# Interval-Valued Hesitant Fuzzy Linguistic Multiattribute Decision-Making Method Based on Three-Parameter Heronian Mean Operators

Authors: Wang Juan; Li Qiang
Journal: Journal of Mathematics
 (2021)

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## Abstract

Numerous variants have been proposed for sets of linguistic terms and the interval-valued hesitant fuzzy set (IVHFS). In particular, the interval-valued hesitant fuzzy linguistic set (IVHFLS) is more suitable for defining the hesitancy and inconsistency inherent in the human cognitive processes of decision making. A key aggregation operator is Heronian mean (HM), based on which the correlation among aggregated arguments can be captured. However, the existing HM operators partially overlook the correlation among more than two arguments and lack the properties of idempotency and reducibility. In this work, the limitations of HM operators are first analyzed. Then, two new HM variants are introduced: three-parameter weighted Heronian mean (TPWHM) and three-parameter weighted geometric Heronian mean (TPWGHM). Thus, the reducibility, idempotency, monotonicity, and boundedness properties are proven for the two computational procedures, and unique situations are mentioned. Furthermore, two more elaborate operators are also introduced which are called the interval-valued hesitant fuzzy linguistic TPWHM (IVHFLTPWHM) and the interval-valued hesitant fuzzy linguistic TPWGHM (IVHFLTPWGHM). The main properties, as well as unique situations of these two computational procedures, are discussed. Finally, the introduced methods are clarified by illustrative examples. In addition, the parameter effects on the decision-making outcomes are discussed and comparisons with other reference methods are made.


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## Body

## 1. Introduction

MADM naturally arises in numerous practical applications, including ones in supplier selection [1], medical diagnosis [2], investment project selection [3], and so on. However, real decision-making problems are complicated by stringent requirements and constraints, including primarily the ability to process fuzzy and vague information. Under these constraints, many methods have been suggested for expressing fuzzy and vague information. Common examples of these methods are those of fuzzy set [4], vague set [5], intuitionistic fuzzy set [6], interval-valued intuitionistic fuzzy set [7, 8], and Pythagorean fuzzy set [9].The traditional methods based on fuzzy sets have been effectively extended by the hesitant fuzzy sets (HFSs) [10, 11], which allow a set of possible membership values. Several HFS variants and enhancements have then emerged. For example, Farhadinia [12, 13] proposed an interval-valued hesitant fuzzy set (IVHFS) and multiple measuring tools. Arithmetic operations for hesitant fuzzy elements (HFEs) were introduced and several aggregation operators were constructed by Xia et al. [14]. Then, Xu et al. [15] proved key HFE properties and defined corresponding distance and correlation metrics. Further, Peng [16] and Xu [17] investigated some HFS measuring tools. Xu [18] and Akram et al. [19] presented variants of the TOPSIS method within the frameworks of the conventional and neutrosophic hesitant fuzzy sets. Zhu et al. [20] developed the operators of the hesitant fuzzy geometric Bonferroni mean (HFGBM) as well as the hesitant fuzzy Choquet geometric Bonferroni mean (HFCGBM). Chen et al. [21] constructed IVHFSs whose membership values could be represented by interval values and hence developed the operator of the interval-valued hesitant fuzzy weighted averaging (IVHFWA). Wei et al. [22] introduced the operation laws for IVHFS and the scoring function. Peng et al. [23] investigated IVHFS computational procedures to aggregate scores in detail. Naz et al. [24] developed an approach based on interval-valued q-rung orthopair dual hesitant fuzzy set.For human reasoning, the interval values are generally more appropriate than crisp values for handling fuzzy and imprecise information. Indeed, decision-making problems might sometimes be too vaguely defined or too complex to employ quantitative reasoning. Alternatively, linguistic qualitative variables could be effectively employed in many applications. A linguistic variable was firstly introduced by Zadeh [4] to express qualitative uncertain information. A single or interval-type expression is denoted by a linguistic variable [25, 26]. Then, HFLSs were investigated by Lin et al. [27] to depict fuzzy object attributes and also indicate potential membership values of a linguistic term. However, under conditions of information fuzziness and complexity, crisp decisions become challenging to make. Interval-type values are generally more suitable for handling information fuzziness and impreciseness. Consequently, IVHFLS was studied by Wang et al. [28] to combine the key advantages of the linguistic approach based on fuzzy concept and IVHFSs. Then, Gong et al. [29] presented an IVHLFS scoring function and proposed a corresponding TOPSIS-based ranking approach. Meng et al. [30] defined the IVHLFS laws of operation and introduced several IVHLFS aggregation operators. Mao et al. [31] addressed MADM problems using cloud computational procedures based on IVHFLNs. Feng et al. [32] proposed a MADM approach with the operator related to the interval-valued q-rung dual hesitant linguistic Maclaurin symmetric mean operators.Nowadays, the interrelationship among aggregated arguments has been accounted for through the design of novel aggregation operators including the Heronian mean (HM) and the Bonferroni mean (BM) [33, 34]. The HM operators were found to have certain advantages over BM operators, which overlook the operator correlation with attributes [35, 36]. The HM operators were first introduced by Beliakov et al. [37], who employed the operators for solving MADM problems. Liu et al. [38] proposed a generalized HM (GHM) operator and examined some special cases of this operator. Furthermore, concrete applications of the HM operators have been explored. Yu [39] proposed two operators: the intuitionistic fuzzy geometric HM (IFGHM) as well as the intuitionistic fuzzy geometric weighted HM (IFGWHM). Liu [40] introduced the generalized Heronian OWA operator, which shares similar characteristics with the BM and Bonferroni OWA operators. Liu et al. [41] developed a MAGDM algorithm utilizing the operator based on the 2-tuple linguistic neutrosophic Dombi power HM. Xu et al. [42] implemented the copula HM when a linguistic interval-valued intuitionistic fuzzy decision making is a concern. Ayub et al. [43] constructed several cubic fuzzy HM Dombi aggregation operators, which were based on combining the Dombi operational laws and HM operators. Lin et al. [44] presented a MADM algorithm with picture fuzzy interactional partitioned HM aggregation operators. Tian et al. [45] developed an MCDM method with the partitioned HM (PHM) operator and the Shapley fuzzy measure with a single-valued neutrosophic set.Although significant progress has been made in the theory and applications of the HM, HFS, HFLS, and IVHFLS operators, some shortcomings persist. (1) The existing HM operators consider only the correlation among pairs of aggregated arguments but ignore higher-order complex correlation patterns among three or more aggregated arguments. (2) The generalized HM (GHM) and weighted geometric HM (WGHM) operators do not satisfy the properties of reducibility and idempotency and hence cannot be effectively employed in addressing real decision-making problems. Thus, in our work, we have two essential contributions. Firstly, we introduce two novel HM operators, namely, the three-parameter weighted HM (TPWHM) and the three-parameter weighted geometric HM (TPWGHM), which are shown to be reducible, idempotent, monotone, and bounded and they do not suffer from deficiencies that are already mentioned. The other sections of the manuscript are organized as follows. Key concepts and operational laws for interval numbers, IVHFSs, and (IVHFLSs) are reviewed in Section2. In Section 3, we equip the HM operators with three parameters and propose the three-parameter weighted HM (TPWHM) and the three-parameter weighted geometric HM (TPWGHM) operators. Section 4 deals with the TPWHM and TPWGHM operators which are extended to aggregate IVHFL information. Section 5 gives computational tools for the decision-making procedure. Section 6 points outs an example based on our proposed method and demonstrates the advantages of our approach through comparisons and sensitivity analysis. Section 7 concludes the research.


## 2. Preliminaries

Here, a brief review is presented containing basic concepts and operations for interval numbers, IVHF numbers, and FL numbers.


### 2.1. Interval Numbers

Definition 1 (see [46]).
a˜ denotes an interval number, i.e., a˜=aL,aU or a˜=x|aL≤x≤aU where the lower and upper endpoints of interval number are denoted by aL and aU in the support of a˜, respectively. Two intervals are denoted by a˜=aL,aU and b˜=bL,bU. For these numbers, the arithmetic of interval numbers is expressed by(1)
a˜⊕b˜=aL+bL,aU+bU.(2)
a˜⊗b˜=aLbL,aUbU.(3)
λ⊗a˜=λaL,λaU, λ≥0.(4)
aλ=aLλ,aUλ, λ≥0.Definition 2 (see [46]).
Two intervals are denoted bya˜=aL,aU and b˜=bL,bU. Then, lena˜=aU−aL and lenb˜=bU−bL are called the lengths of those intervals. The possibility degree for a˜≥b˜ is given as follows:(1)pa˜≥b˜=max1−maxaU−bLlena˜+lenb˜,0,0.
Likewise, the possibility degree thatb˜≥a˜ is given as follows:(2)pb˜≥a˜=max1−maxaU−bLlena˜+lenb˜,0,0.


