The new similarity measure and distance measure between hesitant fuzzy linguistic term sets and their application in multi-criteria decision making

Abstract

Considering that the existing cosine similarity measure between hesitant fuzzy linguistic term sets(HFLTSs) has an impediment as it does not satisfy the axiom of similarity measure, we propose a new similarity measure of HFLTSs in the paper, which is constructed based on the existing cosine similarity measure and Euclidean distance measure of HFLTSs. Then the corresponding distance measure of HFLTSs is obtained according to the relationship between the similarity measure and the distance measure. Furthermore, we develop the TOPSIS method to the proposed distance measure in hesitant fuzzy linguistic decision environment and apply the closeness coefficients to rank the alternatives. The main advantage of the proposed method is that it not only considers the distance measure from the point view of algebra and geometry but also overcomes the disadvantage of the existing cosine similarity measure. Finally, an example is provided to illustrate the feasibility of the proposed method and some comparative analyses are given to show its efficiency.

Keywords

Hesitant fuzzy linguistic term set similarity measure distance measure TOPSIS

1 Introduction

Multi-criteria decision making is a series of procedures in a specific order to help the decision makers find the optimal alternative. Due to the complexity of the decision making environment, there is some uncertainty in multi-criteria decision making problems. In 1965, Zadeh [1] proposed a fuzzy set A = {(x_j, μ_A (x_j)) |x_j ∈ X)} to handle the imprecise and vague information in decision making problems, where μ_A (x_j) is the membership degree of x_j ∈ X to the set A.

Since the fuzzy set was proposed, it has received a lot of attention in many fields, such as pattern recognition, medical diagnosis and so on [2 –5]. However, in some practical decision problems, it is difficult to describe the membership value of an element with a single value. For example, two experts are invited to evaluate the profitability of some airline, one expert believes that the possibility of its profit is 0.7, the other expert believes that the possibility of its profit is 0.6. The airline profit evaluation cannot be represented by the fuzzy set at this time. In order to describe the relevant information better, Torra [6] proposed the concept of hesitant fuzzy set (HFS), which contains all possible memberships of an element in [0, 1]. Then the airline profit evaluation can be represented by a HFS H = {0.6, 0.7}, where 0.6 and 0.7 represent its membership degree, respectively. Since the HFS was proposed, it has attracted a lot of attention from many researchers ([7 –11]). For example, Xu et al. [7] defined the distance and correlation measures between HFSs under the assumption that two hesitant fuzzy elements have the same length and applied these distance measures to multi-criteria decision making problems. Farhadinia [8] investigated the relationship between the entropy, the similarity measure and the distance measure for HFSs and interval-valued hesitant fuzzy sets (IVHFSs), and two clustering algorithms are developed under a hesitant fuzzy environment in which indices of similarity measures of HFSs and IVHFSs are applied in data analysis and classification. Zhang et al. [10] defined the best additive consistency index, the worst additive consistency index and the average additive consistency index to measure the consistency level of hesitant fuzzy preference relation. Garg et al. [11] proposed Maclaurin symmetric mean aggregation operators based on t-norm operations between dual hesitant fuzzy soft set and applied these operators to multi-criteria decision making problems.

But in some practical decision making problems, many criteria should be assessed in a qualitative form. For example, when the car design is evaluated online by a customer, he/she thinks that the design is “very good”, it is suitable to be evaluated in a linguistic term set(LTS) because the linguistic evaluation is very close to human’s cognitive process. Thus, Zadeh ([12 –14]) proposed the LTS to describe the corresponding decision information, the general seven terms of LTS can be expressed as S = {s₀ : very poor, s₁ : poor, s₂ : a little poor, s₃ : medium, s₄ : a littlegood, s₅good, s₆ : very good} So the car design evaluation can be represented as {s₆}. However, if the car design is evaluated online by many customers, some of them think the design is good, the others think its design is just medium, then the car design evaluation cannot be represented by the LTS with a single value. In order to express the above information, Rodríguez et al. [15] proposed the hesitant fuzzy linguistic term set (HFLTS) based on HFS and LTS. Then the car design evaluation can be represented as {s₂, s₃}, where s₃, s₅ represent the possible membership degrees of the car design, they are called hesitant fuzzy linguistic elements(HFLEs). The HFLTSs are highly useful in handling situations where people are hesitant in providing their preferences with regard to objects in a decision-making process, more and more multi-attribute decision making theories and methods under hesitant fuzzy linguistic environment have been developed ([15 –19]). For example, Dong et al. [16] proposed a novel distance-based consensus measure and developed an optimization-based consensus model in the hesitant linguistic group decision making problems. Yu et al. [17] developed a new method to deal with multi-criteria group decision making problems with unbalanced HFLTSs by considering the psychological behavior of decision makers. Liao et al. [18] proposed the hesitant fuzzy linguistic preference utility-TOPSIS method to select the best fire rescue plan. Yu et al. [19] introduced some aggregation operators of HFLTS and proposed a decision method to evaluate the meteorological disaster that occurred in China.

On the other hand, similarity measure and distance measure are important topics in multi-criteria decision making problems, which can describe the similarity degree and difference between two different alternatives. They have been widely studied in the past ten years ([20 –34]). For example, Song et al. [20] took account into the similarity measure between intuitionistic fuzzy sets(IFSs) and proposed the corresponding distance measure between intuitionistic fuzzy belief functions. Liao et al. [21] presented a family of similarity measures and distance measures between HFLTSs and applied them to rank alternatives in multi-criteria decision making problems. Lee et al. [22] presented a similarity measure between HFLTSs based on likelihood relations. There still have a lot of related studies consider the similarity measure between HFLTSs, we can refer to [23 –26]. Furthermore, the cosine similarity measure is also a significant similarity measure, which is defined as the inner product of two vectors divided by the product of their lengths [27], some scholars studied the cosine similarity measure ([28 –31]). For example, Ye [28] introduced a weighted cosine similarity measure between intuitionistic fuzzy sets and applied it to pattern recognition and medical diagnosis. Furthermore, Ye [29] presented the cosine similarity measure between interval-valued fuzzy sets with risk preference and altered its decision making method depending on decision makers’ preference. Liao et al. [30] defined the cosine similarity measure between HFLTSs and extended the TOPSIS method and VIKOR method to the cosine distance measure. Considering the interaction between the pairs of the membership degrees, Garg [31] proposed an improved cosine similarity measure between IFSs and applied it to decision making process. Other studies on distance measures can refer to [32 –34]. Distance measure is often used with TOPSIS method in multi-criteria decision making problems. Hwang et al. [35] originally introduced the TOPSIS method to solve the multi-criteria decision making problem, the basic notion of the TOPSIS is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. A number of papers are devoted to fuzzy TOPSIS method in [36 –40].

