Hesitant fuzzy linguistic aggregation operators based on global vision

Abstract

The hesitant fuzzy linguistic term set is an efficient tool for modeling quantitative information. In recent years, hesitant fuzzy linguistic information aggregation operators have increasingly attracted the attention of scholars. However, most of the existing operators assume that aggregated elements are independent, which overlooks the interconnectedness between elements in decision making situations. In this paper, we introduce several tools for integrating hesitant fuzzy linguistic variables, such as the hesitant fuzzy linguistic reducible weighted Bonferroni mean (HFLRWBM) operator, the hesitant fuzzy linguistic generalized reducible weighted Bonferroni mean (HFLGRWBM) operator, and the hesitant fuzzy linguistic weighted power Bonferroni mean (HFLWPBM) operator. These operators take into consideration the influences of two factors: the relative importance of each individual criterion and the interaction relationships between the values. It should be noted that the HFLWPBM operator is suitable for dealing with hesitant fuzzy linguistic evaluated information provided by decision makers are fused. In addition, several special forms of these operators are investigated, and their properties and advantages are discussed. Using these operators, this paper proposed a method for multiple attribute decision making. The feasibility and validity of this approach are demonstrated using a number of examples.

Keywords

Bonferroni mean hesitant fuzzy linguistic term set HFLGRWBM operator HFLRWBM operator HFLWPBM operator interrelationship

1 Introduction

Multiple attribute decision making (MADM) plays a central role in decision making theory, and has been widely used in many fields, such as supplier evaluation and selection, green industrial engineering and power system planning [2 –4]. In the typical MADM problem, decision makers are needed to provide the attribute evaluation values of the objects. Sometimes, people use qualitative values instead of quantitative values when evaluating the decision information. There are several reasons for this. Due to the characteristics of decision objects, it is difficult to quantitatively evaluate the decision information, but the evaluation can be expressed in linguistic terms. For example, linguistic labels such as “high”, “ordinary” and “poor” are usually used to evaluate the comprehensive quality of a person. Another reason is that it may be extremely difficult to obtain quantitative information or the cost of acquiring the necessary data is very high. For example, linguistic terms such as “big”, “extremely big” and “small” are commonly used to evaluate the size of a vehicle. Therefore, one can find that fuzzy linguistic multiple attribute decision making has been widely used in many different types of fields for decision analysis, including choosing importation strategies [7], selecting medical treatment options [8], and evaluating enterprise innovative ability [9].

However, due to limitations on the time available for evaluation and the level of knowledge about the objects in decision making activities, the evaluator might vacillate between several linguistic terms or provide a relatively sophisticated linguistic term. To model this situation, Rodriguez et al. recently presented the hesitant fuzzy linguistic term set (HFLTS) [11] theory on the basis of the theory of hesitant fuzzy sets [12]. Because the HFLTS can better represent decision makers’ qualitative evaluations, it has received extensive attention from scholars. Zhang et al. developed the hesitant fuzzy linguistic weighted averaging operator based on the 2-tuple linguistic aggregation operators [13]. Wang expanded the theory of HFLTS and introduced the extended hesitant fuzzy linguistic OWA operator [14]. Lee et al. defined the WA operator and likelihood based comparison relations for HFLTS [15].

The Bonferroni Mean (BM) operator is a classical and popular information aggregation method, which can take the mutual relativity of input arguments into account. This is the main reason why it can be widely used in practical applications. Fabio et al. combined the OWAAC operator with BM, introducing a new aggregation operator called BON-OWAAC [18]. Xu et al. proposed some intuitionistic fuzzy BM operators to solve MADM problems [20]. Wei et al. put forward some aggregation formulas of BM operators to solve uncertain linguistic MADM problems [21]. Furthermore, Yu et al. provided some BM operators for multi-criteria group decision making(MCGDM) problems with hesitant fuzzy numbers [22]. He et al. defined new operational laws of intuitionistic fuzzy sets, and then provided the intuitionistic fuzzy interaction BM [23] and the extended Atanassov’s intuitionistic fuzzy interaction BM [24]. However, the weighted forms of these Bonferroni means barely have the property of reducibility, which will be discussed in Section 3. In recent years, many researchers have been devoted to studying the reducible weighted BM operator in different fuzzy environments, such as the single-valued neutrosophic environments [25], hesitant fuzzy environments [26], intuitionistic fuzzy environments [27, 28], simplified neutrosophic linguistic fuzzy environments [29], and gray linguistic environments [30]. However, until now, the reducible weighted BM operators have not obtained wide acceptance. Moreover, in order to model the decision making problems that require simultaneous consideration of the interrelationships between the input arguments and the relationships between the fused values, He et al. defined some power Bonferroni means in hesitant fuzzy environments [31], intuitionistic fuzzy environments [32], and interval-valued hesitant fuzzy environments [33].

Although researchers have introduced some hesitant fuzzy linguistic aggregation operators, there are currently no operators from an overall perspective that consider the relevance of input factors. Mean-while, no research has been reported on the BM in hesitant fuzzy linguistic environments. To fill this gap and enrich decision making methods in hesitant fuzzy linguistic environments, this paper presents a new set of hesitant fuzzy linguistic BM operators. The framework of this paper is as follows: Section 2 reviews some concepts, such as HFLTS, HFLE, the HFLE comparison law and the most common BM operators. In Sections 3 and 4, we design some heistant fuzzy linguistic reducible weighted BM operators and study some of their properties. In Sections 5, we extend the power Bonferroni mean to hesitant fuzzy linguistic environments, and introduce the HFLPBM operator and the HFLWPBM operator. A new method for solving MADM problems with hesitant fuzzy linguistic information is presented in Section 6. In Section 7, the validity and feasibility of this proposed approach are verified using two numerical examples. In Section 8, the comparative analysis with other hesitant fuzzy linguistic aggregation operators is provided and finally, conclusions and advice for future research work are presented in Section 9.

2 Preliminaries

2.1 The linguistic approach

For qualitative analysis in decision processes, the linguistic approach is one kind of technique that provides linguistic terms for the evaluation of decision objects. A linguistic discrete term set can be described by S = {s_α|α = 1, 2, …, t}, where s_α denotes a possible evaluation value expressed in a special language, and t is a positive odd integer. For example, when you evaluate the stiffness of a material, you can define a linguistic discrete term set S = {s₁ : extremely soft; s₂ : soft, s₃ : slightly soft, s₄ : normal, s₅ : slightly hard, s₆ : hard, s₇ : extremely hard}.

The following requirements are the basis of the logical reasoning ability of the linguistic term set S:

s_α < s_β if α < β;

Negation operator: N (s_i) = s_j, j = t + 1 - i.