### 2.2. IVHFS

Definition 3 (see [46]).
A nonempty fixed set is denoted byX. The HFS HX is defined as follows:(3)H=x,h˜Hx|x∈X,where the function h˜Hx:X returns a subset of 0,1, which represents the possible membership degrees of a given element x∈X in H. The basic HFS unit is h˜=h˜Hx.Definition 4 (see [47]).
A nonempty fixed set is denoted by X. IVHFSE on X is(4)E=x,h˜Ex|x∈X,where h˜Ex: X is the set containing all possible interval fuzzy values in 0,1, which extracts the possible degrees of the membership of the element x∈X in E. h˜=h˜Exi=γ˜=γL,γU|γ˜∈h˜Exi is called an IVHFE.Definition 5 (see [46]).
Leth˜=γL,γUh˜1=γ1L,γ1U and h˜2=γ2L,γ2U be three IVHFEs.
The IVHFE operational rules are(1)
h˜λ=∪γ˜∈h˜γLλ,γUλ.(2)
λh˜=∪γ˜∈h˜1−1−γLλ,1−1−γUλ,λ>0.(3)
h˜1⊕h˜2=∪γ˜1∈h˜1,γ˜2∈h˜2γ1L+γ2L−γ1Lγ2L,γ1U+γ2U−γ1Uγ2U.(4)
h˜1⊗h˜2=∪γ˜1∈h˜1,γ˜2∈h˜2γ1Lγ2L,γ1Uγ2U.Definition 6 (see [47])).
The scoring function for any IVHFEh˜ is(5)sh˜=1#h˜∑γ˜∈h˜γ˜,where #h˜ is the number of interval values in h˜. Obviously, sh˜ is an interval value belonging to 0,1.
Leth˜1 and h˜2 be any two IVHFEs. The IVHFE comparison laws are as follows:(1)
Ifsh˜1≥sh˜2, then h˜1≥h˜2.(2)
Ifsh˜1<sh˜2, then h˜1<h˜2.
Two score functions can be compared using equation (5), while the magnitude of two IVHFEs can be evaluated based on Definition 2.


### 2.3. IVHFLSs

IVHFLSs are defined based on linguistic sets and IVHFSs.Definition 7 (see [28]).
Letx=x1,x2,⋯,xn denote a reference set sθx∈S, θx∈0,1,2,…,2t. The IVHFLS is defined by(6)A=x,sθx,ΓAx,x∈X,where ΓAx denotes a set of finite closed intervals of [0, 1]:(7)ΓAx=ai−,ai+,i=1,2,…,#ΓAx,where the count of the interval values of A is expressed by#ΓAx. We call x,sθx,ΓAx an interval-valued hesitant fuzzy linguistic element (IVHFLE).
Likewise, utilizing the IVHFLE and HFLE computational procedures, basic IVHFLE operational laws are given next.Definition 8.
Leta=sθa,Γθa, a1=sθa1,Γθa1, and a2=sθa2,Γθa2 be three IVHFLEs. Then, γθa=γθaL,γθaU∈Γθa, γθa1=γa1L,γa1U∈Γθa1, γθa2=γa2L,γa2U∈Γθa2. The operations on these elements are(1)
nega=s2t−θa,∪γθa∈Γθa1−γaL,1−γaU.(2)
a1⊕a2=〈sθa1+θa2,∪γθa1∈Γθa1,γθa2∈Γθa2γa1L+γa2L−γa1Lγa2L,γa1U+γa2U−γa1Uγa2U〉.(3)
a1⊗a2=〈sθa1⊗θa2,∪γθa1∈Γθa1,γθa2∈Γθa2γa1Lγa2L,γa1Uγa2U〉.(4)
aλ=〈sθaλ,∪γθa∈ΓθaγaLλ,γaUλ.(5)
λa=〈sλθ(a),∪γθa∈Γθ(a)1−1−γaLλ,1−1−γaUλ〉, λ>0.Definition 9 (see [27]).
The scoring function for any IVHFLE a is(8)sa=12#Γθa∑γθa∈Γθa1γ,where the count of the interval values h˜ is denoted by #Γθa and s(a) is an interval value in [0, 1].


## 2.1. Interval Numbers

Definition 1 (see [46]).
a˜ denotes an interval number, i.e., a˜=aL,aU or a˜=x|aL≤x≤aU where the lower and upper endpoints of interval number are denoted by aL and aU in the support of a˜, respectively. Two intervals are denoted by a˜=aL,aU and b˜=bL,bU. For these numbers, the arithmetic of interval numbers is expressed by(1)
a˜⊕b˜=aL+bL,aU+bU.(2)
a˜⊗b˜=aLbL,aUbU.(3)
λ⊗a˜=λaL,λaU, λ≥0.(4)
aλ=aLλ,aUλ, λ≥0.Definition 2 (see [46]).
Two intervals are denoted bya˜=aL,aU and b˜=bL,bU. Then, lena˜=aU−aL and lenb˜=bU−bL are called the lengths of those intervals. The possibility degree for a˜≥b˜ is given as follows:(1)pa˜≥b˜=max1−maxaU−bLlena˜+lenb˜,0,0.
Likewise, the possibility degree thatb˜≥a˜ is given as follows:(2)pb˜≥a˜=max1−maxaU−bLlena˜+lenb˜,0,0.


## 2.2. IVHFS

Definition 3 (see [46]).
A nonempty fixed set is denoted byX. The HFS HX is defined as follows:(3)H=x,h˜Hx|x∈X,where the function h˜Hx:X returns a subset of 0,1, which represents the possible membership degrees of a given element x∈X in H. The basic HFS unit is h˜=h˜Hx.Definition 4 (see [47]).
A nonempty fixed set is denoted by X. IVHFSE on X is(4)E=x,h˜Ex|x∈X,where h˜Ex: X is the set containing all possible interval fuzzy values in 0,1, which extracts the possible degrees of the membership of the element x∈X in E. h˜=h˜Exi=γ˜=γL,γU|γ˜∈h˜Exi is called an IVHFE.Definition 5 (see [46]).
Leth˜=γL,γUh˜1=γ1L,γ1U and h˜2=γ2L,γ2U be three IVHFEs.
The IVHFE operational rules are(1)
h˜λ=∪γ˜∈h˜γLλ,γUλ.(2)
λh˜=∪γ˜∈h˜1−1−γLλ,1−1−γUλ,λ>0.(3)
h˜1⊕h˜2=∪γ˜1∈h˜1,γ˜2∈h˜2γ1L+γ2L−γ1Lγ2L,γ1U+γ2U−γ1Uγ2U.(4)
h˜1⊗h˜2=∪γ˜1∈h˜1,γ˜2∈h˜2γ1Lγ2L,γ1Uγ2U.Definition 6 (see [47])).
The scoring function for any IVHFEh˜ is(5)sh˜=1#h˜∑γ˜∈h˜γ˜,where #h˜ is the number of interval values in h˜. Obviously, sh˜ is an interval value belonging to 0,1.
Leth˜1 and h˜2 be any two IVHFEs. The IVHFE comparison laws are as follows:(1)
Ifsh˜1≥sh˜2, then h˜1≥h˜2.(2)
Ifsh˜1<sh˜2, then h˜1<h˜2.
Two score functions can be compared using equation (5), while the magnitude of two IVHFEs can be evaluated based on Definition 2.


## 2.3. IVHFLSs

IVHFLSs are defined based on linguistic sets and IVHFSs.Definition 7 (see [28]).
Letx=x1,x2,⋯,xn denote a reference set sθx∈S, θx∈0,1,2,…,2t. The IVHFLS is defined by(6)A=x,sθx,ΓAx,x∈X,where ΓAx denotes a set of finite closed intervals of [0, 1]:(7)ΓAx=ai−,ai+,i=1,2,…,#ΓAx,where the count of the interval values of A is expressed by#ΓAx. We call x,sθx,ΓAx an interval-valued hesitant fuzzy linguistic element (IVHFLE).
Likewise, utilizing the IVHFLE and HFLE computational procedures, basic IVHFLE operational laws are given next.Definition 8.
Leta=sθa,Γθa, a1=sθa1,Γθa1, and a2=sθa2,Γθa2 be three IVHFLEs. Then, γθa=γθaL,γθaU∈Γθa, γθa1=γa1L,γa1U∈Γθa1, γθa2=γa2L,γa2U∈Γθa2. The operations on these elements are(1)
nega=s2t−θa,∪γθa∈Γθa1−γaL,1−γaU.(2)
a1⊕a2=〈sθa1+θa2,∪γθa1∈Γθa1,γθa2∈Γθa2γa1L+γa2L−γa1Lγa2L,γa1U+γa2U−γa1Uγa2U〉.(3)
a1⊗a2=〈sθa1⊗θa2,∪γθa1∈Γθa1,γθa2∈Γθa2γa1Lγa2L,γa1Uγa2U〉.(4)
aλ=〈sθaλ,∪γθa∈ΓθaγaLλ,γaUλ.(5)
λa=〈sλθ(a),∪γθa∈Γθ(a)1−1−γaLλ,1−1−γaUλ〉, λ>0.Definition 9 (see [27]).
The scoring function for any IVHFLE a is(8)sa=12#Γθa∑γθa∈Γθa1γ,where the count of the interval values h˜ is denoted by #Γθa and s(a) is an interval value in [0, 1].


## 3. Three-Parameter Heronian Mean (TPHM) Operators

The HM is a key aggregation operator, through which the relations among individual arguments can be captured. Compared with other aggregation operators, the HM operator can more effectively capture the correlation of aggregated arguments. However, existing HM operators consider only pairwise correlation among arguments and also do not satisfy the idempotency and reducibility properties. To overcome these limitations, we propose the operators of the three-parameter weighted HM (TPWHM). Ternary interconnections between arguments could be captured by new operators, and also some desirable properties could be demonstrated.


