# Multicriteria Decision Making Based on Archimedean Bonferroni Mean Operators of Hesitant Fermatean 2-Tuple Linguistic Terms

Authors: Huijuan Wang; Xin Wang; Lidong Wang
Journal: Complexity
 (2019)

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## Abstract

The study is concerned with the representation and aggregation of complex uncertainty information. First, the concept of hesitant Fermatean 2-tuple linguistic sets (HF2TLSs) is introduced for characterizing an individual’s imprecision preferences and assessing information by combining 2-tuple linguistic terms and Fermatean fuzzy sets. The advantage of hesitant Fermatean 2-tuple linguistic information is that it can handle higher levels of uncertainty and express the decision-makers’ hesitancy. Second, we extend Bonferroni mean (BM) operators under the background of HF2TLSs for the sake of their application in information fusion and decision making. The Archimedeant-norm and s-norm- (ATS-) based hesitant Fermatean 2-tuple linguistic weighted Bonferroni mean (A-HF2TLWBM) operator and the ATS-based hesitant Fermatean 2-tuple linguistic weighted geometric Bonferroni mean (A-HF2TLWGBM) operator are developed by considering the interrelationship between any two variables. The main benefit of the proposed operators is that these operators deliver more complete and flexible results compared to existing methods. Moreover, some fundamental properties and special cases are examined by adjusting parameter values. Finally, an approach is designed as a support for handling decision making problems, and an example regarding investment selection is provided to demonstrate the practicality of the designed method with a detailed discussion of parameter influence and comparisons with the existing methods.


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## Body

## 1. Introduction

Multicriteria decision making (MCDM) arises from decision theory, which can help us select the optimal solution from multiple possible alternatives and plays an indispensable role in various fields, such as the economy [1–3] and medicine [4]. MCDM is a complex process subject to external environmental influences and other subjective factors affecting decision-makers. In today’s information-intensive conditions, imprecise information is increasing in volume and complexity, and some sound methods have been established. Linguistic variables, characterized by qualitative words or phrases, offer a natural and direct semantic description of the intuition and preferences of decision-makers [5]. For instance, when assessing teachers’ levels, decision-makers tend to give their judgments in the form of linguistic terms, for example, “good,” “fair,” and “poor.” Generally, state-of-the-art methods for representing linguistic variables are based on membership grades, symbolic computational methods, and combinations thereof.Fuzzy set (FS) theory was introduced by Zadeh [6], in which linguistic terms are characterized in the manner of the membership degree expressed by a real number between 0,1. To better reflect complex decision information in the form of linguistic terms, different types of fuzzy set-based methods have been developed. Orthopair fuzzy sets have been coined for describing imprecision information by introducing a pair of values μz,νz in the unit interval, where μz indicates the support degree (membership degree) and νz records against degree (nonmembership degree) of z belonging to a fuzzy concept. Compared with FS theory, the sum of μz and νz for element z belonging to a fuzzy concept is no more than 1 in Atanassov’s intuitionistic fuzzy sets (IFSs) [7]. In practice, a decision-maker may give his judgment information with μz=0.7 and νz=0.6, which is out of the application scope of IFS. Pythagorean fuzzy set (PFS), developed by Yager [8], is a new extension of IFS, in which the sum of squares of support degrees and against degrees is bounded by 1. PFS is more versatile than IFS. Aiming to solve complex decision making problems, a series of extensions and generalizations of PFS have been explored [9–11]. With the increasing complexity involved in the decision making process, the description of abundant uncertainty still needs to be adequately addressed. If a decision-maker gives his preference using a pair of values with support degree 0.9 and against degree 0.5, respectively, this is inconsistent with the definition of both IFS and PFS. Senapati and Yager [12, 13] coined the Fermatean fuzzy set (FFS) with its comparison measures. FFS can characterize more complex uncertain information by redefining the constraint condition 0≤μFz3+νFz3≤1. In other words, IFS and PFS are two special forms of FFS, which means that the FFSs are able to handle higher levels of uncertainties.Ordinal labels offer an easy and direct way of qualitative linguistic description for solving decision problems, and symbolic computational methods have been established on the basis of defining their operation laws. However, the original ordinal labels were imposed as the operational result, so information loss may be unavoidable during the operation process [14, 15]. To address this problem, some efforts have been made by combining linguistic terms with real numbers (or membership degrees), which may also be better aligned with human decision making processes. Herrera and Martínez [16, 17] cultivated 2-tuple linguistic representation terms, whose element comprises a linguistic term and a real number. This approach possesses the characteristics of present fuzzy linguistic evaluation and also avoids the information loss problems of other fuzzy linguistic models. Subsequently, some decision making models have been developed based on 2-tuple linguistic models. Wang [18] built an appropriate approach with the aid of a 2-tuple linguistic presentation model to assess the performance of a new product development project. Zhang et al. [19] established a consensus reaching approach under 2-tuple linguistic judgment information and incomplete weight information.In reality, because of the external factors of decision making situations and the internal cognitive preferences, some MCDM issues are very complicated and give rise to decision-makers’ hesitancy. In order to depict and handle the decision-makers’ hesitancy, Torra [20] coined the hesitant fuzzy set (HFS), taking multiple values in 0,1 as the possible membership degrees. HFS is very effective in handling the human hesitation involved in the decision-making process. Recently, in the light of HFS, some extended models have been explored intensively. Zhou et al. [21] came up with the concept of the hesitant intuitionistic fuzzy set (HIFS) and proposed a decision making method by expressing complementary preference relationships for uncertain HIFS information. Zadeh [6] presented the thought of hesitant fuzzy linguistic term sets (HFLTSs). Rashid et al. [22] and Zhang et al. [23] introduced hesitant intuitionistic fuzzy linguistic term sets (HIFLTSs). Pythagorean 2-tuple linguistic sets studied by Wei et al. [24] are a combination of the PFSs and 2-tuple linguistic evaluation models for dealing with MCDM problems. Liu et al. [25] introduced the notion of Fermatean fuzzy linguistic term sets and developed the related TOPSIS method using different distance measures and aggregation operators. In order to handle higher levels of uncertainty and express decision-makers’ hesitancy, this study introduces hesitant Fermatean 2-tuple linguistic set (HF2TLS) by combining the FFS and 2-tuple linguistic terms.The fusion of uncertainty information can achieve some overall values of alternatives or group opinions in the decision making domain. To better reflect the preferences of decision-makers and the complexity of the decision making domain, a wide range of aggregation operators have been presented. The Bonferroni mean (BM) [26–28] operator is a mean-type aggregation approach, whose salient capability can depict the interrelationship of the input variables well. Some aggregation operators have been developed for highlighting specific roles or a relationship of aggregated variables within various MCDM situations. An in-depth study of Bonferroni mean operators was explored by Yager [29], and some generalized Bonferroni mean (GBM) operators were established to characterize the interrelationship among any three variables. Subsequently, Beliakov et al. [30] and Xia et al. [31–33] developed the dual GBM (DGBM) operators, which are a new generalization of the classical BM operator by considering the relatedness involved in all variables. These operators have been successfully extended to IFSs [32], PFSs [34], and HFSs [33]. In recent years, the Archimedean t-norm and s-norm (ATS) [35, 36] has been used as a generalization of most existing t-norms and s-norms, like algebraic and Einstein t-norm and s-norm [37], which play important roles in inducing multiple aggregation operators for fuzzy terms. Various applications indicate that ATS contributes to uncertain information fusion in complex decision making situations [38–40]. Lan et al. [38] employed the extended triangular co-norm to deal with linguistic information and proposed some aggregation operators. Xia et al. [39] introduced some new operations on IFSs based on ATS and presented some intuitionistic fuzzy aggregation operators. Tao et al. [40] proposed some new operational laws of 2-tuples based on ATS and gave some aggregation operators using the proposed rules.Although Fermatean fuzzy linguistic terms provide a potential way to capture the cognitive uncertainty during making decisions, there exist some situations where humans may hesitate to express their preferences with a single variable. To better evaluate complex practical problems, it is necessary to combine hesitant fuzzy sets and Fermatean fuzzy linguistic terms to represent the subjective preferences with the higher levels of uncertainties. In this study, HF2TLSs are proposed, and their operational laws between HF2TLSs are developed. In order to allow HF2TLSs to handle higher levels of uncertainties and express the decision-makers’ hesitancy, how to facilitate the aggregation of these HF2TLSs becomes a significant issue. We define two related aggregation operators of the ATS-based hesitant Fermatean 2-tuple linguistic weighted Bonferroni mean. These operators consider the interrelationship between any two variables. In addition, the established methods are two parameterized families of the operational laws and aggregation operators of HF2TLSs, respectively, which cover a wide range of particular forms that exhibit a more flexible way for adapting the specific needs of problem solving. To examine the practicality of the aggregation operators of HF2TLSs, we offer a procedure for solving ranking problems on the strength of the developed operators within hesitant Fermatean 2-tuple linguistic environments.The structure of this study is as follows. Section2 briefly reviewed some basic concepts including definition of FFSs, BM operator, and GBM operator. Section 3 introduces the definition of HF2TLSs. And some new operational rules and related theorems about HF2TLSs based on ATS are developed. Section 4 shows some aggregation operators and examines their properties and particular forms. In Section 5, we propose a ranking scheme under the hesitant Fermatean 2-tuple linguistic environment. In Section 6, an example with some detailed steps and comparisons is given to illustrate the practicality and flexibility of the aforementioned methods. The conclusion is presented in the final section.


## 2. Preliminaries

In the following, some fundamental definitions and functions are introduced.LetSf=sf0,sf1,sf2,…,sfg be a collection of linguistic terms with odd cardinality, where sfr denotes a linguistic term with the order relationship sfr>sft, ifr>t, and the negation operator is negsfr=sfg−r [14]. A 2-tuple linguistic method consists of sfr,αr, where sfr∈Sf and αr∈−0.5,0.5 denotes symbolic translation [14, 16, 17].Definition 1 (see [17]).
LetSf=sfrr=0,1,…,g be a collection of linguistic terms, the function Δ is introduced to get β∈0,1 that is equivalent to a 2-tuple linguistic term (2-TLT):(1)Δ:0,g⟶Sf×−0.5,0.5,Δβ=sfr,αrwithsfr,r=roundβ,αr=β−r,αr∈−0.5,0.5,where the output of the function “round” is a positive number that is close to β. Δ−1 is the inverse function of Δ, and(2)Δ−1:Sf×−0.5,0.5⟶0,g,Δ−1:sfr,αr=r+αr=β.
From Definition 1, a linguistic termsfr has an equivalent element Δsfr=sfr,0 in 2-tuple linguistic set by adding an element as symbolic translation.Definition 2 (see [12]).
A Fermatean fuzzy setF in Z is derived as(3)F=z,μFz,νFzz∈Z,where μFz:Z⟶0,1 and νFz:Z⟶0,1 with the constraint condition(4)0≤μFz3+νFz3≤1,for anyz∈Z,
μ
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z and νFz denote the support (membership) degree and the against (nonmembership) degree of z in the set F, respectively. And z∈Z, πFz=1−μFz3−νFz33 is identified as the indeterminacy degree of z to Z. For convenience, Senapati and Yager called μFz,νFz a Fermatean fuzzy number (FFN).Definition 3 (see [33]).
Ifp,q>0 and urr=1,2,…,n is a set of nonnegative integers, the Bonferroni mean (BM) operator is given as follows:(5)BMp,qu1,u2,…,un=1nn−1∑r,t=1,r≠tnurputq1/p+q.Definition 4 (see [33]).
Ifp,q>0 and urr=1,2,…,n is a set of nonnegative integers, the geometric Bonferroni mean (GBM) operator is described as follows:(6)GBMp,qu1,u2,…,un=1p+q∑r,t=1,r≠tnpur+qut1/nn−1.


