Approach to Multiple Attribute Group Decision Making Based on Hesitant Fuzzy Linguistic Aggregation Operators

Minghua Shi; Qingxian Xiao

doi:10.1515/jisys-2017-0192

Article Open Access

Approach to Multiple Attribute Group Decision Making Based on Hesitant Fuzzy Linguistic Aggregation Operators

Minghua Shi and Qingxian Xiao

Published/Copyright: March 2, 2018

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From the journal Journal of Intelligent Systems Volume 29 Issue 1

Abstract

Inspired by the nonlinear weighted average operator, this paper proposes several generalized power average operators to aggregate hesitant fuzzy linguistic decision information. It is worth noting that the new operators take both the location and date weight information and the relative closeness of the decision-making information into consideration, a characteristic that results in objectivity and fairness in a group decision making. Moreover, we demonstrate some useful properties of the operators and discuss their associations. A new approach based on the designed operators is then proposed for hesitant fuzzy linguistic multiple attribute group decision-making problems, in which the attribute weights are known or unknown. Finally, this paper demonstrates the efficiency and feasibility of the proposed method through a numerical example.

Keywords: Linguistic term; group decision making; operator; hesitant fuzzy linguistic term set

1 Introduction

Multiple attribute group decision making (MAGDM), is at the core of decision making science and has been widely applied in many areas, including economic benefit evaluation [40], information technology improvement project selection [39], supply chain strategy selection [3], and tailing impoundment site selection [5]. Owing to the nature of the objective things, such as assessing an enterprise’s innovation capability and evaluating an operational command system’s operating speed, information may often be expressed through qualitative methods [9, 10, 20, 21]. We are currently living in a complicated and volatile social and economic environment; thus, the decision problems that people encounter have become more complex. For this reason, people may be faced with a difficult choice among several linguistic terms or may have to use very complicated terms to express their viewpoint when evaluating decision-making information [22, 24, 33]. To model such situations, Rodríguez et al. recently presented the hesitant fuzzy linguistic term sets (HFLTSs) theory [28] based on the hesitant fuzzy set (HFS) theory [27]. MAGDM with hesitant fuzzy linguistic information has been a great concern for many scholars, and extensive progress has been made in terms of its theoretical analysis and practical applications.

Determining how to select the correct aggregation operator or aggregate different assessment results into a comprehensive value is an important and critical problem of MAGDM. A large number of fuzzy linguistic aggregation operators have been introduced during the past several decades, such as the linguistic max–min weighted averaging operator [44], the linguistic weighted conjunction operators [8], the linguistic weighted OWA operator [26], and the linguistic hybrid aggregation operator [36]. However, these operators consider only the individual criteria and fail to consider criteria in groups, thus, neglecting the influence of the relationships among the aggregated values. In view of this, Xu et al. [42] introduced the linguistic weighted power average (LWPA) operator based on the power average operator [45]. This operator constructs an index weight coefficient for each individual criterion, according to the relationship between this criterion and other criteria, and the aggregation of the results is, therefore, much more objective. In recent years, many researchers have focused on extending the LWPA operator to different fuzzy linguistic situations; examples include the linguistic interval two-tuple power ordered weighted average (LI2TPOWA) operator [25], the linguistic proportional two-tuple power weighted average (LP2TPWA) operator [11], the intuitionistic trapezoid fuzzy linguistic power weighted average (ITrFLPWA) operator [12], and the intuitionistic linguistic power weighted aggregation (ILPWA) operator [19].

Although research on the classic fuzzy linguistic aggregation operator and its applications has matured, that of hesitant fuzzy linguistics still remains in its initial stages. Zhang and Qi [49] proposed the hesitant fuzzy linguistic weighted averaging (HFLWA) operator and the hesitant fuzzy linguistic weighted geometric (HFLWGA) operator based on the transform function, which can transform HFLTSs into HFSs. Xu et al. [43] extended the ordered weighted distance (OWD) operator to HFLTSs. Wei et al. [31] and Lin et al. [17] respectively, proposed the hesitant fuzzy linguistic hybrid average (HFLHA) operator and the HFLWA operator. Based on the HFLHA and HFLWA operators, Lin et al. [16] further proposed some methods for enterprise information systems. Yu et al. [47] investigated the situations in which the input data are expressed in HFLTSs and introduced some quasi-harmonic averaging operators. All of these hesitant fuzzy linguistic operators consider the aggregated variables independently. However, owing to the complexity of the decision-making system, the aggregated variables always have a correlation. Yu et al. [46] put forward some hesitant fuzzy linguistic Heronian mean (HM) operators, which can consider the interrelationship of the aggregated arguments. Lin [15] investigated the geometric HM considering both the HM and the geometric mean under a hesitant fuzzy linguistic environment. Yu et al. [48] extended the Maclaurin symmetric mean (MSM) to a hesitant fuzzy linguistic environment, which can capture the correlation between any number of arguments.

By analyzing the existing hesitant fuzzy linguistic decision-making research, we found that the following problems exist: (1) Most existing ordered hesitant fuzzy linguistic operators are based on the partial orders of HFLTSs. This can lead to operators that are not increasingly monotonic. As indicated by Wang and Xu [29] if using a partial order in decision making, there may be ties between two different information. (2) Most existing hesitant fuzzy linguistic operators do not consider the influence of outliers. Owing to a lack of expertise or decision-making time, outliers occasionally occur in a group of evaluation values. To obtain reliable aggregation results, utilizing a suitable operator to reduce the influence of outliers in the decision-making process is an important step. (3) The existing generalized power average operators all use an explicit expression x^r to verify their properties, which does not have universality. With an explicit expression, it is difficult for the decision maker to deeply participate in the decision-making process.