### 3.1. Heronian Mean

The Heronian mean (HM) is an essential aggregation operator for capturing pairwise argument correlation [26, 28, 29]. This operator can be mathematically defined as follows.Definition 10 (see [38]).
A nonnegative number set is denoted bycii=1,2,…,n. HM is expressed by(9)HMc1,c2,…,cn=2nn+1∑i=1n∑j=incicj.
The idempotency, monotonicity, and boundedness properties hold for this operator.Definition 11 (see [38]).
A nonnegative number set is denoted bycii=1,2,…,n. For p,q≥0, the generalized HM (GHM) is expressed by(10)GHMp,qc1,c2,…,cn=2nn+1∑i=1n∑j=incipcjq1/p+q.
The idempotency, monotonicity, and boundedness properties again hold for this operator [28].Definition 12 (see [36]).
Letc1,c2,…,cn be a set of nonnegative numbers. Also, a weight vector w=w1,w2,…,wnT is assumed such that wi>0 and ∑i=1nwi=1. For p,q≥0, the generalized weighted HM (GWHM) operator is(11)GWHMp,qc1,c2,…,cn=2nn+1∑i=1n∑j=inwicipwjcjq1/p+q.Definition 13 (see [39]).
Letc1,c2,…,cn be a set of n nonnegative numbers. w=w1,w2,…,wnT is assumed such that wi>0 and ∑i=1nwi=1. For p,q≥0, the weighted geometric HM (WGHM) operator is(12)WGHMp,qc1,c2,…,cn=1p+q∏i=1n∏j=inpciwi+qcjwj2/nn+1.
Although it looks counter-intuitive, the idempotency and reducibility properties are not satisfied by the GWHM and WGHM operators [26, 29]. To address these limitations, alternative computational procedures could be studied further.


### 3.2. Three-Parameter Weighted HM (TPWHM) Operator

As mentioned above, the existing HM operators overlook the ternary and higher-order correlations among aggregated arguments. Also, idempotency and reducibility are not satisfied by these operators. Real-world decision-making scenarios would be negatively affected by these drawbacks. To address these drawbacks, we further develop two novel HM operators and investigate four of their key properties. These operators are the three-parameter weighted HM (TPWHM).Definition 18.
Letc1,c2,…,cn be a set of n nonnegative numbers. The weight vector is set to be w=w1,w2,…,wnT such that wi>0 and ∑i=1nwi=1. For p,q,r≥0, the three-parameter weighted HM (TPWHM) operator is(13)TPWHMp,q,rc1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwkcipcjqckr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
For the TPWHM operator, we show that the reducibility, idempotency, monotonicity, and boundedness properties hold.Theorem 1 (TPWHM is reducible).
The weight vector is set to bew=1/n,1/n,…,1/nT. Hence, the TPWHM is reducible to revised HM (RHM) expressed by(14)RHMp,q,rc1,c2,…,cn=2∑δ=1nδδ+1∑i=1n∑j=in∑k=jncipcjqckr1/p+q+r.Proof.
Based on Definition18 and given that w=1/n,1/n,…,1/nT, we get(15)TPWHMp,q,rc1,c2,…,cn=∑i=1n∑j=in∑k=j1/n3∑i=1n∑j=in∑k=jn1/n31/p+q+r=2∑δ=1nδδ+1∑i=1n∑j=in∑k=jncipcjqckr1/p+q+r=RHMp,q,rc1,c2,…,cn.Theorem 2 (TPWHM is idempotent).
Given the set of nonnegative numbersc1,c2,…,cn,(16)TPWHMp,q,rc1,c2,…,cn=ccould be obtained.Proof.
Givenc=c1,c2,…,cn and based on equation (13), we get(17)TPWHMp,q,rc1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwkcpcqcr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r=c∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r=c.Theorem 3 (TPWHM is monotone).
Two sets of nonnegative numbers are denoted byc=c1,c2,…,c and b=b1,b2,…,bn, respectively. If ci≥bi for all i, then(18)TPWHMp,q,rc1,c2,…,cn≥TPWHMp,q,rb1,b2,…,bn.Proof.
Asp,q,r≥0 and ci≥bi for all i, then according to equation (13), we have(19)wiwjwkcipcjqckr≥wiwjwkbipbjqbkr.
Furthermore,(20)∑i=1n∑j=in∑k=jnwiwjwkcipcjqckr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥∑i=1n∑j=in∑k=jnwiwjwkbipbjqbkr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
Therefore,(21)TPWHMp,q,rc1,c2,…,cn≥TPWHMp,q,rb1,b2,…,bn.Theorem 4 (TPWHM is bounded).
The TPWHM operator output is bounded between highest and lowest scores, i.e.,(22)minc1,c2,…,cn≤TPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn, β=maxc1,c2,…,cn, and utilizing the idempotent property of TPWHM (Theorem 2), TPWHMp,q,rα,α,…,α=α, TPWHMp,q,rβ,β,…,β=β are obtained.
Sinceα≤ci≤β and based on the TPWHM monotonicity (Theorem 3), we have(23)TPWHMp,q,rα,α,…,α≤TPWHMp,q,rc1,c2,…,cn≤TPWHMp,q,rβ,β,…,β≤TPWHMp,q,rβ,β,…,β.
Therefore,α≤TPWHMp,q,rc1,c2,…,cn≤β:(24)minc1,c2,…,cn≤TPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.
Two special TPWHM cases are defined as follows for different settings of the parametersp, q, and r.(1)
Whenr=0, we have(25)TPWHM1,1,0c1,c2,…,cn=∑i=1n∑j=inn+1−jwiwjcipcjq∑i=1n∑j=inn+1−jwiwj1/p+q.(2)
At a value of 1 for each ofp, q, and r, we have(26)TPWHM1,1,1c1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwkcicjck∑i=1n∑j=in∑k=jnwiwjwk1/3.


### 3.3. Three-Parameter Weighted Geometric HM (TPWGHM) Operator

We introduce here the definition, properties, and special cases of TPWGHM.Definition 19.
A set ofn nonnegative numbers is denoted by c1,c2,…,cn. w=w1,w2,…,wnT such that wi>0 and ∑i=1nwi=1. For p,q,r≥0, the three-parameter weighted geometric HM (TPWGHM) operator is(27)TPWGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb.
The TPWGHM operator is reducible, idempotent, monotone, and bounded as we show next.Theorem 5 (TPWGHM is reducible).
Forw=1/n,1/n,…,1/nT, the TPWGHM can be downgraded to revised geometric HM (RGHM):(28)RGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2∑δ=1nδδ+1.Proof.
Forw=1/n,1/n,…,1/nT, equation (26) leads to(29)TPWGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2n+1−j1/n∑δ=1nδδ+1n+1−j1/n=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2∑δ=1nδδ+1=RGHMp,q,rc1,c2,…,cn.
Therefore, the TPWGHM operator is reducible.Theorem 6 (TPWGHM is idempotent).
For the nonnegative number setc1,c2,…,cn, the TPWGHM is idempotent, i.e.,(30)TPWGHMp,q,rc1,c2,…,cn=c.Proof.
For the nonnegative numbersc1,c2,…,cn, equation (26) leads to(31)TPWGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpc+qc+rc2n+1−jwk∑δ=1nδδ+1∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1n∑j=in∑k=jnn+1−jwk∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1n∑j=inn+1−j∑k=jnwk∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1nn+1−i+⋯+1=1p+q+rpc+qc+rc=c.Theorem 7 (TPWGHM is monotone).
For any two sets of nonnegative numbersc1,c2,…,cn and b1,b2,…,bn, if ci≥bi for all i,(32)TPWGHMp,q,rc1,c2,…,cn≥TPWGHMp,q,rb1,b2,…,bnare obtained.Proof.
Asp,q,r≥0 and ci≥bi for all i, equation (26) leads to(33)pci+qcj+rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb≥pbi+qbj+rbk2n+1−jwk∑δ=1nδδ+1∑b=jnwb.
Furthermore,(34)1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb≥1p+q+r∏i=1n∏j=in∏k=jnpbi+qbj+rbk2n+1−jwk∑δ=1nδδ+1∑b=jnwb.
Therefore,(35)TPWGHMp,q,rc1,c2,…,cn≥TPWGHMp,q,rb1,b2,…,bn.Theorem 8 (TPWGHM is bounded).
The TPWGHM operator output is bounded between maximum and minimum values, i.e.,(36)minc1,c2,…,cn≤TPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn and β=maxc1,c2,…,cn. The TPWGHM idempotency (Theorem 6) leads to(37)TPWGHMp,q,rα,α,…,α=α,TPWGHMp,q,rβ,…,β=β.
Sinceα≤ci≤β, the TPWGHM monotonicity (Theorem 7) leads to(38)TPWGHMp,q,rα,α,…,α≤TPWGHMp,q,rc1,c2,…,cn≤TPWGHMp,q,rβ,β,…β.
This is equivalent toα≤TPWGHMp,q,rc1,c2,⋯,cn≤β. Then, we get(39)minc1,c2,…,cn≤TPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.
Based on parametersp, q and r, two special TPWGHM cases follow.(1)
Whenr = 0, we have(40)TPWGHMp,q,0c1,c2,…,cn=1p+q∏i=1n∏j=inpci+qcj2n+1−j∑δ=1nδδ+1.(2)
At a value of 1 for each ofp, q, and r, we have(41)TPWGHM1,1,1c1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwk,ci,cj,ck∑i=1n∑j=in∑k=jnwiwjwk1/3.