## 3. Hesitant Fermatean Fuzzy 2-Tuple Linguistic Sets

In order to handle higher levels of uncertainties and express the decision-makers’ hesitancy, this section introduces hesitant Fermatean 2-tuple linguistic sets (HF2TLSs) on the basis of FFSs and 2-TLT to represent complex uncertainty information. We also deduce some operational laws of HF2TLSs information.


### 3.1. Hesitant Fermatean Fuzzy 2-Tuple Linguistic Sets

Definition 5.
A HF2TLSℱ on Z is defined as follows:(7)ℱ=sfθℱr,αℱr,μℱrz,νℱrzsfθℱr,αℱr∈Sf,where sfθℱr∈Sf, αℱr∈−0.5,0.5, and μℱrz,νℱrz are located in the unit interval 0,1 with 0≤μℱrz3+νℱrz3≤1. μℱrz and νℱrz denote the support degree and against degree of the element z belonging to sfθℱr,αℱr, respectively. 1≤r≤ℱ and ℱ is the number of elements of ℱ. For convenience, sfθℱr,αℱr,μℱrz,νℱrz is called a hesitant Fermatean 2-tuple linguistic number (HF2TLN).Definition 6.
Letℱ=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ be a HF2TLS; the score function of ℱ is defined as follows:(8)SCℱ=∑r=1ℱΔ−1sfθℱr,αℱr×μℱr3−νℱr3ℱ.Definition 7.
Letℱ=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ be a HF2TLS; the accuracy function of ℱ is defined as follows:(9)ACℱ=∑r=1ℱΔ−1sfθℱr,αℱr×μℱr3+νℱr3ℱ.Definition 8.
Letℱ1=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ1 and ℱ2=sfθℱt,αℱt,μℱt,νℱt,t=1,2,…,ℱ2 be two HF2TLSs; then, the order relationships between ℱ1 and ℱ2 are described as follows:(1)
If SCℱ1<SCℱ2,thenℱ1≺ℱ2(2)
If SCℱ1>SCℱ2,thenℱ1≻ℱ2(3)
When SCℱ1=SCℱ2,(a)
ACℱ1=ACℱ2,andℱ1∼ℱ2(b)
ACℱ1<ACℱ2,andℱ1≺ℱ2(c)
ACℱ1>ACℱ2,andℱ1≻ℱ2


### 3.2. New Operational Laws for HF2TLSs

Thet-norm and s-norm are two kinds of binary operations used in probabilistic metrics and fuzzy logic, which generalizes intersection operation and conjunction operation. An Archimedean t-norm is defined as Tc,d=ζ−1ζc+ζd, where ζ is a strictly continuous decreasing function ζ:0,1⟶0,∞ such that ζ1=0. Its dual function s-norm can be externalized as Sc,d=χ−1χc+χd with χc=ζ1−c. By using a different mapping T:0,1×0,1⟶0,1, multiple different operators can be derived for handling fuzzy term aggregation problems. ATS was incorporated into 2-tuple linguistic environments by Tao et al. [40], and some ATS-based operational laws for intuitionistic fuzzy numbers were developed by Xia et al. [31–33]. Enlightened by these works, we deduce the following operational laws of HF2TLSs information.Definition 9.
Letℱ1=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ1 and ℱ2=sfθℱt,αℱt,μℱt,νℱt,t=1,2,…,ℱ2 be any two HF2TLSs; then, some operational laws are defined as follows (λ>0 and Δℱr−1=Δ−1sfθℱr,αℱr):(1)
ℱ1⊕ℱ2=Δχ−1χΔℱr−1+χΔℱt−1,χ−1χμℱr+χμℱt,ζ−1ζνℱr+ζνℱtsfθℱr,αℱr,μℱr,νℱr∈ℱ1,sfθℱt,αℱt,μℱt,νℱt∈ℱ2(2)
ℱ1⊗ℱ2=Δζ−1ζ△ℱr−1+ζ△ℱt−1,ζ−1ζμℱr+ζμℱt,χ−1χνℱr+χνℱtsfθℱr,αℱr,μℱr,νℱr∈ℱ1,sfθℱt,αℱt,μℱt,νℱt∈ℱ2(3)
λℱ1=Δχ−1λχΔℱr−1,χ−1λχμℱr,ζ−1λζνℱrsfθℱr,αℱr,μℱr,νℱr∈ℱ1(4)
ℱ1λ=Δζ−1λζ△ℱr−1,ζ−1λζμℱr,χ−1λχνℱrsfθℱr,αℱr,μℱr,νℱr∈ℱ1
In the sequel, some special operational laws are obtained by taking different forms ofζc and their dual function χc=ζ1−c. Meanwhile, sfθℱr,αℱr,μℱr,νℱr∈ℱ1andsfθℱt,αℱt,μℱt,νℱt∈ℱ2 are simplified, so it is omitted during the operation process of HF2TLSs.Remark 1.
Letζc=−logc and χc=−log1−c; then, the operational laws defined by Definition 9 can be read as the following equations:(1)
ℱ1⊕ℱ2=ΔΔℱr−1+Δℱt−1−Δℱr−1⋅Δℱt−1,μℱr+μℱt−μℱrμℱt,νℱrνℱt(2)
ℱ1⊗ℱ2=ΔΔℱr−1⋅Δℱt−1,μℱrμℱt,νℱr+νℱt−νℱrνℱt(3)
λℱ1=Δ1−1−Δℱr−1λ,1−1−μℱrλ,νℱrλ(4)
ℱ1λ=ΔΔℱr−1λ,μℱrλ,1−1−νℱrλRemark 2.
Ifζc=log2−c/c and χc=log2−1−c/1−c, then the operational laws via the Einstein t-norm and s-norm can be obtained:(10)ℱ1⊕ℱ2=ΔΔℱr−1+Δℱt−11+Δℱr−1⋅Δℱt−1,μℱr+μℱt1+μℱrμℱt,νℱrνℱt1+1−νℱr1−νℱt;ℱ1⊗ℱ2=ΔΔℱr−1⋅Δℱt−11−1−Δℱr−11−Δℱt−1,μℱrμℱt1−1−μℱr1−μℱt,νℱr+νℱt1+νℱrνℱt;λℱ1=Δ1+Δℱr−1λ−1−Δℱr−1λ1+Δℱr−1λ+1−Δℱr−1λ,1+μℱrλ−1−μℱrλ1+μℱrλ+1−μℱrλ,2νℱrλ2−νℱrλ+νℱrλ;ℱ1λ=Δ2Δℱr−1λ2−Δℱr−1λ+Δℱr−1λ,2μℱrλ2−μℱrλ+μℱrλ,1+νℱrλ−1−νℱrλ1+νℱrλ+1−νℱrλ.Remark 3.
Ifζc=logε+1−εc/c, χc=log1−1−εc/1−c, and ε>0, then the operational laws via the Hamacher t-norm and s-norm can be obtained:(11)ℱ1⊕ℱ2=ΔΔℱr−1+Δℱt−1−1−ε⋅Δℱr−1⋅Δℱt−11−1−ε⋅Δℱr−1⋅Δℱt−1,μℱr+μℱt−1−εμℱrμℱt1−1−εμℱrμℱt,νℱrνℱtε+1−ενℱr+νℱt−νℱrνℱt;ℱ1⊗ℱ2=ΔΔℱr−1⋅Δℱt−1ε+1−εΔℱr−1+Δℱt−1−Δℱr−1⋅Δℱt−1,μℱrμℱtε+1−εμℱr+μℱt−μℱrμℱt,νℱr+νℱt−1−ενℱrνℱt1−1−ενℱrνℱt;λℱ1=Δ1+ε−1⋅Δℱr−1λ−1−Δℱr−1λ1+ε−1⋅Δℱr−1λ+ε−11−Δℱr−1λ,1+ε−1μℱrλ−1−μℱrλ1+ε−1μℱrλ+ε−11−μℱrλ,ενℱrλε+1−ενℱrλ+ε−1νℱrλ;ℱ1λ=ΔεΔℱr−1λε+1−ε⋅Δℱr−1λ+ε−1Δℱr−1λ,εμℱrλε+1−εμℱrλ+ε−1μℱrλ,1+ε−1νℱrλ−1−νℱrλ1+ε−1νℱrλ+ε−11−νℱrλ.Remark 4.
Letζc=logε−1/εc−1, χc=logε−1/ε1−c−1, and ε>1, then the operational laws via the Frank t-norm and s-norm can be obtained:(12)ℱ1⊕ℱ2=Δlogεεε−1εΔℱr−1−1−1εΔℱt−1−1−1+ε−1,logεεε−1εμℱr−1−1εμℱt−1−1+ε−1,logε1+ενℱr−1ενℱt−1ε−1;ℱ1⊗ℱ2=Δlogε1+εΔℱr−1−1εΔℱt−1−1ε−1,logε1+εμℱr−1εμℱt−1ε−1,logεεε−1ενℱr−1−1ενℱt−1−1+ε−1;λℱ1=Δ1−logε1+ε1−Δℱr−1−1λε−1λ−1,1−logε1+ε1−μℱr−1λε−1λ−1,logε1+ενℱr−1λε−1λ−1;ℱ1λ=Δlogε1+εΔℱr−1−1λε−1λ−1,logε1+εμℱr−1λε−1λ−1,1−logε1+ε1−νℱr−1λε−1λ−1.Theorem 1.
Letℱ1 and ℱ2 are two HF2TLSs, then the following equations hold (λ,λ1,λ2>0).(1)
ℱ1⊕ℱ2=ℱ2⊕ℱ1(2)
ℱ1⊗ℱ2=ℱ2⊗ℱ1(3)
λℱ1⊕ℱ2=λℱ1⊕λℱ2(4)
ℱ1⊗ℱ2λ=ℱ1λ⊗ℱ2λ(5)
λ1+λ2ℱ1=λ1ℱ1⊕λ2ℱ1(6)
ℱ1λ1+λ2=ℱ1λ1⊗ℱ1λ2Proof.
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μ
ℱ
r
+
λ
ζ
μ
ℱ
t
,
χ
−
1
λ
χ
ν
ℱ
r
+
λ
χ
ν
ℱ
t
;
ℱ
1
λ
⊗
ℱ
2
λ
=
Δ
ζ
−
1
ζ
ζ
−
1
λ
ζ
△
ℱ
r
−
1
+
ζ
−
1
λ
ζ
△
ℱ
t
−
1
,
ζ
−
1
ζ
ζ
−
1
λ
ζ
μ
ℱ
r
+
ζ
−
1
λ
ζ
μ
ℱ
t
,
χ
−
1
χ
χ
−
1
λ
χ
ν
ℱ
r
+
χ
−
1
λ
χ
ν
ℱ
t
=
Δ
ζ
−
1
λ
ζ
Δ
ℱ
r
−
1
+
λ
ζ
Δ
ℱ
t
−
1
,
ζ
−
1
λ
ζ
μ
ℱ
r
+
λ
ζ
μ
ℱ
t
,
χ
−
1
λ
χ
ν
ℱ
r
+
λ
χ
ν
ℱ
t
.
So
ℱ
1
⊗
ℱ
2
λ
=
ℱ
1
λ
⊗
ℱ
2
λ
holds
.
(17)
λ
1
ℱ
1
⊕
λ
2
ℱ
1
=
Δ
χ
−
1
χ
χ
−
1
λ
1
χ
Δ
ℱ
r
−
1
+
λ
2
χ
Δ
ℱ
r
−
1
,
χ
−
1
χ
χ
−
1
λ
1
χ
μ
ℱ
r
+
λ
2
χ
μ
ℱ
r
,
ζ
−
1
ζ
ζ
−
1
λ
1
ζ
ν
ℱ
r
+
λ
2
ζ
ν
ℱ
r
=
Δ
χ
−
1
λ
1
+
λ
2
χ
Δ
ℱ
r
−
1
,
χ
−
1
λ
1
+
λ
2
χ
μ
ℱ
r
,
ζ
−
1
λ
1
+
λ
2
ζ
ν
ℱ
r
=
λ
1
+
λ
2
ℱ
1
.
By using the above equations, we can obtain
λ
1
+
λ
2
ℱ
1
=
λ
1
ℱ
1
⊕
λ
2
ℱ
1
.
(18)
ℱ
1
λ
1
⊗
ℱ
1
λ
2
=
Δ
ζ
−
1
ζ
ζ
−
1
λ
1
ζ
Δ
ℱ
r
−
1
+
ζ
−
1
λ
2
ζ
Δ
ℱ
r
−
1
,
ζ
−
1
ζ
ζ
−
1
λ
1
ζ
μ
ℱ
r
+
ζ
−
1
λ
2
ζ
μ
ℱ
r
,
χ
−
1
χ
χ
−
1
λ
1
χ
ν
ℱ
r
+
χ
−
1
λ
2
χ
ν
ℱ
r
=
Δ
ζ
−
1
λ
r
+
λ
2
ζ
Δ
ℱ
r
−
1
,
ζ
−
1
λ
1
+
λ
2
ζ
μ
ℱ
r
,
χ
−
1
λ
1
+
λ
2
χ
ν
ℱ
r
=
ℱ
1
λ
1
+
λ
2
.


