# A Novel Group Decision-Making Approach for Hesitant Fuzzy Linguistic Term Sets and Its Application to VIKOR

Authors: Xiuli Geng; Yunting Jin; Yongzheng Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper develops a novel group decision-making (GDM) approach for solving multiple-criteria group decision-making (MCGDM) problems with uncertainty. The hesitant fuzzy linguistic term sets (HFLTSs) are applied to elicit the decision makers’ linguistic preferences due to their distinguished efficiency and flexibility in representing uncertainty. However, the existing context-free grammar for linguistic description cannot allow generating the linguistic expressions completely free to limit the richness of HFLTSs, and the related methods for dealing with HFLTSs also have limitations in aggregating HFLTSs with different lengths and types. Therefore, this paper proposes extended context-free grammar and a novel GDM approach for HFLTSs, considering the advantages of the rough set theory and OWA operators. The rough set theory can manage the uncertainty existing in the fuzzy representation and deal with HFLTSs represented by the 2-tuple fuzzy linguistic model to get rough number sets. The OWA operator can aggregate these sets with different numbers of elements into an interval simply and objectively. Then, an extended VIKOR method based on the proposed GDM approach for HFLTSs is presented to solve the MCGDM problems. Finally, two examples are given to illustrate the applicability and validity of the developed GDM approach and the hesitant VIKOR method through sensitivity and comparison analysis with other existing approaches.


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## Body

## 1. Introduction

Decision-making is a common activity for human beings to select the desirable alternatives in many different fields such as evaluation [1], selection [2], and improvement [3]. Such problems are always presented as multicriteria decision-making (MCDM) problems. The complexity and importance of the real-world decision problems make the inclusion of multiple points of view necessary in order to achieve a solution from the knowledge provided by a group of experts [4]. Therefore, group decision-making (GDM) is a usual technique in MCDM practice. These problems having complex processes where several criteria must be satisfied to find the desirable alternative by multiple experts or decision makers (DMs) are called multiple criteria group decision-making (MCGDM) problems. How to solve MCGDM problems under fuzzy environment has been a challenging and attention-attracting topic in recent decades.The judgments of experts are often vague and uncertain and cannot be expressed with exact numerical information. Since the introduction of the fuzzy set by Zadeh [5], the fuzzy set and its extensions have been widely used to express and model the fuzzy and vague information in the decision-making process. Fuzzy sets require a positive membership for each element and support the favoring evidence only. Intuitionistic fuzzy sets (IFSs) introduced by Atanassov [6] support both favoring and opposing evidences by means of the membership function and nonmembership function and have the advantage that permits the experts having a margin of error in establishing the membership for each element. Type-2 fuzzy sets [7] and fuzzy multisets [8] are other extensions of fuzzy sets. Type-2 fuzzy sets have the advantage that permits the membership of an element having some possible distributions on possible values, where the membership of each element is defined as a fuzzy set. When defining the membership of a given element, fuzzy multisets deal with uncertainty by allowing several values. However, the biggest difficulty of establishing the membership degree in the GDM process is that experts may have a set of possible values. Aiming at such a situation, Torra [9] introduced hesitant fuzzy sets (HFSs) in terms of a function that returns a set of membership values for each element in the domain. Recently, HFSs have been widely used in solving MCDM problems due to their distinguished efficiency and flexibility in modeling uncertainty and vagueness in the decision- making process. Zhang [10] presented hesitant fuzzy power aggregation operators for multiple attribute group decision-making. Xia and Xu [11] presented some aggregation operators of hesitant fuzzy information for GDM. Zhang and Wei [12] extended the VIKOR method based on HFSs for the decision-making problem. Some measures for HFSs have been presented for decision-making [13–15].Based on the HFSs and fuzzy linguistic approach, Rodríguez et al. [16] introduced the concept of hesitant fuzzy linguistic term sets (HFLTSs) for richer expressions in MCDM. HFLTS complies with the situation that experts prefer adopting imprecise linguistic terms to express their judgments. It avoids the restriction for preference flexibility caused by using a single term or interval linguistic terms. The aim of this paper is to use and improve the operating method of HFLTSs to solve MCGDM problems under the linguistic environment.The commonly used linguistic description approaches for HFLTSs are the ordered structure approach and the context-free grammar approach. Rodríguez et al. [16] presented how to generate comparative linguistic expressions by using context-free grammar. Context-free grammar can generate different linguistic expressions depending on the specific problem. Based on traditional context-free grammar, we consider similar but extended context-free grammar to support the completely free expression. For the linguistic GDM, the process of computing with words (CWW) is indispensable. In the HFLTS environment, free expression brings convenience for describing experts’ preferences. However, the CWW processes for HFLTSs become more complex due to each element being an arbitrary linguistic term subset. Some distance and similarity measures for HFLTSs were put forward and applied to solve MCDM problems [17–19], but these approaches assume that the HFLTSs have the same length. To aggregate or compute HFLTSs with different lengths in solving decision-making problems, many methods were put forward as shown in Table 1.Table 1
Review of the operating methods of HFLTSs with different lengths.
AuthorsMethods used for operating HFLTSsContributions to the field of the HFLTSRodríguez et al. [16]Introducing the concept ofenvelope for an HFLTS to compare two HFLTSsFirst introducing the concept ofenvelopeFarhadinia [13]Using the envelope for each HFLTS to aggregate the preference of experts based on the 2-tuple linguistic representation modelConsidering extended context-free grammar close to human beings’ cognitive modelsMontes et al. [20]Carrying out the computing with words processes using the envelope of an HFLTS and the 2-tuple linguistic representation modelPresenting a practical application in decision-making of a 2-tuple linguistic fuzzy model with hesitant informationWu et al. [21]Proposing a maximum support degree model based on the envelope of the HFLTSProposing a new linguistic group decision model by combining the linguistic distributions and HFLTSsBoyacı [22]Obtaining the envelope for each HFLTS and proposing the pessimistic and optimistic collective preference relations for comparisonUsing the HFLTS-based additive ratio assessment (ARAS) methodLiu and Rodriguez [23]Introducing a fuzzy envelope for the HFLTS whose representation is a fuzzy membership functionPresenting a new representation of the hesitant fuzzy linguistic term sets using a fuzzy envelopeChen and Hong [24]Aggregating the fuzzy sets in each HFLTS into a fuzzy set and performingα-cut to these aggregated fuzzy sets to get intervalsConsidering the pessimistic attitude and the optimistic attitude of the decision makerLee and Chen [25]Adopting 1-cut of hesitant fuzzy linguistic term sets for dealing with fuzzy decision-makingProposing a new fuzzy group decision-making method based on the proposed likelihood-based comparison relations of HFLTSsDong et al. [26]Proposing a new fuzzy envelope of the HFLTS by using a Bonferroni mean operatorProposing a new fuzzy envelope of the HFLTS and a new cosine similarity measure for HFLTSsZhu and Xu [27]Introducing a method to add linguistic terms to HFLTSs to make sure that two HFLTSs have the same number of linguistic termsDeveloping a concept of hesitant fuzzy linguistic preference relations (HFLPRs) as a tool to collect and present the DMs’ preferencesLiao et al. [17];Liao and Xu [28];Liao et al. [29]Extending the short hesitant fuzzy linguistic element by adding the linguistic term which is the average of the maximal term and the minimal termProposing several different types of correlation coefficients for HFLTSs and cosine distance and similarity measures for HFLTSsLiao et al. [30]Enlarging the shorter HFLTS by adding a linguistic term which is between the maximum and the minimum linguistic termProposing two distinct methods to compare the HFLTSs and investigating the ELECTRE II method in the HFLTS environmentLei et al. [31]Extending the shorter HFLE by adding the maximum valueProposing the behavioral multigranulation decision-theoretic rough set over two universes with HFL informationFrom the above reviews, we can conclude that the existing operating methods of HFLTSs with different lengths can be classified into three main categories. (1) The first category is the envelope-based method, and the envelope of the HFLTS is a linguistic interval. The HFLTSs can be aggregated or compared as intervals. Rodríguez et al. [16] first proposed the envelope concept to compare HFLTSs based on their envelopes, which are numerical intervals. The introduction of the concept of envelope can simplify the comparison operation and other operations. However, it is unreasonable to judge one HFLTS is absolutely superior to another if they have common elements. Although several extended research studies have been conducted, there still exist limitations of the envelope-based method. The one disadvantage of it is that the linguistic interval finally obtains crisp values, losing the initial fuzzy representation. The other disadvantage is that it seems unreasonable to support selecting the preferences from the predefined term sets completely free. If we give hesitant linguistic expressions out of context-free grammar, the envelope-based method may fail to work efficiently. For example, s1,s2,s3 and s1,s3 have the same envelope s1,s3. (2) The second category includes the fuzzy envelope-based method and α-cut method. Liu and Rodriguez [23] proposed a fuzzy envelope for the HFLTS. The fuzzy envelope can retain the vagueness of comparative linguistic expressions to a certain extent, but determining the parameters of the fuzzy membership function is fairly complicated, and considerable requisite calculations are required for an MCDM problem in the context of HFLTSs [32]. (3) The third category is the term-adding method, which is to extend the short HFLTSs by adding some linguistic terms until they have the same length as others. The disadvantage of the term-adding method is that it would change the information of the original hesitant fuzzy elements by filling some artificial values.To support extended context-free grammar and operate HFLTSs with different lengths, we propose a novel GDM approach in the HFLTS environment. The contribution of this approach is transforming the HFLTSs into rough intervals by taking advantage of the 2-tuple linguistic representation model and the rough set theory. In the aggregation phase, the obtained rough numbers are grouped into an interval using the ordered weighted averaging (OWA) operator. In the exploitation phase, an extended VIKOR method based on the proposed GDM approach for HFLTSs is presented.The 2-tuple linguistic representation model proposed by Herrera and Martínez [33] is based on the concept of symbolic translation, which is composed of a linguistic term and a real number. The main advantage of this representation is to be continuous in its domain. Therefore, it can express any counting of information in the universe of the discourse. Herrera and Martínez [33] pointed out that the computational technique based on the symbolic translation can deal with the 2-tuples without loss of information, such as comparison and aggregation. In recent studies, the 2-tuple linguistic representation model has been used in MCGDM problems successfully, e.g., material selection [34], product management [35], and computer network security system evaluation [36]. Rough set theory proposed by Pawlak [37] is a mathematical approach to manage uncertain data or problems of the information systems. Its main advantage is that it requires no external parameters and uses the information presented in the given data only. The OWA operator proposed by Yager [38] provides an aggregation result lying between the max and min operators and has received increasing attention. The weight associated with each data depends on the position it takes in the descending arrangement of the data rather than the particular data. Due to the advantage that the OWA operator can provide a wide family of aggregation functions and aggregate a set of values regardless of their numbers, we apply the OWA operator to aggregate the elements in an HFLTS and multiple HFLTSs in the GDM process.HFLTS has been combined with many MCDM methods, such as ELECTRE [39], extended ELECTRE [40], TOPSIS [41, 42], and VIKOR [28, 43]. The VIKOR method for compromise ranking determines a compromise solution by providing a maximum “group utility” for the “majority” and a minimum of an “individual regret” for the “opponent,” which is an effective tool for MCDM, particularly in a situation where the decision maker is not able or does not know how to express his/her preference at the beginning of system design. We pay attention to apply the proposed group decision-making approach to solve MCGDM problems using the VIKOR method.This paper focuses on dealing with MCGDM problems in the context of linguistic evaluation using HFLTSs and the VIKOR method. The rest of the paper is organized as follows. In Section2, we briefly review the concepts of HFLTSs and 2-tuple linguistic representation models and introduce how to apply the 2-tuple linguistic representation model to compute with the hesitant fuzzy linguistic information. In Section 3, a novel group decision-making approach for HFLTSs is presented based on the rough set theory and the OWA operator. In Section 4, we give out an extended VIKOR method based on the proposed GDM approach for HFLTSs. In Section 5, two application examples are provided to illustrate the efficiency of the proposed GDM approach and the extended VIKOR, respectively, and the results are compared with other existing methods. Finally, conclusions are drawn in Section 6.


## 2. Preliminaries

In this section, some concepts and operations of HFLTSs and the 2-tuple linguistic representation model are briefly reviewed, and then, how to apply the 2-tuple linguistic representation model for computing with the hesitant fuzzy linguistic information is introduced.


### 2.1. Concept and Basic Operations of HFLTSs

Definition 1.
(see [16]). Let S be a linguistic term set, S=s0,…,sg; an HFLTS, Hs, is an ordered finite subset of the consecutive linguistic terms of S.
The empty HFLTS and the full HFLTS for a linguistic variableϑ are defined as follows:(1)
The empty HFLTS:HSϑ=,(2)
The full HFLTS:HSϑ=SDefinition 2.
(see [16]). Let S=s0,…,sg be a linguistic term set; HS0, HS1, and HS2 are three arbitrary HFLTSs on S. HSC is the complement set of HS0. Three operations are defined as follows:(1)
HSC=S−HS0=sisi∈Sandsi∉HS0(2)
HS1∩HS2=sisi∈HS1andsi∈HS2(3)
HS1∪HS2=sisi∈HS1orsi∈HS2
Due to the present decision-making problems having higher uncertainty, experts in the decision-making group might hesitate among different linguistic terms to express their preferences. Context-free grammar is close to human beings’ cognitive model and can generate comparative linguistic expressions. Rodríguez et al. [16] pointed out how to generate comparative linguistic expressions by using context-free grammar. Rodríguez et al. [4] considered similar but extended context-free grammar to that defined in Rodríguez et al. [16] which might generate comparative linguistic expressions similar to the expressions used by experts in GDM problems. Extended context-free grammar refers to a set containing a single term or several adjacent linguistic terms and cannot support the arbitrarily linguistic term mix. One special case is omitted, that is, experts may be hesitant to choose a better evaluation or a worse evaluation. Therefore, this paper improves extended context-free grammar to introduce the binary relation “or.”Definition 3.
LetGH be improved context-free grammar and S=s0,…,sg be a linguistic term set. The elements of GH = (VN, VT, I, P) are defined as follows:VN = {<primary term>, <composite term>, <unary relation>, <binary relation>, <conjunction>}VT = {lower than, greater than, at least, at most, between, or, and, s0, s1, ..., sg}I∈VNFor context-free grammar,GH, the production rules are as follows:P = {I ::= <primary term>|<composite term><composite term> ::== <unary relation><primary term>|<binary relation><primaryterm><conjunction><primary term><primaryterm> ::== s0s1sg<unaryrelation> ::== lower than|greater than|at least|at most<binaryrelation> ::== between|or<conjunction> :==and}Definition 4.
LetEGH be a function that transforms linguistic expressions ll∈Sll obtained by context-free grammar GH into a HFLTS HS, where S is the linguistic term set used by GH and Sll:EGH:Sll⟶HSThe comparative linguistic expressions generated byGH can be converted into HFLTSs by means of the following:(1)
EGH (si) = sisi∈S(2)
EGH (at most si) = sjsj∈Sandsj≤si(3)
EGH (lower than si) = sjsj∈Sandsj<si(4)
EGH (at least si) = sjsj∈Sandsj≥si(5)
EGH (greater than si) = sjsj∈Sandsj>si(6)
EGH (between si and sj) = sksk∈Sandsi≤sk≤sj(7)
EGH (si or sj, …, sk) = si,sj,…,skExample 1.
LetS be a linguistic term set.S=s0:nothing,s1:very low,s2:low,s3:medium,s4:high,s5:very high,s6:perfectThree experts give their opinions aiming at the same evaluation object based on improved context-free grammar:ll1: between low and high; ll2: low or high; and ll3: between medium and very high. According to the function EGH, three different HFLTSs are obtained:HS1 = {s2, s3, s4}, HS2 = {s2, s4}, and HS3 = {s3, s4, s5}


