM operator.(12)HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn2nn+1⨁i=1nsαkip⊗⨁j=insαkjq1/p+q=HFLHMp,qh1,h2,…,hn.Remark 17.
IfI(Bki)=n-i+1(i=1,2,…,n) and q→0, then HFLEHM operator reduces to(13)limq→0⁡HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimq→0⁡1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimq→0⁡2nn+1⨁i=1nsαkip⊗⨁j=1nsαkjq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn2⨁i=1nn-i+1sαkipnnn+11/p=⋃1≤k≤m⨁i=1nn-i+1nn+1/2sαkip1/p=⋃1≤k≤m⨁i=1nwisαkip1/pwhich is called hesitant fuzzy linguistic generalized linear decrease weighted averaging (HFLGLDWA) operator, where wi=(n-i+1)/(n(n+1)/2)(i=1,2,…,n) meet the following conditions: (1) wi≥0; (2) ∑wi=1; (3) w1≥w2≥⋯≥wn.Remark 18.
IfI(Bki)=n-i+1(i=1,2,…,n) and p→0, then HFLEHM operator reduces to(14)limp→0⁡HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimp→01∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimp→02nn+1⨁i=1nsαkip⊗⨁j=insαkjq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn2⨁i=1n⨁j=insαkjqnn+11/q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁j=1njnn+1/2sαkjq1/q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁j=1nwjsαkjq1/qwhich is called hesitant fuzzy linguistic generalized linear increase weighted averaging (HFLGLIWA) operator, where wj=j/(n(n+1)/2)(j=1,2,…,n) meet the following conditions: (1) wj≥0; (2) ∑wj=1; (3) w1≤w2≤⋯≤wn.Property 19.
Lethi(i=1,2,…,n) be a collection of HFLEs; if h1=h2=⋯=hn=sα1,sα2,…,sαm=h, then(15)HFLEHMp,qh1,h2,…,hn=h.
​Property 20.
Lethi=sα1i,sα2i,…,sαmi and hi′=sβ1i,sβ2i,…,sβmi(i=1,2,…,n) be hesitant fuzzy linguistic evaluation values of same attributes, and sαk(i)≤sβk(i),i=1,2,…,n,k=1,2,…,m; then HFLEHMp,q(h1,h2,…,hn)≤HFLEHMp,q(h1′,h2′,…,hn′).Property 21.
Lethi=sα1i,sα2i,…,sαmi(i=1,2,…,n) be a collection of HFLEs, αk+=maxi⁡αki, αk-=mini⁡αki, h+=α1+,α2+,…,αm+, and h-=α1-,α2-,…,αm-; then h-≤HFLEHMp,q(h1,h2,…,hn)≤h+.


### 3.2. HFLWEHM Operator

It should be noted that the HFLEHM operator considers the aggregate variables to have the same importance. In real decision-making it is difficult to satisfy this condition. To reflect the importance of the aggregate variables in the aggregation operator, two weighted forms of hesitant fuzzy linguistic EHM operators are introduced in this subsection.Definition 22.
Lethi=sα1i,sα2i,…,sαmi(i=1,2,…,n) be a collection of HFLEs, with the weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. For any p,q≥0,(16)HFLWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip⊗⨁αkij∈hij,hij∈Bkiλij′sαkijq1/p+qwhich is called the hesitant fuzzy linguistic weighted extended Heronian mean (HFLEHM) operator, where λi=wiI(Bki)/∑i=1nwiI(Bki) and λij′=wij/∑wij.It is worth noting that if the importance of the aggregate variables is given by the form of HFLEs, for example,S^=s^α∣α=0,1,…,6=very unimportant, unimportant, slightly unimportant, medium, slightly important, important, very important, wi=h^i(i=1,2,…,n) being a collection of HFLEs on S^, then wi=μ(h^i)/∑j=1nμ(h^j), λi=wiI(Bki)/∑i=1nwiI(Bki), and λij′=wij/∑wij.Remark 23.
IfI(Bki)=1(i=1,2,…,n;k=1,2,…,m), that is to say, all hi(i=1,2,…,n) are independent, then HFLWEHM operator reduces to(17)HFLWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip⊗⨁αkij∈hij,hij∈Bkiλij′sαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nwisαkip+q1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nwisαkir1/r,r=p+qwhich is called hesitant fuzzy linguistic generalized weighted arithmetic averaging (HFLGWAA) operator.Remark 24.
Ifq=0(i=1,2,…,n), then(18)HFLWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip⊗⨁αkij∈hij,hij∈Bkiλij′sαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip1/p.
​Property 25.
Ifwi=(1/n)(i=1,2,…,n), then(19)HFLWEHMp,qh1,h2,…,hn=HFLEHMp,qh1,h2,…,hn.Property 26.
Lethi(i=1,2,…,n) be a collection of HFLEs with the weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. If h1=h2=⋯=hn=sα1,sα2,…,sαm=h, then(20)HFLWEHMp,qh1,h2,…,hn=h.The proofs of Properties27 and 28 are similar to those of Properties 20 and 21; therefore we omit the details.Property 27.
Lethi={sα1(i),sα2(i),…,sαm(i)} and hi′={sβ1(i),sβ2(i),…,sβm(i)}(i=1,2,…,n) be two collections of HFLEs, with the same weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. If sαk(i)≤sβk(i), i=1,2,…,n, k=1,2,…,m, then HFLWEHMp,q(h1,h2,…,hn)≤HFLWEHMp,q(h1′,h2′,…,hn′).Property 28.
Lethi={sα1(i),sα2(i),…,sαm(i)}(i=1,2,…,n) be a collection of HFLEs with the weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. αk+=maxi⁡αki, αk-=mini⁡αki, α+={α1+,α2+,…,αm+}, and α-={α1-,α2-,…,αm-}; then α-≤HFLWEHMp,q(α1,α2,…,αn)≤α+.Due to the complexity of the decision-making system, people may encounter a situation wherein the importance of the aggregate variables is unknown. Inspired by the work of Xu and Yager [46], we introduce a new operator to deal with this kind of problem in decision-making.Definition 29.
Lethi={sα1(i),sα2(i),…,sαm(i)}(i=1,2,…,n) be a collection of HFLEs. For any p,q>0,(21)HFLLSDWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nwisαkip⊗⨁αkij∈hij,hij∈Bkiwij′sαkijq1/p+qwhich is called the hesitant fuzzy linguistic linear support degree weighted extended Heronian mean (HFLLSDWEHM) operator, where wi=g(Bi/TV)-g(Bi-1/TV) is the support degree of hi in h1,h2,…,hn, T(hi)=∑j=1,j≠insup⁡(hi,hj), TV=∑i=1n(1+T(hi)), Bi=∑j=1i(1+T(hj)), and g(x) has the following properties. (1) g(0)=0, g(1)=1; (2) ∀x,y∈[0,1], g(x)≥g(y), if x>y. Similarly, wij′=g(Bi′/TV′)-g(Bi-1′/TV′) is the support degree of hij in Bki.Property 30.
Lethii=1,2,…,n be a collection of HFLEs. If h1=h2=⋯=hn=sα1,sα2,…,sαm=h, then HFLLSDWEHMp,q(h1,h2,…,hn)=h.Property 31.
Lethi={sα1(i),sα2(i),…,sαm(i)}(i=1,2,…,n) be a collection of HFLEs, αk+=maxi⁡αki, αk-=mini⁡αki, α+={α1+,α2+,…,αm+}, and α-={α1-,α2-,…,αm-}; then(22)α-≤HFLLSDWEHMp,qα1,α2,…,αn≤α+.We should emphasize that, unlike the HFLWEHM operator, the HFLLSDWEHM operator does not have the monotonicity property.