The results of calculations with linguistic information probably do not belong to the linguistic term set, which results in a calculation that is not rigorous. In light of this limitation, Xu [36] presented a general continuum set $\bar{S} = {s_{α} | α \in [0, p]},$ p > t. If s_α ∈ S, then s_α is called an original linguistic term (often given by the decision makers); otherwise, we call it a virtual linguistic term (which only appears in the calculation process) [19].

Let s_α, $s_{β} \in \bar{S}$ and λ, λ₁, λ₂ > 0. The Following operation principles were introduced by Xu [36]:

s_α ⊕ s_β = s_β ⊕ s_α = s_α+β;

λs_α = s_λα;

(λ₁ + λ₂) s_α = λ₁s_α ⊕ λ₂s_α;

s_α ⊗ s_β = s_β ⊗ s_α = s_αβ;

(s_α) ^λ = s_α^λ.

If the decision making information received is insufficient (or subject to some restrictive conditions), the evaluators may vacillate between several linguistic terms. Rodriguez et al. proposed the concept of HFLTS to handle this situation, enlightened by the theory of hesitant fuzzy sets (HFSs).

Definition 1. [37] Let X be fixed, and S = {s_α|α = 1, 2, …, t} be a linguistic term set. Then $H_{S} = {< x_{i}, h_{S} (x_{i}) > | x_{i} \in X, i = 1, 2, \dots, n}$ is called a hesitant fuzzy linguistic term set on X, where h_S (x_i) is a subset of S and is commonly referred to as the hesitant fuzzy linguistic element (HFLE).

However, in decision making, we always find that HFLEs have different numbers of elements. Hence, for convenience of comparison and computation, we suppose that all HFLEs have the same number of elements. If the numbers of elements are unequal, then the HFLE with fewer elements is revised by adding the element $0.5 h_{S}^{+} (x) \oplus 0.5 h_{S}^{-} (x)$ until the lengths are equal. The maximal and minimal terms of h_S (x) are denoted $h_{S}^{+} (x)$ and $h_{S}^{-} (x)$ , respectively.

Example 1. Complicated as the clinical manifestations of diseases may be, a doctor always gives a qualitative description of a patient’s symptoms, such as cough, pulse, and appetite. For example, suppose a doctor evaluates the cough symptoms of three patients, x₁, x₂ and x₃. The linguistic evaluation scale is defined as S = {s₁ : very serious, s₂ : serious, s₃ : slightly serious, s₄ : fair, s₅ : slightly mild, s₆ : mild, s₇ : very mild}. The doctor provides his evaluation as H_S = {< x₁, h_S (x₁) = {s₅, s₆, s₇} > , <x₂, h_S (x₂) = {s₄, s₅} > , <x₃, h_S (x₃) = {s₂, s₃, s₅} >}, where H_S is an HFLTS, and h_S (x₁), h_S (x₂) and h_S (x₃) are three HFLEs. To make them have the same length, h_S (x₂) is extended to ${\bar{h}}_{S} (x_{2}) = {s_{4}, s_{4.5}, s_{5}}$ .

Definition 2. [37] Let S = {s_α|α = 1, 2, 3, …, t} be a linguistic term set, and let h_S = {s_{η
_l}|s_{η
_l} ∈ S, l = 1, 2, …, L} be an HFLE. $μ (h_{S}) = \frac{1}{L} \sum_{i = 1}^{L} η_{i}$ is called the score of h_S, and $υ (h_{S}) = \frac{1}{L} \sqrt{\sum_{i \neq k}^{L} (η_{i} - η_{k})^{2}}$ is called the deviation degree of h_s. Then, the comparison law between two HFLEs ( $h_{s}^{1} = {s_{η_{l}^{1}} | s_{η_{l}^{1}} \in S, l = 1, 2, \dots, L}$ and $h_{s}^{2} = {s_{η_{l}^{2}} | s_{η_{l}^{2}} \in S, l = 1, 2, \dots, L}$ ) can be defined as

If $μ (h_{s}^{1}) > μ (h_{s}^{2})$ , then $h_{s}^{1} > h_{s}^{2}$ ;

If $μ (h_{s}^{1}) = μ (h_{s}^{2})$ , then;

if $υ (h_{s}^{1}) = υ (h_{s}^{2})$ , then $h_{s}^{1} = h_{s}^{2}$ ;

if $υ (h_{s}^{1}) > υ (h_{s}^{2})$ , then $h_{s}^{1} < h_{s}^{2}$ .

Example 2. Let $h_{S}^{1} = {s_{3}, s_{4}, s_{8}}$ , $h_{S}^{2} = {s_{4}, s_{6}}$ , $h_{S}^{3} = {s_{4}, s_{6}, s_{7}}$ be three HFLEs, then $μ (h_{S}^{1}) = 5$ , $μ (h_{S}^{2}) = 5$ , $μ (h_{S}^{3}) \approx 5.67$ . Hence $h_{S}^{3} > h_{S}^{2}$ , $h_{S}^{3} > h_{S}^{1}$ . Because $υ (h_{S}^{1}) \approx 2.2 > 2 = υ (h_{S}^{2})$ . Therefore, $h_{S}^{3} > h_{S}^{2} > h_{S}^{1}$ .

2.2 Bonferroni mean

Definition 3. [20] Let p, q ≥ 0, and {a₁, a₂, …, a_n} be a set of nonnegative real numbers. If ${BM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) = {(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}}$ then BM^p,q is called a Bonferroni Mean (BM) operator.

The BM operator can only aggregate decision information that is represented by real numbers. Xu et al. (2011) and Zhu et al. (2013) extended the BM operator to intuitionistic fuzzy values (IFVs) and the hesitant fuzzy sets environment, respectively.

Definition 4. [20] Let α_i (i = 1, 2, …, n) be a collection of IFVs. For any p, q ≥ 0, $\begin{matrix} {IFBM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (α_{i}^{p} \otimes α_{j}^{q}))}^{\frac{1}{p + q}} \end{matrix}$ and $\begin{matrix} {IFWBM}^{p, q} (α_{1}, \dots, α_{n}) \\ = {(\frac{\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ((w_{i} α_{i}^{p}) \otimes (w_{j} α_{j}^{q}))}{n (n - 1)})}^{\frac{1}{p + q}} \end{matrix}$

We call IFBM^p,q an intuitionistic fuzzy BM (IFBM) operator and IFWBM^p,q an intuitionistic fuzzy weighted BM (IFWBM) operator.