### 2.2. Computing with HFLTSs Using the 2-Tuple Fuzzy Linguistic Representation Model

LetS = si:i=0,1,2,…,g be a finite and ordered discrete linguistic term set, where si represents a possible value for a linguistic variable. The 2-tuple fuzzy linguistic representation model deals with linguistic information by introducing a new parameter called symbolic translation. The concept of symbolic translation is described in Definition 5. It is used to make the information representation continuous in its domain, and it is the foundation of the computation techniques of the 2 tuples. The concept and basic operations of the 2-tuple fuzzy linguistic representation model are as follows.Definition 5.
(see [33]). Let β be a value representing the result of an aggregation of the indices of a set of labels assessed in the linguistic term set S, i.e., the result of a symbolic aggregation operation β∈0,g, being g + 1, the cardinality of S. Let i = roundβ and α=β−i be two values such that i∈0,g and α∈−0.5,0.5; then, α is called a symbolic translation.
The linguistic representation model 2-tuplesi,αi, si∈S and αi∈−0.5,0.5, is developed from the above concept:(1)
si represents the linguistic label center of the information(2)
αi is a numerical value expressing the value of the translation from the original result β to the closest index label, i, in the linguistic term set S, i.e., the symbolic translationDefinition 6.
(see [33]). Let S = s0,s1,…,sg be a linguistic term set and β∈0,g be a value representing the result of a symbolic aggregation; then, the 2-tuple that expresses the equivalent information to β is obtained with the function Δ:(1)Δ:0,g⟶S×0.5,−0.5,Δβ=si,αi,si,i=roundβ,αi=β−i,αi∈−0.5,0.5,where round⋅ is the usual round operation, si has the closest index label to β, and αi is the value of the symbolic translation.
Contrarily, letS = s0,s1,…,sg be a linguistic term set and si,αi be a 2-tuple. There is always a Δ−1 function:(2)Δ−1:S×0.5,−0.5⟶0,g,Δ−1si,αi=i+αi=β.
The original linguistic evaluation variable can be converted into a linguistic 2-tuple by adding value zero as symbolic translation:si∈S⟹si,0.Example 2.
The decision information in Example1 can be transformed into the following 2-tuple information:HS1 = {(s2, 0),(s3, 0),(s4, 0)}, HS2 = {(s2, 0),(s4, 0)}, HS3 = {(s3, 0),(s4, 0),(s5, 0)}Δ−1HS1 = {2, 3, 4}, Δ−1HS2 = {2, 4}, Δ−1HS3 = {3, 4, 5}


## 2.1. Concept and Basic Operations of HFLTSs

Definition 1.
(see [16]). Let S be a linguistic term set, S=s0,…,sg; an HFLTS, Hs, is an ordered finite subset of the consecutive linguistic terms of S.
The empty HFLTS and the full HFLTS for a linguistic variableϑ are defined as follows:(1)
The empty HFLTS:HSϑ=,(2)
The full HFLTS:HSϑ=SDefinition 2.
(see [16]). Let S=s0,…,sg be a linguistic term set; HS0, HS1, and HS2 are three arbitrary HFLTSs on S. HSC is the complement set of HS0. Three operations are defined as follows:(1)
HSC=S−HS0=sisi∈Sandsi∉HS0(2)
HS1∩HS2=sisi∈HS1andsi∈HS2(3)
HS1∪HS2=sisi∈HS1orsi∈HS2
Due to the present decision-making problems having higher uncertainty, experts in the decision-making group might hesitate among different linguistic terms to express their preferences. Context-free grammar is close to human beings’ cognitive model and can generate comparative linguistic expressions. Rodríguez et al. [16] pointed out how to generate comparative linguistic expressions by using context-free grammar. Rodríguez et al. [4] considered similar but extended context-free grammar to that defined in Rodríguez et al. [16] which might generate comparative linguistic expressions similar to the expressions used by experts in GDM problems. Extended context-free grammar refers to a set containing a single term or several adjacent linguistic terms and cannot support the arbitrarily linguistic term mix. One special case is omitted, that is, experts may be hesitant to choose a better evaluation or a worse evaluation. Therefore, this paper improves extended context-free grammar to introduce the binary relation “or.”Definition 3.
LetGH be improved context-free grammar and S=s0,…,sg be a linguistic term set. The elements of GH = (VN, VT, I, P) are defined as follows:VN = {<primary term>, <composite term>, <unary relation>, <binary relation>, <conjunction>}VT = {lower than, greater than, at least, at most, between, or, and, s0, s1, ..., sg}I∈VNFor context-free grammar,GH, the production rules are as follows:P = {I ::= <primary term>|<composite term><composite term> ::== <unary relation><primary term>|<binary relation><primaryterm><conjunction><primary term><primaryterm> ::== s0s1sg<unaryrelation> ::== lower than|greater than|at least|at most<binaryrelation> ::== between|or<conjunction> :==and}Definition 4.
LetEGH be a function that transforms linguistic expressions ll∈Sll obtained by context-free grammar GH into a HFLTS HS, where S is the linguistic term set used by GH and Sll:EGH:Sll⟶HSThe comparative linguistic expressions generated byGH can be converted into HFLTSs by means of the following:(1)
EGH (si) = sisi∈S(2)
EGH (at most si) = sjsj∈Sandsj≤si(3)
EGH (lower than si) = sjsj∈Sandsj<si(4)
EGH (at least si) = sjsj∈Sandsj≥si(5)
EGH (greater than si) = sjsj∈Sandsj>si(6)
EGH (between si and sj) = sksk∈Sandsi≤sk≤sj(7)
EGH (si or sj, …, sk) = si,sj,…,skExample 1.
LetS be a linguistic term set.S=s0:nothing,s1:very low,s2:low,s3:medium,s4:high,s5:very high,s6:perfectThree experts give their opinions aiming at the same evaluation object based on improved context-free grammar:ll1: between low and high; ll2: low or high; and ll3: between medium and very high. According to the function EGH, three different HFLTSs are obtained:HS1 = {s2, s3, s4}, HS2 = {s2, s4}, and HS3 = {s3, s4, s5}


## 2.2. Computing with HFLTSs Using the 2-Tuple Fuzzy Linguistic Representation Model

LetS = si:i=0,1,2,…,g be a finite and ordered discrete linguistic term set, where si represents a possible value for a linguistic variable. The 2-tuple fuzzy linguistic representation model deals with linguistic information by introducing a new parameter called symbolic translation. The concept of symbolic translation is described in Definition 5. It is used to make the information representation continuous in its domain, and it is the foundation of the computation techniques of the 2 tuples. The concept and basic operations of the 2-tuple fuzzy linguistic representation model are as follows.Definition 5.
(see [33]). Let β be a value representing the result of an aggregation of the indices of a set of labels assessed in the linguistic term set S, i.e., the result of a symbolic aggregation operation β∈0,g, being g + 1, the cardinality of S. Let i = roundβ and α=β−i be two values such that i∈0,g and α∈−0.5,0.5; then, α is called a symbolic translation.
The linguistic representation model 2-tuplesi,αi, si∈S and αi∈−0.5,0.5, is developed from the above concept:(1)
si represents the linguistic label center of the information(2)
αi is a numerical value expressing the value of the translation from the original result β to the closest index label, i, in the linguistic term set S, i.e., the symbolic translationDefinition 6.
(see [33]). Let S = s0,s1,…,sg be a linguistic term set and β∈0,g be a value representing the result of a symbolic aggregation; then, the 2-tuple that expresses the equivalent information to β is obtained with the function Δ:(1)Δ:0,g⟶S×0.5,−0.5,Δβ=si,αi,si,i=roundβ,αi=β−i,αi∈−0.5,0.5,where round⋅ is the usual round operation, si has the closest index label to β, and αi is the value of the symbolic translation.
Contrarily, letS = s0,s1,…,sg be a linguistic term set and si,αi be a 2-tuple. There is always a Δ−1 function:(2)Δ−1:S×0.5,−0.5⟶0,g,Δ−1si,αi=i+αi=β.
The original linguistic evaluation variable can be converted into a linguistic 2-tuple by adding value zero as symbolic translation:si∈S⟹si,0.Example 2.
The decision information in Example1 can be transformed into the following 2-tuple information:HS1 = {(s2, 0),(s3, 0),(s4, 0)}, HS2 = {(s2, 0),(s4, 0)}, HS3 = {(s3, 0),(s4, 0),(s5, 0)}Δ−1HS1 = {2, 3, 4}, Δ−1HS2 = {2, 4}, Δ−1HS3 = {3, 4, 5}


## 3. A Novel Group Decision-Making Approach for Hesitant Fuzzy Linguistic Term Sets

This section describes a novel GDM approach based on HFLTSs. Aggregation operators are the most widely used tool for combining individual preference information into overall preference information in the GDM process. The traditional operators are arithmetic average operators and geometric average operators. These operators consider the DMs’ preferences, and the weights are always determined subjectively. The OWA operator is a parameterized way of aggregating from “and” to “or.” The associated weights can be determined objectively. The classic method for determining the weights is quantifier-guided aggregation. Three fuzzy linguistic preferences, for the most (fuzzy majority), at least half, and as much as possible, are considered in this paper.After obtaining the 2-tuple sets in Section2.2, the rough set theory is introduced to transform these sets into rough numbers sets, and the obtained rough numbers sets can be aggregated into an interval using an OWA operator. Then, the GDM problem in the context of HFLTSs degenerates into an information aggregation problem for interval numbers. The framework of the proposed group decision-making approach for hesitant fuzzy linguistic term sets is shown in Figure 1.Figure 1
The framework of the proposed group decision-making approach for HFLTSs.


### 3.1. Elicitation of Linguistic Expressions in Decision-Making

LetX be a set of evaluation objects, X = xi1≤i≤n, let C be a set of evaluation criteria, C = cj1≤j≤m, and let E be a set of experts, E = ek1≤k≤l. According to the given linguistic term set, expert ek uses proposed context-free grammar to give out the linguistic expression lljkxi concerning the criterion cj for evaluating xi. The linguistic expression lljkxi can be transformed into an HFLTS HSjkxi using the transformation function EGH. The hesitant GDM information is presented as shown in Table 2.Table 2
Hesitant fuzzy linguistic term sets given by the experts.
ObjectxiCriteriac1c2…cj….cme1HS11xiHS21xi…HSj1xi…HSm1xie2HS12xiHS22xi…HSj2xi…HSm2xi⋮⋮⋮⋮⋮⋮⋮ekHS1kxiHS2kxi…HSjkxi…HSmkxi⋮⋮⋮⋮⋮⋮⋮elHS1lxiHS2lxi…HSjlxi….HSmlxi


### 3.2. Rough Number Enabled HFLTS Information Processing

Experts in the decision-making group have diversified opinions on the evaluated objects. Moreover, hesitant linguistic information given by all experts may have different lengths. Therefore, translating all the HFLTSs into the information with the same length is a critical procedure for information aggregation. Computing the average value of all the elements in an HFLTS is unreasonable obviously, ignoring the uncertainty of each element. Each linguistic term in the predefined linguistic term set can be deemed as a class. The rough numbers can give the lower and upper approximations of the target class to describe the uncertainty of the class appearing in a group decision-making problem.Example 3.
Let HFLTSs in Example1 be the information given by three experts with respect to HSjkxi. HSj1xi = {s2, s3, s4}, HSj2xi = {s2, s4}, and HSj3xi = {s3, s4, s5}.
Takings2 for an example, s2 in this group decision-making information has fuzziness and uncertainty for the inconsistent judgments of all experts. The boundary region of s2, i.e., the difference between the lower and upper approximations of s2, can imply that the knowledge about this term is better. Rough number is a concept proposed by Zhai et al. [44] for managing the imprecise design information, which is derived from the basic notions of the rough set. The basic notions of rough sets are as follows.
Rough set theory (RST) is an effective mathematical tool to deal with subjective and vague information using only the given information, which does not require any external information or additional subjective adjustment for data analysis. Furthermore, RST excels in handling imprecise information especially when the data set is small in size and other tools like statistics are not suitable [45]. RST uses the lower and upper approximations to form the approximation of a target set and expresses vagueness using the boundary region of a set. This is indeed the unique advantage of the rough set theory in dealing with vagueness and uncertainty.
LetU be a universe containing all the objects, and all the objects can be categorized into n classes. Assume that set R is the collection of these classes, R=C1,C2,…,Cn. Let Y be an arbitrary object of U. If these classes are ordered in the manner of C1<C2<⋯<Cn, then for any class Ci∈R, 1≤i≤n, the lower approximation of Ci can be defined as(3)Apr¯Ci=∪Y∈URY≤Ci.
The upper approximation ofCi can be defined as(4)Apr¯Ci=∪Y∈URY≥Ci.
The boundary region ofCi can be expressed as(5)BndCi=∪Y∈URY≠Ci=Y∈URY>Ci∪Y∈URY<Ci.
Lim¯Ci and Lim¯Ci represents the lower limit and upper limit for Ci, respectively, which are defined as follows:(6)Lim¯Ci=1ML∑RYY∈Apr¯Ci,where ML is the number of objects contained in the lower approximation of Ci.(7)Lim¯Ci=1MU∑RYY∈Apr¯Ci,where MU is the number of objects contained in the upper approximation of Ci.
The rough boundary interval ofCi is the interval between the lower limit Lim¯Ci and the upper limit Lim¯Ci, which is denoted as RBndCi:(8)RBndCi=Lim¯Ci−Lim¯Ci.
Accordingly, the vague classCi can be expressed by its lower limit and upper limit as follows:(9)RNCi=Lim¯Ci,Lim¯Ci.
The above definitions of the rough boundary interval and rough number can be used to deal with the imprecise evaluation information in group decision-making problems.Example 4.
HSj1xi = {s2, s3, s4}, HSj2xi = {s2, s4}, and HSj3xi = {s3, s4, s5} in Example 3 can be represented by the 2-tuple linguistic representation model first, and then, each element can be defined by its rough number to quantify and analyze the subjective evaluations:HSj1xi = {(s2, 0),(s3, 0), (s4, 0)}, HSj2xi = {(s2, 0), (s4, 0)}, and HSj3xi = {(s3, 0), (s4, 0),(s5, 0)}Δ−1HSj1xi = {2, 3, 4}, Δ−1HSj2xi = {2, 4}, and Δ−1HSj3xi = {3, 4, 5}Δ−1HSj1xi = {Lim¯2,Lim¯2, Lim¯3,Lim¯3, Lim¯4,Lim¯4}whereLim¯2 = (2 + 2)/2 = 2; Lim¯2 = (2 + 3 + 4 + 2 + 4 + 3 + 4 + 5)/8 = 3.375Lim¯3 = (2 + 3 + 2 + 3)/4 = 2.5; Lim¯3 = (3 + 4 + 4 + 4 + 5)/5 = 4Lim¯4 = (2 + 3 + 4 + 2 + 4 + 3 + 4)/7 = 3.143; Lim¯4 = (4 + 4 + 4 + 5)/4 = 4.25Lim¯5 = (2 + 3 + 4 + 2 + 4 + 3 + 4 + 5)/8 = 3.375; Lim¯5 = 5/1 = 5Δ−1HSj1xi = {[2, 3.375], [2.5, 4], [3.143, 4.25]}Δ−1HSj2xi = {[2, 3.375], [3.143, 4.25]}Δ−1HSj3xi = {[2.5, 4], [3.143, 4.25], [3.375, 5]}