## 3.1. Heronian Mean

The Heronian mean (HM) is an essential aggregation operator for capturing pairwise argument correlation [26, 28, 29]. This operator can be mathematically defined as follows.Definition 10 (see [38]).
A nonnegative number set is denoted bycii=1,2,…,n. HM is expressed by(9)HMc1,c2,…,cn=2nn+1∑i=1n∑j=incicj.
The idempotency, monotonicity, and boundedness properties hold for this operator.Definition 11 (see [38]).
A nonnegative number set is denoted bycii=1,2,…,n. For p,q≥0, the generalized HM (GHM) is expressed by(10)GHMp,qc1,c2,…,cn=2nn+1∑i=1n∑j=incipcjq1/p+q.
The idempotency, monotonicity, and boundedness properties again hold for this operator [28].Definition 12 (see [36]).
Letc1,c2,…,cn be a set of nonnegative numbers. Also, a weight vector w=w1,w2,…,wnT is assumed such that wi>0 and ∑i=1nwi=1. For p,q≥0, the generalized weighted HM (GWHM) operator is(11)GWHMp,qc1,c2,…,cn=2nn+1∑i=1n∑j=inwicipwjcjq1/p+q.Definition 13 (see [39]).
Letc1,c2,…,cn be a set of n nonnegative numbers. w=w1,w2,…,wnT is assumed such that wi>0 and ∑i=1nwi=1. For p,q≥0, the weighted geometric HM (WGHM) operator is(12)WGHMp,qc1,c2,…,cn=1p+q∏i=1n∏j=inpciwi+qcjwj2/nn+1.
Although it looks counter-intuitive, the idempotency and reducibility properties are not satisfied by the GWHM and WGHM operators [26, 29]. To address these limitations, alternative computational procedures could be studied further.


## 3.2. Three-Parameter Weighted HM (TPWHM) Operator

As mentioned above, the existing HM operators overlook the ternary and higher-order correlations among aggregated arguments. Also, idempotency and reducibility are not satisfied by these operators. Real-world decision-making scenarios would be negatively affected by these drawbacks. To address these drawbacks, we further develop two novel HM operators and investigate four of their key properties. These operators are the three-parameter weighted HM (TPWHM).Definition 18.
Letc1,c2,…,cn be a set of n nonnegative numbers. The weight vector is set to be w=w1,w2,…,wnT such that wi>0 and ∑i=1nwi=1. For p,q,r≥0, the three-parameter weighted HM (TPWHM) operator is(13)TPWHMp,q,rc1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwkcipcjqckr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
For the TPWHM operator, we show that the reducibility, idempotency, monotonicity, and boundedness properties hold.Theorem 1 (TPWHM is reducible).
The weight vector is set to bew=1/n,1/n,…,1/nT. Hence, the TPWHM is reducible to revised HM (RHM) expressed by(14)RHMp,q,rc1,c2,…,cn=2∑δ=1nδδ+1∑i=1n∑j=in∑k=jncipcjqckr1/p+q+r.Proof.
Based on Definition18 and given that w=1/n,1/n,…,1/nT, we get(15)TPWHMp,q,rc1,c2,…,cn=∑i=1n∑j=in∑k=j1/n3∑i=1n∑j=in∑k=jn1/n31/p+q+r=2∑δ=1nδδ+1∑i=1n∑j=in∑k=jncipcjqckr1/p+q+r=RHMp,q,rc1,c2,…,cn.Theorem 2 (TPWHM is idempotent).
Given the set of nonnegative numbersc1,c2,…,cn,(16)TPWHMp,q,rc1,c2,…,cn=ccould be obtained.Proof.
Givenc=c1,c2,…,cn and based on equation (13), we get(17)TPWHMp,q,rc1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwkcpcqcr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r=c∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r=c.Theorem 3 (TPWHM is monotone).
Two sets of nonnegative numbers are denoted byc=c1,c2,…,c and b=b1,b2,…,bn, respectively. If ci≥bi for all i, then(18)TPWHMp,q,rc1,c2,…,cn≥TPWHMp,q,rb1,b2,…,bn.Proof.
Asp,q,r≥0 and ci≥bi for all i, then according to equation (13), we have(19)wiwjwkcipcjqckr≥wiwjwkbipbjqbkr.
Furthermore,(20)∑i=1n∑j=in∑k=jnwiwjwkcipcjqckr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥∑i=1n∑j=in∑k=jnwiwjwkbipbjqbkr∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
Therefore,(21)TPWHMp,q,rc1,c2,…,cn≥TPWHMp,q,rb1,b2,…,bn.Theorem 4 (TPWHM is bounded).
The TPWHM operator output is bounded between highest and lowest scores, i.e.,(22)minc1,c2,…,cn≤TPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn, β=maxc1,c2,…,cn, and utilizing the idempotent property of TPWHM (Theorem 2), TPWHMp,q,rα,α,…,α=α, TPWHMp,q,rβ,β,…,β=β are obtained.
Sinceα≤ci≤β and based on the TPWHM monotonicity (Theorem 3), we have(23)TPWHMp,q,rα,α,…,α≤TPWHMp,q,rc1,c2,…,cn≤TPWHMp,q,rβ,β,…,β≤TPWHMp,q,rβ,β,…,β.
Therefore,α≤TPWHMp,q,rc1,c2,…,cn≤β:(24)minc1,c2,…,cn≤TPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.
Two special TPWHM cases are defined as follows for different settings of the parametersp, q, and r.(1)
Whenr=0, we have(25)TPWHM1,1,0c1,c2,…,cn=∑i=1n∑j=inn+1−jwiwjcipcjq∑i=1n∑j=inn+1−jwiwj1/p+q.(2)
At a value of 1 for each ofp, q, and r, we have(26)TPWHM1,1,1c1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwkcicjck∑i=1n∑j=in∑k=jnwiwjwk1/3.


## 3.3. Three-Parameter Weighted Geometric HM (TPWGHM) Operator

We introduce here the definition, properties, and special cases of TPWGHM.Definition 19.
A set ofn nonnegative numbers is denoted by c1,c2,…,cn. w=w1,w2,…,wnT such that wi>0 and ∑i=1nwi=1. For p,q,r≥0, the three-parameter weighted geometric HM (TPWGHM) operator is(27)TPWGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb.
The TPWGHM operator is reducible, idempotent, monotone, and bounded as we show next.Theorem 5 (TPWGHM is reducible).
Forw=1/n,1/n,…,1/nT, the TPWGHM can be downgraded to revised geometric HM (RGHM):(28)RGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2∑δ=1nδδ+1.Proof.
Forw=1/n,1/n,…,1/nT, equation (26) leads to(29)TPWGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2n+1−j1/n∑δ=1nδδ+1n+1−j1/n=1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2∑δ=1nδδ+1=RGHMp,q,rc1,c2,…,cn.
Therefore, the TPWGHM operator is reducible.Theorem 6 (TPWGHM is idempotent).
For the nonnegative number setc1,c2,…,cn, the TPWGHM is idempotent, i.e.,(30)TPWGHMp,q,rc1,c2,…,cn=c.Proof.
For the nonnegative numbersc1,c2,…,cn, equation (26) leads to(31)TPWGHMp,q,rc1,c2,…,cn=1p+q+r∏i=1n∏j=in∏k=jnpc+qc+rc2n+1−jwk∑δ=1nδδ+1∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1n∑j=in∑k=jnn+1−jwk∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1n∑j=inn+1−j∑k=jnwk∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1nn+1−i+⋯+1=1p+q+rpc+qc+rc=c.Theorem 7 (TPWGHM is monotone).
For any two sets of nonnegative numbersc1,c2,…,cn and b1,b2,…,bn, if ci≥bi for all i,(32)TPWGHMp,q,rc1,c2,…,cn≥TPWGHMp,q,rb1,b2,…,bnare obtained.Proof.
Asp,q,r≥0 and ci≥bi for all i, equation (26) leads to(33)pci+qcj+rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb≥pbi+qbj+rbk2n+1−jwk∑δ=1nδδ+1∑b=jnwb.
Furthermore,(34)1p+q+r∏i=1n∏j=in∏k=jnpci+qcj+rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb≥1p+q+r∏i=1n∏j=in∏k=jnpbi+qbj+rbk2n+1−jwk∑δ=1nδδ+1∑b=jnwb.
Therefore,(35)TPWGHMp,q,rc1,c2,…,cn≥TPWGHMp,q,rb1,b2,…,bn.Theorem 8 (TPWGHM is bounded).
The TPWGHM operator output is bounded between maximum and minimum values, i.e.,(36)minc1,c2,…,cn≤TPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn and β=maxc1,c2,…,cn. The TPWGHM idempotency (Theorem 6) leads to(37)TPWGHMp,q,rα,α,…,α=α,TPWGHMp,q,rβ,…,β=β.
Sinceα≤ci≤β, the TPWGHM monotonicity (Theorem 7) leads to(38)TPWGHMp,q,rα,α,…,α≤TPWGHMp,q,rc1,c2,…,cn≤TPWGHMp,q,rβ,β,…β.
This is equivalent toα≤TPWGHMp,q,rc1,c2,⋯,cn≤β. Then, we get(39)minc1,c2,…,cn≤TPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.
Based on parametersp, q and r, two special TPWGHM cases follow.(1)
Whenr = 0, we have(40)TPWGHMp,q,0c1,c2,…,cn=1p+q∏i=1n∏j=inpci+qcj2n+1−j∑δ=1nδδ+1.(2)
At a value of 1 for each ofp, q, and r, we have(41)TPWGHM1,1,1c1,c2,…,cn=∑i=1n∑j=in∑k=jnwiwjwk,ci,cj,ck∑i=1n∑j=in∑k=jnwiwjwk1/3.


