Hesitant fuzzy linguistic TOPSIS method using a possibility-based comparison approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets

Abstract

The purpose of this study is to develop a hesitant fuzzy linguistic TOPSIS (The technique for order preference by similarity to ideal solution) method with a possibility-based comparison approach for addressing multi-criteria decision-making (MCDM) problems within the environment of hesitant fuzzy linguistic term sets (HFLTSs). This paper firstly analyses the existing comparison methods for HFLTSs and develops a new possibility degree formula which can address the issues in the previous ones. Then, based on the possibilities of the HFLTS binary relations, this paper defines the possibility-based outranking index to determine hesitant fuzzy linguistic positive ideal and negative ideal solutions. Subsequently, this paper introduces the concept of possibility-based comparison indices to establish a possibility-based closeness coefficient of each alternative relative to the ideal solutions. Based on a possibility-based comparison approach with the ideal solutions, this paper develops a hesitant fuzzy linguistic TOPSIS method for handling MCDM problems in which both the evaluative ratings of alternatives and the importance weights of criteria are expressed by HFLTSs. Finally, a numerical example is furnished to verify the feasibility and practicality of the proposed method and a comparative analysis with the existing methods is provided to illustrate the effectiveness and advantages of the proposed method.

Keywords

Multi-criteria decision-making hesitant fuzzy linguistic terms set TOPSIS possibility-based closeness coefficient possibility-based comparison approach

1 Introduction

Hesitant fuzzy set (HFS), initially proposed by Torra [30], is a powerful tool to model the quantitative settings in which experts hesitate between several numerical values to assess an alternative in the process of multi-criteria decision-making (MCDM) [21 , 36]. However, the attribute values regarding alternatives are usually uncertain or fuzzy due to the increasing complexity of the socio-economic environment and the vagueness of the inherent subjective nature of human thinking; thus, it may be appropriate and sufficient to assess the information in a qualitative form rather than a quantitative form. To deal with such cases, Zadeh [40] introduced the fuzzy linguistic approach, which represents qualitative information as linguistic values by means of linguistic variables and has been extended to several different models, such as the linguistic model based on type-2 fuzzy sets [31, 39], 2-tuple fuzzy linguistic representation model [6 , 23], the proportional 2-tuple model [32], and so on [5, 10]. However, the fuzzy linguistic approach and its several variants still have the serious drawback that they assess a linguistic variable only by using a single linguistic term. However, a single term is not suitable or sufficient to represent human’s more comprehensive cognition. In a real-life MCDM situation, an expert may not easily use a single linguistic term to express the evaluative ratings of alternatives based on his/her knowledge, where he/she would like to use several linguistic terms simultaneously or would like to look for a more complex linguistic expression as the assessment values of alternatives for each criterion. In order to deal with such situations, Rodriguez et al. [28] developed hesitant fuzzy linguistic term sets (HFLTSs) to increase the flexibility and richness of linguistic elicitation based on the fuzzy linguistic approach and HFSs. HFLTSs allow the DMs to hesitate about several possible linguistic preference values to assess a linguistic variable. In comparison with the fuzzy linguistic approach which only assesses a single linguistic term for the linguistic variables, HFLTSs can address situations in which the DMs think of several possible linguistic terms at the same time or richer expressions than a single term for an indicator, alternative, variable, etc. [13–16 , 42].

Recently, the studies on MCDM problems within the environment of HFLTSs have made great progresses. For example, Beg and Rashid [1] proposed the distance between two HFLTSs and developed the modified TOPSIS method for HFLTSs. Liao et al. [12] proposes a family of distance and similarity measures between two HFLTSs, based on which the satisfaction degrees for different alternatives are established and are then used to rank alternatives in MCDM. Liao and Xu [11] proposed a family of cosine distance and similarity measures for HFLTSs from the geometric point of view, such as the cosine distance and similarity measures, the weighted cosine distance and similarity measures, the order weighted cosine distance and similarity measures, and the continuous cosine distance and similarity measures. Then, based on the proposed hesitant fuzzy linguistic cosine distance measures, the cosine distance-based HFL-TOPSIS method and the cosine-distance-based HFL-VIKOR method are developed to dealing with hesitant fuzzy linguistic MCDM problems. Lee and Chen [9] proposed a new fuzzy decision making method based on the proposed likelihood-based comparison relations of HFLTSs and the proposed hesitant fuzzy linguistic operators of HFLTSs. Lee and Chen [8] defined the 0-cut of HFLTSs, utilized the membership functions of the linguistic terms, and finally proposed thecomparison relations of HFLTSs. Wei et al. [38] defined operations on HFLTSs and gave possibility degree formulas for comparing HFLTSs. They presented the hesitant fuzzy linguistic weighted averaging (LWA) operator and the hesitant fuzzy linguistic-weighted OWA (LOWA) operator and used these operators and the comparison methods to deal with MCDM problems with different situations in which importance weights of criteria or experts are known or unknown. Liu and Rodríguez [17] introduced a fuzzy envelope for HFLTS whose representation is a fuzzy membership function obtained from aggregating the fuzzy membership functions of the linguistic terms of the HFLTS.