## 3.1. HFLEHM Operator

Definition 14.
Lethi=sα1i,sα2i,…,sαmi(i=1,2,…,n) be a collection of HFLEs, for any p,q≥0,(10)HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+qWe call HFLEHMp,q a hesitant fuzzy linguistic extended Heronian mean (HFLEHM) operator, where set Bki consists of some elements of αki,αki+1,…,αkn which have correlation with αk(i), and I(Bki) represents the cardinality of Bki.Remark 15.
IfI(Bki)=1(i=1,2,…,n;k=1,2,…,m), that is to say, all hi(i=1,2,…,n) are independent, then HFLEHM operator reduces to(11)HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1n⨁i=1nsαkip⊗sαkiq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1n⨁i=1nsαkip+q1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1n⨁i=1nsαkir1/r,r=p+q,which is called hesitant fuzzy linguistic generalized arithmetic averaging (HFLGAA) operator.Remark 16.
IfI(Bki)=n-i+1(i=1,2,…,n), that is to say, all hi(i=1,2,…,n) are related to each other, then HFLEHM operator reduces to HFLHM operator.(12)HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn2nn+1⨁i=1nsαkip⊗⨁j=insαkjq1/p+q=HFLHMp,qh1,h2,…,hn.Remark 17.
IfI(Bki)=n-i+1(i=1,2,…,n) and q→0, then HFLEHM operator reduces to(13)limq→0⁡HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimq→0⁡1∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimq→0⁡2nn+1⨁i=1nsαkip⊗⨁j=1nsαkjq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn2⨁i=1nn-i+1sαkipnnn+11/p=⋃1≤k≤m⨁i=1nn-i+1nn+1/2sαkip1/p=⋃1≤k≤m⨁i=1nwisαkip1/pwhich is called hesitant fuzzy linguistic generalized linear decrease weighted averaging (HFLGLDWA) operator, where wi=(n-i+1)/(n(n+1)/2)(i=1,2,…,n) meet the following conditions: (1) wi≥0; (2) ∑wi=1; (3) w1≥w2≥⋯≥wn.Remark 18.
IfI(Bki)=n-i+1(i=1,2,…,n) and p→0, then HFLEHM operator reduces to(14)limp→0⁡HFLEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimp→01∑i=1nIBki⨁i=1nsαkip⊗⨁αkij∈hij,hij∈Bkisαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hnlimp→02nn+1⨁i=1nsαkip⊗⨁j=insαkjq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn2⨁i=1n⨁j=insαkjqnn+11/q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁j=1njnn+1/2sαkjq1/q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁j=1nwjsαkjq1/qwhich is called hesitant fuzzy linguistic generalized linear increase weighted averaging (HFLGLIWA) operator, where wj=j/(n(n+1)/2)(j=1,2,…,n) meet the following conditions: (1) wj≥0; (2) ∑wj=1; (3) w1≤w2≤⋯≤wn.Property 19.
Lethi(i=1,2,…,n) be a collection of HFLEs; if h1=h2=⋯=hn=sα1,sα2,…,sαm=h, then(15)HFLEHMp,qh1,h2,…,hn=h.
​Property 20.
Lethi=sα1i,sα2i,…,sαmi and hi′=sβ1i,sβ2i,…,sβmi(i=1,2,…,n) be hesitant fuzzy linguistic evaluation values of same attributes, and sαk(i)≤sβk(i),i=1,2,…,n,k=1,2,…,m; then HFLEHMp,q(h1,h2,…,hn)≤HFLEHMp,q(h1′,h2′,…,hn′).Property 21.
Lethi=sα1i,sα2i,…,sαmi(i=1,2,…,n) be a collection of HFLEs, αk+=maxi⁡αki, αk-=mini⁡αki, h+=α1+,α2+,…,αm+, and h-=α1-,α2-,…,αm-; then h-≤HFLEHMp,q(h1,h2,…,hn)≤h+.