## 3.1. Hesitant Fermatean Fuzzy 2-Tuple Linguistic Sets

Definition 5.
A HF2TLSℱ on Z is defined as follows:(7)ℱ=sfθℱr,αℱr,μℱrz,νℱrzsfθℱr,αℱr∈Sf,where sfθℱr∈Sf, αℱr∈−0.5,0.5, and μℱrz,νℱrz are located in the unit interval 0,1 with 0≤μℱrz3+νℱrz3≤1. μℱrz and νℱrz denote the support degree and against degree of the element z belonging to sfθℱr,αℱr, respectively. 1≤r≤ℱ and ℱ is the number of elements of ℱ. For convenience, sfθℱr,αℱr,μℱrz,νℱrz is called a hesitant Fermatean 2-tuple linguistic number (HF2TLN).Definition 6.
Letℱ=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ be a HF2TLS; the score function of ℱ is defined as follows:(8)SCℱ=∑r=1ℱΔ−1sfθℱr,αℱr×μℱr3−νℱr3ℱ.Definition 7.
Letℱ=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ be a HF2TLS; the accuracy function of ℱ is defined as follows:(9)ACℱ=∑r=1ℱΔ−1sfθℱr,αℱr×μℱr3+νℱr3ℱ.Definition 8.
Letℱ1=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ1 and ℱ2=sfθℱt,αℱt,μℱt,νℱt,t=1,2,…,ℱ2 be two HF2TLSs; then, the order relationships between ℱ1 and ℱ2 are described as follows:(1)
If SCℱ1<SCℱ2,thenℱ1≺ℱ2(2)
If SCℱ1>SCℱ2,thenℱ1≻ℱ2(3)
When SCℱ1=SCℱ2,(a)
ACℱ1=ACℱ2,andℱ1∼ℱ2(b)
ACℱ1<ACℱ2,andℱ1≺ℱ2(c)
ACℱ1>ACℱ2,andℱ1≻ℱ2


## 3.2. New Operational Laws for HF2TLSs

Thet-norm and s-norm are two kinds of binary operations used in probabilistic metrics and fuzzy logic, which generalizes intersection operation and conjunction operation. An Archimedean t-norm is defined as Tc,d=ζ−1ζc+ζd, where ζ is a strictly continuous decreasing function ζ:0,1⟶0,∞ such that ζ1=0. Its dual function s-norm can be externalized as Sc,d=χ−1χc+χd with χc=ζ1−c. By using a different mapping T:0,1×0,1⟶0,1, multiple different operators can be derived for handling fuzzy term aggregation problems. ATS was incorporated into 2-tuple linguistic environments by Tao et al. [40], and some ATS-based operational laws for intuitionistic fuzzy numbers were developed by Xia et al. [31–33]. Enlightened by these works, we deduce the following operational laws of HF2TLSs information.Definition 9.
Letℱ1=sfθℱr,αℱr,μℱr,νℱr,r=1,2,…,ℱ1 and ℱ2=sfθℱt,αℱt,μℱt,νℱt,t=1,2,…,ℱ2 be any two HF2TLSs; then, some operational laws are defined as follows (λ>0 and Δℱr−1=Δ−1sfθℱr,αℱr):(1)
ℱ1⊕ℱ2=Δχ−1χΔℱr−1+χΔℱt−1,χ−1χμℱr+χμℱt,ζ−1ζνℱr+ζνℱtsfθℱr,αℱr,μℱr,νℱr∈ℱ1,sfθℱt,αℱt,μℱt,νℱt∈ℱ2(2)
ℱ1⊗ℱ2=Δζ−1ζ△ℱr−1+ζ△ℱt−1,ζ−1ζμℱr+ζμℱt,χ−1χνℱr+χνℱtsfθℱr,αℱr,μℱr,νℱr∈ℱ1,sfθℱt,αℱt,μℱt,νℱt∈ℱ2(3)
λℱ1=Δχ−1λχΔℱr−1,χ−1λχμℱr,ζ−1λζνℱrsfθℱr,αℱr,μℱr,νℱr∈ℱ1(4)
ℱ1λ=Δζ−1λζ△ℱr−1,ζ−1λζμℱr,χ−1λχνℱrsfθℱr,αℱr,μℱr,νℱr∈ℱ1
In the sequel, some special operational laws are obtained by taking different forms ofζc and their dual function χc=ζ1−c. Meanwhile, sfθℱr,αℱr,μℱr,νℱr∈ℱ1andsfθℱt,αℱt,μℱt,νℱt∈ℱ2 are simplified, so it is omitted during the operation process of HF2TLSs.Remark 1.
Letζc=−logc and χc=−log1−c; then, the operational laws defined by Definition 9 can be read as the following equations:(1)
ℱ1⊕ℱ2=ΔΔℱr−1+Δℱt−1−Δℱr−1⋅Δℱt−1,μℱr+μℱt−μℱrμℱt,νℱrνℱt(2)
ℱ1⊗ℱ2=ΔΔℱr−1⋅Δℱt−1,μℱrμℱt,νℱr+νℱt−νℱrνℱt(3)
λℱ1=Δ1−1−Δℱr−1λ,1−1−μℱrλ,νℱrλ(4)
ℱ1λ=ΔΔℱr−1λ,μℱrλ,1−1−νℱrλRemark 2.
Ifζc=log2−c/c and χc=log2−1−c/1−c, then the operational laws via the Einstein t-norm and s-norm can be obtained:(10)ℱ1⊕ℱ2=ΔΔℱr−1+Δℱt−11+Δℱr−1⋅Δℱt−1,μℱr+μℱt1+μℱrμℱt,νℱrνℱt1+1−νℱr1−νℱt;ℱ1⊗ℱ2=ΔΔℱr−1⋅Δℱt−11−1−Δℱr−11−Δℱt−1,μℱrμℱt1−1−μℱr1−μℱt,νℱr+νℱt1+νℱrνℱt;λℱ1=Δ1+Δℱr−1λ−1−Δℱr−1λ1+Δℱr−1λ+1−Δℱr−1λ,1+μℱrλ−1−μℱrλ1+μℱrλ+1−μℱrλ,2νℱrλ2−νℱrλ+νℱrλ;ℱ1λ=Δ2Δℱr−1λ2−Δℱr−1λ+Δℱr−1λ,2μℱrλ2−μℱrλ+μℱrλ,1+νℱrλ−1−νℱrλ1+νℱrλ+1−νℱrλ.Remark 3.
Ifζc=logε+1−εc/c, χc=log1−1−εc/1−c, and ε>0, then the operational laws via the Hamacher t-norm and s-norm can be obtained:(11)ℱ1⊕ℱ2=ΔΔℱr−1+Δℱt−1−1−ε⋅Δℱr−1⋅Δℱt−11−1−ε⋅Δℱr−1⋅Δℱt−1,μℱr+μℱt−1−εμℱrμℱt1−1−εμℱrμℱt,νℱrνℱtε+1−ενℱr+νℱt−νℱrνℱt;ℱ1⊗ℱ2=ΔΔℱr−1⋅Δℱt−1ε+1−εΔℱr−1+Δℱt−1−Δℱr−1⋅Δℱt−1,μℱrμℱtε+1−εμℱr+μℱt−μℱrμℱt,νℱr+νℱt−1−ενℱrνℱt1−1−ενℱrνℱt;λℱ1=Δ1+ε−1⋅Δℱr−1λ−1−Δℱr−1λ1+ε−1⋅Δℱr−1λ+ε−11−Δℱr−1λ,1+ε−1μℱrλ−1−μℱrλ1+ε−1μℱrλ+ε−11−μℱrλ,ενℱrλε+1−ενℱrλ+ε−1νℱrλ;ℱ1λ=ΔεΔℱr−1λε+1−ε⋅Δℱr−1λ+ε−1Δℱr−1λ,εμℱrλε+1−εμℱrλ+ε−1μℱrλ,1+ε−1νℱrλ−1−νℱrλ1+ε−1νℱrλ+ε−11−νℱrλ.Remark 4.
Letζc=logε−1/εc−1, χc=logε−1/ε1−c−1, and ε>1, then the operational laws via the Frank t-norm and s-norm can be obtained:(12)ℱ1⊕ℱ2=Δlogεεε−1εΔℱr−1−1−1εΔℱt−1−1−1+ε−1,logεεε−1εμℱr−1−1εμℱt−1−1+ε−1,logε1+ενℱr−1ενℱt−1ε−1;ℱ1⊗ℱ2=Δlogε1+εΔℱr−1−1εΔℱt−1−1ε−1,logε1+εμℱr−1εμℱt−1ε−1,logεεε−1ενℱr−1−1ενℱt−1−1+ε−1;λℱ1=Δ1−logε1+ε1−Δℱr−1−1λε−1λ−1,1−logε1+ε1−μℱr−1λε−1λ−1,logε1+ενℱr−1λε−1λ−1;ℱ1λ=Δlogε1+εΔℱr−1−1λε−1λ−1,logε1+εμℱr−1λε−1λ−1,1−logε1+ε1−νℱr−1λε−1λ−1.Theorem 1.
Letℱ1 and ℱ2 are two HF2TLSs, then the following equations hold (λ,λ1,λ2>0).(1)
ℱ1⊕ℱ2=ℱ2⊕ℱ1(2)
ℱ1⊗ℱ2=ℱ2⊗ℱ1(3)
λℱ1⊕ℱ2=λℱ1⊕λℱ2(4)
ℱ1⊗ℱ2λ=ℱ1λ⊗ℱ2λ(5)
λ1+λ2ℱ1=λ1ℱ1⊕λ2ℱ1(6)
ℱ1λ1+λ2=ℱ1λ1⊗ℱ1λ2Proof.
(13)
ℱ
1
⊕
ℱ
2
=
Δ
χ
−
1
χ
Δ
ℱ
r
−
1
+
χ
Δ
ℱ
t
−
1
,
χ
−
1
χ
μ
ℱ
r
+
χ
μ
ℱ
t
,
ζ
−
1
ζ
ν
ℱ
r
+
ζ
ν
ℱ
t
=
Δ
χ
−
1
χ
Δ
ℱ
t
−
1
+
χ
Δ
ℱ
r
−
1
,
χ
−
1
χ
μ
ℱ
t
+
χ
μ
ℱ
r
,
ζ
−
1
ζ
ν
ℱ
t
+
ζ
ν
ℱ
r
=
ℱ
2
⊕
ℱ
1
.
(14)
ℱ
1
⊗
ℱ
2
=
Δ
ζ
−
1
ζ
Δ
ℱ
r
−
1
+
ζ
Δ
ℱ
t
−
1
,
ζ
−
1
ζ
μ
ℱ
r
+
ζ
μ
ℱ
t
,
χ
−
1
χ
ν
ℱ
r
+
χ
ν
ℱ
t
=
Δ
ζ
−
1
ζ
Δ
ℱ
t
−
1
+
ζ
Δ
ℱ
r
−
1
,
ζ
−
1
ζ
μ
ℱ
t
+
ζ
μ
ℱ
r
,
χ
−
1
χ
ν
ℱ
t
+
χ
ν
ℱ
r
=
ℱ
2
⊗
ℱ
1
.
(15)
λ
ℱ
1
⊕
ℱ
2
=
Δ
χ
−
1
λ
χ
χ
−
1
χ
Δ
ℱ
r
−
1
+
χ
Δ
ℱ
t
−
1
,
χ
−
1
λ
χ
χ
−
1
χ
μ
ℱ
r
+
χ
μ
ℱ
t
,
ζ
−
1
λ
ζ
ζ
−
1
ζ
ν
ℱ
r
+
ζ
ν
ℱ
t
=
Δ
χ
−
1
λ
χ
Δ
ℱ
r
−
1
+
λ
χ
Δ
ℱ
t
−
1
,
χ
−
1
λ
χ
μ
ℱ
r
+
λ
χ
μ
ℱ
t
,
ζ
−
1
λ
ζ
ν
ℱ
r
+
λ
ζ
ν
ℱ
t
;
λ
ℱ
1
⊕
λ
ℱ
2
=
Δ
χ
−
1
χ
χ
−
1
λ
χ
Δ
ℱ
r
−
1
+
λ
χ
Δ
ℱ
t
−
1
,
χ
−
1
χ
χ
−
1
λ
χ
μ
ℱ
r
+
λ
χ
μ
ℱ
t
,
ζ
−
1
ζ
ζ
−
1
λ
ζ
ν
ℱ
r
+
λ
ζ
ν
ℱ
t
=
Δ
χ
−
1
λ
χ
Δ
ℱ
r
−
1
+
λ
χ
Δ
ℱ
t
−
1
,
χ
−
1
λ
χ
μ
ℱ
r
+
λ
χ
μ
ℱ
t
,
ζ
−
1
λ
ζ
ν
ℱ
r
+
λ
ζ
ν
ℱ
t
.
So
λ
ℱ
1
⊕
ℱ
2
=
λ
ℱ
1
⊕
λ
ℱ
2
holds
.
(16)
ℱ
1
⊗
ℱ
2
λ
=
Δ
ζ
−
1
λ
ζ
ζ
−
1
ζ
Δ
ℱ
r
−
1
+
ζ
Δ
ℱ
t
−
1
,
ζ
−
1
λ
ζ
ζ
−
1
ζ
μ
ℱ
r
+
ζ
μ
ℱ
t
,
χ
−
1
λ
χ
χ
−
1
χ
ν
ℱ
r
+
χ
ν
ℱ
t
=
Δ
ζ
−
1
λ
ζ
Δ
ℱ
r
−
1
+
λ
ζ
Δ
ℱ
t
−
1
,
ζ
−
1
λ
ζ
μ
ℱ
r
+
λ
ζ
μ
ℱ
t
,
χ
−
1
λ
χ
ν
ℱ
r
+
λ
χ
ν
ℱ
t
;
ℱ
1
λ
⊗
ℱ
2
λ
=
Δ
ζ
−
1
ζ
ζ
−
1
λ
ζ
△
ℱ
r
−
1
+
ζ
−
1
λ
ζ
△
ℱ
t
−
1
,
ζ
−
1
ζ
ζ
−
1
λ
ζ
μ
ℱ
r
+
ζ
−
1
λ
ζ
μ
ℱ
t
,
χ
−
1
χ
χ
−
1
λ
χ
ν
ℱ
r
+
χ
−
1
λ
χ
ν
ℱ
t
=
Δ
ζ
−
1
λ
ζ
Δ
ℱ
r
−
1
+
λ
ζ
Δ
ℱ
t
−
1
,
ζ
−
1
λ
ζ
μ
ℱ
r
+
λ
ζ
μ
ℱ
t
,
χ
−
1
λ
χ
ν
ℱ
r
+
λ
χ
ν
ℱ
t
.
So
ℱ
1
⊗
ℱ
2
λ
=
ℱ
1
λ
⊗
ℱ
2
λ
holds
.
(17)
λ
1
ℱ
1
⊕
λ
2
ℱ
1
=
Δ
χ
−
1
χ
χ
−
1
λ
1
χ
Δ
ℱ
r
−
1
+
λ
2
χ
Δ
ℱ
r
−
1
,
χ
−
1
χ
χ
−
1
λ
1
χ
μ
ℱ
r
+
λ
2
χ
μ
ℱ
r
,
ζ
−
1
ζ
ζ
−
1
λ
1
ζ
ν
ℱ
r
+
λ
2
ζ
ν
ℱ
r
=
Δ
χ
−
1
λ
1
+
λ
2
χ
Δ
ℱ
r
−
1
,
χ
−
1
λ
1
+
λ
2
χ
μ
ℱ
r
,
ζ
−
1
λ
1
+
λ
2
ζ
ν
ℱ
r
=
λ
1
+
λ
2
ℱ
1
.
By using the above equations, we can obtain
λ
1
+
λ
2
ℱ
1
=
λ
1
ℱ
1
⊕
λ
2
ℱ
1
.
(18)
ℱ
1
λ
1
⊗
ℱ
1
λ
2
=
Δ
ζ
−
1
ζ
ζ
−
1
λ
1
ζ
Δ
ℱ
r
−
1
+
ζ
−
1
λ
2
ζ
Δ
ℱ
r
−
1
,
ζ
−
1
ζ
ζ
−
1
λ
1
ζ
μ
ℱ
r
+
ζ
−
1
λ
2
ζ
μ
ℱ
r
,
χ
−
1
χ
χ
−
1
λ
1
χ
ν
ℱ
r
+
χ
−
1
λ
2
χ
ν
ℱ
r
=
Δ
ζ
−
1
λ
r
+
λ
2
ζ
Δ
ℱ
r
−
1
,
ζ
−
1
λ
1
+
λ
2
ζ
μ
ℱ
r
,
χ
−
1
λ
1
+
λ
2
χ
ν
ℱ
r
=
ℱ
1
λ
1
+
λ
2
.