However, the above hesitant fuzzy linguistic MCDM methods have some drawbacks, which are clarified as follows:

The existing comparison methods in [8 , 38] are not suitable to compare two HFLTSs in some situations. Concretely, Rodríguez et al.’s envelope-based comparison [28, 29] and Lee and Chen’s likelihood-based comparison relations of HFLTSs [8, 9] may yield unreasonable results when two HFLTSs have some common elements or each of two HFLTSs has only one element (see a further analysis in Example 3.1). Although Wei et al.’s possibility degree formulas [38] for comparing HFLTSs can address this issue, Wei et al.’s formulas are too complicated and are not convenient to use.

Liao and Xu’s method [15] and Liao et al.’s HFL-TOPSIS method [16] were developed on the basis of the proposed distance measures. However, both of Liao and Xu’s distance measures [11] and Liao et al.’s distance measures [12] are based on the assumption that each pair of HFLTSs has equal length. This is not in accordance with real cases because it is impossible to make sure that all HFLTSs have equal length. If the lengths of different HFLTSs are different, a normalization method [43] is used to fill some artificial values into the short HFLTS until both of them have the same length. Although this method can obtain their distance measure, the rationality can not be guaranteed, because different results are obtained by adding different values in the shorter HFLTS. In addition, it should be stated that filling some artificial values into a HFLTS would change the information of the original HFLTS. Thus, such an approach is less well justified theoretically and less reliable practically.

The likelihood-based hesitant fuzzy linguistic decision makings proposed in [8 , 28, 38] cannot distinguish the preference order of alternatives in some situations.

The existing hesitant fuzzy linguistic MCDM methods in [8 , 38] are only suitable to deal with problems in which the importance weights of criteria take the form of numerical values and are determined in advance, and they fail in addressing MCDM problems in which the importance weights of criteria take the form of HFLTSs.

To overcome the aforesaid drawbacks in [8 , 38], we develop a new possibility degree formula that can avoid the disadvantages of the previous possibility degree formulas. This new possibility degree formula is more simplified and does not need the associated two HFLTSs to have the same length. Instead of the distance measures in the classical TOPSIS method [7] and the existing hesitant fuzzy linguistic TOPSIS methods in the literature [11, 12], the concept of possibilities is introduced to establish a possibility-based closeness coefficient. Then, inspired by Chen [2, 3], this paper presents a hesitant fuzzy linguistic TOPSIS method with a possibility-based comparison approach for addressing the MCDM problem in which both the assessments of alternatives on criteria and the weights of criteria are expressed by HFLTSs. Finally, through an illustrative example of car evaluation problem, the proposed possibility-based TOPSIS method is easy to implement and can overcome the drawbacks of the existing hesitant fuzzy linguistic decision-making methods in the literature.

The rest of paper is organized as follows. Section 2 reviews some basic concepts related to HFLTSs. Section 3 analyses the drawbacks of some existing comparison methods for HFLTSs and then proposes a new possibility degree formula for ranking HFLTSs. Section 4 first formulates a MCDM problem within the context of HFLTSs. Then, Section 4 develops a hesitant fuzzy linguistic TOPSIS method for addressing MCDM problems in a hesitant fuzzy linguistic framework involving HFLTS importance weights. Section 5 provides a car evaluation problem to demonstrate the feasibility and the applicability of the proposed method. Section 6 conducts a comparative analysis with other hesitant fuzzy linguistic MCDM methods to illustrate the advantages of the proposed method. Section 7 proposes the conclusions of this paper.

2 Preliminaries

2.1 Linguistic term sets

Let S ={ s₀, s₁, ⋯ , s_2τ } be a finite and totally ordered linguistic term set with odd cardinality and the midterm representing and assessment of “approximately” 0.5, where s_i represents a possible value for a linguistic variable, and the following characteristics should be satisfied [28 , 40]:

There is a negation operator: neg (s_i) = s_2τ-i;

The set is ordered: if i ≥ j, then s_i ≥ s_j;

There exists a maximization operator: max(s_i, s_j) = s_i if s_i ≥ s_j;

There exists a minimization operator: min(s_i, s_j) = s_i if s_i ≤ s_j.