If two HFLTSs are not equal, the value of the cosine similarity measure between HFLTSs(Liao et al. [30]) is 1 (the example can be seen in section 3). That is to say, the cosine similarity measure defined by them does not satisfy the axiom of similarity measure, it is not a regular similarity measure. This flaw will certainly limit its applicability and lead to the initial information distortion in decision process. Motivated by this, the objective of the paper is to propose an innovative method to construct a new similarity measure of HFLTSs, which not only includes the cosine similarity measure of HFLTSs in Liao et al. [30] and the Euclidean distance measure of HFLTSs but also satisfies the axiom of similarity measure. As a result, the contributions of the paper are summarized as follows:

The proposed new similarity measure improves the cosine similarity measure of HFLTSs in Liao et al. [30] and overcomes its disadvantage.

The TOPSIS method is developed to the proposed distance measure, which could improve the adaptability of HFLTSs in practice and the effectiveness of processing decision information.

The rest of the paper is organized as follows: In Section 2, the concepts of HFS, LTS, HFLTS and some related knowledge about HFLTSs are briefly reviewed. In Section 3, we propose an innovative method to construct a new similarity measure and distance measure of HFLTSs, the relevant properties are also discussed. In Section 4, we develop the TOPSIS method to the proposed distance measure in hesitant fuzzy linguistic environment and give a multi-criteria decision making method. In Section 5, a practical example is given to illustrate the feasibility of the proposed method and some comparative analyses with other methods are conducted to show its effectiveness. Finally, the conclusions and recommendations for future research are presented in Section 6.

2 Preliminaries

In this section, we review and discuss some related knowledge, including HFS, LTS, HFLTS, the score function of HFLTS. Some existing distance measures and similarity measures between HFLTSs are also given. Throughout this paper, let X = {x₁, x₂, . . . , x_n} be a discrete and finite discourse set.

2.1 Hesitant fuzzy set

In practical fuzzy decision making problems, the HFS provides a better representation of reality and uncertainty to express the preferences of the decision makers. Torra [6] firstly introduce the concept of HFS.

Definition 1. [6] Let X = {x₁, x₂, . . . , x_n} be a fixed set, a HFS E on X is defined as: $E = {(x_{j}, e (x_{j})) | x_{j} \in X},$ where e (x_j) is a set of some values between 0 and 1, denoting the possible membership degree of x_j ∈ X to the set E. Furthermore, we call e (x_j) a hesitant fuzzy element.

2.2 Linguistic term set

LTS has been studied in depth and applied to many fields. Some extension forms of LTSs have been developed for handling more complicated multi-criteria decision making problems.

Definition 2. [41, 42] Let S′ = {s_i|i = - t, . . . , -1, 0, 1, . . . , t} be a finite and totally ordered discrete linguistic term set, where s_i is a linguistic variable, t is a positive integer, it satisfies the following properties:

(1) The set is ordered: s_i ≤ s_j if i ≤ j; max(s_i, s_j) = s_i if s_i ≥ s_j;min(s_i, s_j) = s_i if s_i ≤ s_j;

The negation operator is defined: neg (s_i) = s_j satisfying with i + j = 2t.

2.3 Hesitant fuzzy linguistic term set

HFLTS was used to deal with the situations where decision makers think of several possible linguistic values than a single linguistic term for an alternative, etc. It is more convenient to reflect the decision-makers’ preferences.

Definition 3. [15] Given a fixed set X, assume S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS, then a HFLTS on X can be defined as: $\begin{matrix} H_{s} = {(x_{j}, h_{s} (x_{j})) | x_{j} \in X}, \end{matrix}$ where h_s (x_j) ⊆ s represents the possible membership degrees of the element x_j ∈ X. For convenience, the element of h_s (x_j) is called hesitant fuzzy linguistic element (HFLE).

For two HFLTSs $H_{s}^{1}$ and $H_{s}^{2}$ , if the numbers of linguistic terms in $H_{s}^{1}$ and $H_{s}^{2}$ are not equal, it is difficult to calculate the similarity measure between $H_{s}^{1}$ and $H_{s}^{2}$ . In order to deal with this situation, we assume L = max {L₁, L₂}, where L₁, L₂ represent the number of elements in $H_{s}^{1}$ and $H_{s}^{2}$ respectively. Zhu et al. [43] proposed the rules of regulation: the set of fewer number of elements is extended by adding the linguistic term ${\bar{S}}_{δ_{1}} (x_{j}) = S_{δ}^{+} (x_{j}) + (1 - ζ) S_{δ}^{-} (x_{j}) (0 \leq ζ \leq 1)$ until they have the same number of elements, where $S_{δ}^{+} (x_{j}) = {max}_{l = L_{1}}^{L_{k}} {S_{δ_{1}} (x_{j})}$ , $S_{δ}^{-} (x_{j}) = {min}_{l = L_{1}}^{L_{k}} {S_{δ_{1}} (x_{j})}$ j = 1, 2, . . . , n). In practical decision problems, the optimized parameter is related to the risk preference of the decision makers. The optimists want to add the maximum value $S_{δ}^{+} (x_{j})$ , while the pessimists want to add the minimum value $S_{δ}^{-} (x_{j})$ . In this paper, we assume that the maximum element is added to the set with fewer number of elements.