## 4. ATS-Based HF2TLSs Aggregation Operators

In this section, using the operational laws of HF2TLSs defined in the previous section and extending the Bonferroni mean (BM) to the context of HF2TLSs, we introduce the related definitions of ATS-based HF2TLSs aggregation operators.


### 4.1. A-HF2TLWBM Aggregation Operator

Definition 10.
Letℱrr=1,2,…,n be a collection of HF2TLNs, w=w1,w2,…,wn with wr∈0,1 be a weight vector, and p,q>0:(19)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn1nn−1⊕r,t=1,r≠tnwrℱrp⊗wtℱtq1/p+q,is called an ATS-based hesitant Fermatean 2-tuple linguistic weighted Bonferroni mean aggregation operator.Theorem 2.
Letℱrr=1,2,…,n be a collection of HF2TLNs, then the result derived from Definition 10 is formulated as follows (N=1/nn−1):(20)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnΔζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχΔℱr−1+qζχ−1wtχΔℱt−1,ζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχμℱr+qζχ−1wtχμℱt,χ−11p+qχζ−1N∑r,t=1,r≠tnζχ−1pχζ−1wrζνℱr+qχζ−1wtζνℱt.Theorem 3.
Letℱr∣r=1,2,…,n be a collection of HF2TLNs and p,q>0; three properties of the A-HF2TLWBM operator can be obtained as follows:(1)
Idempotency: ifℱr=ℱ for all r, then(21)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=ℱ.(2)
Monotonicity: assume thatsfθℱr,αℱr,μℱr,νℱr∈ℱr,sfθℱr′,αℱr′,μℱr′,νℱr′∈ℱr′. If ℱr and ℱr′ satisfy sfθℱr,αℱr,μℱr,νℱr≤sfθℱr′,αℱr′,μℱr′,νℱr′, then sfθℱr,αℱr≤sfθℱr′,αℱr′,μℱr≤μℱr′,νℱr≥νℱr′:(22)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn≤A‐HF2TLWBMp,qℱ1′,ℱ2′,…,ℱn′.(3)
Boundedness: denoteℱ+=maxℱr=maxsfθℱr,αℱr,maxμℱr,minνℱrand ℱ−=minℱr=minsfθℱr,αℱr,minμℱr,maxνℱr, then(23)ℱ−≤A‐HF2TLWBMp,qℱr,ℱ2,…,ℱn≤ℱ+.Proof.
(24)
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
,
ℱ
2
,
…
,
ℱ
n
⟹
A
−
HF
2
TLWBM
p
,
q
ℱ
,
ℱ
,
…
,
ℱ
=
⋃
ℱ
1
,
…
,
ℱ
n
Δ
ζ
−
1
1
p
+
q
ζ
χ
−
1
N
1
N
χ
ζ
−
1
p
ζ
χ
−
1
w
χ
Δ
ℱ
−
1
+
q
ζ
χ
−
1
w
χ
Δ
ℱ
−
1
,
ζ
−
1
1
p
+
q
ζ
χ
−
1
χ
ζ
−
1
p
ζ
χ
−
1
w
χ
μ
ℱ
+
q
ζ
χ
−
1
w
χ
μ
ℱ
,
χ
−
1
1
p
+
q
χ
ζ
−
1
ζ
χ
−
1
p
ζ
χ
−
1
w
ζ
ν
ℱ
+
q
χ
ζ
−
1
w
ζ
ν
ℱ
,
=
∪
ℱ
,
…
,
ℱ
Δ
ζ
−
1
1
p
+
q
ζ
ζ
−
1
p
+
q
ζ
χ
−
1
w
χ
Δ
ℱ
−
1
,
ζ
−
1
1
p
+
q
ζ
ζ
−
1
p
+
q
ζ
χ
−
1
w
χ
μ
ℱ
,
χ
−
1
1
p
+
q
χ
χ
−
1
p
+
q
χ
ζ
−
1
w
ζ
ν
ℱ
,
=
ℱ
.
ζ and ζ−1 are strictly decreasing functions; χ and χ−1 are strictly increasing functions. From Theorem 2, we can obtain that(25)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnΔζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχΔℱr−1+qζχ−1wtχΔℱt−1,ζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχμℱr+qζχ−1wtχμℱt,χ−11p+qχζ−1N∑r,t=1,r≠tnζχ−1pχζ−1wrζνℱr+qχζ−1wtζνℱt,≤⋃ℱ1′,…,ℱn′Δζ−11p+qζχ−1N×∑r,t=1,r≠tnχζ−1pζχ−1wr′χΔℱr′−1+qζχ−1wt′χΔℱt′−1,ζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wr′χμℱr′+qζχ−1wt′χμℱt′,χ−11p+qχζ−1N∑r,t=1,r≠tnζχ−1pχζ−1wr′ζνℱr′+qχζ−1wt′ζνℱt′,=A‐HF2TLWBMp,qℱ1′,ℱ2′,…,ℱn′,where sfθℱr,αℱr,μℱr,νℱr∈ℱr,Δℱr−1=Δ−1sfθℱr,αℱr, sfθℱr′,αℱr′,μℱr′,νℱr′∈ℱr′,Δℱr′−1=Δ−1sfθℱr′,αℱr′, and N=1/nn−1.
According to the idempotency property, we obtainA‐HF2TLWBMp,qℱ1+,ℱ2+,…,ℱn+=ℱ+and A‐HF2TLWBMp,qℱ1−,ℱ2−,…,ℱn−=ℱ−. Using the monotonicity, we obtain the following inequalities:
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
−
,
ℱ
2
−
,
…
,
ℱ
n
−
≤
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
,
ℱ
2
,
…
,
ℱ
n
≤
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
+
,
ℱ
2
+
,
…
,
ℱ
n
+
. Thus, ℱ−≤A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn≤ℱ+.
Furthermore, some special cases are examined by taking different parameters values (N=1/nn−1).Case 1.
Whenq⟶0, then we have(26)A‐HF2TLWBMp,0ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnN⊕r=1nwrℱrp1/p=⋃ℱ1,…,ℱnΔζ−11pζχ−1N∑r=1nχζ−1pζχ−1wrχΔℱr−1,ζ−11pζχ−1N∑r=1nχζ−1pζχ−1wrχμℱr,χ−11pχζ−1N∑r=1nζχ−1pχζ−1wrζνℱr.Case 2.
Whenp=1,q⟶0, then we obtain ATS-based hesitant Fermatean 2-tuple linguistic unit weighted Bonferroni mean aggregation operator as follows:(27)A‐HF2TLWBM1,0ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnN⊕r=1nwrℱr=⋃ℱ1,…,ℱnΔχ−1N∑r=1nwrχΔℱr−1,χ−1N∑r=1nwrχμℱr,ζ−1N∑r=1nwrζνℱr.Case 3.
Whenp=1,q=1, then we get ATS-based hesitant Fermatean 2-tuple linguistic interrelated square weighted Bonferroni mean aggregation operator as follows:(28)A‐HF2TLWBM1,1ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnN⊕r,t=1,r≠tnwrℱr⊗wtℱt1/2=⋃ℱ1,…,ℱnΔζ−112ζχ−1N∑r,t=1,r≠tnwrχΔℱr−1+wtχΔℱt−1,ζ−112ζχ−1N∑r,t=1,r≠tnwrχμℱr+wtχμℱt,χ−112χζ−1N∑r,t=1,r≠tnwrζνℱr+wtζνℱt.