Definition 5. [38] Let h_i (i = 1, 2, …, n) be a collection of HFEs. For any p, q ≥ 0, $\begin{matrix} {HFBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (h_{i}^{p} \otimes h_{j}^{q}))}^{\frac{1}{p + q}} \end{matrix}$ and $\begin{matrix} {HFWBM}^{p, q} (h_{1}, \dots, h_{n}) \\ = {(\frac{\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ((w_{i} h_{i}^{p}) \otimes (w_{j} h_{j}^{q}))}{n (n - 1)})}^{\frac{1}{p + q}} \end{matrix}$

We call HFBM^p,q a hesitant fuzzy BM (HFBM) operator and HFWBM^p,q a hesitant fuzzy weighted BM (HFWBM) operator.

3 Hesitant fuzzy linguistic weighted BMs

The abovementioned BM operators cannot be used to accommodate the situation in which the decision information is represented by HFLEs. In this section, based on the HFBM and HFWBM operators, some BM operators are developed for aggregating hesitant fuzzy linguistic information.

Definition 6. Let S = {s_α|α = 1, 2, …, t} be a linguistic term set, and h₁, h₂, …, h_n be HFLEs on S with the weight vector w = (w₁, w₂, …, w_n) ^T, where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . For any p,q ≥ 0, $\begin{matrix} {HFLSBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} {{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}))}^{\frac{1}{p + q}}} \\ {HFLWSBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} {{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ((w_{i} s_{α_{i}}^{p}) \otimes (w_{j} s_{α_{j}}^{q})))}^{\frac{1}{p + q}}} \end{matrix}$

We call HFLSBM^p,q a hesitant fuzzy linguistic standard BM operator and HFLWSBM^p,q a hesitant fuzzy linguistic weighted standard BM operator.

It is generally known that when w_i = 1/n (i = 1, …, n), the classic weighted averaging (WA)operator reduces to the arithmetic averaging (AA) operator. For example, if w_i = 1/n(i = 1, …, n), then the linguistic WA operator reduces to the linguistic AA operator: $\begin{matrix} LWA (s_{α_{1}}, s_{α_{2}}, \dots, s_{α_{n}}) \\ = w_{1} s_{α_{1}} \oplus w_{2} s_{α_{2}} \oplus \dots \oplus w_{n} s_{α_{n}} \\ = \frac{1}{n} (s_{α_{1}} \oplus s_{α_{2}} \oplus \dots \oplus s_{α_{n}}) \\ = LAA (s_{α_{1}}, s_{α_{2}}, \dots, s_{α_{n}}) \end{matrix}$

However, it is obvious that the HFLWSBM^p,q can-not be reduced to the HFLSBM^p,q. In recent years, many researchers have been devoted to studying an improved BM operator that has the desired reducibility [25 –30]. Based on the hesitant fuzzy reducible weighted BM [26] defined by Zhou, we introduce the following BM operator with the desired reducibility in the hesitant fuzzy linguistic environment.

Definition 7. Let S = {s_α|α = 1, 2, …, t} be a linguistic term set, and h₁, h₂, …, h_n are HFLEs on S with the weight vector w = (w₁, w₂, …, w_n) ^T, where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . For anyp, q ≥ 0,

$\begin{matrix} {HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}))}^{\frac{1}{p + q}}} \end{matrix}$ (1)

We call HFLRWBM^p,q a hesitant fuzzy linguistic reducible weighted BM operator. Specifically, if w_i = 1/n (i = 1, …, n), then HFLRWBM^p,q reduces to HFLBM^p,q.

If we look at this in another light, the construction of the formula is easy to understand. $\begin{matrix} {HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{i = 1}^{n} w_{i} (s_{α_{i}}^{p} (\oplus_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} \frac{w_{j}}{1 - w_{i}} s_{α_{j}}^{q})))}^{\frac{1}{p + q}}} \end{matrix}$

We see that the term $\oplus_{j = 1, j \neq i}^{n} \frac{w_{j}}{1 - w_{i}} s_{α_{j}}^{q}$ is the weighted average of $s_{α_{j}}^{q}$ (j ≠ i), where $\sum_{j = 1, j \neq i} \frac{w_{j}}{1 - w_{i}} = 1$ , $\frac{w_{j}}{1 - w_{i}}$ indicates the importance degree of s_{α_i} for s_{α_j}, and $\oplus_{i = 1}^{n} w_{i} (s_{α_{i}}^{p} (\oplus_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} \frac{w_{j}}{1 - w_{i}} s_{α_{j}}^{q}))$ is the weighted average of $s_{α_{i}}^{p}$ and $\oplus_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{n} \frac{w_{j}}{1 - w_{i}} s_{α_{j}}^{q}$ . Therefore the HFLRWBM operator reflects the two layer weighted average calculation which is the key feature of the Bonferroni mean [11]. Through the above analysis, we know that the HFLRWBM operator not only reflects the importance degrees of the decision criteria but also considers the interrelationships between the criteria.

It is worth noting that if we base our construction on another common reducible form of weighted Bonferroni mean [25] defined by Tian et al., then we can obtain the following aggregation formula. $\underset{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}}{\cup} {{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (({nw}_{i} s_{α_{i}})^{p} \otimes ({nw}_{j} s_{α_{j}})^{q}))}^{\frac{1}{p + q}}}$

If h₁ = h₁ (i = 1, 2, …, n), then $\begin{matrix} \underset{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}}{\cup} {{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (({nw}_{i} s_{α_{i}})^{p} \otimes ({nw}_{j} s_{α_{j}})^{q}))}^{\frac{1}{p + q}}} \\ = \underset{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}}{\cup} {s_{α_{1}} {(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({nw}_{i})^{p} ({nw}_{j})^{q})}^{\frac{1}{p + q}}} \neq h_{1} \end{matrix}$ (if $\sum_{i, j = 1; i \neq j}^{n} ({nw}_{i})^{p} ({nw}_{j})^{q} \neq n (n - 1)$ ).

It seems to be unreasonable to declare that the weighted average value of n equivalent HFLTSs varies with their weights.

Remark 1. If q = 0, then the HFLRWBM operator degenerates to the HFLGWA operator: $\begin{matrix} {HFLRWBM}^{p, 0} (h_{1}, h_{2}, \dots, h_{n}) \\ = \underset{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}}{\cup} {{(\oplus_{i = 1}^{n} w_{i} s_{α_{i}}^{p})}^{1 / p}} \\ = HFLGWA (h_{1}, h_{2}, \dots, h_{n}) \end{matrix}$

Remark 2. If p = 2, q = 0, then the HFLRWBM operator degenerates to the following. $\begin{matrix} {HFLRWBM}^{2, 0} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {{(\oplus_{i = 1}^{n} w_{i} s_{α_{i}}^{2})}^{\frac{1}{2}}} \end{matrix}$ which we call the hesitant fuzzy linguistic weighted quadratic mean operator.