The score function of HFLTSs is defined as follows:

Definition 4. [44] Let S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS, $H_{s} = {S_{δ_{1}^{k}} (x_{j}) | S_{δ_{1}^{k}} (x_{j}) \in S^{'}}$ (L = 1, 2, . . . , L_j, j = 1, , 2, . . . , n) be the HFLTS on X, then the score function of H_S is: $\begin{matrix} F (H_{s}) = \frac{1}{n} \sum_{j = 1}^{n} \bar{δ} (x_{j}) \\ - \frac{\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(δ_{1} (x_{j}) - \bar{δ} (x_{j}))}^{2})}{Var (2 t)}, \end{matrix}$ where $\bar{δ} = \frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} δ_{l} (x_{j})$ , $Var (2 t) = \frac{\sum_{i = 0}^{2 t} (i - t)^{2}}{2 t + 1}$ .

Lemma 1. [44] For two HFLTSs $H_{s}^{1}$ and $H_{s}^{2}$ , the comparison rules between them are defined as follows:

$H_{s}^{1} > H_{s}^{2}$ if and only if $F (H_{s}^{1}) > F (H_{s}^{2})$ ;

$H_{s}^{1} = H_{s}^{2}$ if and only if $F (H_{s}^{1}) = F (H_{s}^{2})$ .

2.4 Existing distance measures and similarity measures between hesitant fuzzy linguistic term sets

Distance and similarity measures are the two most important measures for HFLTSs, they are the base of the TOPSIS method. Here we give the Euclidean distance measure as follows.

Definition 5. [21] Given a fixed set X, assume that S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS. For any two HFLTSs $H_{s}^{1} = {(x_{j}, h_{s}^{1} (x_{j})) | x_{j} \in}$ and $H_{s}^{2} = {(x_{j}, h_{s}^{2} (x_{j})) | x_{j} \in} (j = 1, 2, . ., n)$ , where $h_{s}^{k} (x_{j}) = {S_{δ_{1}^{k}} (x_{j}) | S_{δ_{1}^{k}} (x_{j}) S^{'}, l = 1, 2, . . ., L_{j}, k = 1, 2}$ , is the maximum number of linguistic terms in $h_{s}^{1} (x_{j})$ and $h_{s}^{2} (x_{j})$ , assume the weight of x_j is ω_j (j = 1, 2, . . . , n), then the weighted Euclidean distance measure between $H_{s}^{1}$ and $H_{s}^{2}$ can be defined as:

$\begin{matrix} D_{ω HFL} (H_{s}^{1}, H_{s}^{2}) = \\ {(\sum_{j = 1}^{n} \frac{ω_{j}}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{1} (x_{j}) - δ_{l}^{2} (x_{j}) |}{2 t + 1})}^{2})}^{\frac{1}{2}} \end{matrix}$ (1)

Remark 1. For all j = 1, 2, . . . , n, if $ω_{j} = \frac{1}{n}$ , then the weighted Euclidean distance measure $D_{ω HFL} (H_{s}^{1}, H_{s}^{2})$ is reduced to the normalized Euclidean distance measure $D_{HFL} (H_{s}^{1}, H_{s}^{2})$ : $\begin{matrix} D_{HFL} (H_{s}^{1}, H_{s}^{2}) = \\ {(\frac{1}{n} (\sum_{j = 1}^{n} \frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{1} (x_{j}) - δ_{l}^{2} (x_{j}) |}{2 t + 1})}^{2}))}^{\frac{1}{2}} . \end{matrix}$

Liao et al. [30] defined a cosine similarity measure between HFLTSs as follows:

Definition 6. [30] Given a fixed set X, assume S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS. For any two HFLTSs $H_{s}^{1} = {(x_{j}, h_{s}^{1} (x_{j})) | x_{j} \in}$ and $H_{s}^{2} = {(x_{j}, h_{s}^{2} (x_{j})) | x_{j} \in} (j = 1, 2, . ., n)$ , where $h_{s}^{k} (x_{j}) = {S_{δ_{1}^{k}} (x_{j}) | S_{δ_{1}^{k}} (x_{j}) S^{'}, l = 1, 2, . . ., L_{j}, k = 1, 2}$ , is the maximum number of linguistic terms in $h_{s}^{1} (x_{j})$ and $h_{s}^{2} (x_{j})$ , assume that the weight of different element x_j is ω_j (j = 1, 2, . . . , n), then the weighted cosine similarity measure can be defined as:

${Cos}_{ω HFL} (H_{s}^{1}, H_{s}^{2}) = \frac{\sum_{j = 1}^{n} (\frac{ω_{j}}{L_{j}} \sum_{l = 1}^{L_{j}} (\frac{| δ_{l}^{1} (x_{j}) |}{2 t + 1} . \frac{| δ_{l}^{2} (x_{j}) |}{2 t + 1}))}{{(\sum_{j = 1}^{n} (\frac{ω_{j}}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{1} (x_{j}) |}{2 t + 1})}^{2}) . \sum_{j = 1}^{n} (\frac{ω_{j}}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{2} (x_{j}) |}{2 t + 1})}^{2}))}^{\frac{1}{2}}}$ (2)

Remark 2. For all j = 1, 2, . . . , n, if $ω_{j} = \frac{1}{n}$ , then the weighted cosine similarity measure ${Cos}_{ω HFL} (H_{s}^{1}, H_{s}^{2})$ is reduced to the normalized cosine similarity measure ${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2})$ :

${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) = \frac{\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} (\frac{| δ_{l}^{1} (x_{j}) |}{2 t + 1} . \frac{| δ_{l}^{2} (x_{j}) |}{2 t + 1}))}{{(\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{1} (x_{j}) |}{2 t + 1})}^{2}) . \sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{2} (x_{j}) |}{2 t + 1})}^{2}))}^{\frac{1}{2}}}$ (3)

3 The new similarity measure and distance measure between hesitant fuzzy linguistic term sets

In this section, we define a new similarity measure of HFLTSs based on the weighted cosine similarity measure Cos_ωHFL and the weighted Euclidean distance measure D_ωHFL.

It is already known that the regular similarity measure should satisfy the following Lemma 2.