## 3.2. HFLWEHM Operator

It should be noted that the HFLEHM operator considers the aggregate variables to have the same importance. In real decision-making it is difficult to satisfy this condition. To reflect the importance of the aggregate variables in the aggregation operator, two weighted forms of hesitant fuzzy linguistic EHM operators are introduced in this subsection.Definition 22.
Lethi=sα1i,sα2i,…,sαmi(i=1,2,…,n) be a collection of HFLEs, with the weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. For any p,q≥0,(16)HFLWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip⊗⨁αkij∈hij,hij∈Bkiλij′sαkijq1/p+qwhich is called the hesitant fuzzy linguistic weighted extended Heronian mean (HFLEHM) operator, where λi=wiI(Bki)/∑i=1nwiI(Bki) and λij′=wij/∑wij.It is worth noting that if the importance of the aggregate variables is given by the form of HFLEs, for example,S^=s^α∣α=0,1,…,6=very unimportant, unimportant, slightly unimportant, medium, slightly important, important, very important, wi=h^i(i=1,2,…,n) being a collection of HFLEs on S^, then wi=μ(h^i)/∑j=1nμ(h^j), λi=wiI(Bki)/∑i=1nwiI(Bki), and λij′=wij/∑wij.Remark 23.
IfI(Bki)=1(i=1,2,…,n;k=1,2,…,m), that is to say, all hi(i=1,2,…,n) are independent, then HFLWEHM operator reduces to(17)HFLWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip⊗⨁αkij∈hij,hij∈Bkiλij′sαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nwisαkip+q1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nwisαkir1/r,r=p+qwhich is called hesitant fuzzy linguistic generalized weighted arithmetic averaging (HFLGWAA) operator.Remark 24.
Ifq=0(i=1,2,…,n), then(18)HFLWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip⊗⨁αkij∈hij,hij∈Bkiλij′sαkijq1/p+q=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nλisαkip1/p.
​Property 25.
Ifwi=(1/n)(i=1,2,…,n), then(19)HFLWEHMp,qh1,h2,…,hn=HFLEHMp,qh1,h2,…,hn.Property 26.
Lethi(i=1,2,…,n) be a collection of HFLEs with the weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. If h1=h2=⋯=hn=sα1,sα2,…,sαm=h, then(20)HFLWEHMp,qh1,h2,…,hn=h.The proofs of Properties27 and 28 are similar to those of Properties 20 and 21; therefore we omit the details.Property 27.
Lethi={sα1(i),sα2(i),…,sαm(i)} and hi′={sβ1(i),sβ2(i),…,sβm(i)}(i=1,2,…,n) be two collections of HFLEs, with the same weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. If sαk(i)≤sβk(i), i=1,2,…,n, k=1,2,…,m, then HFLWEHMp,q(h1,h2,…,hn)≤HFLWEHMp,q(h1′,h2′,…,hn′).Property 28.
Lethi={sα1(i),sα2(i),…,sαm(i)}(i=1,2,…,n) be a collection of HFLEs with the weight wi(i=1,2,…,n), wi∈[0,1], ∑i=1nwi=1. αk+=maxi⁡αki, αk-=mini⁡αki, α+={α1+,α2+,…,αm+}, and α-={α1-,α2-,…,αm-}; then α-≤HFLWEHMp,q(α1,α2,…,αn)≤α+.Due to the complexity of the decision-making system, people may encounter a situation wherein the importance of the aggregate variables is unknown. Inspired by the work of Xu and Yager [46], we introduce a new operator to deal with this kind of problem in decision-making.Definition 29.
Lethi={sα1(i),sα2(i),…,sαm(i)}(i=1,2,…,n) be a collection of HFLEs. For any p,q>0,(21)HFLLSDWEHMp,qh1,h2,…,hn=⋃1≤k≤msαk1∈h1,sαk2∈h2,…,sαkn∈hn⨁i=1nwisαkip⊗⨁αkij∈hij,hij∈Bkiwij′sαkijq1/p+qwhich is called the hesitant fuzzy linguistic linear support degree weighted extended Heronian mean (HFLLSDWEHM) operator, where wi=g(Bi/TV)-g(Bi-1/TV) is the support degree of hi in h1,h2,…,hn, T(hi)=∑j=1,j≠insup⁡(hi,hj), TV=∑i=1n(1+T(hi)), Bi=∑j=1i(1+T(hj)), and g(x) has the following properties. (1) g(0)=0, g(1)=1; (2) ∀x,y∈[0,1], g(x)≥g(y), if x>y. Similarly, wij′=g(Bi′/TV′)-g(Bi-1′/TV′) is the support degree of hij in Bki.Property 30.
Lethii=1,2,…,n be a collection of HFLEs. If h1=h2=⋯=hn=sα1,sα2,…,sαm=h, then HFLLSDWEHMp,q(h1,h2,…,hn)=h.Property 31.
Lethi={sα1(i),sα2(i),…,sαm(i)}(i=1,2,…,n) be a collection of HFLEs, αk+=maxi⁡αki, αk-=mini⁡αki, α+={α1+,α2+,…,αm+}, and α-={α1-,α2-,…,αm-}; then(22)α-≤HFLLSDWEHMp,qα1,α2,…,αn≤α+.We should emphasize that, unlike the HFLWEHM operator, the HFLLSDWEHM operator does not have the monotonicity property.


## 4. An Approach to MAGDM with Hesitant Fuzzy Linguistic Information

The critical components of solving MAGDM problems include attribute weights, expert weights, and attribute evaluation values. In this section, we utilize the proposed operators to solve MAGDM problems in which attribute weights are known or unknown, and the evaluation values are expressed by HFLTSs.For a MAGDM problem, assume thatD=d1,d2,…,dm are decision experts with the weight vectors ω=(ω1,ω2,…,ωm)T and X=x1,x2,…,xn is a finite set of n alternatives, and G=g1,g2,…,gl is a set of attributes, whose weight vector is w=(w1,w2,…,wl)T. Set Gc={gc1,gc2,…,gcd} is constructed using the elements of gc,gc+1,…,gn which have a correlation with gc and ⋄(Gc)=c1,c1,…,cd. Suppose that the decision experts dk∈D provide the hesitant fuzzy linguistic information to evaluate the characteristics of the alternatives xi∈X under attribute gj∈G, denoted by a HFLE hij(k); then the hesitant fuzzy linguistic decision matrix H(k)=(hij(k))n×l is constructed.An approach to multiple attribute group decision-making problems is provided in the following steps:Step 1 (normalize the evaluation information matrices).
For benefit attributes, higher evaluation values indicate the better alternative. However, for cost attributes, smaller evaluation values indicate the better alternative. All attributes of the alternative should have the same judgment standard in decision-making [47]. Therefore, decision matrix H(k)=(hij(k))n×l is transformed into the normalized decision matrix H-(k)=(h-ij(k))n×l, where h-ij(k)=hij(k), for benefit criteria gj; h-ij(k)=N(hij(k)), for cost criteria gj, i=1,2,…,n; j=1,2,…,l; and k=1,2,…,m.Step 2.
Calculate each expert’s comprehensive values of the alternatives.Case 1.
If the weighting vectors of the attributes are known, then the HFLWEHM operator is used to calculate thedk(k=1,2,…,m) comprehensive evaluation value hi(k) of xi(i=1,2,⋯,n).(23)hik=HFLWEHMp,qh-i1k,h-i2k…,h-ilk=⋃1≤t≤rαtkj1∈h-i1k,sαtkj2∈h-i2k,…,sαtkjl∈h-ilk⨁s=1lλssαtkisp⊗⨁sαtkjs∈h-jsk,js∈⋄Gsλjs′sαtkjsq1/p+qwhere λs=wsI(Gs)/∑i=1nwiI(Gs) and λjs′=wjs/∑wjs.Case 2.
If the weighting vectors of the attributes are unknown, then the HFLLSDWEHM operator is used to calculatedk(k=1,2,…,m) comprehensive evaluation value hi(k) of xi(i=1,2,…,n).(24)hik=HFLLSDWEHMp,qh-i1k,h-i2k…,h-ilk=⋃1≤t≤rαtki1∈h-i1k,sαtki2∈h-i2k,…,sαtkil∈h-ilk⨁s=1lwsksαtkisp⊗⨁sαtkjs∈h-jsk,js∈⋄Gsw~jsksαtkjsq1/p+qwhere wsk=g(Bsk/TV)-g(Bs-1k/TV) is the support degree of h-s(k) in h-sk,h-s+1k,…,h-lk, h-s(k) denotes the s column of H-(k), Th-sk=∑b=1,b≠slsup⁡(h-s(k),h-b(k)), TV=∑s=1l(1+T(h-s(k))), and Bjk=∑s=1j(1+T(h-s(k))). Similarly, w~js(k)=g(B~jsk/TV′)-g(B~j-1sk/TV′) is the support degree of h-js(k) in h-jsk,h-j+1sk,…,h-dsk, and h-js(k) denotes the js column of H-(k). In this paper, without loss of generality, we let sup⁡(h-s(k),h-b(k))=μρl(h-s(k),h-b(k))/t.Step 3.
Calculate the group comprehensive evaluation value.
Case (i). If the weighting vectors of the expert are known, then the HFLGWAA operator (in this paper we let r=1) is used to calculate the group comprehensive evaluation value hi of each alternative xi(i=1,2,…,n).(25)hi=HFLGWAAhi1,hi2…,him=⋃1≤k≤rsαk1∈hi1,sαk2∈hi2,…,sαkm∈him⨁i=1mωisαki
Case (ii). If the weighting vectors of the expert are unknown, then we first calculate the linguistic similarity lsmef between any two experts de and dfe,f=1,2,…,m.(26)lsmef=∑i=1nρlhide,hidfThese linguistic similarity values are summarized as follows:(27)LSM=lsm11lsm12⋯lsm1mlsm21lsm22⋯lsm2m⋮⋮⋮⋮lsmm1lsmm2⋯lsmmmNext, the average similarity of each expertde(e=1,2,…,m) in the group is calculated as lsme=(1/(l-1))⨁f=1,f≠ellsmef, and the weight of each expert de is obtained as ωe=μ(lsme)/∑e=1mμ(lsme). Finally, the group comprehensive evaluation values can be obtained using (25).Step 4.
Rankxi(i=1,2,…,n) in ascending order according to hi(i=1,2,…,n) by the ranking method defined in Definition 3.The flow chart of the proposed MAGDM method is shown in Figure1.Figure 1
The flow chart of the proposed MAGDM method.