Remark 3. If p = 1, q = 0, then the HFLRWBM operator degenerates to the HFLWA operator: $\begin{matrix} {HFLRWBM}^{1, 0} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {\oplus_{i = 1}^{n} w_{i} s_{α_{i}}} \end{matrix}$

Remark 4. If p = q = 1, then the HFLBWBM operator degenerates to the hesitant fuzzy linguistic bilateral weighted geometric BM operator: $\begin{matrix} {HFLRWBM}^{1, 1} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}} \otimes s_{α_{j}}))}^{\frac{1}{2}}} \end{matrix}$

Theorem 1.LetS = {s_α|α = 1, 2, …, t} be a linguistic term set, and supposeh₁, h₂, …, h_nare HFLEs onSwith the weight vectorw = (w₁, w₂, …, w_n) ^T, where w_i ∈ [0, 1], and $\sum_{i = 1}^{n} w_{i} = 1$ . For p, q ≥ 0, the following properties are satisfied.

(Commutativity) Let $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ be any permutation of (h₁, h₂, …, h_n). Then ${HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) = {HFLRWBM}^{p, q} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ .

(Monotonicity) Let $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ and (h₁, h₂, …, h_n) be two HFLEs on S. If s_{α_i} ≤ s_{β_i} for all s_{α_i} ∈ h_i, $s_{β_{i}} \in h_{i}^{'}$ (i = 1, 2, …, n), then ${HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \leq {HFLRWBM}^{p, q} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ .

(Boundedness) Let h_max = {s_t, s_t, ⋯ , s_t} and h_min = {s_-t, s_-t, …, s_-t}. Then h_min ≤ HFLRWBM^p,q (h₁, h₂, …, h_n) ≤ h_max

(Idempotency) If h₁ = h₂ = ⋯ = h_n = {s_α₁, s_α₂, …, s_{α_m}}, then HFLRWBM^p, q (h₁, h₂, …, h_n) = h₁.

Proof. (1) Let $\begin{matrix} {HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}))}^{\frac{1}{p + q}}} \\ {HFLRWBM}^{p, q} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'}) \\ = ⋃_{s_{β_{1}} \in h_{1}^{'}, s_{β_{2}} \in h_{2}^{'}, \dots, s_{β_{n}} \in h_{n}^{'}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i}^{'} w_{j}^{'}}{1 - w_{i}^{'}} (s_{β_{i}}^{p} \otimes s_{β_{j}}^{q}))}^{\frac{1}{p + q}}} \end{matrix}$

Since $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ is any permutation of (h₁, h₂, …, h_n), then we have $\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}) = \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i}^{'} w_{j}^{'}}{1 - w_{i}^{'}} (s_{β_{i}}^{p} \otimes s_{β_{j}}^{q})$

Thus, ${HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) = {HFLRWBM}^{p, q} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ .

(2) Because s_{α_i} ≤ s_{β_i} for all s_{α_i} ∈ h_i, $s_{β_{i}} \in h_{i}^{'}$ , i = 1, 2, …, n, then $s_{α_{i}}^{p} \otimes s_{α_{j}}^{q} \leq s_{β_{i}}^{p} \otimes s_{β_{j}}^{q}$ , and $\begin{matrix} {(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}))}^{\frac{1}{p + q}} \\ \leq {(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{β_{i}}^{p} \otimes s_{β_{j}}^{q}))}^{\frac{1}{p + q}} \end{matrix}$

Hence, ${HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \leq {HFLRWBM}^{p, q} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'}) .$

(3) Because ∀i ∈ {1, 2, …, n}, s_-t ≤ s_{α_i} ≤ s_t, then we have $\begin{matrix} h_{min} & = & ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} {s_{- t}} \\ = & ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{- t})^{p + q})}^{\frac{1}{p + q}}} \\ \leq & ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}))}^{\frac{1}{p + q}}} \\ \leq & ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{t})^{p + q})}^{\frac{1}{p + q}}} \\ = & ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} {s_{t}} = h_{max} \end{matrix}$

That is $h_{min} \leq {HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \leq h_{max}$

(4) Because h₁ = h₂ = ⋯ = h_n = {s_α₁, s_α₂, …, s_{α_m}}, then $\begin{matrix} {HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}))}^{\frac{1}{p + q}}} \\ = \cup {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{k}}^{p} \otimes s_{α_{k}}^{q}))}^{\frac{1}{p + q}}} \\ = \cup {{((s_{k})^{p + q} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}})}^{\frac{1}{p + q}}} = \cup {s_{k}} \end{matrix}$

This completes the proof of Theorem 1.

4 Hesitant fuzzy linguistic GBMs

Both the HFLBM and HFLRWBM operators consider the correlations between any two decision criteria but not the correlations between more than two criteria. Inspired by the generalized BM (GBM), we propose the following definitions.

Definition 8. Let S = {s_α|α = 1, 2, …, t} be a linguistic term set, and suppose h₁, h₂, …, h_n are HFLEs on S with the weight vector w = (w₁, w₂, …, w_n) ^T, where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . For any p, q, r ≥ 0,

$\begin{matrix} {HFLGBM}^{p, q, r} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\frac{\oplus_{\begin{matrix} i, j, l = 1 \\ i \neq j \neq l \end{matrix}}^{n} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q} \otimes s_{α_{l}}^{r})}{n (n - 1) (n - 2)})}^{\frac{1}{p + q + r}}} \\ {HFLGRWBM}^{p, q, r} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j, l = 1 \\ i \neq j \neq l \end{matrix}}^{n} \frac{w_{i} w_{j} w_{l} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q} \otimes s_{α_{l}}^{r})}{(1 - w_{i}) (1 - w_{i} - w_{j})})}^{\frac{1}{p + q + r}}} \end{matrix}$ (2)

We call HFLGBM^p, q, r a hesitant fuzzy linguistic generalized BM operator and HFLGRWBM^p, q, r a hesitant fuzzy linguistic generalized reducible weighted BM operator. Specifically, if w = (1/n, …, 1/n) ^T, then HFLGRWBM^p, q, r degenerates to HFLGBM^p, q, r.

Remark 1. If r = 0, then the HFLGRWBM operator degenerates to the HFLRWBM operator: $\begin{matrix} {HFLGRWBM}^{p, q, 0} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}} (s_{α_{i}}^{p} \otimes s_{α_{j}}^{q}))}^{\frac{1}{p + q}}} \\ = {HFLRWBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \end{matrix}$

Remark 2. If p = 1, q = 1, and r = 1, then we have $\begin{matrix} {HFLGRWBM}^{1, 1, 1} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} {{(\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \neq l \end{matrix}}^{n} \frac{w_{i} w_{j} w_{l}}{(1 - w_{i}) (1 - w_{i} - w_{j})} (s_{α_{i}} \otimes s_{α_{j}} \otimes s_{α_{l}}))}^{\frac{1}{3}}} \end{matrix}$ which we call the hesitant fuzzy linguistic triple weighted mean operator.