Based on these reasons, in this study, we introduce some new generalized weighted power-averaging aggregation operators to deal with hesitant fuzzy linguistic information and propose a new approach to solve MAGDM problems. Our operators have the following attractive advantages: (1) When using the proposed operators to aggregate input arguments, HFLTS does not need to be converted into other fuzzy sets, reducing information loss, and, thereby, alleviating the influence of excessively large (or small) input arguments on the aggregation results. (2) The ordered hesitant fuzzy linguistic operators defined in this paper are based on the total orders of HFLTSs. Therefore, they satisfy the increasing monotonicity and have more practical use. (3) The proposed operators are using composite aggregation functions that can give the participants the right to the free choice of an appropriate function in the decision-making process.

The remainder of this paper is organized as follows. Section 2 reviews some concepts related to HFLTS, hesitant fuzzy linguistic elements (HFLEs), the comparison rule among HFLEs, and the LWPA operator. Section 3 develops some generalized power average operators and also discusses the relative characteristics of these operators. Section 4 develops an approach to handle hesitant fuzzy linguistic MAGDM problems. Section 5 demonstrates the efficiency and feasibility of the proposed group decision-making method. In Section 6, we present a study comparing the proposed operator to other common aggregation methods. Finally, some concluding remarks are provided in Section 7.

2 Preliminaries

2.1 Linguistic Approach

Let S={s_α|α=1, 2,…,t; t=2n−1, n∈N} be a linguistic term set, where s_α represents a possible linguistic term, satisfying the following requirements [33, 35]:

s_α<s_β if α<β;
Negation operator: N(s_i)=s_j, j=t+1−i.

For example, when evaluating water temperature, one set may be defined as follows.

S={s₁: very cool, s₂: cool, s₃: slightly cool, s₄: fair, s₅: slightly hot, s₆: hot, s₇: very hot}.

Unfortunately, because the linguistic term set S is discrete, it is inconvenient to calculate and analyze. In view of this, Xu extended S to a continuous set [34]:

S¯={sα|α∈[0, p]}, p(p>t).

An element of set S is known as an original linguistic term (often provided by the decision makers), and an element of set S¯\S is known as a virtual linguistic term (appearing only in the calculation process) [37].

Let sα, sβ∈S¯ and λ, λ₁, λ₂>0. Xu defined the following operation principles [41]:

(1) s_α⊕s_β=s_β⊕s_α=s_α+β ; (2) λs_α=s_λα; (3) (λ₁+λ₂)s_α=λ₁s_α⊕λ₂s_α; (4) s_α⊗s_β=s_β⊗s_α=s_αβ; (5) (sα)λ=sαλ.

Combining HFSs and the linguistic term sets, Rodríguez et al. [22] proposed the HFLTS. Liao et al. [14] further introduced the mathematical form of HFLTS, as follows.

Definition 1 ([47]): Let X be a universe of discourse and S={s_α|α=1, 2,…,t} be a linguistic term set. The hesitant fuzzy linguistic term set H_s on X is defined as

HS={<xi, hS(xi)>|xi∈X, i=1,…,n},

where h_s(x_i) is a set of some linguistic terms in S and is commonly referred to as the hesitant fuzzy linguistic element (HFLE).

In real applications, the numbers of elements in HFLEs are always different. To make the aggregation of information easier, we assume that all HFLEs have the same number of elements. If the numbers are unequal, each HFLE with fewer elements than the longest HFLE is extended by adding 0.5hS+(x)⊕0.5hS−(x) until it has the same length as the longest HFLE [32]. The maximum and minimum elements in h_s(x) are denoted by hS+(x) and hS−(x), respectively.

Example 1: Owing to the complexity of the operation of a control system, qualitative description is often used for evaluation. For example, a mechanical engineer may want to evaluate the operation of control systems x₁ and x₂. The linguistic evaluation scale can be defined as follow:

S={s₁: extremely complex, s₂: complex, s₃: slightly complex, s₄: fair, s₅: slightly easy, s₆: easy, s₇: extremely easy}. In this case, the mechanical engineer provides the following evaluation:

HS={<x1, hS(x1)={s2, s4, s5}>, <x2, hS(x2)={s1, s2}>},

where H_s is an HFLTS, and h_s(x₁) and h_s(x₂) are two HFLEs. To equalize the lengths of h_s(x₁) and h_s(x₂), h_s(x₂) is extended to h¯S(x2)={s1, s1.5, s2}.

To make HFLTSs more convenient to use in MAGDM, Wang and Xu proposed the following total order of HFLTSs.

Definition 2 ([29]): Suppose that hS={sηl|sηl∈S, l=1, 2,…,L} is an HFLE on S={s_α|α=1, 2,…,t}. Let Dk(hS)=sDk, where Dk=1L∑i=1L(I(sηi))k, k=1, 2,…,L, and function I(·) is defined to obtain the subscript of sηi such that I(sηi)=ηi. Then, the relation between the two HFLEs (hS1={sηl1|sηl1∈S, l=1, 2,…,L} and hS2={sηl2|sηl2∈S, l=1, 2,…,L}) defined by hS1≺⌢DnhS2⇔(hS1=hS2)∨(hS1≺DnhS2) is a total order, where hS1≺DnhS2⇔(D1(hS1)<D1(hS2))∨(((D1(hS1)=D1(hS2))∧ (there exists m>2) ((for all i<m) (Dk(hS1)=Dk(hS2)))∧(Dm(hS1)>Dm(hS2))).