Lemma 2. [21] Let S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS, $H_{s}^{1}$ and $H_{s}^{2}$ be any two HFLTSs, if the similarity measure $S (H_{s}^{1}, H_{s}^{2})$ satisfies the following properties:

$0 \leq S (H_{s}^{1}, H_{s}^{2}) \leq 1$ ;

$S (H_{s}^{1}, H_{s}^{2}) = 1$ if and only if $H_{s}^{1} = H_{s}^{2}$ ;

$S (H_{s}^{1}, H_{s}^{2}) = S (H_{s}^{1}, H_{s}^{2})$ ;

then the similarity measure

S (H_{s}^{1}, H_{s}^{2})

is a regular similarity measure.

The cosine similarity measure Cos_ωHFL proposed by Liao et al. [30] is not a regular similarity measure, we can see from the following Example 1.

Example 1. Two experts are invited to evaluate the car design, they provide their evaluations with HFLTSs, for some given LTS S′ = {s_-3 : very poor, s_-2: poor, s_-1 : a little poor, s₀ : medium, s₁: a little good, s₂ : good, s₃: very good}, the car design evaluations can be represented as HFLTSs $H_{s}^{1} = {s_{- 1}, s_{- 2}}$ and $H_{s}^{2} = {s_{- 1}, s_{- 2}}$ , respectively.

According to Lemma 1, it is already known $H_{s}^{1} \neq H_{s}^{2}$ . Using formula (3) to calculate the similarity measure between $H_{s}^{1}$ and $H_{s}^{2}$ , we have ${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) = 1$ . That is to say, the similarity measure Cos_HFL does not satisfy the property (2) in Lemma 2, it is not a regular similarity measure. If it is applied in multi-criteria decision making problem, it may lead to the decision information distortion in decision process. Motivated by this, we propose an innovative method to construct a similarity measure between HFLTSs, which is based on the cosine similarity measure Cos_HFL and the Euclidean distance measure D_HFL. We also know that the proposed new similarity measure overcomes the disadvantage of the similarity measure Cos_HFL defined in Liao et al. [30].

3.1 Definitions and properties

Definition 7. Given a fixed set X, assume S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS. For any two HFLTSs $H_{s}^{1} = {(x_{j}, h_{s}^{1} (x_{j})) | x_{j} \in}$ and $H_{s}^{2} = {(x_{j}, h_{s}^{2} (x_{j})) | x_{j} \in} (j = 1, 2, . ., n)$ , where $h_{s}^{k} (x_{j}) = {S_{δ_{1}^{k}} (x_{j}) | S_{δ_{1}^{k}} (x_{j}) S^{'}, l = 1, 2, . . ., L_{j}, k = 1, 2}$ , is the maximum number of linguistic terms in $h_{s}^{1} (x_{j})$ and $h_{s}^{2} (x_{j})$ , the new similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ can be defined as:

$\begin{matrix} S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = \\ \frac{1}{2} [{Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) + 1 - D_{HFL} (H_{s}^{1}, H_{s}^{2}), \end{matrix}$ where $\begin{matrix} {Cos}_{HFL} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) = \frac{\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} (\frac{| δ_{l}^{1} (x_{j}) |}{2 t + 1} . \frac{| δ_{l}^{2} (x_{j}) |}{2 t + 1}))}{{(\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{1} (x_{j}) |}{2 t + 1})}^{2}) . \sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{2} (x_{j}) |}{2 t + 1})}^{2}))}^{\frac{1}{2}}}, \end{matrix}$ $\begin{matrix} D_{HFL} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) = {(\frac{1}{n} (\sum_{j = 1}^{n} \frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{1} (x_{j}) - δ_{l}^{2} (x_{j}) |}{2 t + 1})}^{2}))}^{\frac{1}{2}} . \end{matrix}$

The similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ is defined based on the cosine similarity measure Cos_HFL and Euclidean distance measure D_HFL, it deal with the decision information not only from the point view of geometry but also from the point view of algebra. It is more efficient to solve the multi-criteria decision problems in hesitant fuzzy decision environment than the existing similarity measures.

Theorem 1. The similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ is a regular similarity measure.

Proof: According to Lemma 2, if the similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ satisfies all properties in Lemma 2, then it is a regular similarity measure. We can prove it by the following steps.

${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2})$ can be regarded as the extension of cosine function (the inner product of two vectors divided by the product of their lengths), then $0 \leq {Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) \leq 1$ . Because $D_{ω HFL} (H_{s}^{1}, H_{s}^{2})$ is an Euclidean distance measure, we have $0 \leq 1 - D_{HFL} (H_{s}^{1}, H_{s}^{2}) \leq 1$ then $0 \leq S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) \leq 1$ .

If $H_{s}^{1} = H_{s}^{2}$ , we have ${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) = 1$ , $D_{HFL} (H_{s}^{1}, H_{s}^{2}) = 0$ , then $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1$ . On the other hand, when $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1$ , we have ${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) + 1 - D_{HFL} (H_{s}^{1}, H_{s}^{2}) = 2$ , that is ${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) = 1 + D_{HFL} (H_{s}^{1}, H_{s}^{2})$ . Because $0 \leq {Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) \leq 1$ and $D_{ω HFL} (H_{s}^{1}, H_{s}^{2}) \geq 0$ should exist at the same time, so they must have $D_{ω HFL} (H_{s}^{1}, H_{s}^{2}) = 0$ and ${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) = 1$ .□

When ${Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) = 1$ , $H_{s}^{1} = {kH}_{s}^{2}$ is obtained, where k is a constant; Furthermore, when, we have $H_{s}^{1} = H_{s}^{2}$ . That is to say, if $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1$ , we have $H_{s}^{1} = H_{s}^{2}$ .

So $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1$ if and only if $H_{s}^{1} = H_{s}^{2}$ .

Because

\begin{matrix} {Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}) = {Cos}_{HFL} (H_{s}^{1}, H_{s}^{2}), \end{matrix}

\begin{matrix} D_{HFL} (H_{s}^{1}, H_{s}^{2}) = D_{HFL} (H_{s}^{1}, H_{s}^{2}), \end{matrix}

then

S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})

can be obtained easily.