The following properties may be proved along the same lines as the proof of Theorem 1, therefore, we omit the details.

Theorem 2.LetS = {s_α|α = 1, 2, …, t} be a linguistic term set, and supposeh₁, h₂, …, h_nare HFLEs onSwith the weight vectorw = (w₁, w₂, …, w_n) ^T, wherew_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Forp, q, r ≥ 0, t he following properties are satisfied.

(Commutativity) Let $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ be any permutation of (h₁, h₂, …, h_n).Then ${HFLGRWBM}^{p, q, r} (h_{1}, h_{2}, \dots, h_{n}) = {HFLGRWBM}^{p, q, r} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ .

(Monotonicity) Let $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ and (h₁, h₂, …, h_n) be two HFLEs on S. If s_{α_i} ≤ s_{β_i} for all s_{α_i} ∈ h_i, $s_{β_{i}} \in h_{i}^{'}$ , i = 1, 2, …, n, then ${HFLGRWBM}^{p, q, r} (h_{1}, h_{2}, \dots, h_{n}) \leq {HFLGRWBM}^{p, q, r} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ .

(Boundedness) Let h_max = {s_t, s_t, …, s_t} and h_min = {s_-t, s_-t, …, s_-t}. Then h_min ≤ HFLGRWBM^p,q,r (h₁, h₂, …, h_n) ≤ h_max.

(Idempotency) If h₁ = h₂ = ⋯ = h_n = {s_α₁, s_α₂, …, s_{α_m}}, then HFLGRWBM^p,q,r (h₁, h₂, …, h_n) = h₁

5 Hesitant fuzzy linguistic PBMs

The abovementioned Bonferroni means only consider the interrelationships between the input arguments, and cannot reflect the relative closeness of decision making information. In some decision situations, we should take all these factors into account. For this reason, we extend the power Bonferroni mean [31], which was defined by He et al., to the hesitant fuzzy linguistic environment.

Definition 9. Let S = {s_α|α = 1, 2, …, t} be a linguistic term set, and suppose h₁, h₂, …, h_n are HFLEs on S. For any p, q ≥ 0,

$\begin{matrix} {HFLPBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(\frac{n (1 + T (s_{α_{i}}))}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}}))} s_{α_{i}})}^{p} \\ \otimes {{(\frac{n (1 + T (s_{α_{j}}))}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}}))} s_{α_{j}})}^{q}))}^{\frac{1}{p + q}} \end{matrix}$

We call HFLPBM^p,q a hesitant fuzzy linguistic power Bonferroni mean (HFLPBM) operator. $T (s_{α_{i}}) = \sum_{j = 1, j \neq i}^{n} Sup (s_{α_{i}}, s_{α_{j}})$ , where Sup (s_{α_i}, s_{α_j}) is the support for s_{α_i} and s_{α_j}, satisfying:

Sup (s_{α_i}, s_{α_j}) ∈ [0, 1];

Sup (s_{α_i}, s_{α_j}) = Sup (s_{α_j}, s_{α_i});

if d (s_{α_i}, s_{α_j}) ≤ d (s_{α_l}, s_{α_k}), then $Sup (s_{α_{i}}, s_{α_{j}}) \geq Sup (s_{α_{l}}, s_{α_{k}})$ In this paper, without loss of generality, we let $Sup (s_{α_{i}}, s_{α_{j}}) = 1 - \frac{| α_{i} - α_{j} |}{t - 1} .$

Remark 1. If q = 0 and $\frac{1 + T (s_{α_{i}})}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}}))} = \frac{1}{n}$ (i = 1, 2, …, n), then $\begin{matrix} {HFLPBM}^{p, q} (h_{1}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {{(\frac{1}{n} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} s_{α_{i}}^{p})}^{\frac{1}{p}}} \end{matrix}$ which we call the generalized hesitant fuzzy linguistic mean operator.

Remark 2. If p = 2, q = 0, and $\frac{1 + T (s_{α_{i}})}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}}))} = \frac{1}{n}$ (i = 1, 2, …, n) then $\begin{matrix} {HFLPBM}^{p, q} (h_{1}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {{(\frac{1}{n} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} s_{α_{i}}^{2})}^{\frac{1}{2}}} \end{matrix}$ which we call the hesitant fuzzy linguistic quadratic mean operator.

Remark 3. If p = 1, q = 0, and $\frac{1 + T (s_{α_{i}})}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}}))} = \frac{1}{n}$ (i = 1, 2, …, n), then $\begin{matrix} {HFLPBM}^{p, q} (h_{1}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {\frac{1}{n} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} s_{α_{i}}} \end{matrix}$ which we call the hesitant fuzzy linguistic averaging operator.

Theorem 3.LetS = {s_α|α = 1, 2, …, t} be a linguistic term set, and supposeh₁, h₂, …, h_nare HFLEs onS. The following properties are satisfied.

(Commutativity) Let $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ be any permutation of (h₁, h₂, …, h_n). Then ${HFLPBM}^{p, q} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'}) = {HFLPBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n})$ .

(Idempotency) If h₁ = h₂ = ⋯ = h_n = {s_α₁, s_α₂, …, s_{α_m}}, then HFLPBM^p, q (h₁, h₂, …, h_n) = h₁.

Proof. (1) Since $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ is any permutation of (h₁, h₂, …, h_n), then we have $\frac{n (1 + T (s_{α_{i}}))}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}}))} s_{α_{i}} = \frac{n (1 + T (s_{α_{i}^{'}}))}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}^{'}}))} s_{α_{i}^{'}}$ and $\frac{n (1 + T (s_{α_{j}}))}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}}))} s_{α_{j}} = \frac{n (1 + T (s_{α_{j}^{'}}))}{\sum_{t = 1}^{n} (1 + T (s_{α_{t}^{'}}))} s_{α_{j}^{'}}$

Thus ${HFLPBM}^{p, q} (h_{1}^{'}, \dots, h_{n}^{'}) = {HFLPBM}^{p, q} (h_{1}, \dots, h_{n})$ .

(2) From h₁ = h₂ = ⋯ = h_n = {s_α₁, s_α₂, …, s_{α_m}}, it follows that $\begin{matrix} {HFLPBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {{(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} s_{α_{1}}^{p + q})}^{\frac{1}{p + q}}} \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {{(s_{α_{1}}^{p + q})}^{\frac{1}{p + q}}} \\ = ⋃_{s_{α_{1}} \in h_{1}, \dots, s_{α_{n}} \in h_{n}} {s_{α_{1}}} = h_{1} \end{matrix}$

This completes the proof of Theorem 3.

If the weighting vector of the HFLEs is given, then we can define the hesitant fuzzy linguistic weighted power BM (HFLWPBM) operator.