## 5. A Numerical Example

With the improvement of people’s health awareness, environmental protection, and energy conservation, an increasing number of “green” products are produced. The corresponding industrial structure also shows a trend toward green development, and this has initiated the concept of a “green supply chain.” The construction of a green supply chain has become the primary challenge and trend for enterprises to provide green products and develop sustainability in a society. The crucial factor for implementing successful green supply chain management lies in the correct choice of suppliers, especially in terms of which meet enterprise requirements and have sustainable development strategies. Therefore, we consider the green supplier selection in the hesitant fuzzy linguistic environment as follows.As we know, electric bicycles, featuring convenience, and energy conservation are widely promoted and utilized in China. It can be concluded from data released by the National Bureau of Statistics of China that the cumulative number of products in China since 2017 has exceeded 60 million. However, problems with batteries are a key bottleneck that limits the development of electric bicycles. Good and poor batteries are intermingled, so their elimination rate is high. Unfortunately, if they are not properly disposed of, the batteries also emit heavy metals, including lead and acid, causing serious damage to the environment. Therefore, it is of particular importance for electric bicycle manufacturers to locate green battery suppliers. Take, for example, that an electric bicycle enterprise has established five possible battery sourcesxi(i=1,2,…,5) and five assessment attributes: (a) g1 environmental damage from production materials; (b) g2 product price; (c) g3 resource consumption; (d) g4 service level; and (e) enterprise reputation. The linguistic term set S=sα∣α=0,1,…,8=extremely low, very low, low, slightly low, average, slightly high, high, very high, extremely high is used to evaluate the five alternatives and S^=s0: extremely unimportant, s1: very unimportant, s2: unimportant, s3: slightly unimportant, s4: average, s5: slightly important, s6: important, s7: very important, s8: extremely important is used to evaluate the attribute weights. To achieve an objective and logical decision, the company invited three experts di(i=1,2,3) to evaluate these five feasible alternatives. Attribute evaluation values provided by the experts are represented by HFLTSs, as presented in Tables 1–3. The weight vectors w(i) of attributes by the experts di(i=1,2,3) are shown in Table 4.Table 1
Expert information matrixH(1) [16].
g 1 g 2 g 3 g 4 g 5 s 1 , s 2 s 2 , s 3 s 3 s 5 , s 6 , s 7 s 7 , s 8 s 1 , s 2 s 3 , s 4 , s 5 s 2 , s 3 , s 4 s 6 , s 7 s 6 , s 7 s 1 , s 2 , s 3 s 2 , s 3 , s 4 s 1 , s 2 , s 3 s 6 s 6 s 3 , s 4 s 1 , s 2 , s 3 s 2 , s 3 , s 4 s 4 , s 5 , s 6 s 4 , s 5 s 3 , s 4 , s 5 s 2 , s 3 s 3 , s 4 s 6 , s 7 s 5 , s 6Table 2
Expert information matrixH(2) [15].
g 1 g 2 g 3 g 4 g 5 { s 0 , s 1 , s 2 } { s 2 , s 3 , s 4 } { s 2 , s 3 } { s 4 , s 5 , s 6 } { s 6 , s 7 } { s 1 } { s 3 , s 4 } { s 2 , s 3 } { s 5 , s 6 } { s 6 } { s 1 , s 2 } { s 0 , s 2 , s 3 } { s 1 , s 3 , s 4 } { s 6 } { s 6 , s 7 } { s 2 , s 3 , s 4 } { s 1 , s 2 } { s 2 , s 3 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 3 , s 4 } { s 1 , s 2 } { s 3 , s 4 } { s 5 , s 6 } { s 6 }Table 3
Expert information matrixH(3) [16].
g 1 g 2 g 3 g 4 g 5 { s 2 , s 3 , s 4 } { s 2 , s 3 } { s 3 , s 4 } { s 5 , s 6 , s 7 } { s 5 , s 6 , s 7 } { s 1 , s 3 } { s 4 , s 5 } { s 2 , s 3 } { s 6 } { s 5 , s 6 } { s 2 , s 3 , s 4 } { s 3 , s 4 , s 5 } { s 1 , s 2 , s 3 } { s 5 , s 6 } { s 5 , s 6 } { s 3 , s 4 } { s 2 , s 3 , s 4 } { s 1 , s 2 , s 3 } { s 2 , s 3 , s 4 } { s 4 , s 5 } { s 3 , s 5 } { s 5 , s 6 , s 7 } { s 2 , s 3 } { s 6 , s 7 , s 8 } { s 4 , s 5 , s 6 }Table 4
Attribute weights information matrix [16].
g 1 g 2 g 3 g 4 g 5 w ( 1 ) { s 1 } { s 2 , s 3 } { s 6 , s 7 } { s 5 , s 6 } { s 5 } w ( 2 ) { s 1 } { s 3 } { s 6 } { s 6 , s 7 } { s 6 } w ( 3 ) { s 1 , s 2 } { s 3 , s 4 } { s 7 , s 8 } { s 7 } { s 7 }