### 4.2. A-HF2TLWGBM Aggregation Operator

In the sequel, the classical GBM operator is extended under the background of HF2TLSs situation, and we derive the following aggregation operator.Definition 11.
Letℱrr=1,2,…,n be a collection of HF2TLNs, w=w1,w2,…,wn be a weight vector, and p,q>0; then,(29)A‐HF2TLWGBMp,qℱ1,ℱ2,…,ℱn=∪ℱ1,…,ℱn1p+q⊗r,t=1,r≠tnpℱrwr⊕qℱtwt1/nn−1,which is called an ATS-based hesitant Fermatean 2-tuple linguistic weighted geometric Bonferroni mean.Theorem 4.
Letℱrr=1,2,…,n be a collection of HF2TLNs; then, the result derived from Definition 11 is also a HF2TLS and (N=1/nn−1):(30)A‐HF2TLWGBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn·Δχ−11p+qχζ−1∑r,t=1,r≠tnNζχ−1pχζ−1wrζΔℱr−1+qχζ−1wtζΔℱt−1,χ−11p+qχζ−1∑r,t=1,r≠tnNζχ−1pχζ−1wrζμℱr+qχζ−1wtζμℱt,ζ−11p+qζχ−1∑r,t=1,r≠tnNχζ−1pζχ−1wrχνℱr+qζχ−1wtχνℱt.
Similar to the A-HF2TLWBM operator, we can obtain that the A-HF2TLWGBM operator has also idempotency, boundedness, and monotonicity. In addition, some special forms of the A-HF2TLWGBM operator can be given according to the changes of parametersp and q as follows.Case 1.
Whenq⟶0, then the following equality holds:(31)A‐HF2TLWGBMp,0ℱ1,ℱ2,…,ℱn=∪ℱ1,…,ℱn1p⊕r=1npℱrwrN=∪ℱ1,…,ℱnΔχ−11pχζ−1∑r=1nNζχ−1pχζ−1wrζ△ℱr−1,χ−11pχζ−1∑r=1nNζχ−1pχζ−1wrζμℱr,ζ−11pζχ−1∑r=1nNχζ−1pζχ−1wrχνℱr.Case 2.
Whenp=1andq⟶0, then the ATS-based hesitant Fermatean 2-tuple linguistic weighted unit geometric Bonferroni mean operator holds:(32)A‐HF2TLWGBM1,0ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn⊕r=1nℱrwrN,=⋃ℱ1,…,ℱnΔζ−1∑r=1nNwrζΔℱr−1,ζ−1∑r=1nNwrζμℱr,χ−1∑r=1nNwrχνℱr.Case 3.
Whenp=1andq=1, then we obtain the ATS-based hesitant Fermatean 2-tuple linguistic weighted interrelated square geometric Bonferroni mean operator:(33)A‐HF2TLWGBM1,1ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn12⊗r,t=1,r≠tnℱrwr⊕ℱtwtN=⋃ℱ1,…,ℱnΔχ−112χζ−1∑r,t=1,r≠tnNζχ−1χζ−1wrζ△ℱr−1+χζ−1wtζ△ℱt−1,χ−112χζ−1∑r,t=1,r≠tnNζχ−1χζ−1wrζμℱr+χζ−1wtζμℱt,ζ−112ζχ−1∑r,t=1,r≠tnNχζ−1ζχ−1wrχνℱr+ζχ−1wtχνℱt.


## 4.1. A-HF2TLWBM Aggregation Operator

Definition 10.
Letℱrr=1,2,…,n be a collection of HF2TLNs, w=w1,w2,…,wn with wr∈0,1 be a weight vector, and p,q>0:(19)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn1nn−1⊕r,t=1,r≠tnwrℱrp⊗wtℱtq1/p+q,is called an ATS-based hesitant Fermatean 2-tuple linguistic weighted Bonferroni mean aggregation operator.Theorem 2.
Letℱrr=1,2,…,n be a collection of HF2TLNs, then the result derived from Definition 10 is formulated as follows (N=1/nn−1):(20)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnΔζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχΔℱr−1+qζχ−1wtχΔℱt−1,ζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχμℱr+qζχ−1wtχμℱt,χ−11p+qχζ−1N∑r,t=1,r≠tnζχ−1pχζ−1wrζνℱr+qχζ−1wtζνℱt.Theorem 3.
Letℱr∣r=1,2,…,n be a collection of HF2TLNs and p,q>0; three properties of the A-HF2TLWBM operator can be obtained as follows:(1)
Idempotency: ifℱr=ℱ for all r, then(21)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=ℱ.(2)
Monotonicity: assume thatsfθℱr,αℱr,μℱr,νℱr∈ℱr,sfθℱr′,αℱr′,μℱr′,νℱr′∈ℱr′. If ℱr and ℱr′ satisfy sfθℱr,αℱr,μℱr,νℱr≤sfθℱr′,αℱr′,μℱr′,νℱr′, then sfθℱr,αℱr≤sfθℱr′,αℱr′,μℱr≤μℱr′,νℱr≥νℱr′:(22)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn≤A‐HF2TLWBMp,qℱ1′,ℱ2′,…,ℱn′.(3)
Boundedness: denoteℱ+=maxℱr=maxsfθℱr,αℱr,maxμℱr,minνℱrand ℱ−=minℱr=minsfθℱr,αℱr,minμℱr,maxνℱr, then(23)ℱ−≤A‐HF2TLWBMp,qℱr,ℱ2,…,ℱn≤ℱ+.Proof.
(24)
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
,
ℱ
2
,
…
,
ℱ
n
⟹
A
−
HF
2
TLWBM
p
,
q
ℱ
,
ℱ
,
…
,
ℱ
=
⋃
ℱ
1
,
…
,
ℱ
n
Δ
ζ
−
1
1
p
+
q
ζ
χ
−
1
N
1
N
χ
ζ
−
1
p
ζ
χ
−
1
w
χ
Δ
ℱ
−
1
+
q
ζ
χ
−
1
w
χ
Δ
ℱ
−
1
,
ζ
−
1
1
p
+
q
ζ
χ
−
1
χ
ζ
−
1
p
ζ
χ
−
1
w
χ
μ
ℱ
+
q
ζ
χ
−
1
w
χ
μ
ℱ
,
χ
−
1
1
p
+
q
χ
ζ
−
1
ζ
χ
−
1
p
ζ
χ
−
1
w
ζ
ν
ℱ
+
q
χ
ζ
−
1
w
ζ
ν
ℱ
,
=
∪
ℱ
,
…
,
ℱ
Δ
ζ
−
1
1
p
+
q
ζ
ζ
−
1
p
+
q
ζ
χ
−
1
w
χ
Δ
ℱ
−
1
,
ζ
−
1
1
p
+
q
ζ
ζ
−
1
p
+
q
ζ
χ
−
1
w
χ
μ
ℱ
,
χ
−
1
1
p
+
q
χ
χ
−
1
p
+
q
χ
ζ
−
1
w
ζ
ν
ℱ
,
=
ℱ
.
ζ and ζ−1 are strictly decreasing functions; χ and χ−1 are strictly increasing functions. From Theorem 2, we can obtain that(25)A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnΔζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχΔℱr−1+qζχ−1wtχΔℱt−1,ζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wrχμℱr+qζχ−1wtχμℱt,χ−11p+qχζ−1N∑r,t=1,r≠tnζχ−1pχζ−1wrζνℱr+qχζ−1wtζνℱt,≤⋃ℱ1′,…,ℱn′Δζ−11p+qζχ−1N×∑r,t=1,r≠tnχζ−1pζχ−1wr′χΔℱr′−1+qζχ−1wt′χΔℱt′−1,ζ−11p+qζχ−1N∑r,t=1,r≠tnχζ−1pζχ−1wr′χμℱr′+qζχ−1wt′χμℱt′,χ−11p+qχζ−1N∑r,t=1,r≠tnζχ−1pχζ−1wr′ζνℱr′+qχζ−1wt′ζνℱt′,=A‐HF2TLWBMp,qℱ1′,ℱ2′,…,ℱn′,where sfθℱr,αℱr,μℱr,νℱr∈ℱr,Δℱr−1=Δ−1sfθℱr,αℱr, sfθℱr′,αℱr′,μℱr′,νℱr′∈ℱr′,Δℱr′−1=Δ−1sfθℱr′,αℱr′, and N=1/nn−1.
According to the idempotency property, we obtainA‐HF2TLWBMp,qℱ1+,ℱ2+,…,ℱn+=ℱ+and A‐HF2TLWBMp,qℱ1−,ℱ2−,…,ℱn−=ℱ−. Using the monotonicity, we obtain the following inequalities:
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
−
,
ℱ
2
−
,
…
,
ℱ
n
−
≤
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
,
ℱ
2
,
…
,
ℱ
n
≤
A
‐
HF
2
TLWBM
p
,
q
ℱ
1
+
,
ℱ
2
+
,
…
,
ℱ
n
+
. Thus, ℱ−≤A‐HF2TLWBMp,qℱ1,ℱ2,…,ℱn≤ℱ+.
Furthermore, some special cases are examined by taking different parameters values (N=1/nn−1).Case 1.
Whenq⟶0, then we have(26)A‐HF2TLWBMp,0ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnN⊕r=1nwrℱrp1/p=⋃ℱ1,…,ℱnΔζ−11pζχ−1N∑r=1nχζ−1pζχ−1wrχΔℱr−1,ζ−11pζχ−1N∑r=1nχζ−1pζχ−1wrχμℱr,χ−11pχζ−1N∑r=1nζχ−1pχζ−1wrζνℱr.Case 2.
Whenp=1,q⟶0, then we obtain ATS-based hesitant Fermatean 2-tuple linguistic unit weighted Bonferroni mean aggregation operator as follows:(27)A‐HF2TLWBM1,0ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnN⊕r=1nwrℱr=⋃ℱ1,…,ℱnΔχ−1N∑r=1nwrχΔℱr−1,χ−1N∑r=1nwrχμℱr,ζ−1N∑r=1nwrζνℱr.Case 3.
Whenp=1,q=1, then we get ATS-based hesitant Fermatean 2-tuple linguistic interrelated square weighted Bonferroni mean aggregation operator as follows:(28)A‐HF2TLWBM1,1ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱnN⊕r,t=1,r≠tnwrℱr⊗wtℱt1/2=⋃ℱ1,…,ℱnΔζ−112ζχ−1N∑r,t=1,r≠tnwrχΔℱr−1+wtχΔℱt−1,ζ−112ζχ−1N∑r,t=1,r≠tnwrχμℱr+wtχμℱt,χ−112χζ−1N∑r,t=1,r≠tnwrζνℱr+wtζνℱt.