### 3.3. Rough Information Aggregation Based on the OWA Operator

The following job is to aggregate the elements in a rough number set into an interval number. As the elements in the rough number sets for allΔ−1HSjkxi, 1≤k≤l, are different, the traditional averaging operators with given weights are not flexible and reasonable. The OWA operator provides a parameterized family of aggregation operators that includes the maximum (or), the minimum (and), and the average, as special cases. The basic notions of OWA operators are as follows.Definition 7.
(see [38]). A mapping F from In⟶I is called an OWA operator of dimension n if associated with F is a weighting vector W:(10)W=w1,w2,…,wnTsuch that (1) wi∈0,1, 1≤i≤n, and (2) ∑i=1nwi=1.
And(11)Fa1,a2,…,an=∑j=1nwjbj=w1b1+w2b2+⋯+wnbn,where bj is the jth largest element in the collection a1,a2,…,an.
The most important issue of applying OWA operators is to determine the associated weights. Yager [38] presented a formula to calculate the weighting function for the OWA aggregation operator by using the linguistic quantifier proposed by Zadeh [46]. Yager [47] distinguished three categories of relative quantifiers: regular increasing Monotone (RIM) quantifier, regular decreasing monotone (RDM) quantifier, and regular unimodal (RUM) quantifier. The procedure used for generating the weights from the quantifier depends upon the type of the quantifier provided. In the case of the RIM quantifier, the weights for OWA operators are generated as(12)wj=Qjn−Qj−1nforj=1,2,…,n,where wj is associated with bj, which is the jth largest element in the collection a1,a2,…,an.
We consider three fuzzy linguistic preferences: for the most (fuzzy majority), at least half, and as much as possible. These preferences indicate the degree to which the decision maker is satisfied with the number of criteria solved. The linguistic quantifierQ(x) is shown as Figure 2.
For the “for the most,” the linguistic quantifierQ(x) is defined as [48](13)Qx=0,forx≤0.3,2x−0.6,for0.3<x<0.8,1,forx≥0.8.
For the “at least half,” the linguistic quantifierQ(x) is defined as(14)Qx=0,forx≤0,2x,for0<x<0.5,1,forx≥0.5.
For the “as much as possible,” the linguistic quantifierQ(x) is defined as(15)Qx=0,forx≤0.5,2x−1,for0.5<x<1,1,forx≥1.
Assume that the transformed informationΔ−1HSjkxi, which indicates information expert ek given concerning the criterion cj for evaluating the object xi, is in the form of a¯ijk1,a¯ijk1,a¯ijk2,a¯ijk2,…,a¯ijkr,a¯ijkr. The elements in the rough number set can be aggregated into a¯ijk,a¯ijk based on the OWA operator. Since the elements in the rough number set are arranged in the ascending order, a¯ijk and a¯ijk can be determined by the following equations:(16)a¯ijk=∑h=1rwr−h⋅a¯ijkh,1≤h≤r,(17)a¯ijk=∑h=1rwr−h⋅a¯ijkh,1≤h≤r,where wr−h can be obtained according to equation (12).Figure 2
The linguistic quantifierQ(x). (a) “Most.” (b) “At least half.” (c) “As much as possible.”
(a)(b)(c)Example 5.
Δ−1HSj1xi = {[2, 3.375], [2.5, 4], [3.143, 4.25]},Δ−1HSj2xi = {[2, 3.375], [3.143, 4.25]},Δ−1HSj3xi = {[2.5, 4], [3.143, 4.25], [3.375, 5]} in Example 4.Considering the fuzzy linguistic preference “for the most,” the above information can be transformed into the following:If the rough number set has three elements,w1=1−Q2/3 = 0.27, w2=Q2/3−Q2/3 = 0.67, and w3=Q1/3 = 0.06.If the rough number set has two elements,w1=1−Q1/2 = 0.6 and w2=Q1/2 = 0.4.Δ−1HSj1xi = [2.20358, 3.59625],Δ−1HSj2xi = [2.4572, 3.725],Δ−1HSj3xi = [2.72611, 4.1275].Concerning criterioncj for evaluating object xi, the group decision-making information can be obtained by considering the experts’ weights and weights associated with OWA operators.ForΔ−1HSjkxi=a¯ijk,a¯ijk, let wek be the weight of expert ek and wokij be the weight associated with the OWA operator according to the order of information ek given. wkij is the normalized weight for the sum of wek and wokij:(18)wkij=wek+wokij∑k=1lwek+wokij,1≤i≤n,1≤j≤m.The aggregated evaluation information is defined asΔ−1HSjxi, Δ−1HSjxi=a¯ij,a¯ij.(19)a¯ij=∑k=1lwkij⋅a¯ijk,(20)a¯ij=∑k=1lwkij⋅a¯ijk.The final decision matrixA for the MCGDM problem is as follows:(21)An×m=a¯11,a¯11a¯12,a¯12⋯a¯1m,a¯1ma¯21,a¯21a¯22,a¯22⋯a¯2m,a¯2m⋮⋮⋮⋮a¯n1,a¯n1a¯n2,a¯n2⋯a¯nm,a¯nm.


## 3.1. Elicitation of Linguistic Expressions in Decision-Making

LetX be a set of evaluation objects, X = xi1≤i≤n, let C be a set of evaluation criteria, C = cj1≤j≤m, and let E be a set of experts, E = ek1≤k≤l. According to the given linguistic term set, expert ek uses proposed context-free grammar to give out the linguistic expression lljkxi concerning the criterion cj for evaluating xi. The linguistic expression lljkxi can be transformed into an HFLTS HSjkxi using the transformation function EGH. The hesitant GDM information is presented as shown in Table 2.Table 2
Hesitant fuzzy linguistic term sets given by the experts.
ObjectxiCriteriac1c2…cj….cme1HS11xiHS21xi…HSj1xi…HSm1xie2HS12xiHS22xi…HSj2xi…HSm2xi⋮⋮⋮⋮⋮⋮⋮ekHS1kxiHS2kxi…HSjkxi…HSmkxi⋮⋮⋮⋮⋮⋮⋮elHS1lxiHS2lxi…HSjlxi….HSmlxi


## 3.2. Rough Number Enabled HFLTS Information Processing

Experts in the decision-making group have diversified opinions on the evaluated objects. Moreover, hesitant linguistic information given by all experts may have different lengths. Therefore, translating all the HFLTSs into the information with the same length is a critical procedure for information aggregation. Computing the average value of all the elements in an HFLTS is unreasonable obviously, ignoring the uncertainty of each element. Each linguistic term in the predefined linguistic term set can be deemed as a class. The rough numbers can give the lower and upper approximations of the target class to describe the uncertainty of the class appearing in a group decision-making problem.Example 3.
Let HFLTSs in Example1 be the information given by three experts with respect to HSjkxi. HSj1xi = {s2, s3, s4}, HSj2xi = {s2, s4}, and HSj3xi = {s3, s4, s5}.
Takings2 for an example, s2 in this group decision-making information has fuzziness and uncertainty for the inconsistent judgments of all experts. The boundary region of s2, i.e., the difference between the lower and upper approximations of s2, can imply that the knowledge about this term is better. Rough number is a concept proposed by Zhai et al. [44] for managing the imprecise design information, which is derived from the basic notions of the rough set. The basic notions of rough sets are as follows.
Rough set theory (RST) is an effective mathematical tool to deal with subjective and vague information using only the given information, which does not require any external information or additional subjective adjustment for data analysis. Furthermore, RST excels in handling imprecise information especially when the data set is small in size and other tools like statistics are not suitable [45]. RST uses the lower and upper approximations to form the approximation of a target set and expresses vagueness using the boundary region of a set. This is indeed the unique advantage of the rough set theory in dealing with vagueness and uncertainty.
LetU be a universe containing all the objects, and all the objects can be categorized into n classes. Assume that set R is the collection of these classes, R=C1,C2,…,Cn. Let Y be an arbitrary object of U. If these classes are ordered in the manner of C1<C2<⋯<Cn, then for any class Ci∈R, 1≤i≤n, the lower approximation of Ci can be defined as(3)Apr¯Ci=∪Y∈URY≤Ci.
The upper approximation ofCi can be defined as(4)Apr¯Ci=∪Y∈URY≥Ci.
The boundary region ofCi can be expressed as(5)BndCi=∪Y∈URY≠Ci=Y∈URY>Ci∪Y∈URY<Ci.
Lim¯Ci and Lim¯Ci represents the lower limit and upper limit for Ci, respectively, which are defined as follows:(6)Lim¯Ci=1ML∑RYY∈Apr¯Ci,where ML is the number of objects contained in the lower approximation of Ci.(7)Lim¯Ci=1MU∑RYY∈Apr¯Ci,where MU is the number of objects contained in the upper approximation of Ci.
The rough boundary interval ofCi is the interval between the lower limit Lim¯Ci and the upper limit Lim¯Ci, which is denoted as RBndCi:(8)RBndCi=Lim¯Ci−Lim¯Ci.
Accordingly, the vague classCi can be expressed by its lower limit and upper limit as follows:(9)RNCi=Lim¯Ci,Lim¯Ci.
The above definitions of the rough boundary interval and rough number can be used to deal with the imprecise evaluation information in group decision-making problems.Example 4.
HSj1xi = {s2, s3, s4}, HSj2xi = {s2, s4}, and HSj3xi = {s3, s4, s5} in Example 3 can be represented by the 2-tuple linguistic representation model first, and then, each element can be defined by its rough number to quantify and analyze the subjective evaluations:HSj1xi = {(s2, 0),(s3, 0), (s4, 0)}, HSj2xi = {(s2, 0), (s4, 0)}, and HSj3xi = {(s3, 0), (s4, 0),(s5, 0)}Δ−1HSj1xi = {2, 3, 4}, Δ−1HSj2xi = {2, 4}, and Δ−1HSj3xi = {3, 4, 5}Δ−1HSj1xi = {Lim¯2,Lim¯2, Lim¯3,Lim¯3, Lim¯4,Lim¯4}whereLim¯2 = (2 + 2)/2 = 2; Lim¯2 = (2 + 3 + 4 + 2 + 4 + 3 + 4 + 5)/8 = 3.375Lim¯3 = (2 + 3 + 2 + 3)/4 = 2.5; Lim¯3 = (3 + 4 + 4 + 4 + 5)/5 = 4Lim¯4 = (2 + 3 + 4 + 2 + 4 + 3 + 4)/7 = 3.143; Lim¯4 = (4 + 4 + 4 + 5)/4 = 4.25Lim¯5 = (2 + 3 + 4 + 2 + 4 + 3 + 4 + 5)/8 = 3.375; Lim¯5 = 5/1 = 5Δ−1HSj1xi = {[2, 3.375], [2.5, 4], [3.143, 4.25]}Δ−1HSj2xi = {[2, 3.375], [3.143, 4.25]}Δ−1HSj3xi = {[2.5, 4], [3.143, 4.25], [3.375, 5]}


## 3.3. Rough Information Aggregation Based on the OWA Operator

The following job is to aggregate the elements in a rough number set into an interval number. As the elements in the rough number sets for allΔ−1HSjkxi, 1≤k≤l, are different, the traditional averaging operators with given weights are not flexible and reasonable. The OWA operator provides a parameterized family of aggregation operators that includes the maximum (or), the minimum (and), and the average, as special cases. The basic notions of OWA operators are as follows.Definition 7.
(see [38]). A mapping F from In⟶I is called an OWA operator of dimension n if associated with F is a weighting vector W:(10)W=w1,w2,…,wnTsuch that (1) wi∈0,1, 1≤i≤n, and (2) ∑i=1nwi=1.
And(11)Fa1,a2,…,an=∑j=1nwjbj=w1b1+w2b2+⋯+wnbn,where bj is the jth largest element in the collection a1,a2,…,an.
The most important issue of applying OWA operators is to determine the associated weights. Yager [38] presented a formula to calculate the weighting function for the OWA aggregation operator by using the linguistic quantifier proposed by Zadeh [46]. Yager [47] distinguished three categories of relative quantifiers: regular increasing Monotone (RIM) quantifier, regular decreasing monotone (RDM) quantifier, and regular unimodal (RUM) quantifier. The procedure used for generating the weights from the quantifier depends upon the type of the quantifier provided. In the case of the RIM quantifier, the weights for OWA operators are generated as(12)wj=Qjn−Qj−1nforj=1,2,…,n,where wj is associated with bj, which is the jth largest element in the collection a1,a2,…,an.
We consider three fuzzy linguistic preferences: for the most (fuzzy majority), at least half, and as much as possible. These preferences indicate the degree to which the decision maker is satisfied with the number of criteria solved. The linguistic quantifierQ(x) is shown as Figure 2.
For the “for the most,” the linguistic quantifierQ(x) is defined as [48](13)Qx=0,forx≤0.3,2x−0.6,for0.3<x<0.8,1,forx≥0.8.
For the “at least half,” the linguistic quantifierQ(x) is defined as(14)Qx=0,forx≤0,2x,for0<x<0.5,1,forx≥0.5.
For the “as much as possible,” the linguistic quantifierQ(x) is defined as(15)Qx=0,forx≤0.5,2x−1,for0.5<x<1,1,forx≥1.
Assume that the transformed informationΔ−1HSjkxi, which indicates information expert ek given concerning the criterion cj for evaluating the object xi, is in the form of a¯ijk1,a¯ijk1,a¯ijk2,a¯ijk2,…,a¯ijkr,a¯ijkr. The elements in the rough number set can be aggregated into a¯ijk,a¯ijk based on the OWA operator. Since the elements in the rough number set are arranged in the ascending order, a¯ijk and a¯ijk can be determined by the following equations:(16)a¯ijk=∑h=1rwr−h⋅a¯ijkh,1≤h≤r,(17)a¯ijk=∑h=1rwr−h⋅a¯ijkh,1≤h≤r,where wr−h can be obtained according to equation (12).Figure 2
The linguistic quantifierQ(x). (a) “Most.” (b) “At least half.” (c) “As much as possible.”
(a)(b)(c)Example 5.
Δ−1HSj1xi = {[2, 3.375], [2.5, 4], [3.143, 4.25]},Δ−1HSj2xi = {[2, 3.375], [3.143, 4.25]},Δ−1HSj3xi = {[2.5, 4], [3.143, 4.25], [3.375, 5]} in Example 4.Considering the fuzzy linguistic preference “for the most,” the above information can be transformed into the following:If the rough number set has three elements,w1=1−Q2/3 = 0.27, w2=Q2/3−Q2/3 = 0.67, and w3=Q1/3 = 0.06.If the rough number set has two elements,w1=1−Q1/2 = 0.6 and w2=Q1/2 = 0.4.Δ−1HSj1xi = [2.20358, 3.59625],Δ−1HSj2xi = [2.4572, 3.725],Δ−1HSj3xi = [2.72611, 4.1275].Concerning criterioncj for evaluating object xi, the group decision-making information can be obtained by considering the experts’ weights and weights associated with OWA operators.ForΔ−1HSjkxi=a¯ijk,a¯ijk, let wek be the weight of expert ek and wokij be the weight associated with the OWA operator according to the order of information ek given. wkij is the normalized weight for the sum of wek and wokij:(18)wkij=wek+wokij∑k=1lwek+wokij,1≤i≤n,1≤j≤m.The aggregated evaluation information is defined asΔ−1HSjxi, Δ−1HSjxi=a¯ij,a¯ij.(19)a¯ij=∑k=1lwkij⋅a¯ijk,(20)a¯ij=∑k=1lwkij⋅a¯ijk.The final decision matrixA for the MCGDM problem is as follows:(21)An×m=a¯11,a¯11a¯12,a¯12⋯a¯1m,a¯1ma¯21,a¯21a¯22,a¯22⋯a¯2m,a¯2m⋮⋮⋮⋮a¯n1,a¯n1a¯n2,a¯n2⋯a¯nm,a¯nm.


## 4. Extended VIKOR Method for MCGDM Based on Hesitant Fuzzy Linguistic Term Sets

The VIKOR method introduces the multicriteria ranking index based on the particular measure of closeness to the ideal solution [49]. According to Opricovic and Tzeng [50], the multicriteria measure for compromise ranking is developed from the Lp-metric utilized as an aggregating function in a compromise programming method. For an alternative xi1≤i≤n, the evaluating value of the jth criterion 1≤j≤m is denoted as fij. The Lp-metric has the following form [51]:(22)Lp,i=∑j=1nwjfj∗−fijfj∗−fj−p1/p,1≤p≤∞,i=1,2,…,n,where fj∗=maxifij and fj−=minifij.In the VIKOR method,L1,i (or Si) and L∞,i (or Ri) are used to formulate ranking measurements. The solution gained by min Si is with a maximum group utility, and the solution gained by min Ri is with a minimum individual regret of the opponent [50]. The compromise solution is a feasible solution that is the closest to the ideal, and compromise means an agreement established by mutual concessions. v is introduced as the compromise parameter between the group utility and the individual regret. Qi = vf(Si) + (1 − v)f(Ri). v>0.5 represents concerning the group utility (or the majority). v<0.5 represents concerning the individual regret.The VIKOR method is used to treat the decision matrix to calculateS, R, and Q and then obtain the candidate ranking order. According to the summarized steps of the VIKOR method [50], the extended VIKOR approach proposed in this paper has the following five steps.Step 1: determine the positive ideal solutionA∗ and the negative ideal solution A− of the final decision matrix A=a¯ij,a¯ijn×m.Criteria setB represents the “larger-the-better” category, and criteria set C represents the “small-the-better” category.(23)A∗=aj∗1≤j≤m,whereaj∗=maxi=1na¯ijcj∈B,mini=1na¯ijcj∈C,A−=aj−1≤j≤m,whereaj−=mini=1na¯ijcj∈B,maxi=1na¯ijcj∈C.Step 2: compute the values ofS¯i,S¯i and R¯i,R¯i by the following formulas:(24)S¯i=∑cj∈Bwcjaj∗−a¯ijaj∗−aj−+∑cj∈Cwcja¯ij−aj∗aj−−aj∗,(25)S¯i=∑cj∈Bwcjaj∗−a¯ijaj∗−aj−+∑cj∈Cwcja¯ij−aj∗aj−−aj∗,(26)R¯i=maxjwcjaj∗−a¯ijaj∗−aj−,forcj∈B,wcja¯ij−aj∗aj−−aj∗,forcj∈C.(27)R¯i=maxjwcjaj∗−a¯ijaj∗−aj−,forcj∈B,wcja¯ij−aj∗aj−−aj∗,forcj∈C,wherewcj is the weight of criteria cj.Step 3: compute the values ofQ¯i,Q¯i:(28)Q¯i=vS¯i−S−S∗−S−+1−vR¯i−R−R∗−R−,(29)Q¯i=vS¯i−S−S∗−S−+1−vR¯i−R−R∗−R−,whereS−=miniS¯i, S∗=maxiS¯iR−=miniR¯i, R∗=maxiR¯i, and v is the weight of the decision-making strategy of the maximum group utility. v>0.5 represents “voting by majority rule,” v≈0.5 represents “by consensus,” and v<0.5 represents “with veto.” Selection of v depends on the decision strategy of experts, and it may influence the compromise solution.Step 4: rank the alternatives, sorting the values ofQ¯i,Q¯i, S¯i,S¯i, and R¯i,R¯i in the ascending order, and then obtain three ranking lists.For any two rough numbers,RN1 = [L1, U1] and RN2 = [L2, U2], where L1 and L2 represent their lower limits and U1 andU2 represent their upper limits, the ranking rules of two rough numbers are given as follows [52]:(1)
(a)
IfU1 > U2 and L1 ≥ L2 orU1 ≥ U2 and L1 > L2, then RN1 > RN2(b)
IfU1 = U2 and L1 = L2, then RN1 = RN2(2)
LetM1 = (L1 + U1)/2 and M2 = (L2 + U2)/2.(a)
IfL2 > L1 and U1 > U2: if M1 ≤ M2, then RN1 < RN2; if M1 > M2, then RN1 > RN2(b)
IfL1 > L2 and U2 > U1: if M1 ≤ M2, then RN1 < RN2; if M1 > M2, then RN1 > RN2Step 5: propose a compromise solution:Definition 8.
For any two interval numbersA=A¯,A¯ and B=B¯,B¯, the distance between A and B, DA,B, is defined as(30)DA,B=12B¯−A¯2+B¯−A¯2.(1)
If the following two conditions are satisfied,x1 is the best compromise solution. x1 is the object ranked first in the Q¯i,Q¯i list.Condition 1: acceptable advantage:DQ¯i1,Q¯i1,Q¯i2,Q¯i2≥DQ, DQ=1/n−1Condition 2: acceptable stability in decision-making:x1 must also be the best object ranked according to S¯i,S¯i or/and R¯i,R¯i(2)
If one of the conditions is not satisfied, then a set of compromise solutions is obtained:①
If only condition 2 is not satisfied,x1 and x2 are both compromise solutions②
If condition 1 is not satisfied, maximizedX can be obtained according to DQ¯i1,Q¯i1,Q¯iX,Q¯iX<1/n−1, and x1,x2,…,xX are all near to the best compromise solution