### 5.1. Using the Proposed Method

Step 1 (normalize the evaluation information matrices).
It is easy to see thatg1-g3 are cost criteria and g4, g5 are benefit criteria. Therefore, the experts’ information matrices need to be normalized, as shown in Tables 5–7.Table 5
Normalize information matrixH-(1).
g 1 g 2 g 3 g 4 g 5 { s 6 , s 7 } { s 5 , s 6 } { s 5 } { s 5 , s 6 , s 7 } { s 7 , s 8 } { s 6 , s 7 } { s 3 , s 4 , s 5 } { s 4 , s 5 , s 6 } { s 6 , s 7 } { s 6 , s 7 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } { s 6 } { s 6 } { s 4 , s 5 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 3 , s 4 , s 5 } { s 5 , s 6 } { s 4 , s 5 } { s 6 , s 7 } { s 5 , s 6 }Table 6
Normalize information matrixH-(2).
g 1 g 2 g 3 g 4 g 5 { s 6 , s 7 , s 8 } { s 4 , s 5 , s 6 } { s 5 , s 6 } { s 4 , s 5 , s 6 } { s 6 , s 7 } { s 7 } { s 4 , s 5 } { s 5 , s 6 } { s 5 , s 6 } { s 6 } { s 6 , s 7 } { s 5 , s 6 , s 8 } { s 4 , s 5 , s 7 } { s 6 } { s 6 , s 7 } { s 4 , s 5 , s 6 } { s 6 , s 7 } { s 5 , s 6 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 6 , s 7 } { s 4 , s 5 } { s 5 , s 6 } { s 6 }Table 7
Normalize information matrixH-(3).
g 1 g 2 g 3 g 4 g 5 { s 4 , s 5 , s 6 } { s 5 , s 6 } { s 4 , s 5 } { s 5 , s 6 , s 7 } { s 5 , s 6 , s 7 } { s 5 , s 7 } { s 3 , s 4 } { s 5 , s 6 } { s 6 } { s 5 , s 6 } { s 4 , s 5 , s 6 } { s 3 , s 4 , s 5 } { s 5 , s 6 , s 7 } { s 5 , s 6 } { s 5 , s 6 } { s 4 , s 5 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 3 , s 5 } { s 5 , s 6 , s 7 } { s 5 , s 6 } { s 6 , s 7 , s 8 } { s 4 , s 5 , s 6 }Step 2 (calculate each expert’s comprehensive values of the alternatives).
Based on the interconnectedness of the attribute variables, we obtainG1=g1,g2,g3, G2=g2,g3, G3=g3, G4=g4,g5, and G5=g5. Hence, by using the HFLWEHM operator (p=q=1), the individual comprehensive values hik(i=1,2,…,5;k=1,2,3) are obtained, as shown in Table 8.Table 8
Individual comprehensive evaluation valuesH.
d 1 d 2 d 3 h 1 ( 1 ) = { s 5.586 , s 6.09 , s 6.606 } h 1 ( 2 ) = { s 4.894 , s 5.612 , s 6.336 } h 1 ( 3 ) = { s 4.692 , s 5.503 , s 6.318 } h 2 ( 1 ) = { s 5.144 , s 5.807 , s 6.488 } h 2 ( 2 ) = { s 5.217 , s 5.554 , s 5.897 } h 2 ( 3 ) = { s 5.051 , s 5.418 , s 5.792 } h 3 ( 1 ) = { s 5.445 , s 5.886 , s 6.371 } h 3 ( 2 ) = { s 5.406 , s 5.909 , s 6.844 } h 3 ( 3 ) = { s 4.716 , s 5.431 , s 6.159 } h 4 ( 1 ) = { s 4.121 , s 4.971 , s 8.827 } h 4 ( 2 ) = { s 4.847 , s 5.638 , s 6.441 } h 4 ( 3 ) = { s 4.295 , s 5.143 , s 5.994 } h 5 ( 1 ) = { s 4.919 , s 5.436 , s 5.956 } h 5 ( 2 ) = { s 5.156 , s 5.503 , s 5.855 } h 5 ( 3 ) = { s 4.936 , s 5.797 , s 6.662 }Step 3 (calculate the weight of the experts).
We obtain the experts’ similarity matrixLSM using the linguistic similarity measure formula (26). (28)LSM=~lsm12=s7.647lsm13=s7.604lsm21=s7.647~lsm23=s7.640lsm31=s7.604lsm32=s7.640~Next, the average similarity of each expertde(e=1,2,3) in the group is calculated as d1=7.625, d2=7.643, and d3=7.622, and the weight of each expert ωe(e=1,2,3) is given as ω1=0.333, ω2=0.334, and ω3=0.333.Step 4 (calculate the group comprehensive evaluation value).
Aggregate the individual comprehensive valueshi(k)(i=1,2,…,5;k=1,2,3) using Eq. (25) to obtain the group comprehensive values hi(i=1,2,…,5) of the alternatives xi(i=1,2,…,5), as h1=s5.057,s5.735,s6.178; h2=s5.137,s5.593,s5.945; h3=s5.189,s5.742,s6.146; h4=s4.421,s5.251,s5.819; and h5=s5.004,s5.579,s6.04.Step 5 (rank all the alternatives).
By calculating the score values ofhi(i=1,2,…,5), we can obtain (29)ηh1=5.657;ηh2=5.558;ηh3=5.692;ηh4=5.164;ηh5=5.541According to the binary relation of HFLETs described in Definition 3, the final ranking is x4<x5<x2<x1<x3. Therefore, the optimal choice is x3.