## 4.2. A-HF2TLWGBM Aggregation Operator

In the sequel, the classical GBM operator is extended under the background of HF2TLSs situation, and we derive the following aggregation operator.Definition 11.
Letℱrr=1,2,…,n be a collection of HF2TLNs, w=w1,w2,…,wn be a weight vector, and p,q>0; then,(29)A‐HF2TLWGBMp,qℱ1,ℱ2,…,ℱn=∪ℱ1,…,ℱn1p+q⊗r,t=1,r≠tnpℱrwr⊕qℱtwt1/nn−1,which is called an ATS-based hesitant Fermatean 2-tuple linguistic weighted geometric Bonferroni mean.Theorem 4.
Letℱrr=1,2,…,n be a collection of HF2TLNs; then, the result derived from Definition 11 is also a HF2TLS and (N=1/nn−1):(30)A‐HF2TLWGBMp,qℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn·Δχ−11p+qχζ−1∑r,t=1,r≠tnNζχ−1pχζ−1wrζΔℱr−1+qχζ−1wtζΔℱt−1,χ−11p+qχζ−1∑r,t=1,r≠tnNζχ−1pχζ−1wrζμℱr+qχζ−1wtζμℱt,ζ−11p+qζχ−1∑r,t=1,r≠tnNχζ−1pζχ−1wrχνℱr+qζχ−1wtχνℱt.
Similar to the A-HF2TLWBM operator, we can obtain that the A-HF2TLWGBM operator has also idempotency, boundedness, and monotonicity. In addition, some special forms of the A-HF2TLWGBM operator can be given according to the changes of parametersp and q as follows.Case 1.
Whenq⟶0, then the following equality holds:(31)A‐HF2TLWGBMp,0ℱ1,ℱ2,…,ℱn=∪ℱ1,…,ℱn1p⊕r=1npℱrwrN=∪ℱ1,…,ℱnΔχ−11pχζ−1∑r=1nNζχ−1pχζ−1wrζ△ℱr−1,χ−11pχζ−1∑r=1nNζχ−1pχζ−1wrζμℱr,ζ−11pζχ−1∑r=1nNχζ−1pζχ−1wrχνℱr.Case 2.
Whenp=1andq⟶0, then the ATS-based hesitant Fermatean 2-tuple linguistic weighted unit geometric Bonferroni mean operator holds:(32)A‐HF2TLWGBM1,0ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn⊕r=1nℱrwrN,=⋃ℱ1,…,ℱnΔζ−1∑r=1nNwrζΔℱr−1,ζ−1∑r=1nNwrζμℱr,χ−1∑r=1nNwrχνℱr.Case 3.
Whenp=1andq=1, then we obtain the ATS-based hesitant Fermatean 2-tuple linguistic weighted interrelated square geometric Bonferroni mean operator:(33)A‐HF2TLWGBM1,1ℱ1,ℱ2,…,ℱn=⋃ℱ1,…,ℱn12⊗r,t=1,r≠tnℱrwr⊕ℱtwtN=⋃ℱ1,…,ℱnΔχ−112χζ−1∑r,t=1,r≠tnNζχ−1χζ−1wrζ△ℱr−1+χζ−1wtζ△ℱt−1,χ−112χζ−1∑r,t=1,r≠tnNζχ−1χζ−1wrζμℱr+χζ−1wtζμℱt,ζ−112ζχ−1∑r,t=1,r≠tnNχζ−1ζχ−1wrχνℱr+ζχ−1wtχνℱt.


## 5. MCDM Approach

We aim at applying the developed A-HF2TLWBM operator and A-HF2TLWGBM operator to solve MCDM problems with hesitant Fermatean 2-tuple linguistic information. SupposeA=A1,A2,…,Am denotes a collection of m alternatives and C=C1,C2,…,Cn denotes a set of n criteria, and its corresponding weight vector is w=w1,w2,…,wn, where wt∈0,1 is the weight of Ct and ∑t=1nwt=1. Suppose that H=ℱrtm×n is a hesitant Fermatean 2-tuple linguistic decision matrix, in which ℱrt is an HF2TLN that indicates the assessment values of Ar with regard to Ct. Next, we employ the proposed operators to solve the MCDM problems involving hesitant Fermatean 2-tuple linguistic information. The steps are summarized as follows:Step 1. Obtain the decision matrices H=ℱrtm×n, where ℱrt represents the decision-maker’s preferences.Step 2. Normalize the decision information matrices H. For ease of comparison and information aggregation, the hesitant Fermatean 2-tuple linguistic decision information matrix H=ℱrtm×n should be transformed the normalized matrix expressed by H¯=ℱrt¯m×n:(34)ℱrt¯=ℱrt,benefit criteria,minrℱrt⊘ℱrt,cost criteria,whereminrℱrt⊘ℱrt=ΔΔ−1minrsfθℱrt,αℱrt/Δ−1sfθℱrt,αℱrt,minrμℱrt/μℱrt,maxrνℱrt3−νℱrt3/1−νℱrt33:(35)minℱij=minisθℱij,αℱij,miniμℱij,maxiνℱij.Step 3. Utilize the A-HF2TLWBM operator or the A-HF2TLWGBM operator to get the overall preference valuesℱr,r=1,2,…,m corresponding to alternative Ar. In Theorems 2 and 4, we take ζc=−logc and χc=−log1−c in A-HF2TLWBM operator and A-HF2TLWGBM operator.Step 4. Rank all the alternativesA1,A2,…,Am according to Definitions 6–8 and identify the best one(s).


## 6. Numerical Example

Because of the uncertainty among companies, it is difficult for experts to use just one linguistic term with a single symbolic translation parameter to provide their preferences. To facilitate the elicitation of their preferences, they can use hesitant Fermatean fuzzy 2-tuple linguistic terms, which are more akin to human beings’ cognitive processes.


### 6.1. Background

In this part, we study an example to demonstrate the applicability of the proposed approach in solving a practical MCDM problem with hesitant Fermatean 2-tuple linguistic information. Let us assume that an investment company wants to invest a sum of money to the optimal industry. There are four parallel alternativesA1,A2,A3,A4, which are a car industry, a food company, a computer company, and an arms industry, respectively. Suppose that a company committee composed of professional experts must select the best industry in accordance with the following four criteria: C1 is the mobility analysis; C2 is the risk analysis; C3 is the feasibility of the project analysis; C4 is the profitability analysis; and their weight vector is w=0.2,0.1,0.3,0.4. The experts evaluate four companies by virtue of the linguistic term set Sf=sf0=extremely poor,sf1=poor,sf2=medium,sf3=good,sf4=extremely good.


### 6.2. Illustration of the Proposed Method

Next, we employ an approach designed in Section5 to identify the optimal alternative(s).Step 1. Obtain the decision matrices H=ℱrtm×n, which are shown in Table 1. Considering all criteria are beneficial types; the matrix H=ℱrtm×n maintains its original form.Step 2. Use the A-HF2TLWBM operator to utilize all preference values of the alternative Arr=1,2,3,4 with p=1,q=1, ζc=−logc, and χc=−log1−c, the results are listed as follows:(36)ℱ1=sf1,0.0053,0.4942,0.0086,sf1,0.0034,0.5409,0.0107,ℱ2=sf1,0.0012,0.4885,0.0060,sf1,0.0103,0.5118,0.0071,ℱ3=sf1,0.0108,0.5192,0.0064,sf1,0.0023,0.5946,0.0097,ℱ4=sf1,0.0027,0.5016,0.0077,sf1,0.0093,0.6287,0.0072,sf1,0.0101,0.4616,0.0059,sf1,0.0073,0.6035,0.0055.Step 3. From Definition 6, the values of score functionSCℱr,r=1,2,3,4 are shown as follows:(37)SCℱ1=0.1401,SCℱ2=0.1261,SCℱ3=0.1761,SCℱ4=0.1745.Step 4. According to Definition 8, the ranking is obtained as follows:(38)A3≻A4≻A1≻A2.Table 1
HF2TLTSs decision matrix H.
C
1
C
2
C
3
C
4
A
1
s
f
2
,
0
,
0.38
,
0.52
,
s
f
3
,
0
,
0.68
,
0.35
s
f
4
,
0
,
0.62
,
0.22
s
f
1
,
0
,
0.24
,
0.80
s
f
3
,
0
,
0.51
,
0.47
A
2
s
f
1
,
0
,
0.73
,
0.28
s
f
4
,
0
,
0.56
,
0.87
s
f
3
,
0
,
0.25
,
0.47
,
s
f
4
,
0
,
0.39
,
0.33
s
f
4
,
0
,
0.25
,
0.71
A
3
s
f
3
,
0
,
0.78
,
0.23
s
f
1
,
0
,
0.08
,
0.81
s
f
4
,
0
,
0.21
,
0.65
,
s
f
3
,
0
,
0.65
,
0.34
s
f
2
,
0
,
0.46
,
0.48
A
4
s
f
2
,
0
,
0.57
,
0.44
,
s
f
3
,
0
,
0.29
,
0.63
s
f
1
,
0
,
0.35
,
0.65
s
f
3
,
0
,
0.42
,
0.57
,
s
f
4
,
0
,
0.86
,
0.61
s
f
4
,
0
,
0.40
,
0.51So,A3 is identified as the optimal alternative. Therefore, the best company for the investment is A3.


### 6.3. Influence Discussion of Parametersp and q

To examine the influence of parametersp and q in the A-HF2TLWBM operator, the changes of score values with parameters p and q increasing are shown in Figures 1–4, where the score values SCℱr are shown as p and q changing from 0 to 3.Figure 1
SC
ℱ
1 of A1 based on the A-HF2TLWBM operator.Figure 2
SC
ℱ
2 of A2 based on the A-HF2TLWBM operator.Figure 3
SC
ℱ
3 of A3 based on the A-HF2TLWBM operator.Figure 4
SC
ℱ
4 of A4 based on the A-HF2TLWBM operator.Figure5 shows SCℱr and rankings of the alternatives when q is considered as the independent variable. As we observe from Figure 5, when q∈0,0.1225, the ranking result is A4≻A3≻A1≻A2, and the optimal alternative is A4, while with q∈0.1225,3, the ranking is A3≻A4≻A1≻A2, and the optimal alternative is A3. Figures 1–5 reflect the changes of the ranking of the alternatives with different decision-makers’ investment preference risk.Figure 5
Variation ofSCℱr based on the A-HF2TLWBM operator (p=1,q∈0,3).Furthermore, different score function values and ranking results are calculated with the changes ofp and q in the A-HF2TLWGBM operator in Step 3. Figures 6–9 illustrate the score values SCℱr obtained using the A-HF2TLWGBM operator as p and q increase from 0 to 3. Figure 10 shows that different score values SCℱr and ranking results with a fixed value of p. From Figure 10, the best alternative is A4 when q∈0,0.1845; there are different ranking of the alternatives, and A1 is the best alternative when q∈0.1845,2.6145, while the best alternative becomes A3 when q∈2.6145,3. Through the above analysis, we see that the ranking result varies more as parameter q changes.Figure 6
SC
ℱ
1 of A1 based on the A-HF2TLWGBM operator.Figure 7
SC
ℱ
2 of A2 based on the A-HF2TLWGBM operator.Figure 8
SC
ℱ
3 of A3 based on the A-HF2TLWGBM operator.Figure 9
SC
ℱ
4 of A4 based on the A-HF2TLWGBM operator.Figure 10
Variation ofSCℱr based on the A-HF2TLWGBM operator (p=1,q∈0,3).From the view point of Liu et al. [27], it is worth noting that the score values SCℱr are smaller as p and q increase from 0 to 3 in the A-HF2TLWBM operator than those in the A-HF2TLWGBM operator. Figures 5 and 10 indicate that the A-HF2TLWBM operator may cause more disadvantageous (or pessimistic) preferences and that the A-HF2TLWGBM operator possesses more advantageous (or optimistic) preferences. Hence, the A-HF2TLWBM operator can be regarded as the pessimistic operator and the A-HF2TLWGBM operator as the optimistic operator, while parameters p and q can be considered as pessimistic or optimistic preference levels. To some extent, it can be concluded that a decision-maker delivering a pessimistic assessment of the prospect prefers to select the A-HF2TLWBM operator with the smaller parameter values p and q, and an optimistic decision-maker can use the A-HF2TLWGBM operator to select the larger parameter values. Neutral decision making results are obtained using the arithmetic averages of the score values SCℱr as shown in Figures 11–14.Figure 11
SC
ℱ
1 of A1 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.Figure 12
SC
ℱ
2 of A2 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.Figure 13
SC
ℱ
3 of A3 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.Figure 14
SC
ℱ
4 of A4 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.