## 5. Illustrative Examples

### 5.1. Example 1

In this section, a numerical example adopted from Rodríguez et al. [4] is provided to validate the effectiveness of the proposed GDM approach based on HFLTSs. A conference committee, composed of 3 researchers E = {e1, e2, e3}, wants to grant the best paper award in an international conference. There are four selected papers, X = {John’s paper, Mike’s paper, David’s paper, Frank’s paper}. The linguistic term set suitable to express such assessments shown in Rodríguez et al. [4] can be given as follows:S = {neither (s0); very low (s1); low (s2); medium (s3); high (s4); very high (s5); absolute (s6)}Step 1: transform the preferences provided by experts in Rodríguez et al. [4] into HFLTSs:(31)P1=−s0,s1s5s0,s1s5,s6−s4,s5s0,s1,s2,s3s2s0,s1,s2−s5,s6s4,s5,s6s4,s5,s6s0,s1,s2,s3−,P2=−s0,s1,s2s4,s5,s6s0,s1,s2s4,s5,s6−s4s1s0,s1s2−s5,s6s4,s5s5s0,s1,s2−,P3=−s4,s5,s6s4,s5s2s0,s1,s2−s4,s5,s6s4,s5,s6s0,s1,s2s0,s1,s2−s4s4s0,s1,s2s1−.Step 2: deal with the assessment information based on the HFLTS using the 2-tuple linguistic representation model:(32)Δ−1p1=−0,150,15,6−4,50,1,2,320,1,2−5,64,5,64,5,60,1,2,3−,Δ−1p2=0,1,24,5,60,1,24,5,6410,125,64,550,1,2,Δ−1p3=−4,5,64,520,1,2−4,5,64,5,60,1,20,1,2−440,1,21−.Step 3: quantify the uncertainty in GDM information based on the rough set theory:(33)Δ−1p1=−0,2.38,0.5,3.174,4.83,4.6,5.25,4.83,60,1,0.5,1.52.83,5.5,3.63,6−4,4.67,4.4,5.330,2.75,0.67,3.14,1,4,1.4,4.51,20,1.14,0.5,1.6,1.14,2−4.75,5.5,5.29,64,4.67,4.4,5.33,4.67,61.75,5,2.83,5.33,3.29,60,1.25,0.6,1.67,1,2.33,1.25,3−,Δ−1p2=−0,2.38,0.5,3.17,0.8,4.254.6,5.250,1,0.5,1.5,1,21.75,5.2,2.83,5.5,3.63,6−4,4.670.67,3.140,1,0.5,1.51.14,2−4.75,5.5,5.29,64,4.67,4.4,5.332.83,5.330,1.25,0.6,1.67,1,2.33−,Δ−1p3=−1.33,5,1.86,5.5,2.38,64,4.83,4.6,5.251,20,3.63,0.5,4.14,1,4.67−4,4.67,4.4,5.33,4.67,61.83,5,2.29,5.5,2.75,60,1,0.5,1.5,1,20,1.14,0.5,1.6,1.14,2−4,5.29,4.75,5.5,5.29,64,4.670,3.29,0.5,3.83,1,4.40.6,1.67−.Step 4: obtain the assessment information with the same length using the OWA operator.Considering the fuzzy linguistic preference “for the most,” the final assessment information obtained corresponding to the three researchers is as follows:(34)Δ−1p1=−0.2,2.74.6,5.250.2,1.23.15,5.7−4.16,4.930.74,3.451,20.4,1.5−5,5.74.31,5.192.57,5.280.7,1.89−,Δ−1p2=−0.38,3.024.45,5.180.4,1.42.59,5.45−4,4.670.67,3.140.2,1.21.14,2−5,5.74.16,4.932.83,5.330.46,1.6−,Δ−1p3=−1.75,5.44.24,51,20.4,4.03−4.31,5.112.19,5.40.4,1.40.4,1.5−4.58,5.474,4.670.42,3.690.6,1.67−.Step 5: obtain the preference relation based on the OWA operator according to the approach proposed in Rodríguez et al. [4]:(35)PC=−0.41,3.084.4,5.140.38,1.382.03,5.08−4.13,4.870.81,3.480.38,1.380.44,1.53−4.89,5.644.13,4.882,4.850.57,1.66−.Step 6: compute the pessimistic and optimistic collective preference for each alternative. The linguistic interval for each alternative is shown in Table3.Table 3
Linguistic intervals for all alternatives in Example1.
JohnMikeDavidFrankp1Rp2Rp3Rp4R[(s2, −0.27), (s3, 0.2)][(s2, 0.32), (s4, −0.52)][(s2, −0.1), (s3, −0.15)][(s1, 0.23), (s4, −0.2)]Step 7: order the set of alternatives and select the best one as the solution to the GDM problem:x2 > x4 > x1 > x3.The best solution is similar with the work of Rodríguez et al. [4]. This example can illustrate the effectiveness of the proposed GDM approach for HFLTSs. However, the envelope-based method for computing with HFLTSs in Rodríguez et al. [4] cannot support experts to select nonadjacent linguistic terms. Next, we changed the preference provided by the three experts and compared the results obtained by the commonly used envelope-based method and the proposed GDM approach. We changed the preference referring to more than two linguistic terms to contain only two linguistic terms using the binary relation “or.” The revised preference provided by the first expert is as follows:(36)p1′=−at mostvlvhat mostvlat leastvh−betweenhandvhnormlnorl−greater thanhhorahoranorm−.The corresponding HFLTSs of the three experts are as follows:(37)p1′=−s0,s1s5s0,s1s5,s6−s4,s5s0,s3s2s0,s2−s5,s6s4,s6s4,s6s0,s3−,p2′=−s0,s2s4,s6s0,s2s4,s6−s4s1s0,s1s2−s5,s6s4,s5s5s0,s2−,p3′=−s4,s6s4,s5s2s0,s2−s4,s6s4,s6s0,s2s0,s2−s4s4s0,s2s1−.The envelope ofp1′, p2′, and p3′ does not change, and also, the ranking result of the envelope-based method does not change. However, the ranking result of the proposed GDM approach for HFLTSs changes as the change of preference given by the three experts. The linguistic interval for each alternative after preference change is shown in Table 4. The revised alternative order is x2 > x4 > x3 > x1. By contrast, we can see that the ranking position of x3 and the position of x1 have changed. The comparison illustrates that the proposed GDM approach for HFLTSs has higher sensitivity to the change of preference information and has much wider applicability under different linguistic decision-making environments.


### 5.2. Example 2

A real example of selecting logistics service suppliers is adopted in this section. Company W is a small and medium-sized electric product manufacturer. Its main products are refrigerators, freezers, and air conditioners. To focus on the core competition ability, reduce cost, and improve customer service, the company decides to adopt a logistics outsourcing strategy. After preliminary screening, five candidates (i.e., alternatives),x1, x2, x3, x4, and x5, remain for further evaluation. Seven evaluation criteria are considered: C = {c1 = quality assurance, c2 = operation efficiency, c3 = logistics technology level, c4 = logistics facility level, c5 = price, c6 = management ability, c7 = development potential level}. c5 is the “small-the-better” criterion, and the other criteria are in the “larger-the-better” category. The weight vector of the criteria set is W = (0.21, 0.19, 0.12, 0.14, 0.17, 0.09, 0.08). A committee composed of 3 experts E = {e1, e2, e3} evaluated the alternative service suppliers. The weights of the 3 experts are {we1 = 0.4, we2 = 0.3, we3 = 0.3}.Step 1: obtain the preferences provided by experts based on proposed context-free grammar, and transform the linguistic expressions into HFLTSs according to Definition4. The predefined linguistic terms set is S. The hesitant evaluation information given by three experts is shown in Tables 5–7. S = {neither (s0); very low (s1); low (s2); medium (s3); high (s4); very high (s5); absolute (s6)}.Table 4
Linguistic intervals for all alternatives after preference change in Example1.
JohnMikeDavidFrankp1Rp2Rp3Rp4R[(s2, −0.32), (s3, 0.21)][(s2, 0.45), (s4, 0.62)][(s2, 0.06), (s3, 0)][(s1, 0.34), (s4, 0.05)]Table 5
The HFLTS information given by experte1 in Example 2.
c1c2c3c4c5c6c7x1s2s0,s1,s2s3,s4s4,s5s3,s4s4s2,s3,s4x2s3,s4s2s2,s3,s4s0,s1,s2s4,s5s2,s3,s4s2,s3x3s4,s5s4,s5,s6s0,s1,s2,s3s5,s6s3s2,s3,s4s3,s4x4s2,s3,s4s5,s6s3,s4s0,s1,s2s3s4,s5s5x5s2,s3s0,s1,s2s2,s3,s4,s5s4,s5s2,s3s0,s1,s2s2,s3Table 6
The HFLTS information given by experte2 in Example 2.
c1c2c3c4c5c6c7x1s3,s4s2,s3,s4s2s0,s1,s2s2,s3s0,s1,s2s3,s4x2s5,s6s0,s1,s2s2,s3s2,s3s0,s1s3,s4s0,s1,s2x3s6s5,s6s4,s5,s6s2,s3s2s5,s6s2,s3,s4x4s0,s1,s2s2,s3s2,s3,s4,s5s5,s6s3,s4s2s2,s3,s4x5s2,s3,s4s3s2,s3s0,s1,s2s3,s4s0,s1,s2s5,s6Table 7
The HFLTS information given by experte3 in Example 2.
c1c2c3c4c5c6c7x1s0,s1,s2,s3s2,s3s2,s3,s4s3,s4s2,s3,s4s5,s6s3x2s3,s4s3s4,s5,s6s4,s5s0,s1,s2s3,s4s4,s5x3s4,s5,s6s3,s4s2,s3s2,s3,s4s1,s2s2,s3s0,s1,s2x4s2,s3,s4,s5s2,s3s2,s3,s4s3,s4s0,s1,s2s2,s3,s4s5x5s2,s3,s4s4s4,s5s2,s3,s4s3,s4s5,s6s0,s1,s2Step 2: deal with the evaluation information based on the HFLTS using the 2-tuple linguistic representation model. Taking Table4 as an example, the information of Δ−1HSj1xi corresponding to this table is shown in Table 8.Table 8
The information ofΔ−1HSj1xi corresponding to Table 4.
c1c2c3c4c5c6c7x1{2}{0, 1, 2}{3, 4}{4, 5}{3, 4}{4}{2, 3, 4}x2{3, 4}{2}{2, 3, 4}{0, 1, 2}{4, 5}{2, 3, 4}{2, 3}x3{4, 5}{4, 5, 6}{0, 1, 2, 3}{5, 6}{3}{2, 3, 4}{3, 4}x4{2, 3, 4}{5, 6}{3, 4}{0, 1, 2}{3}{4, 5}{5}x5{2, 3}{0, 1, 2}{2, 3, 4, 5}{4, 5}{2, 3}{0, 1, 2}{2, 3}Step 3: transform the information ofΔ−1HSjkxi into unified interval numbers for simplifying the group information aggregation.The information ofΔ−1HSjkxi is treated by using the rough set theory. The experts’ evaluation information represented as rough numbers is shown in Tables 9–11. The set of rough numbers can be integrated into a rough number based on the OWA operator. The possible numbers of elements in the rough number set in Tables 9–11 are 2, 3, and 4. Since the elements in the rough number set are arranged in the ascending order, the corresponding weights associated with elements for different fuzzy linguistic preferences are shown in Table 12.Table 9
Evaluation information of experte1 represented as rough numbers.
c1c2c3c4c5c6c7x1{[1.25, 2.8]}{[0, 2.13], [0.5, 2.43], [1.4, 2.67]}{[2.5, 3.5], [3, 4]}{[2.33, 4.33], [2.71, 5]}{[2.6, 3.4], [3, 4]}{[4, 5]}{[2, 3.17], [2.75, 3.4], [3.17, 4]}x2{[3, 4.17], [3.5, 4.75]}{[1.25, 2.33]}{[2, 3.63], [2.5, 4.17], [3, 4.75]}{[0, 2.43], [0.5, 2.83], [1.25, 3.2]}{[1.33, 4.5], [1.86, 5]}{[2, 3.29], [2.75, 3.5], [3.29, 4]}{[1.25, 3.2], [1.6, 4]}x3{[4, 5], [4.5, 5.5]}{[3.67, 5], [4.2, 5.5], [4.71, 6]}{[0, 2.89], [0.5, 3.25], [1.25, 3.57], [1.83, 4.2]}{[3.17, 5.5], [3.57, 6]}{[2, 3]}{[2, 3.57], [2.5, 4.2], [2.8, 5]}{[1.83, 3.5], [2.38, 4]}x4{[1.4, 3.13], [1.86, 3.8], [2.33, 4.33]}{[3, 5.5], [3.5, 6]}{[2.6, 3.71], [3.13, 4.25]}{[0, 3], [ 0.5, 3.5], [1, 4]}{[1.8, 3.33]}{[3, 4.33], [3.33, 5]}{[3.8, 5]}x5{[2, 2.88], [2.5, 3.4]}{[0, 2], [0.5, 2.5], [1, 3]}{[2, 3.5], [2.5, 4], [3, 4.5], [3.5, 5]}{[2.29, 4.33], [2.63, 5]}{[2, 3.17], [2.75, 3.4]}{[0, 2.13], [0.5, 2.83], [1, 3.75]}{[1.25, 3.6], [1.6, 4.67]}Table 10
Evaluation information of experte2 represented as rough numbers.
c1c2c3c4c5c6c7x1{[1.83, 3.33], [2.14, 4]}{[1.4, 2.67], [1.86, 3.33], [2.125, 4]}{[2, 3]}{[0, 2.71], [0.5, 3.17], [1, 3.6]}{[2, 3], [2.6, 3.4}{[0, 3], [0.5, 3.6], [1, 4.25]}{[2.75, 3.4], [3.17, 4]}x2{[3.8, 5.5], [4.17, 6]}{[0, 1.6], [0.5, 2], [1.25, 2.33]}{[2, 3.63], [2.5, 4.17]}{[1.25, 3.2], [1.6, 4]}{[0, 1.86], [0.5, 2.6]}{[2.75, 3.5], [3.29, 4]}{[0, 2.43], [0.5, 2.83], [1.25, 3.2]}x3{[5, 6]}{[4.2, 5.5], [4.71, 6]}{[2.14, 5], [2.5, 5.5], [2.89, 6]}{[2, 3.57], [2.5, 4.2]}{[1.67, 2.33]}{[3.17, 5.5], [3.57, 6]}{[1.25, 3], [1.83, 3.5], [2.38, 4]}x4{[0, 2.6], [0.5, 2.89], [1.4, 3.13]}{[2, 3.5], [2.5, 4.25]}{[2, 3.33], [2.6, 3.71], [3.13, 4.25], [3.33, 5]}{[2.5, 5.5], [3, 6]}{[1.8, 3.33], [2.17, 4]}{[2, 3.33]}{[2, 3.8], [2.5, 4.25], [3, 4.67]}x5{[2, 2.88], [2.5, 3.4], [2.88, 4]}{[1.5, 3.5]}{[2, 3.5], [2.5, 4]}{[0, 2.63], [0.5, 3], [1.25, 3.33]}{[2.75, 3.4], [3.17, 4]}{[0, 2.13], [0.5, 2.83], [1, 3.75]}{[2.17, 5.5], [2.71, 6]}Table 11
Evaluation information of experte3 represented as rough numbers.
c1c2c3c4c5c6c7x1{[0, 2.14], [0.5, 2.5], [1.25, 2.8], [1.83, 3.33]}{[1.4, 2.67], [1.86, 3.33]}{[2, 3], [2.5, 3.5], [3, 4]}{[1.5, 4], [2.33, 4.33]}{[2, 3], [2.6, 3.4], [3, 4]}{[2.4, 5.5], [3, 6]}{[2.75, 3.4]}x2{[3, 4.17], [3.5, 4.75]}{[1.6, 3]}{[3, 4.75], [3.29, 5.5], [3.63, 6]}{[2, 4.5], [2.43, 5]}{[0, 1.86], [0.5, 2.6], [0.8, 3.67]}{[2.75, 3.5], [3.29, 4]}{[2, 4.5], [2.43, 5]}x3{[4, 5], [4.5, 5.5], [5, 6]}{[3, 4.71], [3.67, 5]}{[1.25, 3.57], [1.83, 4.2]}{[2, 3.57], [2.5, 4.2], [2.8, 5]}{[1, 2], [1.67, 2.33]}{[2, 3.57], [2.5, 4.2]}{[0, 2.38], [0.5, 2.71], [1.25, 3]}x4{[1.4, 3.13], [1.86, 3.8], [2.33, 4.33], [2.6, 5]}{[2, 3.5], [2.5, 4.25]}{[2, 3.33, [2.6, 3.71], [3.13, 4.25]}{[1.5, 4.5], [2, 5]}{[0, 2.17], [0.5, 2.6], [1, 3]}{[2, 3.33], [2.33, 4], [3, 4.33]}{[3.8, 5]}x5{[2, 2.88], [2.5, 3.4], [2.88, 4]}{[2, 4]}{[3, 4.5], [3.5, 5]}{[1.25, 3.33], [1.6, 4], [2.29, 4.33]}{[2.75, 3.4], [3.17, 4]}{[1.57, 5.5], [2.13, 6]}{[0, 2.71], [ 0.5, 3.17], [1.25, 3.6]}Table 12
The successive element’s weight in one set for different fuzzy linguistic preferences.
Set with 2 elementsSet with 3 elementsSet with 4 elementsFor the most0.6, 0.40.27, 0.67, 0.060.1, 0.5, 0.4, 0At least half0, 10, 0.33, 0.670, 0, 0.5, 0.5As much as possible1, 00.67, 0.33, 00.5, 0.5, 0, 0Considering the fuzzy linguistic preference “for the most,” the evaluation information in the form of interval numbers is shown in Tables13–15.Table 13
The final evaluation information obtained frome1 for “for the most.”
c1c2c3c4c5c6c7x1[1.25, 2.8][0.42, 2.36][2.70, 3.70][2.48, 4.60][2.76, 3.64][4.00, 5.00][2.57, 3.37]x2[3.20, 4.40][1.25, 2.33][2.40, 4.06][0.41, 2.74][1.54, 4.70][2.58, 3.47][1.39, 3.52]x3[4.20, 5.20][4.09, 5.40][0.75, 3.34][3.33, 5.70][2.00, 3.00][2.38, 4.08][2.05, 3.70]x4[1.76, 3.65][3.20, 5.70][2.81, 3.93][0.40, 3.40][1.80, 3.33][3.13, 4.60][3.80, 5.00]x5[2.20, 3.09][0.40, 2.40][2.65, 4.15][2.43, 4.60][2.30, 3.26][0.40, 2.70][1.39, 4.03]Table 14
The final evaluation information obtained frome2 for “for the most.”
c1c2c3c4c5c6c7x1[1.95, 3.60][1.75, 3.19][2, 3][0.40, 3.07][2.24, 3.16][0.40, 3.48][2.92, 3.64]x2[3.95, 5.70][0.41, 1.91][2.20, 3.85][1.39, 3.52][0.20, 2.16][2.97, 3.70][0.41, 2.74]x3[5.00, 6.00][4.40, 5.70][2.43, 5.40][2.20, 3.82][1.67, 2.33][3.33, 5.70][1.71, 3.40]x4[0.42, 2.83][2.20, 3.80][2.75, 3.89][2.70, 5.70][1.95, 3.60][2.00, 3.33][2.40, 4.15]x5[2.39, 3.30][1.50, 3.50][2.20, 3.70][0.41, 2.92][2.92, 3.64][0.40, 2.70][2.39, 5.70]Table 15
The final evaluation information obtained frome3 for “for the most.”
c1c2c3c4c5c6c7x1[0.75, 2.58][1.58, 2.93][2.40, 3.40][1.83, 4.13][2.46, 3.33][2.64, 5.70][2.75, 3.40]x2[3.47, 4.00][1.60, 3.00][3.23, 5.33][2.17, 4.70][0.38, 2.46][2.97, 3.70][2.17, 4.70]x3[4.40, 5.40][3.68, 4.20][1.48, 3.82][2.38, 4.08][1.27, 2.13][2.20, 3.82][0.41, 2.64]x4[2.00, 3.95][2.20, 3.80][2.47, 3.64][1.70, 4.70][0.40, 2.51][2.28, 3.84][3.80, 5.00]x5[2.39, 3.30][2.00, 4.00][3.20, 4.70][1.55, 3.84][2.92, 3.64][1.79, 5.70][0.41, 3.07]Step 4: aggregate the information in Tables13–15 into the final decision matrix considering both the expert’s weights and the OWA weights associated with each a¯ijk,a¯ijk for 1≤i≤n and 1≤j≤m. The aggregated decision information for different fuzzy linguistic preferences is shown in Tables 16–18, respectively.