Example 2: Let hS1={s2, s3, s4}, hS2={s1, s3}, hS3={s1, s2, s4} be three HFLEs; then, D1(hS1)=s2+3+43=s3, D1(hS2)=s1+32=s2, D1(hS3)=s1+2+43=s73

hence, hS2≺⌢DnhS3≺⌢DnhS1.

2.2 LWPA operator

To enable the aggregated values to support and reinforce one another, Yager developed the power average (PA) operator, which is shown in Definition 3.

Definition 3 ([45]): Let PA be Rⁿ→R, if

PA (a1, a2,…,an)=∑i=1n(1+T(ai))ai∑i=1n(1+T(ai))

where T(ai)=∑j=1,j≠inSup(ai, aj), then, PA is called a power average operator.

Motivated by the PA operator, Xu et al. defined the LWPA operator.

Definition 4 ([38, 42]): A linguistic weighted power average (LWPA) operator is a mapping LWPA:

S¯n→S¯. Such that

LWPA (sα1, sα2,…,sαn)=∑i=1nwi(1+T(sαi))sαi∑i=1nwi(1+T(sαi)),

where T(sαi)=∑j=1,j≠inSup(sαi,sαj), w=(w₁, w₂,…,w_n)^T is the weight vector of (sα1, sα2,…,sαn) with w_i∈[0, 1] and ∑i=1nwi=1. In addition, Sup(sαi, sαj) is the support for sαi and sαj under the following conditions:

(1) Sup(sαi, sαj)∈[0, 1]; (2) Sup(sαi, sαj)=Sup(sαj, sαi);

(3) If d(sαi, sαj)≤d(sαl, sαk), then Sup(sαi, sαj)≥Sup(sαl, sαk).

3 Generalized Linguistic Power Aggregation Operators

To aggregate hesitant fuzzy linguistic decision information, based on the LWPA operator, we develop the following some hesitant fuzzy linguistic weighted PA operators.

Definition 5: Let h_i (i=1, 2,…,n) be a collection of HFLEs with the weight vector (w₁, w₂,…,w_n)^T, w_i∈[0, 1], and ∑i=1nwi=1. Then

HFLGWPA (h1, h2,…,hn)=∪sα1∈h1,sα2∈h2,…,sαn∈hn{f(⊕j=1nwj(1+T(sαj))f(sαj)∑j=1nwj(1+T(sαj)))−}

is called the hesitant fuzzy linguistic generalized weighted power average (HFLGWPA) operator, where T(sαi)=∑j=1,j≠inSup(sαi, sαj), f(·)⁻ is the inverse function of f(·), and f(sαj)=sf(αj), which has the property (1) ∀x, y∈[0, p], f(x)≥f(y), if x≥y; (2) ∀x∈[0, p], f(x)≥0.

Remark 1: If f(x)=x, the HFLGWPA operator reduces to the hesitant fuzzy linguistic weighted power average (HFLWPA) operator.

HFLGWPA (h1, h2,…,hn)=∪sα1∈h1,sα2∈h2,…,sαn∈hn{⊕j=1nwj(1+T(sαj))sαj∑j=1nwj(1+T(sαj))}=HFLWPA (h1, h2,…,hn)

Remark 2: If f(x)=x and Sup(sαi, sαj)=k, for all i≠j, the HFLGWPA operator reduces to the hesitant fuzzy linguistic weighted average (HFLWA) operator.

HFLGWPA (h1, h2,…,hn)=∪sα1∈h1,sα2∈h2,…,sαn∈hn{⊕j=1nwj(1+T(sαj))sαj∑j=1nwj(1+T(sαj))}=∪sα1∈h1,sα2∈h2,…,sαn∈hn{⊕j=1nwj(1+(n−1)k)sαj∑j=1nwj(1+(n−1)k)}=∪sα1∈h1,sα2∈h2,…,sαn∈hn{⊕j=1nwjsαj}=HFLWA (h1, h2,…,hn).

Remark 3: It is that x^λ is a special form of f(x); therefore, the HFLGWPA operator, defined herein, is more general than the weighted power averaging operator proposed in Refs. [18] and [51].

Definition 6: Let h_i (i=1, 2,…,n) be a collection of HFLEs with weight vector w=(w₁, w₂,…,w_n)^T, w_i∈[0, 1] and ∑i=1nwi=1, and (h_σ₍₁₎, h_σ₍₂₎,…,h_σ_{(_n)}) is a permutation of (nw₁h₁, nw₂h₂,…,nw_nh_n) such that hσ(n)≺⌢Dnhσ(n−1)≺⌢Dn⋯≺⌢Dnhσ(2)≺⌢Dnhσ(1). Then, a hesitant fuzzy linguistic generalized ordered weighted power average (HFLGOWPA) operator is defined as

HFLGOWPA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1nωj(1+T(sασ(j)))f(sασ(j))∑j=1nωj(1+T(sασ(j))))−}

where T(sασ(j))=∑i=1,i≠jnSup(sασ(i), sασ(j)), ω=(ω₁, ω₂,…,ω_n)^T is the weight vector of h_σ_{(_i)}(i=1, 2,…,n), and f(·) is as shown in Definition 5. If w=(1/n, 1/n,…,1/n)^T, the HFLGOWPA operator reduces to the HFLGWPA operator. In addition, if ω=(1/n, 1/n,…,1/n)^T, the HFLGOWPA operator reduces to the hesitant fuzzy linguistic generalized ordered power average (HFLGOPA) operator.