Definition 10. Let S = {s_α|α = 1, 2, …, t} be a linguistic term set, and suppose h₁, h₂, …, h_n are HFLEs on S with the weight vector w = (w₁, w₂, …, w_n) ^T, where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . For any p, q ≥ 0,

$\begin{matrix} {HFLWPBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n}) \\ = ⋃_{s_{α_{1}} \in h_{1}, s_{α_{2}} \in h_{2}, \dots, s_{α_{n}} \in h_{n}} \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(\frac{{nw}_{i} (1 + T (s_{α_{i}}))}{\sum_{t = 1}^{n} w_{t} (1 + T (s_{α_{t}}))} s_{α_{i}})}^{p} \\ \otimes {{(\frac{{nw}_{j} (1 + T (s_{α_{j}}))}{\sum_{t = 1}^{n} w_{t} (1 + T (s_{α_{t}}))} s_{α_{j}})}^{q}))}^{\frac{1}{p + q}} \end{matrix}$ (3)

We call HFLWPBM^p, q a hesitant fuzzy linguistic weighted power BM (HFLWPBM) operator. $T (s_{α_{i}}) = \sum_{j = 1, j \neq i}^{n} w_{j} Sup (s_{α_{i}}, s_{α_{j}})$ , where Sup (s_{α_i}, s_{α_j}) satisfies conditions i)-iii).

The following properties may be proved along the same lines as the proof of Theorem 3; therefore, we omit the details.

Theorem 4.LetS = {s_α|α = 1, 2, …, t} be a linguistic term set, and supposeh₁, h₂, …, h_nare HFLEs onSwith the weight vectorw = (w₁, w₂, …, w_n) ^T, wherew_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . For anyp, q ≥ 0, the following properties are satisfied.

(Commutativity) Let $(h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'})$ be any permutation of (h₁, h₂, …, h_n). Then ${HFLWPBM}^{p, q} (h_{1}^{'}, h_{2}^{'}, \dots, h_{n}^{'}) = {HFLWPBM}^{p, q} (h_{1}, h_{2}, \dots, h_{n})$ .

(Idempotency) If h₁ = h₂ = ⋯ = h_n = {s_α₁, s_α₂, …, s_{α_m}}, then HFLWPBM^p, q (h₁, h₂, …, h_n) = h₁.

The particular forms of the HFLWPBM operator are similar to those of the HFLPBM operator; therefore, they are omitted here. It should be noted that, unlike the abovementioned Bonferroni means, the power Bonferroni means do not have the reducibility and monotonicity properties.

6 Multiple criteria decision making with HFLTS information

In this section, we shall utilize the HFLRWBM operator (along with the HFLGRWBM operator and the HFLWPBM operator) to handle MADM with hesitant fuzzy linguistic information.

Step 1. A hesitant fuzzy linguistic MADM problem is considered in this paper. Let X = (x₁, x₂, …, x_n} be a set of alternatives, C = (c₁, c₂, …, c_n} be a set of criteria, and w = (w₁, w₂, …, w_n) ^T be a set of weights, where w_i denotes the weight of criterion c_i, and $\sum_{i = 1}^{n} w_{i} = 1$ , w_i ∈ [0, 1]. Let S = {s_α|α = 1, 2, …, t} be a linguistic evaluation scale, H_i(i = 1, 2, …, n) be an HFLTS on S, and suppose $H_{i} = {< c_{j}, h_{S}^{i} (c_{j}) > | c_{j} \in C, j = 1, 2, \dots, m}$ , where $h_{S}^{i} (c_{j}) = {s_{η_{l}} | s_{η_{l}} \in S, l = 1, 2, \dots, L}$ denotes the hesitant fuzzy linguistic value of x_i under c_j, abbreviated h_ij. The hesitant fuzzy linguistic decision matrix H is defined as $H = (\begin{matrix} h_{11} & h_{12} & \dots & h_{1 m} \\ h_{21} & h_{22} & \dots & h_{2 m} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ h_{n 1} & h_{n 2} & \dots & h_{nm} \end{matrix})$

Step 2. Utilize the HFLRWBM operator (in general, we can take p = q = 1) or the HFLGRWBM operator (in general, we can take p = q = r = 1) or the HFLW-PBM operator (in general, we can take p = q = 1) to derive the hesitant fuzzy linguistic preference values h_k (k = 1, 2, ⋯ , n) of the alternative x_i.

Step 3. Order the alternatives x_i according to h_i (i = 1, 2, …, n) and choose the ideal solution.

7 Illustrative examples

Example 3. The department of Human Resource Management (HRM) is primarily responsible for personnel recruitment and employee assessment. Suppose the HRM department of a company wants to look for a full-time project manager and there are three candidates x_i (i = 1, 2, 3) available in the talent market. After repeated discussions, they took into account four main considerations (with weighting vector w = (0.3, 0.4, 0.1, 0.2) ^T): g₁- Past experience; g₂-Self-confidence; g₃-Proficiency in project management; g₄-Personality. The linguistic term set S is defined as S = {s₁ : very terrible, s₂ : terrible, s₃ : slightly terrible, s₄ : fair, s₅ : slightly prominent, s₆ : prominent s₇ : very prominent}.

Step 1. Based on the decision makers’ evaluations, the hesitant fuzzy linguistic decision matrix H is constructed (see Table 5.1).

Table 5.1
Decision matrix H

g ₁ g ₂ g ₃ g ₄

x ₁ {s₃, s₄} {s₅, s₆, s₇} {s₁, s₂, s₃} {s₅, s₆}

x ₂ {s₄} {s₃, s₄, s₅} {s₃, s₄} {s₇}

x ₃ {s₄, s₅} {s₄, s₅} {s₅} {s₁, s₂, s₃, s₄}

	g ₁	g ₂	g ₃	g ₄
x ₁	{s₃, s₄}	{s₅, s₆, s₇}	{s₁, s₂, s₃}	{s₅, s₆}
x ₂	{s₄}	{s₃, s₄, s₅}	{s₃, s₄}	{s₇}
x ₃	{s₄, s₅}	{s₄, s₅}	{s₅}	{s₁, s₂, s₃, s₄}

Step 2. Utilize the defined operator (here, we take p = q = r = 1) to aggregate the evaluation information from H. The aggregation results are shown in Table 5.2.

Table 5.2

The comprehensive evaluation values

	HFLRWBM	HFLGRWBM	HFLWPBM
x ₁	{s_2.75, s_3.42, s_4.44}	{s_2.23, s_2.97, s_3.65}	{s_2.64, s_3.21, s_4.17}
x ₂	{s_3.73, s_4.15, s_4.55}	{s_3.1, s_3.44, s_3.76}	{s_3.45, s_3.85, s_4.23}
x ₃	{s_3.96, s_4.37, s_4.49, s_4.88}	{s_2.96, s_3.36, s_3.55, s_3.89}	{s_3.83, s_4.18, s_4.28, s_4.61}

Step 3. Order the alternatives x_i according to h_i (i = 1, 2, …, n) and choose the ideal solution. The orderings of the alternatives are shown in Table 5.3.