Remark 3. If the similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ is a regular similarity measure, we can obtain a new distance measure $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ based on the relationship between the distance measure and similarity measure, which can be given as follows: $\begin{matrix} D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1 - S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}), \end{matrix}$ where $\begin{matrix} S_{HFL}^{*} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) = \\ \frac{1}{2} ({Cos}_{HFL} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) + 1 - D_{HFL} (H_{s^{'}}^{1}, H_{s^{'}}^{2})) \end{matrix}$

Theorem 2. The distance measure $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ satisfies the following properties:

$0 \leq D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) \leq 1$ ;

$D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 0$ if and only if $H_{s}^{1} = H_{s}^{2}$ ;

$D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$

Proof. Properties (1) and (3) are obvious, we only give the proof of property (2) here.□

If $H_{s}^{1} = H_{s}^{2}$ , we have $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1$ , then $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1 - S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1$ .

On the other hand, when $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 0$ , we have $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1 - D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 1$ .

Because $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ is a regular similarity measure, according to Lemma 2, we have $H_{s}^{1} = H_{s}^{2}$ .

Thus $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 0$ if and only if $H_{s}^{1} = H_{s}^{2}$ .

Next we give the definition about the weighted similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ as follows:

Definition 8. Given a fixed set X, assume S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS. For any two HFLTSs $H_{s}^{1} = {(x_{j}, h_{s}^{1} (x_{j})) | x_{j} \in}$ and $H_{s}^{2} = {(x_{j}, h_{s}^{2} (x_{j})) | x_{j} \in} (j = 1, 2, . ., n)$ , where $h_{s}^{k} (x_{j}) = {S_{δ_{1}^{k}} (x_{j}) | S_{δ_{1}^{k}} (x_{j}) S^{'}, l = 1, 2, . . ., L_{j}, k = 1, 2}$ is the maximum number of linguistic terms in $h_{s}^{1} (x_{j})$ and $h_{s}^{2} (x_{j})$ , the associated weight vector ω = (ω₁, ω₂, . . . , ω_n), satisfying with $\sum_{j = 1}^{n} ω_{j} = 1 (0 \leq ω_{j} \leq 1)$ , then the weighted similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ can be defined as: $\begin{matrix} S_{HFL}^{*} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) = \\ \frac{1}{2} ({Cos}_{HFL} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) + 1 - D_{HFL} (H_{s^{'}}^{1}, H_{s^{'}}^{2})) \end{matrix}$

Theorem 3. The weighted similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ is a regular similarity measure.

Proof. The proof is similar to Theorem 1, we omit it here.□

Remark 4. Similar to Remark 3, if we assume that the weight of element x_j is ω_j = 1 (0 ≤ ω_j ≤ 1), and satisfying with $\sum_{j = 1}^{n} ω_{j} = 1 (0 \leq ω_{j} \leq 1)$ , then the weighted distance measure $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ can be given as follows: $\begin{matrix} D_{ω HFL}^{*} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) = 1 - S_{ω HFL}^{*} (H_{s^{'}}^{1}, H_{s^{'}}^{2}) \end{matrix}$

Theorem 4. The weighted distance measure $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ also satisfies the following properties:

$0 \leq D_{ω HFL}^{*} (H_{s}^{1}, H_{s}^{2}) \leq 0$ ;

$D_{ω HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = 0$ if and only if $H_{s}^{1} = H_{s}^{2}$ ;

$D_{ω HFL}^{*} (H_{s}^{1}, H_{s}^{2}) = D_{ω HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ .

Proof. The proof is similar to Theorem 2, we omit it here.□

Remark 5. For all j = 1, 2, . . . , n), if $ω_{j} = \frac{1}{n}$ , then the weighted distance measure $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ and the weighted similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ are reduced to the distance measure $D_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ and the similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ respectively.

3.2 Comparison with the existing similarity measure

In order to illustrate the advantage of the proposed similarity measure $S_{HFL}^{*} (H_{s}^{1}, H_{s}^{2})$ , a numerical example is provided to compare with the similarity measure Cos_HFL proposed by Liao et al. [30].

Example 2. Assume that there exist three patterns A₁, A₂, A₃, and the unknown pattern B. The following three factors C₁: the depth of colour, C₂: the shape,C₃: the size are considered. The factors are expressed with different LTSs, which are given as follows respectively:

$S_{1}^{'} = {s_{- 3} : very light, s_{- 2} : light,$ s_-1 : alittle light, s₀ : medium, s₁ : a little dark, s₂ : dark, s₃ : very dark}

$S_{2}^{'} = {s_{- 3} : very round, s_{- 2} : round,$ s_-1 : alittle round, s₀ : medium, s₁ : a little square, s₂ : square, s₃ : very squre} ,

$S_{3}^{'} = {s_{- 3} : very small, s_{- 2} : small, s_{- 1} :$ alittle small, s₀ : medium, s₁ : a little l arg e, s₂ : l arg e, s₃ : very l arg e} .

The concrete evaluations about eaSs, which can be represented as: $\begin{matrix} A_{1} = {(C_{1}, s_{1}, s_{1}), (C_{2}, s_{1}), (C_{3}, s_{2}, s_{3})}; \end{matrix}$ $\begin{matrix} A_{2} = {(C_{1}, s_{- 1}), (C_{2}, s_{- 1}, s_{- 2}), (C_{3}, s_{- 3})}; \end{matrix}$ $\begin{matrix} A_{3} = {(C_{1}, s_{2}, s_{3}), (C_{2}, s_{1}), (C_{3}, s_{0})}, \end{matrix}$ the unknown pattern $\begin{matrix} B = {(C_{1}, s_{- 1}, s_{- 2}), (C_{2}, s_{- 1},), (C_{3}, s_{- 2}, s_{- 3})}, \end{matrix}$ now we want to make sure which one that the pattern B belongs to, so we calculate the similarity measure between B and A_i (i = 1, 2, 3), respectively.