## 4. Three-Parameter Heronian Mean Operators for IVHFL Information

While crisp values can be aggregated with the TPWHM and TPWGHM, these operators fail to combine IVHFL information. Here, the limitation could be addressed through the extension of these two operators to novel HM operators, namely, IVHFLTPWHM and IVHFLTPWHM.


### 4.1. Interval-Valued Hesitant Fuzzy Linguistic TPWHM (IVHFLTPWHM) Operator

Definition 20.
A set of IVHFLEs is defined byc1,c2,…,cn. The weight is set to be w=w1,w2,…,wnT such that wi>0 and ∑i=1nwi=1. For p,q,r≥0, the operator of (IVHFLTPWHM) is defined as(42)IVHFLTPWHMp,q,rc1,c2,…,cn=1∑i=1n∑j=in∑k=jnwiwjwk⊕i=1n⊕j=in⊕k=jnwiwjwkcip⊗cjq⊗ckr1/p+q+r.Theorem 9.
A set of IVHFLEs is defined byci=sθci,Γθcii=1,2,…,n. A weight vector is set to be w=w1,w2,…,wnTsatisfying wi>0 and ∑i=1nwi=1. Utilizing IVHFLTPWHM, the combined outcome is an IVHFLE. For p,q,r≥0,(43)IVHFLTPWHMp,q,rc1,c2,…,cn=s∑i=1n∑j=in∑k=jnwiwjwkθcipθcjqθckr1/p+q+r,∪γθai∈Γθci,γθcj∈Γθcj,γθck∈Γθck,1−1−1−∏i=1n∏j=in∏k=jn1−γθciLpγθcjLqγθckLrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r,(44)1−1−1−∏i=1n∏j=in∏k=jn1−γθciUpγθcjUqγθckUrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
We show that the IVHFLTPWHM operator is idempotent, monotone, and bounded.Theorem 10 (IVHFLTPWHM is idempotent).
Set the nonnegative numbersc=c1,c2,…,cn such that c=sθc,Γθc. Then, we have(45)IVHFLTPWHMp,q,rc1,c2,…,cn=c.Proof.
Asa=c1,c2,…,cn, equation (41) leads to(46)IVHFLTPWHMp,q,rc1,c2,…,cn=1∑i=1n∑j=in∑k=jnwiwjwk⊕i=1n⊕j=in⊕k=jnwiwjwkcp+q+r1/p+q+r=c∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r=c.Theorem 11 (IVHFLTPWHM is monotone).
Letci=sθci,Γθci, and bi=sθbi,Γθbi be two IVHFLEs, γθci=γθciL,γθciU∈Γθci, and γθbi=γθbiL,γθbiU∈Γθbi. If ci≥bi for all i, then(47)IVHFLTPWHMp,q,rc1,c2,…,cn≥IVHFLTPWHMp,q,rb1,b2,…,bn.Proof.
AsγθciL≥γθbiL and γθciU≥γθbiU for all i and p,q,r≥0, according to equation (42), we have(48)γθciLpγθcjLqγθckLr≥γθbiLpγθbjLqγθbkLr,(49)1−γθciLpγθcjLqγθckLrwiwjwk≤1−γθbiLpγθbjLqγθbkLrwiwjwk.
Furthermore,(50)1−1−1−∏i=1n∏j=in∏k=jn1−γθaiLpγθajLqγθakLrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥1−1−1−∏i=1n∏j=in∏k=jn1−γθbiLpγθbjLqγθbkLrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
Similarly, the inequality is expressed as(51)1−1−1−∏i=1n∏j=in∏k=jn1−γθaiUpγθajUqγθakUrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥1−1−1−∏i=1n∏j=in∏k=jn1−γθbiUpγθbjUqγθbkUrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
In addition, the summation is expressed by(52)s∑i=1n∑j=in∑k=jnwiwjwkθcipθcjqθckr1/p+q+r≥s∑i=1n∑j=in∑k=jnwiwjwkθbipθbjqθbkr1/p+q+r.
So, operators are expressed by(53)IVHFLTPWHMp,q,rc1,c2,…,cn≥IVHFLTPWHMp,q,rb1,b2,…,bn.Theorem 12 (IVHFLTPWHM is bounded).
The IVHFLTPWHM operator output is bounded between highest and lowest scores, i.e.,(54)minc1,c2,…,cn≤IVHFLTPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn, β=maxc1,c2,…,cn. According to Theorem 10, IVHFLTPWHMp,q,rα,α,…,α=α, and IVHFLTPWHMp,q,rβ,β,…,β=β.
Sinceα≤ci≤β and based on Theorem 11, we have(55)IVHFLTPWHMp,q,rα,α,…,α≤IVHFLTPWHMp,q,rc1,c2,…,cn≤IVHFLTPWHMp,q,rβ,β,…,β.
Therefore,α≤IVHFLTPWHMp,q,rc1,c2,…,cn≤β, and thus(56)minc1,c2,…,cn≤IVHFLTPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.


### 4.2. Interval-Valued Hesitant Fuzzy Linguistic TPWGHM (IVHFLTPWGHM) Operator

Definition 21.
A set of IVHFLEs is defined byc=c1,c2,…,cn, and let w=w1,w2,…,wnT denote a weight vector satisfying wi>0 and ∑i=1nwi=1. For p,q,r≥0, the operator called IVHFLTPWGHM is(57)IVHFLTPWGHMp,q,rc1,c2,…,cn=1p+q+r⊗i=1n⊗j=in⊗k=jnpci⊕qcj⊕rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb.Theorem 13.
Letp,q,r≥0 and ci=sθci,Γθcii=1,2,…,n denote a set of IVHFLEs with w=w1,w2,…,wnT satisfying wi>0 and ∑i=1nwi=1. For the IVHFLTPWGHM, the total is also an IVHFLE. Then,(58)IVHFLTPWGHMp,q,rc1,c2,…,cn=s∏i=1n∏j=in∏k=jnpθci+qθcj+rθck2n+1−jwk/∑δ=1nδδ+1∑b=jnwbp+q+r,∪γθci∈Γθci,γθcj∈Γθcj,γθck∈Γθck1−1−∏i=1n∏j=in∏k=jn1−1−γθciLp1−γθcjLq1−γθckLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r,(59)1−1−∏i=1n∏j=in∏k=jn1−1−γθciUp1−γθcjUq1−γθckUr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r,
Moreover, the IVHFLTPWGHM operator is idempotent, monotone, and bounded as we show next.Theorem 14 (IVHFLTPWGHM is idempotent).
Letc=c1,c2,…,cn, where c=sθc,Γθc. We have(60)IVHFLTPWGHMp,q,rc1,c2,⋯,cn=c.Proof.
Asc=c1,c2,…,cn, equation (55) leads to(61)IVHFLTPWGHMp,q,rc1,c2,…,cn=1p+q+r⊗i=1n⊗j=in⊗k=jnpc⊕qc⊕rc2n+1−jwk∑δ=1nδδ+1∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1n∑j=in∑k=jnn+1−jwk∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1nn+1−i+⋯+1=1p+q+rpc+qc+rc=c.Theorem 15 (IVHFLTPWGHM is monotone).
Letci=sθci,Γθci, and bi=sθbi,Γθbi be two IVHFLEs, γθci=γθciL,γθciU∈Γθbi, and γθbi=γθbiL,γθbiU∈Γθbi. If ci≥bi for all i, then(62)IVHFLTPWHMp,q,rc1,c2,…,cn≥IVHFLTPWHMp,q,rb1,b2,…,bn.Proof.
AsγθciL≥γθbiL and γθciU≥γθbiU for all i and p,q,r≥0, according to equation (57), we have(63)1−γθciLp1−γθcjLq1−γθckLr≤1−γθbiLp1−γθbjLq1−γθbkLr,(64)1−1−γθciLp1−γθcjLq1−γθckLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb≥1−1−γθbiLp1−γθbjLq1−γθbkLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb,from which we conclude that(65)1−1−1−∏i=1n∏j=in∏k=jn1−γθciLpγθcjLqγθckLrwiwjwk1/∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥1−1−1−∏i=1n∏j=in∏k=jn1−γθbiLpγθbjLqγθbkLrwiwjwk1/∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
In the same way,(66)1−1−∏i=1n∏j=in∏k=jn1−1−γθciLp1−γθcjLq1−γθckLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r≥1−1−∏i=1n∏j=in∏k=jn1−1−γθbiLp1−γθbjLq1−γθbkLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r.
In addition,(67)s∏i=1n∏j=in∏k=jnpθci+qθcj+rθck2n+1−jwk/∑δ=1nδδ+1∑b=jnwbp+q+r≥s∏i=1n∏j=in∏k=jnpθbi+qθbj+rθbk2n+1−jwk/∑δ=1nδδ+1∑b=jnwbp+q+r.
So, we have(68)IVHFLTPWGHMp,q,rc1,c2,…,cn≥IVHFLTPWGHMp,q,rb1,b2,…,bn.Theorem 16 (IVHFLTPWGHM is bounded).
The IVHFLTPWGHM operator outcomet is contained between highest and lowest scores, i.e.,(69)minc1,c2,…,cn≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn, β=maxc1,c2,…,cn. On the basis of Theorem 14, we have(70)IVHFLTPWGHMp,q,rα,α,…,α=α,IVHFLTPWGHMp,q,rβ,β,…,β=β.
Sinceα≤ci≤β and based on Theorem 15, we have(71)IVHFLTPWGHMp,q,rα,α,…,α≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤IVHFLTPWGHMp,q,rβ,β,…,β.
Therefore,α≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤β, and(72)minc1,c2,…,cn≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.