Table 5.3

Decision making results

Operator	Index value			Ordering
HFLRWBM	3.534	4.145	4.427	x₃ ≻ x₂ ≻ x₁
HFLGRWBM	2.953	3.435	3.44	x₃ ≻ x₂ ≻ x₁
HFLWPBM	3.34	3.845	4.276	x₃ ≻ x₂ ≻ x₁

As we can see from Table 5.3, no matter what kind of weighted BM operator is used, the ordering is same and the best alternative is x₃. However, the comprehensive evaluation values obtained using the HFLGRWBM operator are smaller than the corresponding values obtained using other operators. This is because the HFLGRWBM operator performs a three-layer association calculation, and the HFLRWBM operator and the HFLWPBM operator perform two-layer association calculations. Because more criteria are used by the HFLGRWBM operator in the aggregation process, the comprehensive evaluation values are smaller. This could lead to a reduction of identification ability. For this reason, we propose the HFLGRWBM operator,which contains three decision variables.

It is easy to see from formulas (1), (2), and (3) that the computation of the HFLWPBM operator is more complex than those of the HFLRWBM operator and the HFLGRWBM operator. That is the main reason why we do not combine the HFLGBM operator and the power operator. Furthermore, the HFLWPBM operator emphasizes the overall performance and the other two operators emphasize the average level. In the decision making process, we should choose the operator based on the actual instance and need.

Example 4. With the popularization of the Internet and rapid development of network marketing, it has become difficult to find satisfactory goods among the large amount of advertisements. With the help of recommendation systems, the buying procedure can become very smooth. Suppose that a movie recommendation system takes into account the following main considerations(with weighting vector w = (0.4, 0.2, 0.2, 0.2) ^T) to rate movies: c₁-movie script; ·c₂-acting skill; ·c₃-video production; c₄- director’s ability. The linguistic label term set is defined as S = {s₁ : very terrible, s₂ : terrible, s₃ : slightly terrible, s₄ : fair, s₅ : slightly prominent, s₆ : prominent, s₇ : very prominent}. To recommend five movies x_i(i = 1, 2, …, 5) to viewers, the following evaluation process is used.

Step 1. Based on the decision makers’ evaluations, the hesitant fuzzy linguistic decision matrix H is constructed (see Table 5.4).

Table 5.4

Evaluation information matrix

	c ₁	c ₂	c ₃	c ₄
x ₁	{s₂, s₃, s₄}	{s₄, s₅}	{s₄, s₅, s₆}	{s₅, s₆}
x ₂	{s₄, s₅, s₆}	{s₅, s₆}	{s₄, s₅}	{s₄, s₅, s₆}
x ₃	{s₆, s₇}	{s₅, s₆, s₇}	{s₅, s₆}	{s₆}
x ₄	{s₄, s₅, s₆}	{s₃, s₄, s₅}	{s₅, s₆, s₇}	{s₅, s₆}
x ₅	{s₃, s₄}	{s₄, s₅, s₆}	{s₄, s₅, s₆}	{s₄, s₅}

Step 2. Utilize the defined operator to aggregate the evaluation information. (Without loss of generality, here we take the HFLRWBM operator (see Table 5.5)).

Table 5.5

The comprehensive evaluation values obtained using the HFLRWBM operator

p, q	x ₁	x ₂	x ₃	x ₄	x ₅
p = 100, q = 100	h₁ = {s_4.44, s_5.18, s_5.93}	h₂ = {s_4.45, s_5.22, s_5.98}	h₃ = {s_5.95, s_6.22, s_6.95}	h₄ = {s_4.94, s_5.68, s_6.45}	h₅ = {s_3.97, s_4.94, s_5.93}
p = 10, q = 100	h₁ = {s_4.14, s_4.77, s_5.44}	h₂ = {s_4.32, s_5.07, s_5.87}	h₃ = {s_5.64, s_6.11, s_6.67}	h₄ = {s_4.51, s_5.34, s_6.20}	h₅ = {s_3.74, s_4.60, s_5.44}
p = 2, q = 2	h₁ = {s_3.55, s_4.31, s_5.09}	h₂ = {s_4.22, s_5.01, s_5.79}	h₃ = {s_5.43, s_6.07, s_6.57}	h₄ = {s_4.13, s_5.13, s_6.02}	h₅ = {s_3.38, s_4.37, s_5.09}
p = 1, q = 1	h₁ = {s_3.42, s_4.24, s_5.05}	h₂ = {s_4.21, s_5.00, s_5.78}	h₃ = {s_5.3, s_6.06, s_6.56}	h₄ = {s_3.99, s_5.10, s_5.99}	h₅ = {s_3.10, s_4.34, s_5.05}
p = 0.5, q = 0.5	h₁ = {s_3.35, s_4.21, s_5.03}	h₂ = {s_4.21, s_4.99, s_5.77}	h₃ = {s_5.07, s_6.06, s_6.55}	h₄ = {s_3.79, s_5.08, s_5.98}	h₅ = {s_2.62, s_4.33, s_5.03}
p = 1, q = 0.001	h₁ = {s_3.4, s_4.2, s_5.0}	h₂ = {s_4.2, s_5.0, s_5.8}	h₃ = {s_5.12, s_6.1, s_6.6}	h₄ = {s_3.857, s_5.1, s_6.0}	h₅ = {s_2.641, s_4.3, s_5.0}
p = 0.001, q = 1	h₁ = {s_3.63, s_4.4, s_5.17}	h₂ = {s_4.23, s_5.0, s_5.77}	h₃ = {s_5.04, s_6.03, s_6.53}	h₄ = {s_3.82, s_5.12, s_6.0}	h₅ = {s_2.64, s_4.43, s_5.17}

Step 3. Order the alternatives x_i according to h_i (i = 1, 2, …, n) and choose the ideal solution. The orderings of the alternatives are shown in Table 5.6.