At first, we apply the similarity measure Cos_HFL proposed by Liao et al. [30] to calculate it, we can get $\begin{matrix} {Cos}_{HFL} (A_{2}, B) = 0.9092, {Cos}_{HFL} (A_{2}, B) = 0.5196 \end{matrix}$

Then the pattern B belongs to A₁. Intuitively, the pattern B cannot belong to A₁. The reason that Cos_HFL (A₁, B) =1 is the subscripts of the linguistic terms A₁ and B have the linear relationship. But if we use the similarity measure $S_{HFL}^{*} (A_{1}, B) (i = 1, 2, 3)$ proposed in this paper to calculate it, we have $S_{HFL}^{*} (A_{1}, B) = 0.7392$ , $S_{HFL}^{*} (A_{2}, B) = 0.9041$ , $S_{HFL}^{*} (A_{1}, B) = 0.5436$ . It can be observed that the pattern B belongs to A₂ the result is more reasonable intuitively. So we can say that the similarity measure $S_{HFL}^{*}$ shows a better performance than the similarity measure Cos_HFL.

4 The TOPSIS method with proposed distance measure

In this section, we extend the TOPSIS [45] method to the proposed distance measure $D_{ω HFL}^{*}$ between HFLTSs. For a multi-criteria decision making problem, the experts decide to choose the best one from m alternatives according to n criteria. Let S = {s_i|i = - t, . . . , -1, 0, 1, . . , t} be a LTS,H = {H₁, H₂, . . . , H_m} be a set of alternatives and C = {C₁, C₂, . . . , C_m} be a set of criteria. For the alternative H_i with respect to the criterion C_j, the experts provide their evaluations represented by HFLTSs $H_{s}^{ij} = {S_{δ_{1}}^{ij} | l = 1, 2, . . ., L_{j}} (i = 1, 2, . . ., m; j = 1, 2, . . ., n)$ , ω_j (j = 1, 2, . . . , n) is the corresponding weight of criteria satisfying with 0 ≤ ω_j ≤ 1 (j = 1, 2, . . . , n) and $\sum_{j = 1}^{n} ω_{j} = 1$ , the decision matrix H with hesitant fuzzy linguistic information can be given as follows: $\begin{matrix} H = (\begin{matrix} H_{s^{'}}^{11} & H_{s^{'}}^{11} & \dots & H_{s^{'}}^{1 n} \\ H_{s^{'}}^{21} & H_{s^{'}}^{11} & \dots & H_{s^{'}}^{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ H_{s^{'}}^{m 1} & H_{s^{'}}^{m 2} & \dots & H_{s^{'}}^{mn} \end{matrix}), \end{matrix}$ where $H_{s}^{ij} (i = 1, 2, . . ., m; j = 1, 2, . . ., n)$ are HFLTSs.

In the following, we present the steps of the developed TOPSIS method in hesitant fuzzy linguistic environment for multi-criteria decision making problems, which are given as follows:

Step 1. Normalize the decision matrix H, for the benefit-type criteria, we need not do anything; for the cost-type criteria, we should use the operator neg (s_i) = s_i (i + j = 2t) to normalize the decision matrix.

Step 2. Find the hesitant fuzzy linguistic positive ideal solution $H^{+} = {H_{s^{'}}^{1 +}, H_{s^{'}}^{2 +}, . . ., H_{s^{'}}^{n +}}$ and hesitant fuzzy linguistic negative ideal solution $H^{-} = {H_{s^{'}}^{1 -}, H_{s^{'}}^{2 -}, . . ., H_{s^{'}}^{n -}}$ . For i = 1, 2, . . . , m ; j = 1, 2, . . . , n), the two type of ideal solutions are given as follows: $\begin{matrix} H_{s}^{j +} = {max}_{i = 1}^{m} H_{s^{'}}^{ij +}}, H_{s}^{j -} = {min}_{i = 1}^{m} H_{s^{'}}^{ij -}} \end{matrix}$

Under the criteria C_j (j = 1, 2, . . . , n), we can get the value of $F (H_{s}^{ij}) (i = 1, 2, . . ., m)$ by the score function in Definition 3. According to Theorem 1, the order relationship between HFLTSs can be given as follows: if $F (H_{s}^{1}) > F (H_{s}^{2})$ , we have $H_{s}^{1} > H_{s}^{2}$ . So the ideal solution $H_{s}^{j +}$ and $H_{s}^{j -}$ can be obtained.

Step 3. Calculate the separation of each alternative between the hesitant fuzzy linguistic positive ideal solution $H^{+} = {H_{s}^{1 +}, H_{s}^{2 +}, . . ., H_{s}^{n +}}$ and hesitant fuzzy linguistic negative ideal solution $H^{-} = {H_{s}^{1 -}, H_{s}^{2 -}, . . ., H_{s}^{n -}}$ , respectively.

The distance measure between H_i (i = 1, 2, . . . , m) and H⁺ is: $D_{i}^{+} = \sum_{j = 1}^{n} D_{ω HFL}^{*} (H_{s^{'}}^{ij}, H^{+})$ ; the distance measure between H_i (i = 1, 2, . . . , m) and H^- is: $D_{i}^{-} = \sum_{j = 1}^{n} D_{ω HFL}^{*} (H_{s^{'}}^{ij}, H^{-})$ .