Theorem 1: (1) (Commutativity) If (h′1, h′2,…,h′n) is any permutation of (h₁, h₂,…,h_n), then, HFLGOWPA (h1, h2,…,hn)=HFLGOWPA (h′1, h′2,…,h′n).

(2) (Boundedness) Let h_max={s_t, s_t,…,s_t} and h_min={s_{−_t} , s_{−_t} ,…,s_{−_t} }, then, hmin≺⌢DnHFLGOWPA (h1, h2,…,hn)≺⌢Dnhmax.

(3) (Idempotency) If h1=h2=⋯=hn={sα1, sα2,…,sαk}, then, HFLGOWPA (h₁, h₂,…,h_n)=h₁.

Proof: (1) Let HFLGOWPA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1nωj(1+T(sασ(j)))f(sασ(j))∑j=1nωj(1+T(sασ(j))))−}

HFLGOWPA (h′1, h′2,…,h′n)=∪sα′σ(1)∈h′σ(1),sα′σ(2)∈h′σ(2),…,sα′σ(n)∈h′σ(n){f(⊕j=1nωj(1+T(sα′σ(j)))f(sα′σ(j))∑j=1nωj(1+T(sα′σ(j))))−}

Because (h′1, h′2,…,h′n) is any permutation of (h₁, h₂,…,h_n), we then have hσ(j)=h′σ(j) and T(saσ(j))=T(sα′σ(j)). Thus, HFLGOWPA (h1, h2,…,hn)=HFLGOWPA (h′1, h′2,…,h′n).

(2) s−t=f(f(s−t))−=f(⊕j=1nωj(1+T(sασ(j)))f(s−t)∑j=1nωj(1+T(sασ(j))))−≺⌢Dnf(⊕j=1nωj(1+T(sασ(j)))f(sασ(j))∑j=1nωj(1+T(sασ(j))))−

≺⌢Dnf(⊕j=1nωj(1+T(sασ(j)))f(st)∑j=1nωj(1+T(sασ(j))))−=f(f(st))−=st, that is,

hmin≺⌢DnHFLGOWPA (h1, h2,…,hn)≺⌢Dnhmax.

(3) HFLGOWPA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1nωj(1+T(sασ(j)))f(sασ(j))∑j=1nωj(1+T(sασ(j))))−}

=∪{f(⊕j=1nωj(1+T(sασ(j)))f(sαi)∑j=1nωj(1+T(sασ(j))))−}=∪{sαi}=h1.

Remark: Similar to the proof of Theorem 1, we can prove that the HFLGWPA operator has the three properties mentioned above.

Unfortunately, decision makers always encounter MAGDM problems with completely unknown weights. Based on the HFLGWPA and HFLGOWPA operators, we propose the following new operator to solve this problem.

Definition 7: Let h_i (i=1, 2,…,n) be a collection of HFLEs; then, a hesitant fuzzy linguistic generalized power ordered weighted average (HFLGPOWA) operator is defined as

HFLGPOWA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sασ(j)))−}

where (h_σ₍₁₎, h_σ₍₂₎,…,h_σ_{(_n)}) is any permutation of (h₁, h₂,…,h_n) such that hσ(n)≺⌢Dnhσ(n−1)≺⌢Dn⋯≺⌢Dn

hσ(2)≺⌢Dnhσ(1), T(saσ(j))=∑i=1,i≠jnSup(saσ(j), saσ(i)), Vσ(j)=1+T(saσ(j)), TV=∑j=1nVσ(j), Bj=∑i=1jVσ(i), f(·) is as shown in Definition 5, and g(x) has the following properties.

1) g(0)=0, g(1)=1;

2) ∀x, y∈[0, 1], g(x)≥g(y), if x>y.

Theorem 2: (1) (Commutativity) If (h′1, h′2,…,h′n) is any permutation of (h₁, h₂,…,h_n), then,

HFLGPOWA (h1, h2,…,hn)=HFLGPOWA (h′1, h′2,…,h′n).

(2) (Boundedness) Let h_max={s_t, s_t,…,s_t} and h_min={s_{−_t} , s_{−_t} ,…,s_{−_t} }, then,

hmin≺⌢DnHFLGPOWA (h1, h2,…,hn)≺⌢Dnhmax.

(3) (Idempotency) If h1=h2=⋯=hn={sα1, sα2,…,sαk}, then,

HFLGPOWA (h1, h2,…,hn)=h1.

Proof: (1) Let

HFLGPOWA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sασ(j)))−}

HFLGPOWA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1n{g[B′j(TV)′]−g[B′j−1(TV)′]}f(sα′σ(j)))−}

Because (h′1, h′2,…,h′n) is any permutation of (h₁, h₂,…,h_n), we have TV=(TV)′ and Bj=B′j. Thus,

HFLPOWA (h1, h2,…,hn)=HFLPOWA (h′1, h′2,…,h′n).

(2) Because g(x) increases within the interval [0, 1], we have g(BjTV)−g(Bj−1TV)>0. Then,

sα−t=f(f(sα−t))−=f(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sα−t))−≺⌢Dnf(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sασ(j)))− ≺⌢Dnf(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sαt))−=f(f(sαt))−=sαt,

that is,

hmin≺⌢DnHFLPOWA (h1, h2,…,hn)≺⌢Dnhmax

(3) HFLPOWA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sασ(j)))−}

=∪{f(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sαi))−}=∪{f([g(BnTV)−g(Bn−1TV)+g(Bn−2TV)−⋯−g(B0TV)]f(sαi))−}=∪{f(f(sαi))−}=∪{sαi}

That is, HFLPOWA (h₁, h₂,…,h_n)=h₁.