### 6.4. Comparison Analysis with the Existing Methods

To illustrate the practicality and advantages of the proposed method, we compare the proposed MCDM approach with the existing methods [12, 13, 25, 26].


#### 6.4.1. Comparison Analysis of the MCDM Method with DHFS Information

For comparing the proposed MCDM approach with the existing method [26], we use the same practical MCDM problem decision background as shown in Table 2 and the same criteria weight vector 0.2,0.1,0.3,0.4. We apply the DHFWGBM operator and DHFCGBM operator conceived in [26] to cope with Table 2 without the linguistic terms. The corresponding dual hesitant fuzzy matrix is shown in Table 3. The obtained ranking result of alternatives is A4≻A2≻A3≻A1 when m1=m2=1. We also identify the optimal alternative(s) by applying the approach designed in Section 5, where the A-HF2TLWGBM operator with p=q=1, ζc=−logc, and χc=−log1−c is employed to fuse all preference values. The ranking order of alternatives is also A4≻A2≻A1≻A3. The differences of two ranking results arise from the difference of the interrelationship of the input variables. Through comparison with the existing method [26], we observe that when ζc=−logc and χc=−log1−c, the DHFWGBM operator is the general form of A-HF2TLWGBM operator. The ranking result by using the DHFWGBM operator [26] is the same as our ranking result. The established methods are two parameterized families of the aggregation operators of HF2TLSs, which cover a wide range of particular forms that exhibit a more flexible way of adapting the specific needs of problem solving.Table 2
HF2TLTSs decision matrix.
C
1
C
2
C
3
C
4
A
1
s
f
2
,
0
,
0.31
,
0.41
s
f
2
,
0
,
0.62
,
0.22
,
s
f
2
,
0
,
0.25
,
0.40
s
f
2
,
0
,
0.24
,
0.70
s
f
2
,
0
,
0.51
,
0.37
,
A
2
s
f
2
,
0
,
0.53
,
0.21
s
f
2
,
0
,
0.56
,
0.37
s
f
2
,
0
,
0.25
,
0.61
s
f
2
,
0
,
0.25
,
0.47
,
s
f
2
,
0
,
0.39
,
0.33
A
3
s
f
2
,
0
,
0.28
,
0.53
s
f
2
,
0
,
0.08
,
0.81
s
f
2
,
0
,
0.31
,
0.30
,
s
f
2
,
0
,
0.65
,
0.24
s
f
2
,
0
,
0.46
,
0.48
A
4
s
f
2
,
0
,
0.57
,
0.24
,
s
f
2
,
0
,
0.61
,
0.23
s
f
2
,
0
,
0.75
,
0.15
s
f
2
,
0
,
0.42
,
0.10
,
s
f
2
,
0
,
0.77
,
0.21
s
f
2
,
0
,
0.60
,
0.30Table 3
Dual hesitant fuzzy matrix.
C
1
C
2
C
3
C
4
A
1
0.31
,
0.41
0.62
,
0.25
,
0.22
,
0.40
0.24
,
0.70
0.51
,
0.37
A
2
0.53
,
0.21
0.56
,
0.37
0.25
,
0.61
0.25
,
0.39
,
0.47
,
0.33
A
3
0.28
,
0.53
0.08
,
0.81
0.31
,
0.65
,
0.30
,
0.24
0.46
,
0.48
A
4
0.57
,
0.61
,
0.24
,
0.23
0.75
,
0.15
0.42
,
0.77
,
0.10
,
0.21
0.60
,
0.30


#### 6.4.2. Comparison Analysis with the Existing Methods

In the literature, there are some concepts and aggregation operators involving Fermatean fuzzy information. Senapti and Yager [12, 13] proposed an approach for dealing with the MCDM problem with Fermatean fuzzy information by establishing some aggregation operators. Liu et al. [25] presented the FFLWA operator, the FFLWG operator, and the Fermatean fuzzy linguistic distance measures to address MCDM problems with FFLTSs. Jamil and Rashid [26] developed the dual hesitant fuzzy geometric Bonferroni mean method for handling the MCDM problems with DHFSs. Although these concepts and methods are able to improve the ability of representing and processing complex data from different perspectives, they fail to deal with HF2TLTSs information systems. Compared with the method in [26], the pair of membership degree μz and nonmembership degree νz are involved in two methods; however, they perform difference roles. We observe that the method in [26] adopts DHFSs, and the preference information is characterized by the pair of membership degree μz and nonmembership degree νz with μz+νz≤1. It can only solve the MCDM problem described by using μz and νz. However, our new concept of HF2TLSs by combining the FFS and 2-tuple linguistic terms, in which μz and degree νz are with a more loose constraint 0≤μFz3+νFz3≤1, is employed to represent the uncertainty of the linguistic preference information. HF2TLSs can better align with human decision making presentation and characterize more complex uncertainty information.


## 6.1. Background

In this part, we study an example to demonstrate the applicability of the proposed approach in solving a practical MCDM problem with hesitant Fermatean 2-tuple linguistic information. Let us assume that an investment company wants to invest a sum of money to the optimal industry. There are four parallel alternativesA1,A2,A3,A4, which are a car industry, a food company, a computer company, and an arms industry, respectively. Suppose that a company committee composed of professional experts must select the best industry in accordance with the following four criteria: C1 is the mobility analysis; C2 is the risk analysis; C3 is the feasibility of the project analysis; C4 is the profitability analysis; and their weight vector is w=0.2,0.1,0.3,0.4. The experts evaluate four companies by virtue of the linguistic term set Sf=sf0=extremely poor,sf1=poor,sf2=medium,sf3=good,sf4=extremely good.


## 6.2. Illustration of the Proposed Method

Next, we employ an approach designed in Section5 to identify the optimal alternative(s).Step 1. Obtain the decision matrices H=ℱrtm×n, which are shown in Table 1. Considering all criteria are beneficial types; the matrix H=ℱrtm×n maintains its original form.Step 2. Use the A-HF2TLWBM operator to utilize all preference values of the alternative Arr=1,2,3,4 with p=1,q=1, ζc=−logc, and χc=−log1−c, the results are listed as follows:(36)ℱ1=sf1,0.0053,0.4942,0.0086,sf1,0.0034,0.5409,0.0107,ℱ2=sf1,0.0012,0.4885,0.0060,sf1,0.0103,0.5118,0.0071,ℱ3=sf1,0.0108,0.5192,0.0064,sf1,0.0023,0.5946,0.0097,ℱ4=sf1,0.0027,0.5016,0.0077,sf1,0.0093,0.6287,0.0072,sf1,0.0101,0.4616,0.0059,sf1,0.0073,0.6035,0.0055.Step 3. From Definition 6, the values of score functionSCℱr,r=1,2,3,4 are shown as follows:(37)SCℱ1=0.1401,SCℱ2=0.1261,SCℱ3=0.1761,SCℱ4=0.1745.Step 4. According to Definition 8, the ranking is obtained as follows:(38)A3≻A4≻A1≻A2.Table 1
HF2TLTSs decision matrix H.
C
1
C
2
C
3
C
4
A
1
s
f
2
,
0
,
0.38
,
0.52
,
s
f
3
,
0
,
0.68
,
0.35
s
f
4
,
0
,
0.62
,
0.22
s
f
1
,
0
,
0.24
,
0.80
s
f
3
,
0
,
0.51
,
0.47
A
2
s
f
1
,
0
,
0.73
,
0.28
s
f
4
,
0
,
0.56
,
0.87
s
f
3
,
0
,
0.25
,
0.47
,
s
f
4
,
0
,
0.39
,
0.33
s
f
4
,
0
,
0.25
,
0.71
A
3
s
f
3
,
0
,
0.78
,
0.23
s
f
1
,
0
,
0.08
,
0.81
s
f
4
,
0
,
0.21
,
0.65
,
s
f
3
,
0
,
0.65
,
0.34
s
f
2
,
0
,
0.46
,
0.48
A
4
s
f
2
,
0
,
0.57
,
0.44
,
s
f
3
,
0
,
0.29
,
0.63
s
f
1
,
0
,
0.35
,
0.65
s
f
3
,
0
,
0.42
,
0.57
,
s
f
4
,
0
,
0.86
,
0.61
s
f
4
,
0
,
0.40
,
0.51So,A3 is identified as the optimal alternative. Therefore, the best company for the investment is A3.


## 6.3. Influence Discussion of Parametersp and q

To examine the influence of parametersp and q in the A-HF2TLWBM operator, the changes of score values with parameters p and q increasing are shown in Figures 1–4, where the score values SCℱr are shown as p and q changing from 0 to 3.Figure 1
SC
ℱ
1 of A1 based on the A-HF2TLWBM operator.Figure 2
SC
ℱ
2 of A2 based on the A-HF2TLWBM operator.Figure 3
SC
ℱ
3 of A3 based on the A-HF2TLWBM operator.Figure 4
SC
ℱ
4 of A4 based on the A-HF2TLWBM operator.Figure5 shows SCℱr and rankings of the alternatives when q is considered as the independent variable. As we observe from Figure 5, when q∈0,0.1225, the ranking result is A4≻A3≻A1≻A2, and the optimal alternative is A4, while with q∈0.1225,3, the ranking is A3≻A4≻A1≻A2, and the optimal alternative is A3. Figures 1–5 reflect the changes of the ranking of the alternatives with different decision-makers’ investment preference risk.Figure 5
Variation ofSCℱr based on the A-HF2TLWBM operator (p=1,q∈0,3).Furthermore, different score function values and ranking results are calculated with the changes ofp and q in the A-HF2TLWGBM operator in Step 3. Figures 6–9 illustrate the score values SCℱr obtained using the A-HF2TLWGBM operator as p and q increase from 0 to 3. Figure 10 shows that different score values SCℱr and ranking results with a fixed value of p. From Figure 10, the best alternative is A4 when q∈0,0.1845; there are different ranking of the alternatives, and A1 is the best alternative when q∈0.1845,2.6145, while the best alternative becomes A3 when q∈2.6145,3. Through the above analysis, we see that the ranking result varies more as parameter q changes.Figure 6
SC
ℱ
1 of A1 based on the A-HF2TLWGBM operator.Figure 7
SC
ℱ
2 of A2 based on the A-HF2TLWGBM operator.Figure 8
SC
ℱ
3 of A3 based on the A-HF2TLWGBM operator.Figure 9
SC
ℱ
4 of A4 based on the A-HF2TLWGBM operator.Figure 10
Variation ofSCℱr based on the A-HF2TLWGBM operator (p=1,q∈0,3).From the view point of Liu et al. [27], it is worth noting that the score values SCℱr are smaller as p and q increase from 0 to 3 in the A-HF2TLWBM operator than those in the A-HF2TLWGBM operator. Figures 5 and 10 indicate that the A-HF2TLWBM operator may cause more disadvantageous (or pessimistic) preferences and that the A-HF2TLWGBM operator possesses more advantageous (or optimistic) preferences. Hence, the A-HF2TLWBM operator can be regarded as the pessimistic operator and the A-HF2TLWGBM operator as the optimistic operator, while parameters p and q can be considered as pessimistic or optimistic preference levels. To some extent, it can be concluded that a decision-maker delivering a pessimistic assessment of the prospect prefers to select the A-HF2TLWBM operator with the smaller parameter values p and q, and an optimistic decision-maker can use the A-HF2TLWGBM operator to select the larger parameter values. Neutral decision making results are obtained using the arithmetic averages of the score values SCℱr as shown in Figures 11–14.Figure 11
SC
ℱ
1 of A1 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.Figure 12
SC
ℱ
2 of A2 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.Figure 13
SC
ℱ
3 of A3 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.Figure 14
SC
ℱ
4 of A4 based on the A-HF2TLWBM operator and the A-HF2TLWGBM operator.