For the given alternative H_i (i = 1, 2, . . . , m), $\begin{matrix} \begin{matrix} - 255 {ptD}_{i}^{+} = \sum_{j = 1}^{n} D_{ω HFL}^{*} (H_{s^{'}}^{ij}, H^{+}) = 1 - S_{ω HFL}^{*} (H_{s^{'}}^{ij}, H_{s^{'}}^{+}) \\ = \sum_{j = 1}^{n} 1 - \frac{1}{2} [{Cos}_{ω HFL} (H_{s^{'}}^{ij}, H_{s^{'}}^{+}) + 1 - D_{ω HFL} (H_{s^{'}}^{ij}, H_{s^{'}}^{+})] = \sum_{j = 1}^{n} \frac{1}{2} \\ (1 - \frac{\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} (\frac{| δ_{l}^{ij} (x_{j}) |}{2 t + 1} . \frac{| δ_{l}^{+} (x_{j}) |}{2 t + 1}))}{{(\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{ij} (x_{j}) |}{2 t + 1})}^{2}) . \sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{+} (x_{j}) |}{2 t + 1})}^{2}))}^{\frac{1}{2}}} + (\sum_{j = 1}^{n} \frac{ω_{j}}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{ij} (x_{j}) - δ_{l}^{+} (x_{j}) |}{2 t + 1})}^{2})), \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} - 18 {ptD}_{i}^{-} = \sum_{j = 1}^{n} D_{ω HFL}^{*} (H_{s^{'}}^{ij}, H^{-}) = 1 - S_{ω HFL}^{*} (H_{s^{'}}^{ij}, H_{s}^{-}) \\ = \sum_{j = 1}^{n} 1 - \frac{1}{2} [{Cos}_{ω HFL} (H_{s^{'}}^{ij}, H_{s^{'}}^{-}) + 1 - D_{ω HFL} (H_{s^{'}}^{ij}, H_{s^{'}}^{-})] = \sum_{j = 1}^{n} \frac{1}{2} \\ (1 - \frac{\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} (\frac{| δ_{l}^{ij} (x_{j}) |}{2 t + 1} . \frac{| δ_{l}^{-} (x_{j}) |}{2 t + 1}))}{{(\sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{ij} (x_{j}) |}{2 t + 1})}^{2}) . \sum_{j = 1}^{n} (\frac{1}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{-} (x_{j}) |}{2 t + 1})}^{2}))}^{\frac{1}{2}}} + (\sum_{j = 1}^{n} \frac{ω_{j}}{L_{j}} \sum_{l = 1}^{L_{j}} {(\frac{| δ_{l}^{ij} (x_{j}) - δ_{l}^{-} (x_{j}) |}{2 t + 1})}^{2})) \end{matrix} \end{matrix}$

Step 4. Calculate the closeness coefficient Φ_i of each alternative H_i (i = 1, 2, . . . , m): $Φ_{i} = \frac{D_{i}^{-}}{D_{i}^{-} + D_{i}^{-}}$

Step 5. Rank the alternatives according to the overall closeness coefficient Φ_i, the greater value Φ_i is, the better alternative H_i will be.

5 Numerical example

In this section, we give a numerical example about a movie recommender system (adapted from Liao et al. [21]) to illustrate the feasibility of the TOPSIS method with proposed distance measure.

5.1 Background

A company intends to give ratings on five movies (H₁, H₂, H₃, H₄, H₅) with respect to the following criteria: story (C₁), acting (C₂), visuals (C₃) and direction (C₄). The weighing vector of four criteria is ω = (0.4, 0.2, 0.2, 0.2). Since these criteria are all qualitative, it is convenient and feasible for the decision makers to represent their evaluations in linguistic term set. The company uses the following LTS to evaluate the movies with respect to the criteria, which can be represented as: S′ = {s_-3 : terrible, s_-2 : very bad, s_-1 : bad, s_-2 : very bad, s_-1 : bad, s₀ : medium, s₁ : well, s₂ : very well, s₃ : perfect} . For example, the evaluation about the movie H₂ under the criterion C₁ is between medium and very well, then the hesitant fuzzy linguistic evaluation can be represented as {s₀, s₁, s₂}. The hesitant fuzzy

linguistic decision matrix $H = (H_{s}^{ij})_{5 \times 4}$ can be given in Table 1

Table 1
The hesitant fuzzy linguistic decision matrix provided by experts

C ₁ C ₂ C ₃ C ₄

H ₁ {s_-2, s_-1, s₀} {s₀, s₁} {s₀, s₁, s₂} {s₁, s₂}

H ₂ {s₀, s₁, s₂} {s₁, s₂} {s₀, s₁} {s₀, s₁, s₂}

H ₃ {s₂, s₃} {s₁, s₂, s₃} {s₁, s₂} {s₂}

H ₄ {s₀, s₁, s₂} {s_-1, s₀, s₁} {s₁, s₂, s₃} {s₁, s₂}

H ₅ {s_-1, s₀} {s₀, s₁, s₂} {s₀, s₁, s₂} {s₀, s₁}

C ₁	C ₂	C ₃	C ₄
H ₁	{s_-2, s_-1, s₀}	{s₀, s₁}	{s₀, s₁, s₂}	{s₁, s₂}
H ₂	{s₀, s₁, s₂}	{s₁, s₂}	{s₀, s₁}	{s₀, s₁, s₂}
H ₃	{s₂, s₃}	{s₁, s₂, s₃}	{s₁, s₂}	{s₂}
H ₄	{s₀, s₁, s₂}	{s_-1, s₀, s₁}	{s₁, s₂, s₃}	{s₁, s₂}
H ₅	{s_-1, s₀}	{s₀, s₁, s₂}	{s₀, s₁, s₂}	{s₀, s₁}

Step 1. Normalize the decision matrix.

As all the criteria are benefit-type criteria, we need not do anything in this step.

Step 2. Determine the hesitant fuzzy linguistic positive ideal solution H⁺ and the hesitant fuzzy linguistic negative ideal solution H^- respectively, which are given as follows: $\begin{matrix} H^{+} = {{s_{2}, s_{3}}, {s_{1}, s_{2}, s_{3}}, {s_{1}, s_{2}, s_{3}}, {s_{2}}} \end{matrix}$ $\begin{matrix} H^{-} = {{s_{- 2}, s_{- 1}, s_{0}}, {s_{- 1}, s_{0}, s_{1}}, {s_{0}, s_{1}}, {s_{0}, s_{1}}} . \end{matrix}$

Step 3. Calculate the distance measure $D_{ω HFL}^{*}$ $(H_{s}^{ij}, H^{+})$ and $D_{ω HFL}^{*} (H_{s}^{ij}, H^{-})$ for each alternative H_i (i = 1, 2, 3, 4, 5), which are obtained in Table 2.

Table 2

The distance measure of each alternative

$D_{i}^{+}$	$D_{i}^{-}$
H ₁	0.2820	0.1090
H ₂	0.1730	0.3659
H ₃	0.0217	0.3366
H ₄	0.1793	0.3426
H ₅	0.2821	0.1892

Step 4. Calculate the closeness coefficient Φ_i, which are obtained in Table 3.