Theorem 3: If g(x)=x, then, HFLGPOWA (h₁, h₂,…,h_n)=HFLGPA (h₁, h₂,…,h_n)

Proof:

HFLGPOWA (h1, h2,…,hn)=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1n[g(BjTV)−g(Bj−1TV)]f(sασ(j)))−} =∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1n(BjTV−Bj−1TV)f(sασ(j)))−=∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1nVσ(j)TVf(sασ(j)))− =∪sασ(1)∈hσ(1),sασ(2)∈hσ(2),…,sασ(n)∈hσ(n){f(⊕j=1n(1+T(sασ(j)))f(sασ(j))∑j=1n(1+T(sασ(j))))−=∪sα1∈h1,sα2∈h2,…,sαn∈hn{f(⊕j=1n(1+T(sαj))f(sαj)∑j=1n(1+T(sαj)))− =HFLGPA (h1, h2,…,hn).

4 Group Decision Making with Hesitant Fuzzy Linguistic Information

Let D={d₁, d₂,…,d_m} be a set of decision makers, and ω=(ω₁, ω₂,…,ω_m)^T be their weight vector. In addition, let X={x₁, x₂,…,x_n} be a set of n alternatives, and G={g₁, g₂,…,g_l} be a set of attributes with the weight vector w=(w₁, w₂,…,w_l)^T.

Suppose that the decision makers d_k∈D provide hesitant fuzzy linguistic evaluated values under the attribute g_j∈G for the alternative x_i∈X, denoted as HFLE hij(k), and construct the hesitant fuzzy linguistic decision matrix H(k)=(hij(k))n×l. Based on the hesitant fuzzy linguistic PA operators provided, we develop the following three-step method for MAGDM.

Step 1: Aggregate the kth line of H(k)=(hij(k))n×l using the HFLGWPA operator and obtain the comprehensive evaluation value Ai(k) of x_i given byd_k:

Ai(k)=HFLGWPA (h1i(k), h2i(k),…,hli(k)), i=1, 2,…,n; k=1, 2,…,m.

Step 2: Calculate the overall comprehensive evaluation value A_i of x_i using the HFLGOWPA operator:

Ai=HFLGOWPA (Ai(1), Ai(2),…,Ai(m)), i=1, 2,…,n.

Step 3: Rank the alternatives x_i (i=1, 2,…,n) according to A_i (i=1, 2,…,n) using the method described in Definition 2.

Remark: (1) Here, without loss of generality, we let Sup(sαi,sαj)=1−|αi−αj|2t. (2) If the weight vector is not present, we select the HFLGPOWA operator for the calculation.

5 Illustrative Example

Suppose a company wants to buy a smartphone company, and there are five possible types of smartphone companies, x_i (i=1,…,5), available in the market. The company decision makers, d_i (i=1, 2, 3), should adhere to the following principles [the weight vector is w=(0.3, 0.4, 0.1, 0.2)^T]:

·g₁, smartphone price; g₂, smartphone portability;

·g₃, battery life of smartphone; g₄, stability of smartphone system.

Using the linguistic label term set S={s_α|α=1, 2,…,9}, the hesitant fuzzy linguistic decision matrices H(k)=(hij(k))5×4k=1, 2, 3 are constructed as in Tables 1–3.

Table 1:

Decision Matrix H⁽¹⁾.

	g₁	g₂	g₃	g₄
x₁	{s₇, s₈}	{s₈, s₉}	{s₆}	{s₄, s₅}
x₂	{s₇, s₈}	{s₇, s₈, s₉}	{s₄, s₅, s₆}	{s₄, s₇}
x₃	{s₃, s₄}	{s₇}	{s₃, s₄, s₆}	{s₅}
x₄	{s₆, s₈}	{s₄, s₆}	{s₆, s₇, s₈}	{s₂, s₃, s₇}
x₅	{s₇, s₈, s₉}	{s₅, s₆}	{s₄, s₆, s₉}	{s₂, s₃}

Table 2:

Decision Matrix H⁽²⁾.

	g₁	g₂	g₃	g₄
x₁	{s₅, s₆}	{s₈}	{s₃, s₅}	{s₄, s₅, s₆}
x₂	{s₂, s₃, s₄}	{s₆, s₈, s₉}	{s₆, s₇}	{s₈}
x₃	{s₂, s₄, s₆}	{s₆}	{s₅, s₆, s₇}	{s₂, s₃}
x₄	{s₆}	{s₄, s₅}	{s₆, s₇}	{s₂, s₄, s₇}
x₅	{s₈, s₉}	{s₅, s₆, s₇}	{s₇, s₈, s₉}	{s₄}

Table 3:

Decision Matrix H⁽³⁾.

	g₁	g₂	g₃	g₄
x₁	{s₇, s₈, s₉}	{s₆, s₈}	{s₇, s₈}	{s₉}
x₂	{s₆, s₇, s₈}	{s₇, s₈, s₉}	{s₄, s₅}	{s₄, s₅, s₆}
x₃	{s₃, s₄, s₆}	{s₉}	{s₄, s₅}	{s₅, s₆, s₇}
x₄	{s₇, s₈}	{s₃, s₄}	{s₂, s₃}	{s₄, s₅}
x₅	{s₇, s₈, s₉}	{s₆, s₇, s₈}	{s₅, s₆, s₇}	{s₃}

Weighting vector of decision makers is known

Suppose that the weighting vector of the decision makers is ω=(0.3, 0.1, 0.6)^T. Then, the best alternative is obtained according to the following steps (In this case, we let f(x)=x².).