#### 5.2.1. Sensitivity Analysis

Parameterv is the weight of the strategy of the “majority of attributes,” and it plays an important role in determining the set of compromise solutions. As seen in Table 19, Q¯i,Q¯i is determined by the value of v. When v is changed, DQ¯i1,Q¯i1,Q¯i2,Q¯i2 is also changed. Aiming at the aggregated decision information obtained by the fuzzy linguistic preference “for the most,” when 0≤v≤0.8789, DQ¯i1,Q¯i1,Q¯i2,Q¯i2≥0.25, and only x3 is the compromise solution; when 0.8789<v≤1, DQ¯i1,Q¯i1,Q¯i2,Q¯i2<0.25 and DQ¯i1,Q¯i1,Q¯i3,Q¯i3≥0.25, and both x3 and x4 are the compromise solutions. From the results in Table 19, we can see that the threshold values of v are different for different fuzzy linguistic preferences. The value of v can differentiate the possible solution sets {x3} and {x3, x4}.Table 16
The aggregated decision information for “for the most.”
c1c2c3c4c5c6c7x1[1.23, 2.88][1.22, 2.79][2.35, 3.35][1.57, 3.94][2.47, 3.35][2.31, 4.91][2.72, 3.43]x2[3.41, 4.52][1.07, 2.33][2.49, 4.23][1.20, 3.47][0.60, 2.89][2.84, 3.62][1.25, 3.51]x3[4.44, 5.44][4.03, 5.11][1.41, 3.94][2.55, 4.38][1.63, 2.43][2.50, 4.30][1.42, 3.25]x4[1.42, 3.47][2.43, 4.24][2.69, 3.83][1.44, 4.44][1.43, 3.14][2.40, 3.87][3.40, 4.76]x5[2.32, 3.23][1.22, 3.22][2.62, 4.12][1.43, 3.75][2.71, 3.51][0.65, 3.24][1.29, 4.06]Table 17
The aggregated decision information for “at least half.”
c1c2c3c4c5c6c7x1[1.77, 3.47][1.80, 3.40][2.80, 3.80][2.31, 4.56][2.90, 3.85][3.04, 5.34][3.06, 3.84]x2[4.04, 5.36][1.38, 2.64][3.12, 5.12][1.88, 4.30][1.29, 4.11][3.25, 3.97][1.91, 4.35]x3[4.85, 5.85][4.65, 5.79][2.22, 4.93][3.14, 5.33][1.85, 2.69][3.09, 5.27][2.12, 3.78]x4[2.15, 4.24][3.04, 5.19][3.15, 4.41][2.25, 5.25][1.83, 3.59][2.96, 4.50][3.66, 4.93]x5[2.70, 3.72][1.61, 3.61][3.26, 4.76][2.21, 4.49][3.09, 3.88][1.46, 4.68][2.05, 5.13]Table 18
The aggregated decision information for “as much as possible.”
c1c2c3c4c5c6c7x1[0.85, 2.65][0.76, 2.47][2.15, 3.15][1.02, 3.51][2.18, 3.12][1.64, 4.28][2.48, 3.32]x2[3.12, 4.04][0.78, 2.14][2.22, 3.90][0.78, 3.05][0.32, 2.46][2.48, 3.42][0.84, 3.09]x3[4.20, 5.20][3.49, 4.49][0.87, 3.54][2.29, 4.02][1.41, 2.30][2.24, 3.94][0.90, 2.90]x4[0.89, 3.05][2.20, 3.90][2.31, 3.53][0.94, 3.94][1.01, 2.84][2.23, 3.60][3.01, 4.49]x5[2.08, 2.96][0.86, 2.86][2.24, 3.74][0.97, 3.32][2.35, 3.28][0.38, 2.83][0.86, 3.53]Step 5: select the appropriate supplier using the proposed VIKOR method.Aiming at the aggregated decision information according to the “for the most,” the positive ideal solutionA∗ and the negative ideal solution A− are determined first, and then S¯i,S¯i, R¯i,R¯i, and Q¯i,Q¯i are computed according to equations (25)–(30):A∗ = {5.44, 5.11, 4.23, 4.44, 0.60, 4.91, 4.76}A− = {1.23, 1.07, 1.41, 1.20, 3.51, 0.65, 1.25}According to the verification rules of VIKOR, if the acceptable advantage and acceptable ability are satisfied, the best rank can be assigned as a compromise solution. Letv=0.5, which represents selecting the appropriate object in compromise. First, the two conditions of the first case are considered.Condition 1:DQ¯i1,Q¯i1,Q¯i2,Q¯i2=0.5×0.208−0.0002+0.862−0.5192=0.323.DQ = 1/(5 − 1) = 0.25.Condition 1 is satisfied.Condition 2:x1 is the best object ranked according to S¯i,S¯i and R¯i,R¯i, and condition 2 is satisfied.Therefore,x1, i.e., x3, is the compromise solution.Considering all the possible values ofv, the final results are shown in Table 19.Table 19
The results of the VIKOR method based on the proposed GDM approach.
(a) The valuesS, R, and Q and the preference ranking order in the case of “for the most”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.435, 0.859][0.128, 0.210][0.451−0.036v, 1−0.023v]x2[0.274, 0.763][0.131, 0.190][0.471−0.270v, 0.867−0.017v]x3[0.122, 0.536][0.060, 0.120][0, 0.399 + 0.150v]x4[0.227, 0.754][0.098, 0.201][0.254−0.116v, 0.937−0.099v]x5[0.408, 0.876][0.123, 0.183][0.421−0.042v, 0.819 + 0.181v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8789{x3, x4}, 0.8789<v≤1(b) The valuesS, R, and Q and the preference ranking order in the case of “at least half”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.440, 0.829][0.123, 0.210][0.504−0.045v, 1−0.049v]x2[0.254, 0.804][0.136, 0.190][0.578−0.354v, 0.887 + 0.032v]x3[0.077, 0.521][0.034, 0.120][0, 0.489 + 0.072v]x4[0.198, 0.746][0.083, 0.190][0.279−0.125v, 0.889−0.044v]x5[0.376, 0.868][0.110, 0.180][0.431−0.053v, 0.830 + 0.170v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8113{x3, x4}, 0.8113<v≤1(c) The valuesS, R, and Q and the preference ranking order in the case of “as much as possible”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.410, 0.865][0.123, 0.210][0.410−0.024v, 1−0.010v]x2[0.268, 0.740][0.120, 0.189][0.387−0.190v, 0.857−0.033v]x3[0.120, 0.533][0.063, 0.120][0, 0.389 + 0.161v]x4[0.207, 0.745][0.104, 0.208][0.279−0.163v, 0.987−0.156v]x5[0.399, 0.873][0.117, 0.185][0.366−0.005v, 0.830 + 0.170v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8614{x3, x4}, 0.8614<v≤1An obvious advantage of the GDM approach for HFLTSs proposed in this paper is that it can support experts to give arbitrarily term mix to increase the richness of linguistic expressions. In this section, we change some HFLTSs on the evaluation ofx4 given by experts e1 and e2. Suppose the two experts tend to use the binary relation “or” and give nonadjacent linguistic terms due to the uncertainty of the market. The original and changed information on the evaluation of x4 by e1 and e2 is shown in Table 20. Aiming at the changed evaluation information, the aggregation result obtained by the envelope-based approach would not change for that the envelopes of the changed HFLTSs are the same as those of the original HFLTSs. However, the aggregation result obtained by the novel group decision-making approach proposed in this paper is different from Tables 13–15 under conditions of three fuzzy linguistic preferences. Furthermore, the VIKOR results of the proposed method based on the changed information will also be different from Table 19, and the related result is shown in Table 21. We can see that the compromise solutions remain unchanged, but the threshold values of v vary for different fuzzy linguistic preferences.Table 20
The original and changed information on evaluation forx4 by e1 and e2.
c1c2c3c4c5c6c7The original informationx4(e1){s2, s3, s4}{s5, s6}{s3, s4}{s0, s1, s2}{s3}{s4, s5}{s5}x4(e2){s0, s1, s2}{s2, s3}{s2, s3, s4, s5}{s5, s6 }{s3, s4}{s2}{s2, s3, s4}The changed informationx4(e1){s2, s4}{s5, s6}{s3, s4}{s0, s2}{s3}{s4, s5}{s5}x4(e2){s0, s2}{s2, s3}{s2, s5}{s5, s6 }{s3, s4}{s2 }{s2, s4}Table 21
The results of the proposed VIKOR method based on the changed information.
(a) The valuesS, R, and Q and the preference ranking order in the case of “for the most”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.7905{x3, x4}, 0.7505<v≤1(b) The valuesS, R, and Q and the preference ranking order in the case of “at least half”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.4807{x3, x4}, 0.4807<v≤1(c) The valuesS, R, and Q and the preference ranking order in the case of “as much as possible”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8926{x3, x4}, 0.8926<v≤1


#### 5.2.2. Comparisons and Discussion

To illustrate the effectiveness of the proposed group decision-making approach for HFLTSs and the related VIKOR, we use the above case study to analyze the envelope-based approach for HFLTSs. We take the envelope-based approach proposed by Rodríguez et al. [4] as a comparable approach. The HFLTS information given by three experts as shown in Tables 5–7 can be translated into interval information according to the envelope-based approach. The interval information is modeled by the 2-tuple linguistic representation model and then aggregated by the weighted averaging operator. The final decision information of Δ−1HSj1xi is shown in Table 22.Table 22
The aggregated decision information of the comparative method.
c1c2c3c4c5c6c7x1[1.7, 2.9][1.2, 2.9][2.4, 3.4][2.5, 3.8][2.4, 3.7][3.1, 4.0][2.6, 3.7]x2[3.6, 4.6][1.7, 2.3][2.6, 4.3][1.8, 3.2][1.6, 2.9][2.6, 4.0][2.0, 3.3]x3[4.6, 5.6][4.0, 5.4][1.8, 3.9][3.2, 4.5][2.1, 2.4][2.9, 4.3][1.8, 3.4]x4[1.4, 3.7][3.2, 4.2][2.4, 4.3][2.4, 3.8][2.1, 3.0][2.8, 3.8][4.1, 4.7]x5[2.0, 2.6][2.1, 2.9][2.6, 4.4][2.2, 3.8][2.6, 3.6][1.5, 3.2][2.3, 3.6]The decision result of Table20 using the proposed VIKOR approach is shown in Table 23. The ranking order of the alternatives is x3≻x4≻x2≻x5≻x1, which is the same as the ranking order in Table 20. However, the compromise solution set remains unchanged with the change of v from 0 to 1 in Table 23, which has a lower sensitivity.Table 23
The results of the VIKOR method based on the comparative GDM approach.
S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.433, 0.848][0.135,0.195][0.558–0.114v, 0.912–0.085v]x2[0.311, 0.725][0.140, 0.167][0.588–0.307v, 0.749–0.085v]x3[0.099, 0.491][0.040, 0.120][0, 0.469 + 0.052v]x4[0.247, 0.689][0.095, 0.210][0.322–0.126v, 1–0.214v]x5[0.446, 0.850][0.150, 0.180][0.646–0.184v, 0.823 + 0.177v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤1To reveal the features of the proposed VIKOR method, the rough TOPSIS method [53] is also applied in the case study. The rough TOPSIS method is used to deal with the aggregated decision information for different fuzzy linguistic preferences (Tables 16–18). Table 24 shows the final ranking result of the rough TOPSIS.Table 24
The evaluation result of the rough TOPSIS approach.
Closeness coefficient in the rough TOPSISIn the case of “for the most”In the case of “at least half”In the case of “as much as possible”x10.40540.40400.4073x20.48040.47130.4928x30.62900.64630.6242x40.49490.51470.4977x50.40430.41860.4114Ranking orderx3≻x4≻x2≻x1≻x5x3≻x4≻x2≻x5≻x1x3≻x4≻x2≻x5≻x1In the case of “for the most,” if the weight0≤v≤0.8789, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. If the weight 0.8789<v≤1, the compromise solution obtained by the rough TOPSIS is different from the compromise solution of the extended VIKOR. In the case of “at least half,” if the weight 0≤v≤0.8113, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. In the case of “as much as possible,” if the weight 0≤v≤0.8614, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. The rough TOPSIS can only obtain the distinct solution, and it cannot obtain the compromise solutions. The VIKOR method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria, and it determines a compromise solution that could be accepted by the decision makers. Thus, the obtained compromise solution is more robust and could be accepted by the decision makers.