## 4.1. Interval-Valued Hesitant Fuzzy Linguistic TPWHM (IVHFLTPWHM) Operator

Definition 20.
A set of IVHFLEs is defined byc1,c2,…,cn. The weight is set to be w=w1,w2,…,wnT such that wi>0 and ∑i=1nwi=1. For p,q,r≥0, the operator of (IVHFLTPWHM) is defined as(42)IVHFLTPWHMp,q,rc1,c2,…,cn=1∑i=1n∑j=in∑k=jnwiwjwk⊕i=1n⊕j=in⊕k=jnwiwjwkcip⊗cjq⊗ckr1/p+q+r.Theorem 9.
A set of IVHFLEs is defined byci=sθci,Γθcii=1,2,…,n. A weight vector is set to be w=w1,w2,…,wnTsatisfying wi>0 and ∑i=1nwi=1. Utilizing IVHFLTPWHM, the combined outcome is an IVHFLE. For p,q,r≥0,(43)IVHFLTPWHMp,q,rc1,c2,…,cn=s∑i=1n∑j=in∑k=jnwiwjwkθcipθcjqθckr1/p+q+r,∪γθai∈Γθci,γθcj∈Γθcj,γθck∈Γθck,1−1−1−∏i=1n∏j=in∏k=jn1−γθciLpγθcjLqγθckLrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r,(44)1−1−1−∏i=1n∏j=in∏k=jn1−γθciUpγθcjUqγθckUrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
We show that the IVHFLTPWHM operator is idempotent, monotone, and bounded.Theorem 10 (IVHFLTPWHM is idempotent).
Set the nonnegative numbersc=c1,c2,…,cn such that c=sθc,Γθc. Then, we have(45)IVHFLTPWHMp,q,rc1,c2,…,cn=c.Proof.
Asa=c1,c2,…,cn, equation (41) leads to(46)IVHFLTPWHMp,q,rc1,c2,…,cn=1∑i=1n∑j=in∑k=jnwiwjwk⊕i=1n⊕j=in⊕k=jnwiwjwkcp+q+r1/p+q+r=c∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r=c.Theorem 11 (IVHFLTPWHM is monotone).
Letci=sθci,Γθci, and bi=sθbi,Γθbi be two IVHFLEs, γθci=γθciL,γθciU∈Γθci, and γθbi=γθbiL,γθbiU∈Γθbi. If ci≥bi for all i, then(47)IVHFLTPWHMp,q,rc1,c2,…,cn≥IVHFLTPWHMp,q,rb1,b2,…,bn.Proof.
AsγθciL≥γθbiL and γθciU≥γθbiU for all i and p,q,r≥0, according to equation (42), we have(48)γθciLpγθcjLqγθckLr≥γθbiLpγθbjLqγθbkLr,(49)1−γθciLpγθcjLqγθckLrwiwjwk≤1−γθbiLpγθbjLqγθbkLrwiwjwk.
Furthermore,(50)1−1−1−∏i=1n∏j=in∏k=jn1−γθaiLpγθajLqγθakLrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥1−1−1−∏i=1n∏j=in∏k=jn1−γθbiLpγθbjLqγθbkLrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
Similarly, the inequality is expressed as(51)1−1−1−∏i=1n∏j=in∏k=jn1−γθaiUpγθajUqγθakUrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥1−1−1−∏i=1n∏j=in∏k=jn1−γθbiUpγθbjUqγθbkUrwiwjwk1∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
In addition, the summation is expressed by(52)s∑i=1n∑j=in∑k=jnwiwjwkθcipθcjqθckr1/p+q+r≥s∑i=1n∑j=in∑k=jnwiwjwkθbipθbjqθbkr1/p+q+r.
So, operators are expressed by(53)IVHFLTPWHMp,q,rc1,c2,…,cn≥IVHFLTPWHMp,q,rb1,b2,…,bn.Theorem 12 (IVHFLTPWHM is bounded).
The IVHFLTPWHM operator output is bounded between highest and lowest scores, i.e.,(54)minc1,c2,…,cn≤IVHFLTPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn, β=maxc1,c2,…,cn. According to Theorem 10, IVHFLTPWHMp,q,rα,α,…,α=α, and IVHFLTPWHMp,q,rβ,β,…,β=β.
Sinceα≤ci≤β and based on Theorem 11, we have(55)IVHFLTPWHMp,q,rα,α,…,α≤IVHFLTPWHMp,q,rc1,c2,…,cn≤IVHFLTPWHMp,q,rβ,β,…,β.
Therefore,α≤IVHFLTPWHMp,q,rc1,c2,…,cn≤β, and thus(56)minc1,c2,…,cn≤IVHFLTPWHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.


## 4.2. Interval-Valued Hesitant Fuzzy Linguistic TPWGHM (IVHFLTPWGHM) Operator

Definition 21.
A set of IVHFLEs is defined byc=c1,c2,…,cn, and let w=w1,w2,…,wnT denote a weight vector satisfying wi>0 and ∑i=1nwi=1. For p,q,r≥0, the operator called IVHFLTPWGHM is(57)IVHFLTPWGHMp,q,rc1,c2,…,cn=1p+q+r⊗i=1n⊗j=in⊗k=jnpci⊕qcj⊕rck2n+1−jwk∑δ=1nδδ+1∑b=jnwb.Theorem 13.
Letp,q,r≥0 and ci=sθci,Γθcii=1,2,…,n denote a set of IVHFLEs with w=w1,w2,…,wnT satisfying wi>0 and ∑i=1nwi=1. For the IVHFLTPWGHM, the total is also an IVHFLE. Then,(58)IVHFLTPWGHMp,q,rc1,c2,…,cn=s∏i=1n∏j=in∏k=jnpθci+qθcj+rθck2n+1−jwk/∑δ=1nδδ+1∑b=jnwbp+q+r,∪γθci∈Γθci,γθcj∈Γθcj,γθck∈Γθck1−1−∏i=1n∏j=in∏k=jn1−1−γθciLp1−γθcjLq1−γθckLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r,(59)1−1−∏i=1n∏j=in∏k=jn1−1−γθciUp1−γθcjUq1−γθckUr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r,
Moreover, the IVHFLTPWGHM operator is idempotent, monotone, and bounded as we show next.Theorem 14 (IVHFLTPWGHM is idempotent).
Letc=c1,c2,…,cn, where c=sθc,Γθc. We have(60)IVHFLTPWGHMp,q,rc1,c2,⋯,cn=c.Proof.
Asc=c1,c2,…,cn, equation (55) leads to(61)IVHFLTPWGHMp,q,rc1,c2,…,cn=1p+q+r⊗i=1n⊗j=in⊗k=jnpc⊕qc⊕rc2n+1−jwk∑δ=1nδδ+1∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1n∑j=in∑k=jnn+1−jwk∑b=jnwb=1p+q+rpc+qc+rc2∑δ=1nδδ+1∑i=1nn+1−i+⋯+1=1p+q+rpc+qc+rc=c.Theorem 15 (IVHFLTPWGHM is monotone).
Letci=sθci,Γθci, and bi=sθbi,Γθbi be two IVHFLEs, γθci=γθciL,γθciU∈Γθbi, and γθbi=γθbiL,γθbiU∈Γθbi. If ci≥bi for all i, then(62)IVHFLTPWHMp,q,rc1,c2,…,cn≥IVHFLTPWHMp,q,rb1,b2,…,bn.Proof.
AsγθciL≥γθbiL and γθciU≥γθbiU for all i and p,q,r≥0, according to equation (57), we have(63)1−γθciLp1−γθcjLq1−γθckLr≤1−γθbiLp1−γθbjLq1−γθbkLr,(64)1−1−γθciLp1−γθcjLq1−γθckLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb≥1−1−γθbiLp1−γθbjLq1−γθbkLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb,from which we conclude that(65)1−1−1−∏i=1n∏j=in∏k=jn1−γθciLpγθcjLqγθckLrwiwjwk1/∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r≥1−1−1−∏i=1n∏j=in∏k=jn1−γθbiLpγθbjLqγθbkLrwiwjwk1/∑i=1n∑j=in∑k=jnwiwjwk1/p+q+r.
In the same way,(66)1−1−∏i=1n∏j=in∏k=jn1−1−γθciLp1−γθcjLq1−γθckLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r≥1−1−∏i=1n∏j=in∏k=jn1−1−γθbiLp1−γθbjLq1−γθbkLr2n+1−jwk/∑δ=1nδδ+1∑b=jnwb1/p+q+r.
In addition,(67)s∏i=1n∏j=in∏k=jnpθci+qθcj+rθck2n+1−jwk/∑δ=1nδδ+1∑b=jnwbp+q+r≥s∏i=1n∏j=in∏k=jnpθbi+qθbj+rθbk2n+1−jwk/∑δ=1nδδ+1∑b=jnwbp+q+r.
So, we have(68)IVHFLTPWGHMp,q,rc1,c2,…,cn≥IVHFLTPWGHMp,q,rb1,b2,…,bn.Theorem 16 (IVHFLTPWGHM is bounded).
The IVHFLTPWGHM operator outcomet is contained between highest and lowest scores, i.e.,(69)minc1,c2,…,cn≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.Proof.
Letα=minc1,c2,…,cn, β=maxc1,c2,…,cn. On the basis of Theorem 14, we have(70)IVHFLTPWGHMp,q,rα,α,…,α=α,IVHFLTPWGHMp,q,rβ,β,…,β=β.
Sinceα≤ci≤β and based on Theorem 15, we have(71)IVHFLTPWGHMp,q,rα,α,…,α≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤IVHFLTPWGHMp,q,rβ,β,…,β.
Therefore,α≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤β, and(72)minc1,c2,…,cn≤IVHFLTPWGHMp,q,rc1,c2,…,cn≤maxc1,c2,…,cn.