Step 1: Calculate the comprehensive evaluation value (as shown in Table 4).

Table 4:

Comprehensive Evaluations of the Alternatives.

	d₁	d₂	d₃
x₁	A1(1)={s6.895451, s7.322433, s7.752211}	A1(2)={s5.890018, s6.299076, s6.723259}	A1(3)={s7.05091, s7.778371, s8.514693}
x₂	A2(1)={s6.252999, s7.136341, s8.046163}	A2(2)={s5.815889, s6.97029, s7.549834}	A2(3)={s5.897341, s6.849063, s7.807689}
x₃	A3(1)={s5.180354, s5.374591, s5.736049}	A3(2)={s4.281905, s4.945943, s5.727522}	A3(3)={s6.057507, s6.571595, s7.356835}
x₄	A4(1)={s4.707306, s5.666667, s7.084595}	A4(2)={s4.707306, s5.132451, s5.983132}	A4(3)={s4.473999, s4.937948, s5.408327}
x₅	A5(1)={s5.174456, s6.021149, s7.125996}	A5(2)={s6.082763, s6.805969, s7.561512}	A5(3)={s5.813508, s6.741999, s7.700535}

Step 2: Calculate the overall comprehensive evaluation value.

A1=HFLGOWPA (A1(1), A1(2), A1(3))={s7.802708, s8.436883, s9.048668},

A2=HFLGOWPA (A2(1), A2(2), A2(3))={s6.773582, s7.786193, s8.786998},

A3=HFLGOWPA (A3(1), A3(2), A3(3))={s6.454025, s6.878367, s7.52016},

A4=HFLGOWPA (A4(1), A4(2), A4(3))={s5.156195, s5.859866, s6.77772},

A5=HFLGOWPA (A5(1), A5(2), A5(3))={s6.310754, s7.264403, s8.321895}.

Step 3: Rank the alternatives x_i (i=1, 2,…,5).

x4≺⌢Dnx3≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1.

Therefore, the most desirable alternative is x₁.

If we use a different f(x), the degree of satisfaction may differ, as shown in Table 5.

Table 5:

Results of Ranking.

Function	Ordering
f(x)=x³+x²	x4≺⌢Dnx3≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1
f(x)=x²+x	x4≺⌢Dnx3≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1
f(x)=x^1/6+x^1/8	x4≺⌢Dnx3≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1
f(x)=x^1/10+x^1/12	x3≺⌢Dnx4≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1
f(x)=x^1/18+x^1/20	x3≺⌢Dnx4≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1

From Table 5, we can see that when the function f(x) changes, the rankings change slightly: when f(x)=x³+x², the ranking is x4≺⌢Dnx3≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1; however, when f(x)=x²+x; x^1/6+x^1/8; x^1/10+x^1/12; x^1/18+x^1/20, a slight change takes place between x₃ and x₄. All the results indicate that x₁ is the best alternative, which means that the first smartphone company is the best choice.

For convenience, we recommend using simple functions in the proposed operators, such as f(x)=xⁿ, f(x)=n^x. In Figure 1, we can see that using the operators with f(x)=xⁿ can reflect the risk preference of the decision maker. A decision maker with an optimistic decision making outlook can use f(x)=xⁿ with a larger parameter n, whereas a pessimistic decision maker can use f(x)=xⁿ with a smaller parameter n. From Figure 2, we can see that using the operators with f(x)=n^x can reflect the marginal return of risk. If the decision making occurs during an early period of the investment, production, and operating activities, the decision maker can use f(x)=xⁿ with a smaller parameter n, but a larger parameter n in a later period.

Figure 1:

Scores with f(x)=xⁿ.

Figure 2:

Scores with f(x)=n^x.

Weighting vector of decision makers is unknown

However, under some real situations, the weight of the decision makers maybe unknown, in which case the best alternative should be selected according to the following steps. (In Step 2, the HFLGOWPA operator is replaced with the HFLGPOWA operator.)

Step 1⁰: Perform the same steps as in Step 1.

Step 2⁰: Calculate the overall comprehensive evaluation value (as shown in Table 6).

Table 6:

Synthetic Evaluation Value of the Group.

	f(x)=x
	g(x)=x²	g(x)=x
x₁	A₁={s_6.380004, s_6.827676, s_7.28195}	A₁={s_6.812937, s_7.408599, s_8.008821}
x₂	A₂={s_5.890058, s_6.948386, s_7.691494}	A₂={s_6.085858, s_7.036887, s_7.89804}
x₃	A₃={s_4.785307, s_5.251378, s_5.876221}	A₃={s_5.5202911, s_5.96424, s_6.62096}
x₄	A₄={s_4.578997, s_5.080483, s_5.774206}	A₄={s_4.66509, s_5.40117, s_6.497052}
x₅	A₅={s_5.496947, s_6.361629, s_7.370648}	A₅={s_5.855725, s_6.652736, s_7.516591}

Step 3⁰: Rank the alternatives x_i (i=1, 2,…,5).

If f(x)=x and g(x)=x², then

x4≺⌢Dnx3≺⌢Dnx5≺⌢Dnx1≺⌢Dnx2.

Thus, the best choice is x₂.

If f(x)=x and g(x)=x, then

x4≺⌢Dnx3≺⌢Dnx5≺⌢Dnx2≺⌢Dnx1.

Therefore, the best choice is x₁.

Decision makers can, thus, choose the appropriate function according to their preference.