## 5.1. Example 1

In this section, a numerical example adopted from Rodríguez et al. [4] is provided to validate the effectiveness of the proposed GDM approach based on HFLTSs. A conference committee, composed of 3 researchers E = {e1, e2, e3}, wants to grant the best paper award in an international conference. There are four selected papers, X = {John’s paper, Mike’s paper, David’s paper, Frank’s paper}. The linguistic term set suitable to express such assessments shown in Rodríguez et al. [4] can be given as follows:S = {neither (s0); very low (s1); low (s2); medium (s3); high (s4); very high (s5); absolute (s6)}Step 1: transform the preferences provided by experts in Rodríguez et al. [4] into HFLTSs:(31)P1=−s0,s1s5s0,s1s5,s6−s4,s5s0,s1,s2,s3s2s0,s1,s2−s5,s6s4,s5,s6s4,s5,s6s0,s1,s2,s3−,P2=−s0,s1,s2s4,s5,s6s0,s1,s2s4,s5,s6−s4s1s0,s1s2−s5,s6s4,s5s5s0,s1,s2−,P3=−s4,s5,s6s4,s5s2s0,s1,s2−s4,s5,s6s4,s5,s6s0,s1,s2s0,s1,s2−s4s4s0,s1,s2s1−.Step 2: deal with the assessment information based on the HFLTS using the 2-tuple linguistic representation model:(32)Δ−1p1=−0,150,15,6−4,50,1,2,320,1,2−5,64,5,64,5,60,1,2,3−,Δ−1p2=0,1,24,5,60,1,24,5,6410,125,64,550,1,2,Δ−1p3=−4,5,64,520,1,2−4,5,64,5,60,1,20,1,2−440,1,21−.Step 3: quantify the uncertainty in GDM information based on the rough set theory:(33)Δ−1p1=−0,2.38,0.5,3.174,4.83,4.6,5.25,4.83,60,1,0.5,1.52.83,5.5,3.63,6−4,4.67,4.4,5.330,2.75,0.67,3.14,1,4,1.4,4.51,20,1.14,0.5,1.6,1.14,2−4.75,5.5,5.29,64,4.67,4.4,5.33,4.67,61.75,5,2.83,5.33,3.29,60,1.25,0.6,1.67,1,2.33,1.25,3−,Δ−1p2=−0,2.38,0.5,3.17,0.8,4.254.6,5.250,1,0.5,1.5,1,21.75,5.2,2.83,5.5,3.63,6−4,4.670.67,3.140,1,0.5,1.51.14,2−4.75,5.5,5.29,64,4.67,4.4,5.332.83,5.330,1.25,0.6,1.67,1,2.33−,Δ−1p3=−1.33,5,1.86,5.5,2.38,64,4.83,4.6,5.251,20,3.63,0.5,4.14,1,4.67−4,4.67,4.4,5.33,4.67,61.83,5,2.29,5.5,2.75,60,1,0.5,1.5,1,20,1.14,0.5,1.6,1.14,2−4,5.29,4.75,5.5,5.29,64,4.670,3.29,0.5,3.83,1,4.40.6,1.67−.Step 4: obtain the assessment information with the same length using the OWA operator.Considering the fuzzy linguistic preference “for the most,” the final assessment information obtained corresponding to the three researchers is as follows:(34)Δ−1p1=−0.2,2.74.6,5.250.2,1.23.15,5.7−4.16,4.930.74,3.451,20.4,1.5−5,5.74.31,5.192.57,5.280.7,1.89−,Δ−1p2=−0.38,3.024.45,5.180.4,1.42.59,5.45−4,4.670.67,3.140.2,1.21.14,2−5,5.74.16,4.932.83,5.330.46,1.6−,Δ−1p3=−1.75,5.44.24,51,20.4,4.03−4.31,5.112.19,5.40.4,1.40.4,1.5−4.58,5.474,4.670.42,3.690.6,1.67−.Step 5: obtain the preference relation based on the OWA operator according to the approach proposed in Rodríguez et al. [4]:(35)PC=−0.41,3.084.4,5.140.38,1.382.03,5.08−4.13,4.870.81,3.480.38,1.380.44,1.53−4.89,5.644.13,4.882,4.850.57,1.66−.Step 6: compute the pessimistic and optimistic collective preference for each alternative. The linguistic interval for each alternative is shown in Table3.Table 3
Linguistic intervals for all alternatives in Example1.
JohnMikeDavidFrankp1Rp2Rp3Rp4R[(s2, −0.27), (s3, 0.2)][(s2, 0.32), (s4, −0.52)][(s2, −0.1), (s3, −0.15)][(s1, 0.23), (s4, −0.2)]Step 7: order the set of alternatives and select the best one as the solution to the GDM problem:x2 > x4 > x1 > x3.The best solution is similar with the work of Rodríguez et al. [4]. This example can illustrate the effectiveness of the proposed GDM approach for HFLTSs. However, the envelope-based method for computing with HFLTSs in Rodríguez et al. [4] cannot support experts to select nonadjacent linguistic terms. Next, we changed the preference provided by the three experts and compared the results obtained by the commonly used envelope-based method and the proposed GDM approach. We changed the preference referring to more than two linguistic terms to contain only two linguistic terms using the binary relation “or.” The revised preference provided by the first expert is as follows:(36)p1′=−at mostvlvhat mostvlat leastvh−betweenhandvhnormlnorl−greater thanhhorahoranorm−.The corresponding HFLTSs of the three experts are as follows:(37)p1′=−s0,s1s5s0,s1s5,s6−s4,s5s0,s3s2s0,s2−s5,s6s4,s6s4,s6s0,s3−,p2′=−s0,s2s4,s6s0,s2s4,s6−s4s1s0,s1s2−s5,s6s4,s5s5s0,s2−,p3′=−s4,s6s4,s5s2s0,s2−s4,s6s4,s6s0,s2s0,s2−s4s4s0,s2s1−.The envelope ofp1′, p2′, and p3′ does not change, and also, the ranking result of the envelope-based method does not change. However, the ranking result of the proposed GDM approach for HFLTSs changes as the change of preference given by the three experts. The linguistic interval for each alternative after preference change is shown in Table 4. The revised alternative order is x2 > x4 > x3 > x1. By contrast, we can see that the ranking position of x3 and the position of x1 have changed. The comparison illustrates that the proposed GDM approach for HFLTSs has higher sensitivity to the change of preference information and has much wider applicability under different linguistic decision-making environments.


## 5.2. Example 2

A real example of selecting logistics service suppliers is adopted in this section. Company W is a small and medium-sized electric product manufacturer. Its main products are refrigerators, freezers, and air conditioners. To focus on the core competition ability, reduce cost, and improve customer service, the company decides to adopt a logistics outsourcing strategy. After preliminary screening, five candidates (i.e., alternatives),x1, x2, x3, x4, and x5, remain for further evaluation. Seven evaluation criteria are considered: C = {c1 = quality assurance, c2 = operation efficiency, c3 = logistics technology level, c4 = logistics facility level, c5 = price, c6 = management ability, c7 = development potential level}. c5 is the “small-the-better” criterion, and the other criteria are in the “larger-the-better” category. The weight vector of the criteria set is W = (0.21, 0.19, 0.12, 0.14, 0.17, 0.09, 0.08). A committee composed of 3 experts E = {e1, e2, e3} evaluated the alternative service suppliers. The weights of the 3 experts are {we1 = 0.4, we2 = 0.3, we3 = 0.3}.Step 1: obtain the preferences provided by experts based on proposed context-free grammar, and transform the linguistic expressions into HFLTSs according to Definition4. The predefined linguistic terms set is S. The hesitant evaluation information given by three experts is shown in Tables 5–7. S = {neither (s0); very low (s1); low (s2); medium (s3); high (s4); very high (s5); absolute (s6)}.Table 4
Linguistic intervals for all alternatives after preference change in Example1.
JohnMikeDavidFrankp1Rp2Rp3Rp4R[(s2, −0.32), (s3, 0.21)][(s2, 0.45), (s4, 0.62)][(s2, 0.06), (s3, 0)][(s1, 0.34), (s4, 0.05)]Table 5
The HFLTS information given by experte1 in Example 2.
c1c2c3c4c5c6c7x1s2s0,s1,s2s3,s4s4,s5s3,s4s4s2,s3,s4x2s3,s4s2s2,s3,s4s0,s1,s2s4,s5s2,s3,s4s2,s3x3s4,s5s4,s5,s6s0,s1,s2,s3s5,s6s3s2,s3,s4s3,s4x4s2,s3,s4s5,s6s3,s4s0,s1,s2s3s4,s5s5x5s2,s3s0,s1,s2s2,s3,s4,s5s4,s5s2,s3s0,s1,s2s2,s3Table 6
The HFLTS information given by experte2 in Example 2.
c1c2c3c4c5c6c7x1s3,s4s2,s3,s4s2s0,s1,s2s2,s3s0,s1,s2s3,s4x2s5,s6s0,s1,s2s2,s3s2,s3s0,s1s3,s4s0,s1,s2x3s6s5,s6s4,s5,s6s2,s3s2s5,s6s2,s3,s4x4s0,s1,s2s2,s3s2,s3,s4,s5s5,s6s3,s4s2s2,s3,s4x5s2,s3,s4s3s2,s3s0,s1,s2s3,s4s0,s1,s2s5,s6Table 7
The HFLTS information given by experte3 in Example 2.
c1c2c3c4c5c6c7x1s0,s1,s2,s3s2,s3s2,s3,s4s3,s4s2,s3,s4s5,s6s3x2s3,s4s3s4,s5,s6s4,s5s0,s1,s2s3,s4s4,s5x3s4,s5,s6s3,s4s2,s3s2,s3,s4s1,s2s2,s3s0,s1,s2x4s2,s3,s4,s5s2,s3s2,s3,s4s3,s4s0,s1,s2s2,s3,s4s5x5s2,s3,s4s4s4,s5s2,s3,s4s3,s4s5,s6s0,s1,s2Step 2: deal with the evaluation information based on the HFLTS using the 2-tuple linguistic representation model. Taking Table4 as an example, the information of Δ−1HSj1xi corresponding to this table is shown in Table 8.Table 8
The information ofΔ−1HSj1xi corresponding to Table 4.
c1c2c3c4c5c6c7x1{2}{0, 1, 2}{3, 4}{4, 5}{3, 4}{4}{2, 3, 4}x2{3, 4}{2}{2, 3, 4}{0, 1, 2}{4, 5}{2, 3, 4}{2, 3}x3{4, 5}{4, 5, 6}{0, 1, 2, 3}{5, 6}{3}{2, 3, 4}{3, 4}x4{2, 3, 4}{5, 6}{3, 4}{0, 1, 2}{3}{4, 5}{5}x5{2, 3}{0, 1, 2}{2, 3, 4, 5}{4, 5}{2, 3}{0, 1, 2}{2, 3}Step 3: transform the information ofΔ−1HSjkxi into unified interval numbers for simplifying the group information aggregation.The information ofΔ−1HSjkxi is treated by using the rough set theory. The experts’ evaluation information represented as rough numbers is shown in Tables 9–11. The set of rough numbers can be integrated into a rough number based on the OWA operator. The possible numbers of elements in the rough number set in Tables 9–11 are 2, 3, and 4. Since the elements in the rough number set are arranged in the ascending order, the corresponding weights associated with elements for different fuzzy linguistic preferences are shown in Table 12.Table 9
Evaluation information of experte1 represented as rough numbers.
c1c2c3c4c5c6c7x1{[1.25, 2.8]}{[0, 2.13], [0.5, 2.43], [1.4, 2.67]}{[2.5, 3.5], [3, 4]}{[2.33, 4.33], [2.71, 5]}{[2.6, 3.4], [3, 4]}{[4, 5]}{[2, 3.17], [2.75, 3.4], [3.17, 4]}x2{[3, 4.17], [3.5, 4.75]}{[1.25, 2.33]}{[2, 3.63], [2.5, 4.17], [3, 4.75]}{[0, 2.43], [0.5, 2.83], [1.25, 3.2]}{[1.33, 4.5], [1.86, 5]}{[2, 3.29], [2.75, 3.5], [3.29, 4]}{[1.25, 3.2], [1.6, 4]}x3{[4, 5], [4.5, 5.5]}{[3.67, 5], [4.2, 5.5], [4.71, 6]}{[0, 2.89], [0.5, 3.25], [1.25, 3.57], [1.83, 4.2]}{[3.17, 5.5], [3.57, 6]}{[2, 3]}{[2, 3.57], [2.5, 4.2], [2.8, 5]}{[1.83, 3.5], [2.38, 4]}x4{[1.4, 3.13], [1.86, 3.8], [2.33, 4.33]}{[3, 5.5], [3.5, 6]}{[2.6, 3.71], [3.13, 4.25]}{[0, 3], [ 0.5, 3.5], [1, 4]}{[1.8, 3.33]}{[3, 4.33], [3.33, 5]}{[3.8, 5]}x5{[2, 2.88], [2.5, 3.4]}{[0, 2], [0.5, 2.5], [1, 3]}{[2, 3.5], [2.5, 4], [3, 4.5], [3.5, 5]}{[2.29, 4.33], [2.63, 5]}{[2, 3.17], [2.75, 3.4]}{[0, 2.13], [0.5, 2.83], [1, 3.75]}{[1.25, 3.6], [1.6, 4.67]}Table 10
Evaluation information of experte2 represented as rough numbers.
c1c2c3c4c5c6c7x1{[1.83, 3.33], [2.14, 4]}{[1.4, 2.67], [1.86, 3.33], [2.125, 4]}{[2, 3]}{[0, 2.71], [0.5, 3.17], [1, 3.6]}{[2, 3], [2.6, 3.4}{[0, 3], [0.5, 3.6], [1, 4.25]}{[2.75, 3.4], [3.17, 4]}x2{[3.8, 5.5], [4.17, 6]}{[0, 1.6], [0.5, 2], [1.25, 2.33]}{[2, 3.63], [2.5, 4.17]}{[1.25, 3.2], [1.6, 4]}{[0, 1.86], [0.5, 2.6]}{[2.75, 3.5], [3.29, 4]}{[0, 2.43], [0.5, 2.83], [1.25, 3.2]}x3{[5, 6]}{[4.2, 5.5], [4.71, 6]}{[2.14, 5], [2.5, 5.5], [2.89, 6]}{[2, 3.57], [2.5, 4.2]}{[1.67, 2.33]}{[3.17, 5.5], [3.57, 6]}{[1.25, 3], [1.83, 3.5], [2.38, 4]}x4{[0, 2.6], [0.5, 2.89], [1.4, 3.13]}{[2, 3.5], [2.5, 4.25]}{[2, 3.33], [2.6, 3.71], [3.13, 4.25], [3.33, 5]}{[2.5, 5.5], [3, 6]}{[1.8, 3.33], [2.17, 4]}{[2, 3.33]}{[2, 3.8], [2.5, 4.25], [3, 4.67]}x5{[2, 2.88], [2.5, 3.4], [2.88, 4]}{[1.5, 3.5]}{[2, 3.5], [2.5, 4]}{[0, 2.63], [0.5, 3], [1.25, 3.33]}{[2.75, 3.4], [3.17, 4]}{[0, 2.13], [0.5, 2.83], [1, 3.75]}{[2.17, 5.5], [2.71, 6]}Table 11
Evaluation information of experte3 represented as rough numbers.
c1c2c3c4c5c6c7x1{[0, 2.14], [0.5, 2.5], [1.25, 2.8], [1.83, 3.33]}{[1.4, 2.67], [1.86, 3.33]}{[2, 3], [2.5, 3.5], [3, 4]}{[1.5, 4], [2.33, 4.33]}{[2, 3], [2.6, 3.4], [3, 4]}{[2.4, 5.5], [3, 6]}{[2.75, 3.4]}x2{[3, 4.17], [3.5, 4.75]}{[1.6, 3]}{[3, 4.75], [3.29, 5.5], [3.63, 6]}{[2, 4.5], [2.43, 5]}{[0, 1.86], [0.5, 2.6], [0.8, 3.67]}{[2.75, 3.5], [3.29, 4]}{[2, 4.5], [2.43, 5]}x3{[4, 5], [4.5, 5.5], [5, 6]}{[3, 4.71], [3.67, 5]}{[1.25, 3.57], [1.83, 4.2]}{[2, 3.57], [2.5, 4.2], [2.8, 5]}{[1, 2], [1.67, 2.33]}{[2, 3.57], [2.5, 4.2]}{[0, 2.38], [0.5, 2.71], [1.25, 3]}x4{[1.4, 3.13], [1.86, 3.8], [2.33, 4.33], [2.6, 5]}{[2, 3.5], [2.5, 4.25]}{[2, 3.33, [2.6, 3.71], [3.13, 4.25]}{[1.5, 4.5], [2, 5]}{[0, 2.17], [0.5, 2.6], [1, 3]}{[2, 3.33], [2.33, 4], [3, 4.33]}{[3.8, 5]}x5{[2, 2.88], [2.5, 3.4], [2.88, 4]}{[2, 4]}{[3, 4.5], [3.5, 5]}{[1.25, 3.33], [1.6, 4], [2.29, 4.33]}{[2.75, 3.4], [3.17, 4]}{[1.57, 5.5], [2.13, 6]}{[0, 2.71], [ 0.5, 3.17], [1.25, 3.6]}Table 12
The successive element’s weight in one set for different fuzzy linguistic preferences.
Set with 2 elementsSet with 3 elementsSet with 4 elementsFor the most0.6, 0.40.27, 0.67, 0.060.1, 0.5, 0.4, 0At least half0, 10, 0.33, 0.670, 0, 0.5, 0.5As much as possible1, 00.67, 0.33, 00.5, 0.5, 0, 0Considering the fuzzy linguistic preference “for the most,” the evaluation information in the form of interval numbers is shown in Tables13–15.Table 13
The final evaluation information obtained frome1 for “for the most.”
c1c2c3c4c5c6c7x1[1.25, 2.8][0.42, 2.36][2.70, 3.70][2.48, 4.60][2.76, 3.64][4.00, 5.00][2.57, 3.37]x2[3.20, 4.40][1.25, 2.33][2.40, 4.06][0.41, 2.74][1.54, 4.70][2.58, 3.47][1.39, 3.52]x3[4.20, 5.20][4.09, 5.40][0.75, 3.34][3.33, 5.70][2.00, 3.00][2.38, 4.08][2.05, 3.70]x4[1.76, 3.65][3.20, 5.70][2.81, 3.93][0.40, 3.40][1.80, 3.33][3.13, 4.60][3.80, 5.00]x5[2.20, 3.09][0.40, 2.40][2.65, 4.15][2.43, 4.60][2.30, 3.26][0.40, 2.70][1.39, 4.03]Table 14
The final evaluation information obtained frome2 for “for the most.”
c1c2c3c4c5c6c7x1[1.95, 3.60][1.75, 3.19][2, 3][0.40, 3.07][2.24, 3.16][0.40, 3.48][2.92, 3.64]x2[3.95, 5.70][0.41, 1.91][2.20, 3.85][1.39, 3.52][0.20, 2.16][2.97, 3.70][0.41, 2.74]x3[5.00, 6.00][4.40, 5.70][2.43, 5.40][2.20, 3.82][1.67, 2.33][3.33, 5.70][1.71, 3.40]x4[0.42, 2.83][2.20, 3.80][2.75, 3.89][2.70, 5.70][1.95, 3.60][2.00, 3.33][2.40, 4.15]x5[2.39, 3.30][1.50, 3.50][2.20, 3.70][0.41, 2.92][2.92, 3.64][0.40, 2.70][2.39, 5.70]Table 15
The final evaluation information obtained frome3 for “for the most.”
c1c2c3c4c5c6c7x1[0.75, 2.58][1.58, 2.93][2.40, 3.40][1.83, 4.13][2.46, 3.33][2.64, 5.70][2.75, 3.40]x2[3.47, 4.00][1.60, 3.00][3.23, 5.33][2.17, 4.70][0.38, 2.46][2.97, 3.70][2.17, 4.70]x3[4.40, 5.40][3.68, 4.20][1.48, 3.82][2.38, 4.08][1.27, 2.13][2.20, 3.82][0.41, 2.64]x4[2.00, 3.95][2.20, 3.80][2.47, 3.64][1.70, 4.70][0.40, 2.51][2.28, 3.84][3.80, 5.00]x5[2.39, 3.30][2.00, 4.00][3.20, 4.70][1.55, 3.84][2.92, 3.64][1.79, 5.70][0.41, 3.07]Step 4: aggregate the information in Tables13–15 into the final decision matrix considering both the expert’s weights and the OWA weights associated with each a¯ijk,a¯ijk for 1≤i≤n and 1≤j≤m. The aggregated decision information for different fuzzy linguistic preferences is shown in Tables 16–18, respectively.