## 5. MADM with IVHFLTPWHM and IVHFLTPWGHM

Here, we propose a MADM method based on the operators called IVHFLTPWHM and IVHFLTPWGHM. Alternatives and attributes are denoted byY=y1,y2,…,ym and G=g1,g2,…,gn, respectively, for a MADM question. Also, w=w1,w2,…,wnT denotes weights satisfying wj∈0,1, ∑j=1nwj=1.. Individual attribute preferences are given by each decision maker for different options pertinent to the prespecified set S=Si|i=0,1,2,…,2t. In the decision-making process, if several interval values are anonymously provided for alternatives yi, concerning the variable gj, such values could be regarded as an IVHFLE yij. In other cases, if the equal assessments are assigned by two evaluators, it appears just once in the IVHFLE yij. Suppose the decision matrix (DM) is denoted by Y=yijm×n, where yij designates the value of the alternative yi concerning the variable gj gauged by an IVHFLE.Figure1 shows a flowchart of the proposed approach. The optimal candidates could be reached by the following.Step 1. Creation and normalization of the DM.The DM is produced based on the evaluation of the attributes for each candidate. MADM problem generally includes the benefit and cost types of selection attributes. Even though benefit-type attributes need no normalization, cost-type needs it. Converting them could be done as follows:(73)y˜ij=yij,for benefit attributegj,negyij,for cost attributegj,,i=1,2,…,m,…j=1,2,…,n.Thus, the normalized DMY˜=y˜ijm×n can be obtained by equation (73).Step 2. Obtaining the overall performance values.Utilize the IVHFLTPWHM operator(74)ri=IVHFLTPWHMp,q,ry˜i1,y˜i2,…,y˜in=1∑j=1n∑k=jn∑l=knwjwkwl⊕j=1n⊕k=jn⊕l=knwjwkwlyijp⊗yikq⊗yilr1/p+q+r,or the IVHFLTPWGHM operator(75)ri=IVHFLTPWGHMp,q,ry˜i1,y˜i2,…,y˜in=1p+q+r⊗j=1n⊗k=jn⊗l=knpyij⊕qyik⊕ryil2n+1−jwk∑δ=1nδδ+1∑b=jnwb,to obtain the collective scoresrii=1,2,…,m, for p,q,r>0.Step 3. Finding the general scores.Equation (8) helps compute scores expressed as s(yi)(i=1,2,⋯,m) based on the overall performance values of yi.Step 4. Rating all options.All candidates,yii=1,2,…,m, are arranged according to the results of Step 3 following syii=1,2,…,m, and select the best one(s).Figure 1
A flowchart of the proposed MADM method with HM operators.


## 6. Practical Examples

This section provides a practical example which is given based on the proposed approach [28].The ABC company represents a large government-run company specializing mainly in the production and sales of nonferrous metals in China. The company always seeks overseas investments to grow its key business. The department dealing with foreign investments has recently agreed upon utilizing preliminary surveys to generate alternatives based on several countries. After a detailed analysis, the department considered five alternativesy1,y2,y3,y4,y5. Four factors were investigated according to the personnel experience in the department. These factors are g1: resources; g2: politics and policy; g3: economy; and g4: infrastructure. The decision information was assessed by w=0.1,0.3,0.2,0.4T. The differences among the evaluators related to opinions could be clarified. Table 1 depicts decision information implemented by the IVHFLEs .Table 1
The decision matrix for IVHFLS.
g1g2g3g4y1s4,0.3,0.5s5,0.2,0.3,0.4,0.5s4,0.5,0.6,0.7,0.8s3,0.4,0.5y2s2,0.2,0.3,0.5,0.8s4,0.3,0.4,0.4,0.7s3,0.6,0.8s4,0.3,0.4,0.5,0.6y3s3,0.3,0.4,0.5,0.7s4,0.3,0.5s6,0.7,0.8,0.8,0.9s2,0.6,0.7y4s2,0.3,0.4,0.4,0.5,0.5,0.6s3,0.5,0.7s2,0.3,0.5,0.6,0.8s2,0.8,0.9y5s2,0.3,0.6,0.7,0.9s5,0.4,0.7s3,0.5,0.7,0.8,0.9s4,0.7,0.8Table 2
The IVHFLTPWHM-based aggregation results.
Overall evaluation valuesr1s15.58,0.34,0.45,0.38,0.49,0.40,0.51,0.44,0.56r2s10.34,0.34,0.46,0.37,0.57,0.38,0.54,0.41,0.65,0.45,0.61,0.48,0.70,0.42,0.54,0.45,0.64r3s12.77,0.50,0.62,0.51,0.66,0.52,0.65,0.54,0.68r4s7.47,0.61,0.75,0.64,0.79,0.62,0.76,0.65,0.77,0.62,0.80,0.65,0.80r5s13.75,0.54,0.70,0.59,0.74,0.58,0.74,0.63,0.78Table 3
The IVHFLTPWGHM-based aggregation results.
IVHFLTPWGHM-based aggregation resultsr1s3.31,0.43,0.57,0.49,0.62,0.46,0.59,0.53,0.65r2s2.71,0.41,0.54,0.48,0.64,0.44,0.61,0.51,0.70,0.50,0.62,0.57,0.71,0.53,0.69,0.60,0.76r3s2.98,0.57,0.70,0.61,0.76,0.58,0.71,0.62,0.77r4s1.94,0.60,0.74,0.63,0.76,0.65,0.78,0.67,0.80,0.70,0.82,0.72,0.84r5s3.52,0.60,0.79,0.69,0.84,0.64,0.81,0.73,0.86Table 4
The score values associated with the alternatives.
IVHFLTPWHMIVHFLTPWGHMsy16.08311.7999sy25.17941.5730sy37.46291.9826sy45.23011.4092sy5s2.6144Table 5
Ranking results of the alternatives.
RankingIVHFLTPWHMy5≻y3≻y1≻y4≻y2IVHFLTPWGHMy5≻y3≻y1≻y2≻y4


### 6.1. Evaluation Based on IVHFLTPWHM or IVHFLTPWGHM Operators

Step 1. The DM is constructed and normalization is conducted. Assume that no normalization is needed because all variables are benefit-type.Step 2. For each alternative yii=1,2,3,4,5, general evaluation scores ri are combined by the IVHFLTPWHM and IVHFLTPWGHM operators according to equations (32) and (33) (see Tables 2 and 3, and let p = q = r = 1).Step 3. Equation (8) helps compute the scores syi of the collective performance values yi (see Table 4).Step 4. The results of Step 3 help sort out the whole set of options yi (see Table 5).As shown in Table5, the alternative ranking results changed slightly. However, all methods agreed that y5 is the best option, while the worst option is y2y4.