From Figures 3 and 4, we can see that the changing trend in the aggregation values vary with the different combinations of functions. This means that, using the HFLGPOWA operator, a decision maker can deeply participate in the decision-making process. A decision maker with a pessimistic decision-making outlook can use f(x)=x, g(x)=xⁿ with a larger n, whereas an optimistic decision maker can use g(x)=x, f(x)=xⁿ with a larger parameter n.

Figure 3:

Scores with f(x)=x, g(x)=xⁿ.

Figure 4:

Scores with g(x)=x, f(x)=xⁿ.

6 Comparative Analysis

Since the proposal of the HFLTS, various information aggregation methods have been introduced. Rodríguez et al. [23] introduced the likelihood value of HFLTS and applied it to hesitant fuzzy linguistic multiple criteria group decision making. Chen et al. [2] performed an α-cut operation on HFLTS, considering the risk attitude of the decision maker and proposed a method for hesitant fuzzy linguistic multiple attribute decision making. Lee and Chen [13] introduced a hesitant fuzzy linguistic decision-making method based on the likelihood-based comparison relations of HFLTSs and certain aggregation operators. To obtain the likelihood value, these methods convert HFLTS into an easily calculated form, which may lead to a loss of decision-making information. Considerable research has been devoted to changing this status in recent years. Based on unbalanced linguistic term sets, Dong et al. [4] used the two-tuple ordered weighted averaging (TOWA) operator to handle hesitant fuzzy linguistic information. Wei et al. [30] introduced some aggregation operators to solve hesitant fuzzy linguistic decision-making problems based on a convex combination of HFLTSs. Zhang and Wu [50] introduced classical aggregation operators [weighted geometric (WG) operator, weighted averaging (WA) operator, and so on] to the hesitant fuzzy linguistic environment. Based on the equivalent transformation function, Gou et al. defined new operational laws for HFLTSs and utilized the WA [6] and weighted Bonferroni mean (BM) operators [7] to aggregate hesitant fuzzy linguistic information. Wang [28] introduced the concept of extended HFLTSs and proposed some extended hesitant fuzzy linguistic aggregation operators, such as the extended hesitant fuzzy linguistic weighted averaging (EHFLWA) and extended hesitant fuzzy linguistic OWA (EHFLOWA) operators. In general, these methods can be divided into three types: (1) those based on the likelihood-based comparison relation, such as the methods of Rodríguez et al. [22], Wei et al. [30], and Chen and Hong [2]; (2) those based on the transformation function, such as the methods of Gou et al. [6, 7], Dong et al. [4], and Chen and Hong [1, 2]; (3) and those based on all elements of HFLE, such as Zhang and Wu’s method [50] and the method proposed in this paper.

To compare these methods, let us consider the following decision matrix H as shown in Table 7 (the linguistic label term set S={s_α|α=1, 2,…,7}).

Table 7:

Decision Matrix H.

	g₁	g₂	g₃	g₄
x₁	{s₄}	{s₃, s₄, s₅}	{s₁, s₂, s₃}	{s₁, s₃}
x₂	{s₃}	{s₂, s₃, s₄}	{s₃, s₄}	{s₅}
x₃	{s₄, s₅}	{s₄, s₅}	{s₅}	{s₁, s₂, s₃, s₄}
w	0.3	0.4	0.1	0.2

Using the min_upper and max_lower operators, comprehensive evaluation values can be obtained, as follows:

H(x1)={s3, s4}; H(x2)={s4, s5}; H(x3)={s4, s5}.

The preference relation matrix P is

P=(−001−0.510.5−);

hence,

PS=(−001−010−).

The nondominance choice degrees NDQ_i are calculated as follows:

NDQ1=0, NDQ1=1, and NDQ1=1.

The final ranking by the nondominance choice degree is x₁<x₂~x₃.

HFLTSs are converted into trapezoidal fuzzy numbers, and a 0.5-cut operation is applied to these fuzzy numbers to obtain the intervals as shown in Table 8.

Table 8:

The 0.5-Cut Operation Results.

	g₁	g₂	g₃	g₄
x₁	[0.415, 0.585]	[0.25, 0.75]	[0, 0.415]	[0, 0.415]
x₂	[0.25, 0.415]	[0.085, 0.585]	[0.25, 0.585]	[0.585, 0.75]
x₃	[0.415, 0.75]	[0.415, 0.75]	[0.585, 0.75]	[0, 0.585]

The maximum I_max(x_i) (i=1, 2, 3) of each alternative is obtained as follows:

I_max(x₁)=[0.415, 0.75], I_max(x₂)=[0.585, 0.75], and I_max(x₃)=[0.585, 0.75].

Then, the likelihood p(I_max(x_i)≥[0, 1]) of each alternative can be calculated as p(I_max(x₁)≥[0, 1])=0.562, p(I_max(x₂)≥[0, 1])=0.644, and p(I_max(x₃)≥[0, 1])=0.644.

Therefore, the final ranking is x₁<x₂~x₃.

Using the hesitant fuzzy linguistic WA operator, the comprehensive evaluation values can be obtained as Hs1{s3, s4}, Hs2{s3, s4}, and Hs3{s3, s4, s5}.

Then, using the possibility degree ordering methods [22], we obtain Hs3≻0.667Hs2∼Hs1. Therefore, the final ranking is x₂~x₁πx₃.

The likelihood-based comparison relation matrix P is calculated as

P=(0.5710.5560.3330.3330.4290.4440.50.7140.6250.6250.7140.4)

Then, using the HFLWG operator, the integrated scores are as follows:

R(x1)=0.481, R(x2)=0.489, and R(x3)=0.579.

Therefore, the preference order is x₁πx₂πx₃.