## 6.4. Comparison Analysis with the Existing Methods

To illustrate the practicality and advantages of the proposed method, we compare the proposed MCDM approach with the existing methods [12, 13, 25, 26].


### 6.4.1. Comparison Analysis of the MCDM Method with DHFS Information

For comparing the proposed MCDM approach with the existing method [26], we use the same practical MCDM problem decision background as shown in Table 2 and the same criteria weight vector 0.2,0.1,0.3,0.4. We apply the DHFWGBM operator and DHFCGBM operator conceived in [26] to cope with Table 2 without the linguistic terms. The corresponding dual hesitant fuzzy matrix is shown in Table 3. The obtained ranking result of alternatives is A4≻A2≻A3≻A1 when m1=m2=1. We also identify the optimal alternative(s) by applying the approach designed in Section 5, where the A-HF2TLWGBM operator with p=q=1, ζc=−logc, and χc=−log1−c is employed to fuse all preference values. The ranking order of alternatives is also A4≻A2≻A1≻A3. The differences of two ranking results arise from the difference of the interrelationship of the input variables. Through comparison with the existing method [26], we observe that when ζc=−logc and χc=−log1−c, the DHFWGBM operator is the general form of A-HF2TLWGBM operator. The ranking result by using the DHFWGBM operator [26] is the same as our ranking result. The established methods are two parameterized families of the aggregation operators of HF2TLSs, which cover a wide range of particular forms that exhibit a more flexible way of adapting the specific needs of problem solving.Table 2
HF2TLTSs decision matrix.
C
1
C
2
C
3
C
4
A
1
s
f
2
,
0
,
0.31
,
0.41
s
f
2
,
0
,
0.62
,
0.22
,
s
f
2
,
0
,
0.25
,
0.40
s
f
2
,
0
,
0.24
,
0.70
s
f
2
,
0
,
0.51
,
0.37
,
A
2
s
f
2
,
0
,
0.53
,
0.21
s
f
2
,
0
,
0.56
,
0.37
s
f
2
,
0
,
0.25
,
0.61
s
f
2
,
0
,
0.25
,
0.47
,
s
f
2
,
0
,
0.39
,
0.33
A
3
s
f
2
,
0
,
0.28
,
0.53
s
f
2
,
0
,
0.08
,
0.81
s
f
2
,
0
,
0.31
,
0.30
,
s
f
2
,
0
,
0.65
,
0.24
s
f
2
,
0
,
0.46
,
0.48
A
4
s
f
2
,
0
,
0.57
,
0.24
,
s
f
2
,
0
,
0.61
,
0.23
s
f
2
,
0
,
0.75
,
0.15
s
f
2
,
0
,
0.42
,
0.10
,
s
f
2
,
0
,
0.77
,
0.21
s
f
2
,
0
,
0.60
,
0.30Table 3
Dual hesitant fuzzy matrix.
C
1
C
2
C
3
C
4
A
1
0.31
,
0.41
0.62
,
0.25
,
0.22
,
0.40
0.24
,
0.70
0.51
,
0.37
A
2
0.53
,
0.21
0.56
,
0.37
0.25
,
0.61
0.25
,
0.39
,
0.47
,
0.33
A
3
0.28
,
0.53
0.08
,
0.81
0.31
,
0.65
,
0.30
,
0.24
0.46
,
0.48
A
4
0.57
,
0.61
,
0.24
,
0.23
0.75
,
0.15
0.42
,
0.77
,
0.10
,
0.21
0.60
,
0.30


### 6.4.2. Comparison Analysis with the Existing Methods

In the literature, there are some concepts and aggregation operators involving Fermatean fuzzy information. Senapti and Yager [12, 13] proposed an approach for dealing with the MCDM problem with Fermatean fuzzy information by establishing some aggregation operators. Liu et al. [25] presented the FFLWA operator, the FFLWG operator, and the Fermatean fuzzy linguistic distance measures to address MCDM problems with FFLTSs. Jamil and Rashid [26] developed the dual hesitant fuzzy geometric Bonferroni mean method for handling the MCDM problems with DHFSs. Although these concepts and methods are able to improve the ability of representing and processing complex data from different perspectives, they fail to deal with HF2TLTSs information systems. Compared with the method in [26], the pair of membership degree μz and nonmembership degree νz are involved in two methods; however, they perform difference roles. We observe that the method in [26] adopts DHFSs, and the preference information is characterized by the pair of membership degree μz and nonmembership degree νz with μz+νz≤1. It can only solve the MCDM problem described by using μz and νz. However, our new concept of HF2TLSs by combining the FFS and 2-tuple linguistic terms, in which μz and degree νz are with a more loose constraint 0≤μFz3+νFz3≤1, is employed to represent the uncertainty of the linguistic preference information. HF2TLSs can better align with human decision making presentation and characterize more complex uncertainty information.


## 6.4.1. Comparison Analysis of the MCDM Method with DHFS Information

For comparing the proposed MCDM approach with the existing method [26], we use the same practical MCDM problem decision background as shown in Table 2 and the same criteria weight vector 0.2,0.1,0.3,0.4. We apply the DHFWGBM operator and DHFCGBM operator conceived in [26] to cope with Table 2 without the linguistic terms. The corresponding dual hesitant fuzzy matrix is shown in Table 3. The obtained ranking result of alternatives is A4≻A2≻A3≻A1 when m1=m2=1. We also identify the optimal alternative(s) by applying the approach designed in Section 5, where the A-HF2TLWGBM operator with p=q=1, ζc=−logc, and χc=−log1−c is employed to fuse all preference values. The ranking order of alternatives is also A4≻A2≻A1≻A3. The differences of two ranking results arise from the difference of the interrelationship of the input variables. Through comparison with the existing method [26], we observe that when ζc=−logc and χc=−log1−c, the DHFWGBM operator is the general form of A-HF2TLWGBM operator. The ranking result by using the DHFWGBM operator [26] is the same as our ranking result. The established methods are two parameterized families of the aggregation operators of HF2TLSs, which cover a wide range of particular forms that exhibit a more flexible way of adapting the specific needs of problem solving.Table 2
HF2TLTSs decision matrix.
C
1
C
2
C
3
C
4
A
1
s
f
2
,
0
,
0.31
,
0.41
s
f
2
,
0
,
0.62
,
0.22
,
s
f
2
,
0
,
0.25
,
0.40
s
f
2
,
0
,
0.24
,
0.70
s
f
2
,
0
,
0.51
,
0.37
,
A
2
s
f
2
,
0
,
0.53
,
0.21
s
f
2
,
0
,
0.56
,
0.37
s
f
2
,
0
,
0.25
,
0.61
s
f
2
,
0
,
0.25
,
0.47
,
s
f
2
,
0
,
0.39
,
0.33
A
3
s
f
2
,
0
,
0.28
,
0.53
s
f
2
,
0
,
0.08
,
0.81
s
f
2
,
0
,
0.31
,
0.30
,
s
f
2
,
0
,
0.65
,
0.24
s
f
2
,
0
,
0.46
,
0.48
A
4
s
f
2
,
0
,
0.57
,
0.24
,
s
f
2
,
0
,
0.61
,
0.23
s
f
2
,
0
,
0.75
,
0.15
s
f
2
,
0
,
0.42
,
0.10
,
s
f
2
,
0
,
0.77
,
0.21
s
f
2
,
0
,
0.60
,
0.30Table 3
Dual hesitant fuzzy matrix.
C
1
C
2
C
3
C
4
A
1
0.31
,
0.41
0.62
,
0.25
,
0.22
,
0.40
0.24
,
0.70
0.51
,
0.37
A
2
0.53
,
0.21
0.56
,
0.37
0.25
,
0.61
0.25
,
0.39
,
0.47
,
0.33
A
3
0.28
,
0.53
0.08
,
0.81
0.31
,
0.65
,
0.30
,
0.24
0.46
,
0.48
A
4
0.57
,
0.61
,
0.24
,
0.23
0.75
,
0.15
0.42
,
0.77
,
0.10
,
0.21
0.60
,
0.30


## 6.4.2. Comparison Analysis with the Existing Methods

In the literature, there are some concepts and aggregation operators involving Fermatean fuzzy information. Senapti and Yager [12, 13] proposed an approach for dealing with the MCDM problem with Fermatean fuzzy information by establishing some aggregation operators. Liu et al. [25] presented the FFLWA operator, the FFLWG operator, and the Fermatean fuzzy linguistic distance measures to address MCDM problems with FFLTSs. Jamil and Rashid [26] developed the dual hesitant fuzzy geometric Bonferroni mean method for handling the MCDM problems with DHFSs. Although these concepts and methods are able to improve the ability of representing and processing complex data from different perspectives, they fail to deal with HF2TLTSs information systems. Compared with the method in [26], the pair of membership degree μz and nonmembership degree νz are involved in two methods; however, they perform difference roles. We observe that the method in [26] adopts DHFSs, and the preference information is characterized by the pair of membership degree μz and nonmembership degree νz with μz+νz≤1. It can only solve the MCDM problem described by using μz and νz. However, our new concept of HF2TLSs by combining the FFS and 2-tuple linguistic terms, in which μz and degree νz are with a more loose constraint 0≤μFz3+νFz3≤1, is employed to represent the uncertainty of the linguistic preference information. HF2TLSs can better align with human decision making presentation and characterize more complex uncertainty information.


## 7. Conclusions

To further characterize and handle decision-makers’ hesitancy and uncertainty in some complex MCDM problems, this study introduces a new class of uncertain linguistic variables called hesitant Fermatean 2-tuple linguistic sets (HF2TLSs). We extend the ATS and BM operators to the hesitant Fermatean 2-tuple linguistic environment and further proffer some special operational laws for HF2TLSs, by which the A-HF2TLWBM operator and the A-HF2TLWGBM operator are developed. These operators offer an efficient tool in practical applications by capturing the interrelationship of the input variables and avoiding information loss. In addition, we further prove their idempotency, boundedness, and monotonicity. We also discuss some special cases and compare our approach to the existing methods. An example demonstrates the method’s practicality and effectiveness with a detailed discussion under different influences of the parametersp and q. In future research, we will focus on solving multicriteria group decision making problems under the background of HF2TLSs.



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*Source: 5705907-2019-12-20.xml*