### 6.2. Comparison

When both rationality and superiority are a concern, the proposed methods presented in this paper are compared with the methods of Wang et al. [28] who introduced (IVHFLPWA) and (IVHFLPWG) operators. Comparisons were also performed against the method utilized by Gong et al. [29]. Moreover, the suggested distance measures and the aggregation operators are expanded to an IVHFL environment. Table 6 depicts the overall values yi.Table 6
Overall values and ranking information of different approaches.
MethodsScoresRanking resultsWang et al.‘s method [35] by IVHFLPWA operatorsy1=0.280, sy2=0.178, sy3=0.268, sy4=0.171, sy5=0.471y5≻y3≻y1≻y2≻y4Wang et al.‘s method [35] by IVHFLPWG operatorsy1=0.275, sy2=0.167, sy3=0.254, sy4=0.165, sy5=0.461y5≻y3≻y1≻y2≻y4Gong’s method [36] based on similarity degreesy1=0.209, sy2=0.136, sy3=0.324, sy4=0.135, sy5=0.562y5≻y3≻y1≻y2≻y4The proposed method by IVHFLTPWHM operatorsy1=6.803, sy2=5.179, sy3=7.463, sy4=5.230, sy5=9.137y5≻y3≻y1≻y4≻y2The proposed method by IVHFLTPWGHM operatorsy1=1.800, sy2=1.573, sy3=1.983, sy4=1.409, sy5=2.614y5≻y3≻y1≻y4≻y2The results of the comparative analysis can be discussed as follows.(1)
Wang’s [28] methods utilize the power average operator (PA) operator, and thus the attribute interrelations are not accounted for when the corresponding attribute values are aggregated. In brief, Wang’s [28] approach assumes the independence of the attributes, but this is inconsistent with reality. The proposed aggregation operators in this paper are useful and meaningful and can capture the correlations of the aggregated arguments more proficiently when compared to the technique presented in Wang [28].(2)
Gong’s [29] methods depend on the proposed distance measures for IVHFLSs. Indeed, the distance measurement method is realized by a comparison of the values. Also, Gong’s [29] methods do not consider the interrelationship between attributes, and hence this method is not close to reality. In general, the proposed methods not only consider the interrelation between attributes but also extend the input variables to three parameters that can capture more decision-making information than techniques based on the existing HM operators. Analysis pertinent to efficacy and viability of investigated methodology is conducted.


### 6.3. Parameter Influence on Decision Making

When different values of parametersp, q, and r in Step 1 are used to calculate the comprehensive evaluation values, the alternatives are rearranged as shown in Table 7. It is seen from Table 7 that slight modifications are seen in ranking results. It is obvious that y5 or y3 is the best option. Then, p, q, and r can be chosen according to the actual decision-making situation. However, the use of larger or smaller parameter values in the operators can make the calculations more complex and counter-intuitive as shown by Table 7. Therefore, we generally recommend setting the values of the three parameters to unity. This shall make the calculations easier and also fully capture the correlation among aggregated arguments.Table 7
Ranking order by several values ofp, q, and r based on the IVHFLTPWHM operator.
p,q,rsy1sy2sy3sy4sy5Ranking results1,1,24.50924.69615.12253.74496.3960y5≻y3≻y1≻y2≻y41,2,16.54886.59747.79024.66659.3303y5≻y3≻y2≻y1≻y41,2,213.264412.241118.97056.899018.6342y5≻y3≻y1≻y2≻y42,1,19.85299.176311.63735.828013.4952y5≻y3≻y1≻y2≻y42,1,28.55697.92339.89435.294712.2065y5≻y3≻y1≻y2≻y42,2,222.574716.744030.29258.647530.5780y5≻y3≻y1≻y2≻y43,1,16.42756.01546.94024.44249.2085y5≻y3≻y1≻y2≻y43,1,28.66417.92209.34705.289612.8332y5≻y3≻y1≻y2≻y43,3,339.255419.685051.86949.577645.2869y5≻y3≻y1≻y2≻y44,4,4283.370562.1517401.959820.4737305.4695y5≻y3≻y1≻y2≻y4


## 6.1. Evaluation Based on IVHFLTPWHM or IVHFLTPWGHM Operators

Step 1. The DM is constructed and normalization is conducted. Assume that no normalization is needed because all variables are benefit-type.Step 2. For each alternative yii=1,2,3,4,5, general evaluation scores ri are combined by the IVHFLTPWHM and IVHFLTPWGHM operators according to equations (32) and (33) (see Tables 2 and 3, and let p = q = r = 1).Step 3. Equation (8) helps compute the scores syi of the collective performance values yi (see Table 4).Step 4. The results of Step 3 help sort out the whole set of options yi (see Table 5).As shown in Table5, the alternative ranking results changed slightly. However, all methods agreed that y5 is the best option, while the worst option is y2y4.


## 6.2. Comparison

When both rationality and superiority are a concern, the proposed methods presented in this paper are compared with the methods of Wang et al. [28] who introduced (IVHFLPWA) and (IVHFLPWG) operators. Comparisons were also performed against the method utilized by Gong et al. [29]. Moreover, the suggested distance measures and the aggregation operators are expanded to an IVHFL environment. Table 6 depicts the overall values yi.Table 6
Overall values and ranking information of different approaches.
MethodsScoresRanking resultsWang et al.‘s method [35] by IVHFLPWA operatorsy1=0.280, sy2=0.178, sy3=0.268, sy4=0.171, sy5=0.471y5≻y3≻y1≻y2≻y4Wang et al.‘s method [35] by IVHFLPWG operatorsy1=0.275, sy2=0.167, sy3=0.254, sy4=0.165, sy5=0.461y5≻y3≻y1≻y2≻y4Gong’s method [36] based on similarity degreesy1=0.209, sy2=0.136, sy3=0.324, sy4=0.135, sy5=0.562y5≻y3≻y1≻y2≻y4The proposed method by IVHFLTPWHM operatorsy1=6.803, sy2=5.179, sy3=7.463, sy4=5.230, sy5=9.137y5≻y3≻y1≻y4≻y2The proposed method by IVHFLTPWGHM operatorsy1=1.800, sy2=1.573, sy3=1.983, sy4=1.409, sy5=2.614y5≻y3≻y1≻y4≻y2The results of the comparative analysis can be discussed as follows.(1)
Wang’s [28] methods utilize the power average operator (PA) operator, and thus the attribute interrelations are not accounted for when the corresponding attribute values are aggregated. In brief, Wang’s [28] approach assumes the independence of the attributes, but this is inconsistent with reality. The proposed aggregation operators in this paper are useful and meaningful and can capture the correlations of the aggregated arguments more proficiently when compared to the technique presented in Wang [28].(2)
Gong’s [29] methods depend on the proposed distance measures for IVHFLSs. Indeed, the distance measurement method is realized by a comparison of the values. Also, Gong’s [29] methods do not consider the interrelationship between attributes, and hence this method is not close to reality. In general, the proposed methods not only consider the interrelation between attributes but also extend the input variables to three parameters that can capture more decision-making information than techniques based on the existing HM operators. Analysis pertinent to efficacy and viability of investigated methodology is conducted.


## 6.3. Parameter Influence on Decision Making

When different values of parametersp, q, and r in Step 1 are used to calculate the comprehensive evaluation values, the alternatives are rearranged as shown in Table 7. It is seen from Table 7 that slight modifications are seen in ranking results. It is obvious that y5 or y3 is the best option. Then, p, q, and r can be chosen according to the actual decision-making situation. However, the use of larger or smaller parameter values in the operators can make the calculations more complex and counter-intuitive as shown by Table 7. Therefore, we generally recommend setting the values of the three parameters to unity. This shall make the calculations easier and also fully capture the correlation among aggregated arguments.Table 7
Ranking order by several values ofp, q, and r based on the IVHFLTPWHM operator.
p,q,rsy1sy2sy3sy4sy5Ranking results1,1,24.50924.69615.12253.74496.3960y5≻y3≻y1≻y2≻y41,2,16.54886.59747.79024.66659.3303y5≻y3≻y2≻y1≻y41,2,213.264412.241118.97056.899018.6342y5≻y3≻y1≻y2≻y42,1,19.85299.176311.63735.828013.4952y5≻y3≻y1≻y2≻y42,1,28.55697.92339.89435.294712.2065y5≻y3≻y1≻y2≻y42,2,222.574716.744030.29258.647530.5780y5≻y3≻y1≻y2≻y43,1,16.42756.01546.94024.44249.2085y5≻y3≻y1≻y2≻y43,1,28.66417.92209.34705.289612.8332y5≻y3≻y1≻y2≻y43,3,339.255419.685051.86949.577645.2869y5≻y3≻y1≻y2≻y44,4,4283.370562.1517401.959820.4737305.4695y5≻y3≻y1≻y2≻y4


## 7. Conclusions

The HM operator is a very good tool to aggregate fuzzy information, and IVHFLSs exactly describe real situations of human hesitancy and uncertainty. In this work, we firstly propose novel HM operators, which can consider ternary or higher-order correlations among arguments and also satisfy the properties of reducibility and idempotency. Two of these operators are the three-parameter weighted HM (TPWHM) and the three-parameter weighted geometric HM (TPWGHM). We have generalized the novel HM operators within an IVHFL framework. Then, a numerical example pertinent to overseas investments is discussed [28]. The proposed approach is studied with detailed implementation and feasibility in addressing MADM problems. The presented approach is feasible and effective, considers correlations of any three arguments, and has some desirable properties. Finally, the presented method not only captures the correlations of the aggregated arguments, but also has demonstrated desirable performance by sensitivity and comparative analysis.Generally, the decision matrix is excessively dependent on the decision makers. The presented methods do not show how the experts choose the overseas operators from the proposed ones and the associated interactions. We plan to apply the proposed methods to other research areas such as integration with probabilistic dual hesitant values, PFS, HPFS, and CHFS. Besides, it would be extended to other application domains such as machine learning, clustering analysis, performance evaluation, and so on as future research.



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*Source: 3634895-2021-10-19.xml*