### 5.2. The Parametric Analysis of the HFLWEHM Operator

The above decision-making process is repeated multiple times by setting different parameter values (in Step 2) to analyze the influence of the parameters of the HFLWEHM operator on information fusion. From Figures2–6, we can see that the variance of the score values of the group evaluation to the parameters of the HFLWEHM operator is irregular. This means that the HFLWEHM operator can bring about various decision-making possibilities. Through further analysis, we find that if p=q=m, the scores obtained by the HFLWEHM operator trend to increase along with the parameters, as shown in Figure 7. Therefore, the decision-maker’s risk appetite can be reflected by the parameter m. If one takes an aggressive decision-making position, the decision-maker can use the HFLWEHM operator with a larger value of m. However, if one takes defensive decision-making, the decision-maker can use the HFLWEHM operator with a smaller value of m.Figure 2
Group scores of thex1.Figure 3
Group scores of thex2.Figure 4
Group scores of thex3.Figure 5
Group scores of thex4.Figure 6
Group scores of thex5.Figure 7
Group scores of the alternatives.


## 5.1. Using the Proposed Method

Step 1 (normalize the evaluation information matrices).
It is easy to see thatg1-g3 are cost criteria and g4, g5 are benefit criteria. Therefore, the experts’ information matrices need to be normalized, as shown in Tables 5–7.Table 5
Normalize information matrixH-(1).
g 1 g 2 g 3 g 4 g 5 { s 6 , s 7 } { s 5 , s 6 } { s 5 } { s 5 , s 6 , s 7 } { s 7 , s 8 } { s 6 , s 7 } { s 3 , s 4 , s 5 } { s 4 , s 5 , s 6 } { s 6 , s 7 } { s 6 , s 7 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } { s 6 } { s 6 } { s 4 , s 5 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 3 , s 4 , s 5 } { s 5 , s 6 } { s 4 , s 5 } { s 6 , s 7 } { s 5 , s 6 }Table 6
Normalize information matrixH-(2).
g 1 g 2 g 3 g 4 g 5 { s 6 , s 7 , s 8 } { s 4 , s 5 , s 6 } { s 5 , s 6 } { s 4 , s 5 , s 6 } { s 6 , s 7 } { s 7 } { s 4 , s 5 } { s 5 , s 6 } { s 5 , s 6 } { s 6 } { s 6 , s 7 } { s 5 , s 6 , s 8 } { s 4 , s 5 , s 7 } { s 6 } { s 6 , s 7 } { s 4 , s 5 , s 6 } { s 6 , s 7 } { s 5 , s 6 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 6 , s 7 } { s 4 , s 5 } { s 5 , s 6 } { s 6 }Table 7
Normalize information matrixH-(3).
g 1 g 2 g 3 g 4 g 5 { s 4 , s 5 , s 6 } { s 5 , s 6 } { s 4 , s 5 } { s 5 , s 6 , s 7 } { s 5 , s 6 , s 7 } { s 5 , s 7 } { s 3 , s 4 } { s 5 , s 6 } { s 6 } { s 5 , s 6 } { s 4 , s 5 , s 6 } { s 3 , s 4 , s 5 } { s 5 , s 6 , s 7 } { s 5 , s 6 } { s 5 , s 6 } { s 4 , s 5 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 3 , s 5 } { s 5 , s 6 , s 7 } { s 5 , s 6 } { s 6 , s 7 , s 8 } { s 4 , s 5 , s 6 }Step 2 (calculate each expert’s comprehensive values of the alternatives).
Based on the interconnectedness of the attribute variables, we obtainG1=g1,g2,g3, G2=g2,g3, G3=g3, G4=g4,g5, and G5=g5. Hence, by using the HFLWEHM operator (p=q=1), the individual comprehensive values hik(i=1,2,…,5;k=1,2,3) are obtained, as shown in Table 8.Table 8
Individual comprehensive evaluation valuesH.
d 1 d 2 d 3 h 1 ( 1 ) = { s 5.586 , s 6.09 , s 6.606 } h 1 ( 2 ) = { s 4.894 , s 5.612 , s 6.336 } h 1 ( 3 ) = { s 4.692 , s 5.503 , s 6.318 } h 2 ( 1 ) = { s 5.144 , s 5.807 , s 6.488 } h 2 ( 2 ) = { s 5.217 , s 5.554 , s 5.897 } h 2 ( 3 ) = { s 5.051 , s 5.418 , s 5.792 } h 3 ( 1 ) = { s 5.445 , s 5.886 , s 6.371 } h 3 ( 2 ) = { s 5.406 , s 5.909 , s 6.844 } h 3 ( 3 ) = { s 4.716 , s 5.431 , s 6.159 } h 4 ( 1 ) = { s 4.121 , s 4.971 , s 8.827 } h 4 ( 2 ) = { s 4.847 , s 5.638 , s 6.441 } h 4 ( 3 ) = { s 4.295 , s 5.143 , s 5.994 } h 5 ( 1 ) = { s 4.919 , s 5.436 , s 5.956 } h 5 ( 2 ) = { s 5.156 , s 5.503 , s 5.855 } h 5 ( 3 ) = { s 4.936 , s 5.797 , s 6.662 }Step 3 (calculate the weight of the experts).
We obtain the experts’ similarity matrixLSM using the linguistic similarity measure formula (26). (28)LSM=~lsm12=s7.647lsm13=s7.604lsm21=s7.647~lsm23=s7.640lsm31=s7.604lsm32=s7.640~Next, the average similarity of each expertde(e=1,2,3) in the group is calculated as d1=7.625, d2=7.643, and d3=7.622, and the weight of each expert ωe(e=1,2,3) is given as ω1=0.333, ω2=0.334, and ω3=0.333.Step 4 (calculate the group comprehensive evaluation value).
Aggregate the individual comprehensive valueshi(k)(i=1,2,…,5;k=1,2,3) using Eq. (25) to obtain the group comprehensive values hi(i=1,2,…,5) of the alternatives xi(i=1,2,…,5), as h1=s5.057,s5.735,s6.178; h2=s5.137,s5.593,s5.945; h3=s5.189,s5.742,s6.146; h4=s4.421,s5.251,s5.819; and h5=s5.004,s5.579,s6.04.Step 5 (rank all the alternatives).
By calculating the score values ofhi(i=1,2,…,5), we can obtain (29)ηh1=5.657;ηh2=5.558;ηh3=5.692;ηh4=5.164;ηh5=5.541According to the binary relation of HFLETs described in Definition 3, the final ranking is x4<x5<x2<x1<x3. Therefore, the optimal choice is x3.