Using the HFLWG operator, the comprehensive evaluation values can be obtained as follows:

H(x1)={s2.35, s3.25, s4.01}, H(x2)={s2.83, s3.37, s3.84}, and H(x3)={s3.1, s3.87, s4.19, s4.78}.

The calculated scores are μ(H(x₁))=3.204, μ(H(x₂))=3.345, and μ(H(x₃))=3.985.

Therefore, the final ranking is x₁πx₂πx₃.

Using the HFLWA operator, the comprehensive evaluation values can be obtained as follows:

H(x1)={s2.7, s3.4, s4.1}, H(x2)={s3, s3.45, s3.9}, and H(x3)={s3.5, s4.05, s4.25, s4.8}.

The calculated scores are

μ(H(x1))=3.4, μ(H(x2))=3.45, and μ(H(x3))=4.15.

Therefore, the final ranking is x₁πx₂πx₃.

Using the novel operational laws of HFLTS and the weighted averaging operator, the comprehensive evaluation values can be obtained as follows.

H(x1)={s2.86, s2.93, s3.02, s3.18, s3.31,…,s4.10, s4.16, s4.22},

H(x2)={s3.19, s3.30, s3.52, s3.62, s3.90, s3.98},

H(x3)={s3.68, s3.81, s3.94, s4.06, s4.11,…,s4.60, s4.70, s4.82}.

The calculated scores are μ(H(x₁))=3.586, μ(H(x₂))=3.585, and μ(H(x₃))=4.282. Therefore, the final ranking is x₂πx₁πx₃.

From Table 9, the final ranking of the alternatives x₁, x₂, and x₃ for different aggregation methods can be seen. The decision-making results of M1, M2, and M3 lack distinction because M1 and M2 are based on the envelope of the HFLTS and trapezoidal fuzzy numbers, respectively, which may result in a loss of information. For example, under the theory of the HFLTS envelope, h¹={s₁, s₂, s₅} and h²={s₁, s₄, s₅} are equivalent because they have the same lower and upper bounds, which is not in accordance with the facts. Many approximations are involved in the calculation process for M3, which may lead to the decision-making results being unreliable. Although M4, M5, and M6 have solved the information distortion phenomena, these methods do not take global information into consideration and focus solely on individual information. As a result, it can be argued that the alternative x₂ is superior to x₁. By converting HFLTS into HFS and using the novel operational laws of HFLTS, M7 can capture the correlations between aggregated arguments. The HFLGWPA and HFLGOWPA operators proposed in this paper consider the support measure of elements in HFLTS. Therefore M7, M8, and M9 consider alternative x₁ to be superior to alternative x₂. Because M7 and the method described in this paper have different emphases, it is necessary to make a selection according to the actual application situation.

Table 9:

Aggregated Results.

Method	Index value			Result
Method	x₁	x₂	x₃	Result
The min_upper and max_lower operator [23] (M1)	0	1	1	x₁<x₂~x₃
The method based on the α-cut operation using the optimistic attitude of the decision maker [2] (M2)	0.562	0.644	0.644	x₁<x₂~x₃
The hesitant fuzzy linguistic WA operator [30] (M3)	{s₃, s₄}	{s₃, s₄}	{s₃, s₄, s₅}	x₂~x₁πx₃
The HFLWG operator based on the likelihood-based comparison relation [13] (M4)	0.481	0.489	0.579	x₁πx₂πx₃
The HFLWG operator [50] (M5)	3.204	3.345	3.985	x₁πx₂πx₃
The HFLWA operator [50] (M6)	3.4	3.45	4.15	x₁πx₂πx₃
The WA operator based on the novel operational laws of HFLTS [6] (M7)	3.586	3.585	4.282	x₂πx₁πx₃
The HFLGWPA operator (f(x)=x) (M8)	2.195	1.969	2.476	x2≺⌢x1≺⌢x3
The HFLGOWPA operator (f(x)=x) (M9)	2.207	2.094	2.485	x2≺⌢x1≺⌢x3

7 Conclusions

In this paper, we defined some new operators to aggregate hesitant fuzzy linguistic information considering the supports among input arguments, namely, the HFLGWPA, HFLGOW-PA, and HFLGPOWA operators. The desirable properties of the developed operators, such as commutativity, idempotency, and boundedness have been studied. Moreover, we proposed an approach to solving hesitant fuzzy linguistic MAGDM problems. With the developed approach, if the weight vectors are known, we use the HFLGWPA and HFLGOWPA operators to aggregate information; if the weight vectors of the attributes (or decision makers) are unknown, we use the HFLGPOWA operator to aggregate information. The proposed approach can take all individual feature information of the input arguments and the relevant information among them into account. Therefore, the new approach can reduce the influence of outliers (that is, unduly large or unduly small input values) on the aggregation values, thus, making the decision results more reliable. Finally, a case of mergers and acquisitions has been provided to demonstrate the effectiveness of the proposed method.

Acknowledgments

The authors are thankful to the editor and the anonymous reviewers for their constructive comments and suggestions to help improve this paper. This study was funded by the National Natural Science Foundation of China (No. 11171221), the Education Department of Anhui Province Humanities and Social Sciences Key Planning Fund (No. SK2016A0971), the Education Department of Anhui Province Natural Science Key Research Projects (Nos. KJ2016A742 and KJ2017A402), and the Anhui Province College Excellent Young Talents Support Plan Key Projects (No. gxyq2017058).

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Received: 2017-05-04

Published Online: 2018-03-02

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/jisys-2017-0192

Keywords for this article

Linguistic term; group decision making; operator; hesitant fuzzy linguistic term set

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