Table 5.6

Decision making results

p,q	Ordering		HFLGRWBM
	HFLRWBM	HFLWPBM	p,q,r	Ordering
p = 100, q = 100	x₃ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₅	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁	p = 10, q = 10, r = 10	x₃ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₅
p = 10, q = 10	x₃ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₅	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁	p = 1, q = 1, r = 1	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁
p = 2, q = 2	x₃ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₅	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁	p = 1, q = 0.001, r = 0.001	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁
p = 1, q = 1	x₃ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₅	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁	p = 0.001, q = 1, r = 1	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁
p = 0.5, q = 0.5	x₃ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₅	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁	p = 1, q = 0.001, r = 1	x₃ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₅
p = 1, q = 0.001	x₃ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₅	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁	p = 1, q = 1, r = 0.001	x₃ ≻ x₂ ≻ x₄ ≻ x₅ ≻ x₁
p = 0.001, q = 1	x₃ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₅	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁	p = 0.5, q = 0.5, r = 0.5	x₃ ≻ x₄ ≻ x₂ ≻ x₅ ≻ x₁

From Table 5.6, we can see that when the parameters(p, q, r) change, the rankings change slightly. All of the results show that x₃ is the best alternative, which matches the result of Liao’s method in [42]. By analyzing the aggregation values, we can find that the hesitant fuzzy linguistic comprehensive evaluation values derived by the three kinds of operators are influenced by the input parameters p, q, and r. In general, the use of larger parameters for the operators makes the calculations more complex. For this reason, we recommend taking p = q = r = 1, which not only makes the calculations easier but also represents an interrelationship between three individual arguments.

8 Comparative analyses with other operators

Since HFLTS was first put forward, various related aggregation operators have been introduced. Generally, they can be divided into three types: 1) the hesitant fuzzy linguistic aggregation operators based on the envelope of the HFLTS, such as the min_upper and max_lower (MUML) operator [11] and the heist-ant fuzzy linguistic WA (HLWA) operator [40]; 2) the hesitant fuzzy linguistic aggregation operators based on all elements of HFLE, such as the hesitant fuzzy linguistic weighted geometric (HFLWG) operator [37] and the hesitant fuzzy linguistic ordered weighted averaging (HFLWA) operator [37]; 3) the hesitant fuzzy linguistic aggregation operators based on the relationships between HFLEs, such as the HFLRWBM operator and the HFLGRWBM operator. A comparative analysis between them is made based on the same illustrative example.

From Table 5.7, we can see that all of the methods agree that x₃ is the best choice, except for the MUML operator. This is because the MUML operator and HLWA operator use the maximum and minimum elements of the HFLE instead of the full HFLE, which makes it easy to lose decision making information. For example, the MUML operator and HLWA operator consider h¹ = {s₁, s₂, s₅} and h² = {s₁, s₄, s₅} to be equivalent because they have the same maximum and minimum elements. This obviously does not accord with the facts. Although using the envelope of the HFLTS is not accurate, it makes the decision making process simpler and more efficient. The operators in this article (based on the global consistency of HFLTS), along with the HFLWA and HFLWG operators (based on the HFLE), use all of the linguistic terms in the calculation to avoid the loss of decision making information. They focus on different aspects; therefore, it is difficult to say which one is the best choice in general. However, for a specific problem, we should choose the most suitable one.

Table 5.7
Results of Example 3 by different operators

Operator Ranking

MUML x₂ ≻ x₃ ≻ x₁

HLWA x₁ ∼ x₃ ≻ x₂

HFLWA x₃ ≻ x₂ ≻ x₁

HFLWG x₃ ≻ x₂ ≻ x₁

HFLRWBM x₃ ≻ x₂ ≻ x₁

HFLGRWBM x₃ ≻ x₂ ≻ x₁

HFLWPBM x₃ ≻ x₂ ≻ x₁

Operator	Ranking
MUML	x₂ ≻ x₃ ≻ x₁
HLWA	x₁ ∼ x₃ ≻ x₂
HFLWA	x₃ ≻ x₂ ≻ x₁
HFLWG	x₃ ≻ x₂ ≻ x₁
HFLRWBM	x₃ ≻ x₂ ≻ x₁
HFLGRWBM	x₃ ≻ x₂ ≻ x₁
HFLWPBM	x₃ ≻ x₂ ≻ x₁

9 Conclusions

In today’s complex and variable economic and social environment, we often confront vague and uncertain decision making situations. Because people are generally hesitant to choose between several evaluation terms, it is difficult to provide a unique evaluation value. Therefore, the hesitant fuzzy linguistic terms are more practical than the fuzzy language. In this paper, we put forward some operators for HFLTS, including the HFLRWBM operator, the HFLGRWBM operator and the HFLW-PBM operator. Their properties and advantages are discussed and special cases are described. We have also developed a procedure based on these operators to solve hesitant fuzzy linguistic MADM problems. The practicality and effectiveness of the new procedure are illustrated using actual examples. In future work, we expect to develop some extensions and generalizations of Bonferroni means in the hesitant fuzzy linguistic environment and apply them to data mining, weapon system evaluation, information security risk assessment, highway project risk assessment, etc.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11171221), the Education Department of Anhui Province Natural Science Key Research Projects (KJ2016A742), and Anhui Province College Excellent Young Talents Support Plan Key Projects (gxyq2017058).

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Hesitant fuzzy linguistic aggregation operators based on global vision

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 The linguistic approach

2.2 Bonferroni mean

3 Hesitant fuzzy linguistic weighted BMs

7 Illustrative examples

Table 5.1 Decision matrix H g 1 g 2 g 3 g 4 x 1 {s3, s4} {s5, s6, s7} {s1, s2, s3} {s5, s6} x 2 {s4} {s3, s4, s5} {s3, s4} {s7} x 3 {s4, s5} {s4, s5} {s5} {s1, s2, s3, s4}

Table 5.7 Results of Example 3 by different operators Operator Ranking MUML x2 ≻ x3 ≻ x1 HLWA x1 ∼ x3 ≻ x2 HFLWA x3 ≻ x2 ≻ x1 HFLWG x3 ≻ x2 ≻ x1 HFLRWBM x3 ≻ x2 ≻ x1 HFLGRWBM x3 ≻ x2 ≻ x1 HFLWPBM x3 ≻ x2 ≻ x1

Footnotes

Acknowledgments

References

Table 5.1
Decision matrix H

g ₁ g ₂ g ₃ g ₄

x ₁ {s₃, s₄} {s₅, s₆, s₇} {s₁, s₂, s₃} {s₅, s₆}

x ₂ {s₄} {s₃, s₄, s₅} {s₃, s₄} {s₇}

x ₃ {s₄, s₅} {s₄, s₅} {s₅} {s₁, s₂, s₃, s₄}

Table 5.7
Results of Example 3 by different operators

Operator Ranking

MUML x₂ ≻ x₃ ≻ x₁

HLWA x₁ ∼ x₃ ≻ x₂

HFLWA x₃ ≻ x₂ ≻ x₁

HFLWG x₃ ≻ x₂ ≻ x₁

HFLRWBM x₃ ≻ x₂ ≻ x₁

HFLGRWBM x₃ ≻ x₂ ≻ x₁

HFLWPBM x₃ ≻ x₂ ≻ x₁