## 5.2. The Parametric Analysis of the HFLWEHM Operator

The above decision-making process is repeated multiple times by setting different parameter values (in Step 2) to analyze the influence of the parameters of the HFLWEHM operator on information fusion. From Figures2–6, we can see that the variance of the score values of the group evaluation to the parameters of the HFLWEHM operator is irregular. This means that the HFLWEHM operator can bring about various decision-making possibilities. Through further analysis, we find that if p=q=m, the scores obtained by the HFLWEHM operator trend to increase along with the parameters, as shown in Figure 7. Therefore, the decision-maker’s risk appetite can be reflected by the parameter m. If one takes an aggressive decision-making position, the decision-maker can use the HFLWEHM operator with a larger value of m. However, if one takes defensive decision-making, the decision-maker can use the HFLWEHM operator with a smaller value of m.Figure 2
Group scores of thex1.Figure 3
Group scores of thex2.Figure 4
Group scores of thex3.Figure 5
Group scores of thex4.Figure 6
Group scores of thex5.Figure 7
Group scores of the alternatives.


## 6. Comparative Analysis

### 6.1. Comparative Analysis of Operators

In this subsection, the EHMs are compared with other classic aggregation operators, including the hesitant fuzzy linguistic weighted averaging (HFLWA) operator [48], the hesitant fuzzy linguistic weighted Heronian mean (HFLWHM) operator [49], and the hesitant fuzzy linguistic weighted Bonferroni mean (HFLWBM) operator [17]. The decision-making matrix L is used based on the linguistic term set S=sα∣α=0,1,2,…,8 to compare the above hesitant linguistic operators, as shown in Table 9.Table 9
Hesitant fuzzy linguistic information matrixL [4].
g 1 g 2 g 3 g 4 g 5 x 1 { s 5 , s 6 } { s 3 , s 4 } { s 4 , s 5 } { s 5 , s 6 , s 7 } { s 3 , s 4 , s 5 } x 2 { s 6 , s 7 } { s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 4 , s 5 , s 6 } x 3 { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } { s 6 } { s 4 , s 5 } x 4 { s 5 , s 6 } { s 6 , s 7 } { s 5 , s 6 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } w 0.2 0.3 0.2 0.2 0.1From Table10 we can see that the ranking results obtained by the proposed operators are different. This is mainly because they come from different viewpoints to aggregate information. Among the five hesitant linguistic operators, only the HFLWA operator considers argument variables that are independent of each other, which is a limitation of its application to decision-making. In addition, the HFLWA operator is a special form of the other operators. The HFLWHM and HFLWBM operators consider the associated relationship in information infusing. However, they are founded upon the assumption that every argument variable is associated with the rest of the argument variables. This assumption lacks accuracy and may impose an association on two variables that are totally unrelated. This would make the decision-making results unreliable. In view of this, we propose the HFLWEHM and HFLLSDWEHM operators to infuse the hesitant fuzzy linguistic information in which attribute weights are known or unknown.Table 10
Aggregation results by different operators(p=q=1).
Operator Aggregation values Ranking x 1 x 2 x 3 x 4 HFLWA { s 4 , s 4.65 , s 5.3 } { s 5.2 , s 5.75 , s 6.3 } { s 4.8 , s 5.55 , s 6.3 } { s 5.1 , s 5.75 , s 6.4 } x 1 < x 3 < x 2 < x 4 HFLWHM { s 4.04 , s 4.69 , s 5.34 } { s 5.27 , s 5.83 , s 6.38 } { s 4.84 , s 5.60 , s 6.36 } { s 5.18 , s 5.82 , s 6.47 } x 1 < x 3 < x 4 < x 2 HFLWBM { s 3.99 , s 4.93 , s 5.60 } { s 5.54 , s 6.16 , s 6.78 } { s 5.13 , s 5.97 , s 6.81 } { s 5.49 , s 6.17 , s 6.85 } x 1 < x 3 < x 2 < x 4 HFLWEHM { s 4.00 , s 4.65 , s 5.30 } { s 5.26 , s 5.82 , s 6.38 } { s 4.79 , s 5.56 , s 6.35 } { s 5.18 , s 5.79 , s 6.41 } x 1 < x 3 < x 4 < x 2 HFLLSDWEHM { s 3.88 , s 4.58 , s 5.28 } { s 5.20 , s 5.69 , s 6.21 } { s 4.75 , s 5.48 , s 6.23 } { s 5.05 , s 5.74 , s 6.44 } x 1 < x 3 < x 2 < x 4


### 6.2. Methods of Comparative Analysis

We dealt with the same illustrative example by utilizing two existing methods, including Rodríguez et al.’s method [50] and Xu et al.’s method [51], to explore the rationality and flexibility of the proposed hesitant fuzzy linguistic group decision-making method. The aggregation results are presented in Table 11.Table 11
Decision making results.
Method Alternative Value Ranking Xu et al.’s method(HFLOWD, λ=1) x 1 d 1 = 0.098 x 4 < x 5 < x 2 < x 1 < x 3 x 2 d 2 = 0.114 x 3 d 3 = 0.092 x 4 d 4 = 0.151 x 5 d 5 = 0.122  Rodríguez et al.’sMethod (HFLWEHM, p=q=1) x 1 N Q D 1 = 0.935 x 4 < x 2 < x 5 < x 1 < x 3 x 2 N Q D 2 = 0.794 = 0.935 x 3 N Q D 3 = 1 x 4 N Q D 4 = 0.697 x 5 N Q D 5 = 0.799We can see from Table11 that the optimal choice of the three methods is the same when the operators’ parameters have a fixed number of 1. This means that the proposed method in this paper is rational and feasible. Rodríguez et al. calculated the distance between the ideal choice and each alternative using the hesitant fuzzy linguistic fuzzy ordered weighted distance (HFLOWD) operator and then proposed a group decision-making method. However, this method does not consider the interrelationship of input arguments and cannot solve the problem in which the weight vector is represented by HFLTSs. This limits the application of this method. Xu et al.’s method is based on the envelope of HFLEs. It is pointed out in the literature [17, 52] that transforming HFLEs into an envelope may lead to information loss. For example, according to HFLEs envelope theory, h1=s1,s3,s5 and h2={s1,s5} are equivalent, which does not actually correlate with the facts. The method proposed in this paper avoids those disadvantages.