### 5.2.1. Sensitivity Analysis

Parameterv is the weight of the strategy of the “majority of attributes,” and it plays an important role in determining the set of compromise solutions. As seen in Table 19, Q¯i,Q¯i is determined by the value of v. When v is changed, DQ¯i1,Q¯i1,Q¯i2,Q¯i2 is also changed. Aiming at the aggregated decision information obtained by the fuzzy linguistic preference “for the most,” when 0≤v≤0.8789, DQ¯i1,Q¯i1,Q¯i2,Q¯i2≥0.25, and only x3 is the compromise solution; when 0.8789<v≤1, DQ¯i1,Q¯i1,Q¯i2,Q¯i2<0.25 and DQ¯i1,Q¯i1,Q¯i3,Q¯i3≥0.25, and both x3 and x4 are the compromise solutions. From the results in Table 19, we can see that the threshold values of v are different for different fuzzy linguistic preferences. The value of v can differentiate the possible solution sets {x3} and {x3, x4}.Table 16
The aggregated decision information for “for the most.”
c1c2c3c4c5c6c7x1[1.23, 2.88][1.22, 2.79][2.35, 3.35][1.57, 3.94][2.47, 3.35][2.31, 4.91][2.72, 3.43]x2[3.41, 4.52][1.07, 2.33][2.49, 4.23][1.20, 3.47][0.60, 2.89][2.84, 3.62][1.25, 3.51]x3[4.44, 5.44][4.03, 5.11][1.41, 3.94][2.55, 4.38][1.63, 2.43][2.50, 4.30][1.42, 3.25]x4[1.42, 3.47][2.43, 4.24][2.69, 3.83][1.44, 4.44][1.43, 3.14][2.40, 3.87][3.40, 4.76]x5[2.32, 3.23][1.22, 3.22][2.62, 4.12][1.43, 3.75][2.71, 3.51][0.65, 3.24][1.29, 4.06]Table 17
The aggregated decision information for “at least half.”
c1c2c3c4c5c6c7x1[1.77, 3.47][1.80, 3.40][2.80, 3.80][2.31, 4.56][2.90, 3.85][3.04, 5.34][3.06, 3.84]x2[4.04, 5.36][1.38, 2.64][3.12, 5.12][1.88, 4.30][1.29, 4.11][3.25, 3.97][1.91, 4.35]x3[4.85, 5.85][4.65, 5.79][2.22, 4.93][3.14, 5.33][1.85, 2.69][3.09, 5.27][2.12, 3.78]x4[2.15, 4.24][3.04, 5.19][3.15, 4.41][2.25, 5.25][1.83, 3.59][2.96, 4.50][3.66, 4.93]x5[2.70, 3.72][1.61, 3.61][3.26, 4.76][2.21, 4.49][3.09, 3.88][1.46, 4.68][2.05, 5.13]Table 18
The aggregated decision information for “as much as possible.”
c1c2c3c4c5c6c7x1[0.85, 2.65][0.76, 2.47][2.15, 3.15][1.02, 3.51][2.18, 3.12][1.64, 4.28][2.48, 3.32]x2[3.12, 4.04][0.78, 2.14][2.22, 3.90][0.78, 3.05][0.32, 2.46][2.48, 3.42][0.84, 3.09]x3[4.20, 5.20][3.49, 4.49][0.87, 3.54][2.29, 4.02][1.41, 2.30][2.24, 3.94][0.90, 2.90]x4[0.89, 3.05][2.20, 3.90][2.31, 3.53][0.94, 3.94][1.01, 2.84][2.23, 3.60][3.01, 4.49]x5[2.08, 2.96][0.86, 2.86][2.24, 3.74][0.97, 3.32][2.35, 3.28][0.38, 2.83][0.86, 3.53]Step 5: select the appropriate supplier using the proposed VIKOR method.Aiming at the aggregated decision information according to the “for the most,” the positive ideal solutionA∗ and the negative ideal solution A− are determined first, and then S¯i,S¯i, R¯i,R¯i, and Q¯i,Q¯i are computed according to equations (25)–(30):A∗ = {5.44, 5.11, 4.23, 4.44, 0.60, 4.91, 4.76}A− = {1.23, 1.07, 1.41, 1.20, 3.51, 0.65, 1.25}According to the verification rules of VIKOR, if the acceptable advantage and acceptable ability are satisfied, the best rank can be assigned as a compromise solution. Letv=0.5, which represents selecting the appropriate object in compromise. First, the two conditions of the first case are considered.Condition 1:DQ¯i1,Q¯i1,Q¯i2,Q¯i2=0.5×0.208−0.0002+0.862−0.5192=0.323.DQ = 1/(5 − 1) = 0.25.Condition 1 is satisfied.Condition 2:x1 is the best object ranked according to S¯i,S¯i and R¯i,R¯i, and condition 2 is satisfied.Therefore,x1, i.e., x3, is the compromise solution.Considering all the possible values ofv, the final results are shown in Table 19.Table 19
The results of the VIKOR method based on the proposed GDM approach.
(a) The valuesS, R, and Q and the preference ranking order in the case of “for the most”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.435, 0.859][0.128, 0.210][0.451−0.036v, 1−0.023v]x2[0.274, 0.763][0.131, 0.190][0.471−0.270v, 0.867−0.017v]x3[0.122, 0.536][0.060, 0.120][0, 0.399 + 0.150v]x4[0.227, 0.754][0.098, 0.201][0.254−0.116v, 0.937−0.099v]x5[0.408, 0.876][0.123, 0.183][0.421−0.042v, 0.819 + 0.181v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8789{x3, x4}, 0.8789<v≤1(b) The valuesS, R, and Q and the preference ranking order in the case of “at least half”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.440, 0.829][0.123, 0.210][0.504−0.045v, 1−0.049v]x2[0.254, 0.804][0.136, 0.190][0.578−0.354v, 0.887 + 0.032v]x3[0.077, 0.521][0.034, 0.120][0, 0.489 + 0.072v]x4[0.198, 0.746][0.083, 0.190][0.279−0.125v, 0.889−0.044v]x5[0.376, 0.868][0.110, 0.180][0.431−0.053v, 0.830 + 0.170v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8113{x3, x4}, 0.8113<v≤1(c) The valuesS, R, and Q and the preference ranking order in the case of “as much as possible”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.410, 0.865][0.123, 0.210][0.410−0.024v, 1−0.010v]x2[0.268, 0.740][0.120, 0.189][0.387−0.190v, 0.857−0.033v]x3[0.120, 0.533][0.063, 0.120][0, 0.389 + 0.161v]x4[0.207, 0.745][0.104, 0.208][0.279−0.163v, 0.987−0.156v]x5[0.399, 0.873][0.117, 0.185][0.366−0.005v, 0.830 + 0.170v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8614{x3, x4}, 0.8614<v≤1An obvious advantage of the GDM approach for HFLTSs proposed in this paper is that it can support experts to give arbitrarily term mix to increase the richness of linguistic expressions. In this section, we change some HFLTSs on the evaluation ofx4 given by experts e1 and e2. Suppose the two experts tend to use the binary relation “or” and give nonadjacent linguistic terms due to the uncertainty of the market. The original and changed information on the evaluation of x4 by e1 and e2 is shown in Table 20. Aiming at the changed evaluation information, the aggregation result obtained by the envelope-based approach would not change for that the envelopes of the changed HFLTSs are the same as those of the original HFLTSs. However, the aggregation result obtained by the novel group decision-making approach proposed in this paper is different from Tables 13–15 under conditions of three fuzzy linguistic preferences. Furthermore, the VIKOR results of the proposed method based on the changed information will also be different from Table 19, and the related result is shown in Table 21. We can see that the compromise solutions remain unchanged, but the threshold values of v vary for different fuzzy linguistic preferences.Table 20
The original and changed information on evaluation forx4 by e1 and e2.
c1c2c3c4c5c6c7The original informationx4(e1){s2, s3, s4}{s5, s6}{s3, s4}{s0, s1, s2}{s3}{s4, s5}{s5}x4(e2){s0, s1, s2}{s2, s3}{s2, s3, s4, s5}{s5, s6 }{s3, s4}{s2}{s2, s3, s4}The changed informationx4(e1){s2, s4}{s5, s6}{s3, s4}{s0, s2}{s3}{s4, s5}{s5}x4(e2){s0, s2}{s2, s3}{s2, s5}{s5, s6 }{s3, s4}{s2 }{s2, s4}Table 21
The results of the proposed VIKOR method based on the changed information.
(a) The valuesS, R, and Q and the preference ranking order in the case of “for the most”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.7905{x3, x4}, 0.7505<v≤1(b) The valuesS, R, and Q and the preference ranking order in the case of “at least half”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.4807{x3, x4}, 0.4807<v≤1(c) The valuesS, R, and Q and the preference ranking order in the case of “as much as possible”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8926{x3, x4}, 0.8926<v≤1


### 5.2.2. Comparisons and Discussion

To illustrate the effectiveness of the proposed group decision-making approach for HFLTSs and the related VIKOR, we use the above case study to analyze the envelope-based approach for HFLTSs. We take the envelope-based approach proposed by Rodríguez et al. [4] as a comparable approach. The HFLTS information given by three experts as shown in Tables 5–7 can be translated into interval information according to the envelope-based approach. The interval information is modeled by the 2-tuple linguistic representation model and then aggregated by the weighted averaging operator. The final decision information of Δ−1HSj1xi is shown in Table 22.Table 22
The aggregated decision information of the comparative method.
c1c2c3c4c5c6c7x1[1.7, 2.9][1.2, 2.9][2.4, 3.4][2.5, 3.8][2.4, 3.7][3.1, 4.0][2.6, 3.7]x2[3.6, 4.6][1.7, 2.3][2.6, 4.3][1.8, 3.2][1.6, 2.9][2.6, 4.0][2.0, 3.3]x3[4.6, 5.6][4.0, 5.4][1.8, 3.9][3.2, 4.5][2.1, 2.4][2.9, 4.3][1.8, 3.4]x4[1.4, 3.7][3.2, 4.2][2.4, 4.3][2.4, 3.8][2.1, 3.0][2.8, 3.8][4.1, 4.7]x5[2.0, 2.6][2.1, 2.9][2.6, 4.4][2.2, 3.8][2.6, 3.6][1.5, 3.2][2.3, 3.6]The decision result of Table20 using the proposed VIKOR approach is shown in Table 23. The ranking order of the alternatives is x3≻x4≻x2≻x5≻x1, which is the same as the ranking order in Table 20. However, the compromise solution set remains unchanged with the change of v from 0 to 1 in Table 23, which has a lower sensitivity.Table 23
The results of the VIKOR method based on the comparative GDM approach.
S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.433, 0.848][0.135,0.195][0.558–0.114v, 0.912–0.085v]x2[0.311, 0.725][0.140, 0.167][0.588–0.307v, 0.749–0.085v]x3[0.099, 0.491][0.040, 0.120][0, 0.469 + 0.052v]x4[0.247, 0.689][0.095, 0.210][0.322–0.126v, 1–0.214v]x5[0.446, 0.850][0.150, 0.180][0.646–0.184v, 0.823 + 0.177v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤1To reveal the features of the proposed VIKOR method, the rough TOPSIS method [53] is also applied in the case study. The rough TOPSIS method is used to deal with the aggregated decision information for different fuzzy linguistic preferences (Tables 16–18). Table 24 shows the final ranking result of the rough TOPSIS.Table 24
The evaluation result of the rough TOPSIS approach.
Closeness coefficient in the rough TOPSISIn the case of “for the most”In the case of “at least half”In the case of “as much as possible”x10.40540.40400.4073x20.48040.47130.4928x30.62900.64630.6242x40.49490.51470.4977x50.40430.41860.4114Ranking orderx3≻x4≻x2≻x1≻x5x3≻x4≻x2≻x5≻x1x3≻x4≻x2≻x5≻x1In the case of “for the most,” if the weight0≤v≤0.8789, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. If the weight 0.8789<v≤1, the compromise solution obtained by the rough TOPSIS is different from the compromise solution of the extended VIKOR. In the case of “at least half,” if the weight 0≤v≤0.8113, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. In the case of “as much as possible,” if the weight 0≤v≤0.8614, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. The rough TOPSIS can only obtain the distinct solution, and it cannot obtain the compromise solutions. The VIKOR method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria, and it determines a compromise solution that could be accepted by the decision makers. Thus, the obtained compromise solution is more robust and could be accepted by the decision makers.