Table 3

The closeness coefficient of each alternative

H ₁	H ₂	H ₃	H ₄	H ₅
Φ _i	0.2788	0.6790	0.9394	0.6564	0.4014

Step 5. Ranking the alternatives H_i according to the value of closeness coefficient Φ_i (i = 1, 2, 3, 4, 5).

We know that H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁, the best alternative is H₃.

5.2 Comparative analysis

In order to illustrate the feasibility and effectiveness of the proposed method, different methods are used to compare with the same numerical example in section 5.1. The comparison results are displayed in Table 4.

Table 4
Comparison of different methods

Ranking

Approach from Liao et al. [21] H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁

Approach from Liao et al. [30] H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁

Proposed approach based on $D_{ω HFL}^{*}$ H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁

Ranking
Approach from Liao et al. [21]	H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁
Approach from Liao et al. [30]	H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁
Proposed approach based on $D_{ω HFL}^{*}$	H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁

From Table 4, we can see that the proposed method produces the same ranking results as the existing methods, which means the method in this paper is feasible and effective. Compared with other methods, it has some advantages in solving multi-criteria decision making problems.

In Liao et al. [21], they proposed different distance measures between HFLTSs. Firstly, they obtain the hesitant fuzzy linguistic positive ideal solution and hesitant fuzzy linguistic negative ideal solution, then they calculate the distance measure between each alternative and the positive ideal solution, the distance measure between each alternative and the negative ideal solution, respectively. But the distance measures in Liao et al. [21] only consider the HFLTSs from the point view of algebra, which may lead to the decision information loss in decision process. As a result, the proposed method in this paper is superior to the method in Liao et al. [21], because it consider the distance measure not only from the point view of algebra but also from the point view of geometry, and it can avoid the disadvantage of the method in Liao et al. [21].

In Liao et al. [30], they proposed the hesitant fuzzy linguistic TOPSIS method based on the cosine distance measure, which aims at choosing alternative with the shortest distance from the positive ideal solution and the furthest distance from the negative ideal solution. The calculation results indicate that the best alternative is H₃ It is already known that the similarity measure Cos_HFL proposed by Liao et al. [30] is not a regular similarity measure, it cannot deal with the situation that the subscripts of two linguistic terms have the linear relationship, so the result obtained in Liao et al. [30] may be unreliable. The similarity measure proposed in the paper is constructed based on the cosine similarity measure Cos_HFL and the Euclidean distance measure, which can overcome the disadvantage of similarity measure Cos_HFL and improve the adaptability of HFLTSs in practical decision problems; Furthermore, the proposed distance measure $D_{ω HFL}^{*}$ can be applied more widely than the existing similarity measure Cos_HFL, because it shows a better performance in the field of decision making.

6 Conclusions and discussion

The similarity measure is widely used in multi-criteria decision making problems. Considering that the similarity measure Cos_HFL in Liao et al. [30] is not a regular similarity measure, we propose an innovative approach to construct a new similarity measure, which combines the cosine similarity measureCos_HFL and the Euclidean distance together. Then the corresponding distance measure with TOPSIS method is developed, and a numerical example is provided to illustrate the implementation and feasibility of the proposed method.

As a result, the characteristic of the proposed method can be summarized as follows:

The proposed similarity measure considers the similarity degree between two HFLTSs from the point view of algebra and geometry, and it also satisfies the axiom of similarity measure. The corresponding distance measure between HFLTSs is obtained according to the relationship between the similarity measure and distance measure, which can deal with the hesitant fuzzy decision effectively.

The TOPSIS method was developed to the proposed distance measure in hesitant fuzzy linguistic decision environment, which can improve the effectiveness of handling decision information and make the fuzzy set theory perfect.

As future studies, based on the decision results, they can be extended in the following directions:

In the proposed method, the subscript of linguistic terms is calculated directly in process of operations, it does not consider the difference between linguistic terms in different semantic situations. We know that the linguistic scale function [46, 47] can assign different semantic values to the linguistic terms in different circumstances, which can transform the linguistic information more flexibly. So we will apply linguistic scale function to deal with linguistic information under different semantic situations.

Linguistic large-scale group decision making problems are more and more common nowadays [48, 49], the distance measure is a popular and effective tool to make the decision, we will extend our work to deal with large-scale group decision making problems.

It would be very interesting to apply the proposed method to solve the real life decision making problems, such as the site selection for electrical power plant, the energy policy selection, the evaluation quality of the movie and the medical diagnosis, et al.

Conflict of interests

The authors declare no conflict of interests regarding the publication for the paper.

Data availability statement

No additional data are available.

Footnotes

Acknowledgments

This research is fully supported by a grant by National Natural Science foundation of Hunan (2017JJ2096), by National Social Science Foundation of China (15BTJ028), by the outstanding youth project of Hunan Education Department (1713092), by National Natural Science Foundation of Hunan (2018JJ3137), by Philosophy and Social Science Foundation of Hunan(18ZDB012),by Major projects of the National Social Science Foundation of China (17ZDA046).

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The new similarity measure and distance measure between hesitant fuzzy linguistic term sets and their application in multi-criteria decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Hesitant fuzzy set

2.2 Linguistic term set

2.3 Hesitant fuzzy linguistic term set

2.4 Existing distance measures and similarity measures between hesitant fuzzy linguistic term sets

3.1 Definitions and properties

3.2 Comparison with the existing similarity measure

4 The TOPSIS method with proposed distance measure

5 Numerical example

5.1 Background

Table 4 Comparison of different methods Ranking Approach from Liao et al. [21] H3 ≻ H2 ≻ H4 ≻ H5 ≻ H1 Approach from Liao et al. [30] H3 ≻ H2 ≻ H4 ≻ H5 ≻ H1 Proposed approach based on D ω HFL * H3 ≻ H2 ≻ H4 ≻ H5 ≻ H1

Conflict of interests

Data availability statement

Footnotes

Acknowledgments

References

Table 4
Comparison of different methods

Ranking

Approach from Liao et al. [21] H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁

Approach from Liao et al. [30] H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁

Proposed approach based on $D_{ω HFL}^{*}$ H₃ ≻ H₂ ≻ H₄ ≻ H₅ ≻ H₁