## 6.1. Comparative Analysis of Operators

In this subsection, the EHMs are compared with other classic aggregation operators, including the hesitant fuzzy linguistic weighted averaging (HFLWA) operator [48], the hesitant fuzzy linguistic weighted Heronian mean (HFLWHM) operator [49], and the hesitant fuzzy linguistic weighted Bonferroni mean (HFLWBM) operator [17]. The decision-making matrix L is used based on the linguistic term set S=sα∣α=0,1,2,…,8 to compare the above hesitant linguistic operators, as shown in Table 9.Table 9
Hesitant fuzzy linguistic information matrixL [4].
g 1 g 2 g 3 g 4 g 5 x 1 { s 5 , s 6 } { s 3 , s 4 } { s 4 , s 5 } { s 5 , s 6 , s 7 } { s 3 , s 4 , s 5 } x 2 { s 6 , s 7 } { s 6 , s 7 } { s 4 , s 5 , s 6 } { s 4 , s 5 } { s 4 , s 5 , s 6 } x 3 { s 5 , s 6 , s 7 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } { s 6 } { s 4 , s 5 } x 4 { s 5 , s 6 } { s 6 , s 7 } { s 5 , s 6 } { s 4 , s 5 , s 6 } { s 5 , s 6 , s 7 } w 0.2 0.3 0.2 0.2 0.1From Table10 we can see that the ranking results obtained by the proposed operators are different. This is mainly because they come from different viewpoints to aggregate information. Among the five hesitant linguistic operators, only the HFLWA operator considers argument variables that are independent of each other, which is a limitation of its application to decision-making. In addition, the HFLWA operator is a special form of the other operators. The HFLWHM and HFLWBM operators consider the associated relationship in information infusing. However, they are founded upon the assumption that every argument variable is associated with the rest of the argument variables. This assumption lacks accuracy and may impose an association on two variables that are totally unrelated. This would make the decision-making results unreliable. In view of this, we propose the HFLWEHM and HFLLSDWEHM operators to infuse the hesitant fuzzy linguistic information in which attribute weights are known or unknown.Table 10
Aggregation results by different operators(p=q=1).
Operator Aggregation values Ranking x 1 x 2 x 3 x 4 HFLWA { s 4 , s 4.65 , s 5.3 } { s 5.2 , s 5.75 , s 6.3 } { s 4.8 , s 5.55 , s 6.3 } { s 5.1 , s 5.75 , s 6.4 } x 1 < x 3 < x 2 < x 4 HFLWHM { s 4.04 , s 4.69 , s 5.34 } { s 5.27 , s 5.83 , s 6.38 } { s 4.84 , s 5.60 , s 6.36 } { s 5.18 , s 5.82 , s 6.47 } x 1 < x 3 < x 4 < x 2 HFLWBM { s 3.99 , s 4.93 , s 5.60 } { s 5.54 , s 6.16 , s 6.78 } { s 5.13 , s 5.97 , s 6.81 } { s 5.49 , s 6.17 , s 6.85 } x 1 < x 3 < x 2 < x 4 HFLWEHM { s 4.00 , s 4.65 , s 5.30 } { s 5.26 , s 5.82 , s 6.38 } { s 4.79 , s 5.56 , s 6.35 } { s 5.18 , s 5.79 , s 6.41 } x 1 < x 3 < x 4 < x 2 HFLLSDWEHM { s 3.88 , s 4.58 , s 5.28 } { s 5.20 , s 5.69 , s 6.21 } { s 4.75 , s 5.48 , s 6.23 } { s 5.05 , s 5.74 , s 6.44 } x 1 < x 3 < x 2 < x 4


## 6.2. Methods of Comparative Analysis

We dealt with the same illustrative example by utilizing two existing methods, including Rodríguez et al.’s method [50] and Xu et al.’s method [51], to explore the rationality and flexibility of the proposed hesitant fuzzy linguistic group decision-making method. The aggregation results are presented in Table 11.Table 11
Decision making results.
Method Alternative Value Ranking Xu et al.’s method(HFLOWD, λ=1) x 1 d 1 = 0.098 x 4 < x 5 < x 2 < x 1 < x 3 x 2 d 2 = 0.114 x 3 d 3 = 0.092 x 4 d 4 = 0.151 x 5 d 5 = 0.122  Rodríguez et al.’sMethod (HFLWEHM, p=q=1) x 1 N Q D 1 = 0.935 x 4 < x 2 < x 5 < x 1 < x 3 x 2 N Q D 2 = 0.794 = 0.935 x 3 N Q D 3 = 1 x 4 N Q D 4 = 0.697 x 5 N Q D 5 = 0.799We can see from Table11 that the optimal choice of the three methods is the same when the operators’ parameters have a fixed number of 1. This means that the proposed method in this paper is rational and feasible. Rodríguez et al. calculated the distance between the ideal choice and each alternative using the hesitant fuzzy linguistic fuzzy ordered weighted distance (HFLOWD) operator and then proposed a group decision-making method. However, this method does not consider the interrelationship of input arguments and cannot solve the problem in which the weight vector is represented by HFLTSs. This limits the application of this method. Xu et al.’s method is based on the envelope of HFLEs. It is pointed out in the literature [17, 52] that transforming HFLEs into an envelope may lead to information loss. For example, according to HFLEs envelope theory, h1=s1,s3,s5 and h2={s1,s5} are equivalent, which does not actually correlate with the facts. The method proposed in this paper avoids those disadvantages.


## 7. Conclusion

Based on different perspectives on the Heronian mean, we proposed the extended Heronian mean (EHM) operator. Next, we introduced some new operators to infuse hesitant fuzzy linguistic decision-making information, including the HFLWEHM operator and the HFLLSDWEHM operator. New aggregation operators have some good properties, such as monotonicity, boundedness, and idempotency. In addition, by changing the parameters of the operators, some of their particular forms are investigated. In addition, the linguistic axiom definition and calculation formula of the similarity measures for HFLTSs have been given. Based on the above, we presented a new technique for hesitant fuzzy linguistic MAGDM and constructed a practical example study. The advantages and disadvantages of the proposed operators and the decision-making method were discussed using several comparisons.The main contributions of this work are as follows:(1) In an important expansion of the traditional measures theory, we proposed the linguistic axiom definition and calculation formula of similarity measures for HFLTSs.(2) We introduced the HFLWEHM and HFLLSDWEHM operators, which have better characterization for the relation between input arguments compared with the traditional HM operator. Moreover, these extended HM operators can avoid the crossover operation on unrelated input arguments during infusing.(3) The group decision-making method proposed in this paper can deal with MAGDM in which the weight vector is unknown or can be expressed in numerical (or linguistic) form and therefore has more adaptability and flexibility. It should be emphasized that using the HFLLSDWEHM operator to aggregate information with unknown attribute weights can reduce the singular point in the decision-making results.However, any kind of decision method has its own advantages and disadvantages. The decision-making method in this paper is slightly complicated and is suitable only for a hesitant fuzzy linguistic environment. In the future, we will try to simplify the model calculation and extend it to other fuzzy environments. In addition, we will also apply the proposed operators in this paper to data mining, fuzzy clustering, sustainability assessment, figure and pattern recognition, and so on.



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