## 5.2.1. Sensitivity Analysis

Parameterv is the weight of the strategy of the “majority of attributes,” and it plays an important role in determining the set of compromise solutions. As seen in Table 19, Q¯i,Q¯i is determined by the value of v. When v is changed, DQ¯i1,Q¯i1,Q¯i2,Q¯i2 is also changed. Aiming at the aggregated decision information obtained by the fuzzy linguistic preference “for the most,” when 0≤v≤0.8789, DQ¯i1,Q¯i1,Q¯i2,Q¯i2≥0.25, and only x3 is the compromise solution; when 0.8789<v≤1, DQ¯i1,Q¯i1,Q¯i2,Q¯i2<0.25 and DQ¯i1,Q¯i1,Q¯i3,Q¯i3≥0.25, and both x3 and x4 are the compromise solutions. From the results in Table 19, we can see that the threshold values of v are different for different fuzzy linguistic preferences. The value of v can differentiate the possible solution sets {x3} and {x3, x4}.Table 16
The aggregated decision information for “for the most.”
c1c2c3c4c5c6c7x1[1.23, 2.88][1.22, 2.79][2.35, 3.35][1.57, 3.94][2.47, 3.35][2.31, 4.91][2.72, 3.43]x2[3.41, 4.52][1.07, 2.33][2.49, 4.23][1.20, 3.47][0.60, 2.89][2.84, 3.62][1.25, 3.51]x3[4.44, 5.44][4.03, 5.11][1.41, 3.94][2.55, 4.38][1.63, 2.43][2.50, 4.30][1.42, 3.25]x4[1.42, 3.47][2.43, 4.24][2.69, 3.83][1.44, 4.44][1.43, 3.14][2.40, 3.87][3.40, 4.76]x5[2.32, 3.23][1.22, 3.22][2.62, 4.12][1.43, 3.75][2.71, 3.51][0.65, 3.24][1.29, 4.06]Table 17
The aggregated decision information for “at least half.”
c1c2c3c4c5c6c7x1[1.77, 3.47][1.80, 3.40][2.80, 3.80][2.31, 4.56][2.90, 3.85][3.04, 5.34][3.06, 3.84]x2[4.04, 5.36][1.38, 2.64][3.12, 5.12][1.88, 4.30][1.29, 4.11][3.25, 3.97][1.91, 4.35]x3[4.85, 5.85][4.65, 5.79][2.22, 4.93][3.14, 5.33][1.85, 2.69][3.09, 5.27][2.12, 3.78]x4[2.15, 4.24][3.04, 5.19][3.15, 4.41][2.25, 5.25][1.83, 3.59][2.96, 4.50][3.66, 4.93]x5[2.70, 3.72][1.61, 3.61][3.26, 4.76][2.21, 4.49][3.09, 3.88][1.46, 4.68][2.05, 5.13]Table 18
The aggregated decision information for “as much as possible.”
c1c2c3c4c5c6c7x1[0.85, 2.65][0.76, 2.47][2.15, 3.15][1.02, 3.51][2.18, 3.12][1.64, 4.28][2.48, 3.32]x2[3.12, 4.04][0.78, 2.14][2.22, 3.90][0.78, 3.05][0.32, 2.46][2.48, 3.42][0.84, 3.09]x3[4.20, 5.20][3.49, 4.49][0.87, 3.54][2.29, 4.02][1.41, 2.30][2.24, 3.94][0.90, 2.90]x4[0.89, 3.05][2.20, 3.90][2.31, 3.53][0.94, 3.94][1.01, 2.84][2.23, 3.60][3.01, 4.49]x5[2.08, 2.96][0.86, 2.86][2.24, 3.74][0.97, 3.32][2.35, 3.28][0.38, 2.83][0.86, 3.53]Step 5: select the appropriate supplier using the proposed VIKOR method.Aiming at the aggregated decision information according to the “for the most,” the positive ideal solutionA∗ and the negative ideal solution A− are determined first, and then S¯i,S¯i, R¯i,R¯i, and Q¯i,Q¯i are computed according to equations (25)–(30):A∗ = {5.44, 5.11, 4.23, 4.44, 0.60, 4.91, 4.76}A− = {1.23, 1.07, 1.41, 1.20, 3.51, 0.65, 1.25}According to the verification rules of VIKOR, if the acceptable advantage and acceptable ability are satisfied, the best rank can be assigned as a compromise solution. Letv=0.5, which represents selecting the appropriate object in compromise. First, the two conditions of the first case are considered.Condition 1:DQ¯i1,Q¯i1,Q¯i2,Q¯i2=0.5×0.208−0.0002+0.862−0.5192=0.323.DQ = 1/(5 − 1) = 0.25.Condition 1 is satisfied.Condition 2:x1 is the best object ranked according to S¯i,S¯i and R¯i,R¯i, and condition 2 is satisfied.Therefore,x1, i.e., x3, is the compromise solution.Considering all the possible values ofv, the final results are shown in Table 19.Table 19
The results of the VIKOR method based on the proposed GDM approach.
(a) The valuesS, R, and Q and the preference ranking order in the case of “for the most”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.435, 0.859][0.128, 0.210][0.451−0.036v, 1−0.023v]x2[0.274, 0.763][0.131, 0.190][0.471−0.270v, 0.867−0.017v]x3[0.122, 0.536][0.060, 0.120][0, 0.399 + 0.150v]x4[0.227, 0.754][0.098, 0.201][0.254−0.116v, 0.937−0.099v]x5[0.408, 0.876][0.123, 0.183][0.421−0.042v, 0.819 + 0.181v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8789{x3, x4}, 0.8789<v≤1(b) The valuesS, R, and Q and the preference ranking order in the case of “at least half”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.440, 0.829][0.123, 0.210][0.504−0.045v, 1−0.049v]x2[0.254, 0.804][0.136, 0.190][0.578−0.354v, 0.887 + 0.032v]x3[0.077, 0.521][0.034, 0.120][0, 0.489 + 0.072v]x4[0.198, 0.746][0.083, 0.190][0.279−0.125v, 0.889−0.044v]x5[0.376, 0.868][0.110, 0.180][0.431−0.053v, 0.830 + 0.170v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8113{x3, x4}, 0.8113<v≤1(c) The valuesS, R, and Q and the preference ranking order in the case of “as much as possible”S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.410, 0.865][0.123, 0.210][0.410−0.024v, 1−0.010v]x2[0.268, 0.740][0.120, 0.189][0.387−0.190v, 0.857−0.033v]x3[0.120, 0.533][0.063, 0.120][0, 0.389 + 0.161v]x4[0.207, 0.745][0.104, 0.208][0.279−0.163v, 0.987−0.156v]x5[0.399, 0.873][0.117, 0.185][0.366−0.005v, 0.830 + 0.170v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8614{x3, x4}, 0.8614<v≤1An obvious advantage of the GDM approach for HFLTSs proposed in this paper is that it can support experts to give arbitrarily term mix to increase the richness of linguistic expressions. In this section, we change some HFLTSs on the evaluation ofx4 given by experts e1 and e2. Suppose the two experts tend to use the binary relation “or” and give nonadjacent linguistic terms due to the uncertainty of the market. The original and changed information on the evaluation of x4 by e1 and e2 is shown in Table 20. Aiming at the changed evaluation information, the aggregation result obtained by the envelope-based approach would not change for that the envelopes of the changed HFLTSs are the same as those of the original HFLTSs. However, the aggregation result obtained by the novel group decision-making approach proposed in this paper is different from Tables 13–15 under conditions of three fuzzy linguistic preferences. Furthermore, the VIKOR results of the proposed method based on the changed information will also be different from Table 19, and the related result is shown in Table 21. We can see that the compromise solutions remain unchanged, but the threshold values of v vary for different fuzzy linguistic preferences.Table 20
The original and changed information on evaluation forx4 by e1 and e2.
c1c2c3c4c5c6c7The original informationx4(e1){s2, s3, s4}{s5, s6}{s3, s4}{s0, s1, s2}{s3}{s4, s5}{s5}x4(e2){s0, s1, s2}{s2, s3}{s2, s3, s4, s5}{s5, s6 }{s3, s4}{s2}{s2, s3, s4}The changed informationx4(e1){s2, s4}{s5, s6}{s3, s4}{s0, s2}{s3}{s4, s5}{s5}x4(e2){s0, s2}{s2, s3}{s2, s5}{s5, s6 }{s3, s4}{s2 }{s2, s4}Table 21
The results of the proposed VIKOR method based on the changed information.
(a) The valuesS, R, and Q and the preference ranking order in the case of “for the most”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.7905{x3, x4}, 0.7505<v≤1(b) The valuesS, R, and Q and the preference ranking order in the case of “at least half”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.4807{x3, x4}, 0.4807<v≤1(c) The valuesS, R, and Q and the preference ranking order in the case of “as much as possible”S¯i,S¯iR¯i,R¯iQ¯i,Q¯iRanking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤0.8926{x3, x4}, 0.8926<v≤1


## 5.2.2. Comparisons and Discussion

To illustrate the effectiveness of the proposed group decision-making approach for HFLTSs and the related VIKOR, we use the above case study to analyze the envelope-based approach for HFLTSs. We take the envelope-based approach proposed by Rodríguez et al. [4] as a comparable approach. The HFLTS information given by three experts as shown in Tables 5–7 can be translated into interval information according to the envelope-based approach. The interval information is modeled by the 2-tuple linguistic representation model and then aggregated by the weighted averaging operator. The final decision information of Δ−1HSj1xi is shown in Table 22.Table 22
The aggregated decision information of the comparative method.
c1c2c3c4c5c6c7x1[1.7, 2.9][1.2, 2.9][2.4, 3.4][2.5, 3.8][2.4, 3.7][3.1, 4.0][2.6, 3.7]x2[3.6, 4.6][1.7, 2.3][2.6, 4.3][1.8, 3.2][1.6, 2.9][2.6, 4.0][2.0, 3.3]x3[4.6, 5.6][4.0, 5.4][1.8, 3.9][3.2, 4.5][2.1, 2.4][2.9, 4.3][1.8, 3.4]x4[1.4, 3.7][3.2, 4.2][2.4, 4.3][2.4, 3.8][2.1, 3.0][2.8, 3.8][4.1, 4.7]x5[2.0, 2.6][2.1, 2.9][2.6, 4.4][2.2, 3.8][2.6, 3.6][1.5, 3.2][2.3, 3.6]The decision result of Table20 using the proposed VIKOR approach is shown in Table 23. The ranking order of the alternatives is x3≻x4≻x2≻x5≻x1, which is the same as the ranking order in Table 20. However, the compromise solution set remains unchanged with the change of v from 0 to 1 in Table 23, which has a lower sensitivity.Table 23
The results of the VIKOR method based on the comparative GDM approach.
S¯i,S¯iR¯i,R¯iQ¯i,Q¯ix1[0.433, 0.848][0.135,0.195][0.558–0.114v, 0.912–0.085v]x2[0.311, 0.725][0.140, 0.167][0.588–0.307v, 0.749–0.085v]x3[0.099, 0.491][0.040, 0.120][0, 0.469 + 0.052v]x4[0.247, 0.689][0.095, 0.210][0.322–0.126v, 1–0.214v]x5[0.446, 0.850][0.150, 0.180][0.646–0.184v, 0.823 + 0.177v]Ranking orderx3≻x4≻x2≻x5≻x1x3≻x4≻x5≻x2≻x1x3≻x4≻x2≻x5≻x1Compromise solution set{x3}, 0≤v≤1To reveal the features of the proposed VIKOR method, the rough TOPSIS method [53] is also applied in the case study. The rough TOPSIS method is used to deal with the aggregated decision information for different fuzzy linguistic preferences (Tables 16–18). Table 24 shows the final ranking result of the rough TOPSIS.Table 24
The evaluation result of the rough TOPSIS approach.
Closeness coefficient in the rough TOPSISIn the case of “for the most”In the case of “at least half”In the case of “as much as possible”x10.40540.40400.4073x20.48040.47130.4928x30.62900.64630.6242x40.49490.51470.4977x50.40430.41860.4114Ranking orderx3≻x4≻x2≻x1≻x5x3≻x4≻x2≻x5≻x1x3≻x4≻x2≻x5≻x1In the case of “for the most,” if the weight0≤v≤0.8789, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. If the weight 0.8789<v≤1, the compromise solution obtained by the rough TOPSIS is different from the compromise solution of the extended VIKOR. In the case of “at least half,” if the weight 0≤v≤0.8113, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. In the case of “as much as possible,” if the weight 0≤v≤0.8614, the compromise solution obtained by the rough TOPSIS is the same with the compromise solution of the extended VIKOR. The rough TOPSIS can only obtain the distinct solution, and it cannot obtain the compromise solutions. The VIKOR method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria, and it determines a compromise solution that could be accepted by the decision makers. Thus, the obtained compromise solution is more robust and could be accepted by the decision makers.


## 6. Conclusions

Many practical issues in various fields can be formulated into MCGDM problems, which refer to the rank given alternatives by a group of decision makers. The challenge of solving these problems is intensified when considering the uncertain information environment. Decision makers may hesitate among several linguistic terms when expressing their preferences. HFLTS is an efficient fuzzy model to express hesitant information, but the flexibility brings obstacle of aggregating HFLTSs with different lengths. Therefore, reliable GDM methods in the hesitant environment have to be studied and extended to solve the MCGDM problems.In the context of HFLTSs, this paper increases the richness of linguistic expressions by giving out improved context-free grammar and proposes a novel GDM approach based on the 2-tuple fuzzy linguistic representation model, the rough set theory, and the OWA operator. Finally, the GDM approach is extended, and a hesitant VIKOR approach is presented. The case study shows that (1) the novel GDM approach is much more flexible due to the fact that it allows decision makers to select nonadjacent linguistic terms and retains the vagueness in the decision-making information. (2) The novel GDM approach is sensitive to the variation of the information input. (3) Different fuzzy linguistic preferences for the OWA operator affect the associated weights and then contribute to different aggregation results. (4) The weight of strategyv in the hesitant VIKOR has an effect on the compromise solution, and its selection may influence the optimal solution or the number of compromise solutions. Further work may continue to apply the group decision-making approach for HFLTSs to other decision-making methods or solve specific problems.



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*Source: 7682983-2020-07-27.xml*