2.2 Hesitant fuzzy linguistic term sets

Similar to the situations described and managed by hesitant fuzzy sets (HFSs) in [30], where the DMs hesitant about several possible values to assess an alternative, in the qualitative setting it may occur that the DMs hesitate between several linguistic terms to assess a linguistic variable. Hence, motivated by the characteristic of HFSs, Rodríguez et al. [28] proposed a concept of hesitant fuzzy linguistic term sets (HFLTSs), which are utilized to provide a linguistic and computational basis to increase the richness of linguistic elicitation.

Definition 2.1. [28] Let S ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, then, a hesitant fuzzy linguistic term set (HFLTS), H_S on S, is an ordered finite subset of the consecutive linguistic terms of S.

Definition 2.2. [28] Let S ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, and H_S, $H_{S}^{1}$ and $H_{S}^{2}$ be three arbitrary HFLTSs on S.

The complement $H_{S}^{c}$ of H_S is defined as: $H_{S}^{c} = S - H_{S} = {s_{i} | s_{i} \in S, s_{i} \notin H_{S}}$ ;

The intersection $H_{S}^{1} \cap H_{S}^{2}$ of $H_{S}^{1}$ and $H_{S}^{2}$ is defined as: $H_{S}^{1} \cap H_{S}^{2} = {s_{i} | s_{i} \in H_{S}^{1} and s_{i} \in H_{S}^{2}}$ , The union $H_{S}^{1} \cup H_{S}^{2}$ of $H_{S}^{1}$ and $H_{S}^{2}$ is defined as: $H_{S}^{1} \cup H_{S}^{2} = {s_{i} | s_{i} \in H_{S}^{1} or s_{i} \in H_{S}^{2}}$ ;

The upper bound H_{S
⁺} of H_S is defined as: H_{S
⁺} = max{ s_i|s_i ∈ H_S }; The lower bound H_{S
^-} of H_S is defined as: H_{S
^-} = min {s_i|s_i ∈ H_S};

The envelope env (H_S) of H_S is defined as: env (H_S) = [H_{S
^-}, H_{S
⁺}].

Definition 2.3. [34] Let S ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, and then the function of the position index i for the element s_i in S can be defined as follows: $\begin{matrix} I : S \to Z \\ s_{i} \to i \end{matrix}$

Therefore, the number of elements in the HFLTS H_S is l (H_S) = I (H_{S
⁺}) - I (H_{S
^-}) + 1.

2.3 Transforming linguistic expressions into HFLTSs

Although the HFLTS can be used to elicit several linguistic values for a linguistic variable, it is still not similar to the human way of thinking and reasoning. Thus, Rodríguez et al. [28] further proposed a context-free grammar to generate simple but rich linguistic expressions that are more similar to the human expressions and can be easily represented by means of HFLTSs.

Definition 2.4. [28] Let G_H = (V_N, V_T, I, P) be a context-free grammar and let S ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, where V_N denotes a set of nonterminal symbols, V_T denotes a set of terminal symbols, I denotes the starting symbol, and P denotes the production rules, shown as follows: $\begin{matrix} V_{N} & = & {〈 primary term 〉, 〈 composite term 〉, \\ 〈 unary relation 〉, 〈 binary relation 〉, \\ 〈 conjunction 〉}, \\ V_{T} & = & {lower than, greater than, at least, at most, \\ between, and, s_{0}, s_{1}, \dots, s_{2 τ}}, \end{matrix}$

$\begin{matrix} I & \in & V_{N} \\ P & = & {I : : = 〈 primary term 〉 | 〈 composite term 〉, \\ 〈 composite term 〉 : : = 〈 unary relation 〉 \\ 〈 primary term 〉 | 〈 binary relation 〉 \\ 〈 primary term 〉〈 conjunction 〉〈 primary term 〉, \\ 〈 primary term 〉 : : = s_{0} | s_{1} | \dots | s_{2 τ}, \\ 〈 unary relation 〉 : : = lower than | greater than, \\ 〈 binary relation 〉 : : = between, \\ 〈 conjunction 〉 : : = and} \end{matrix}$

Definition 2.5. [28] Let S ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, and E_{G
_H} be a function that transforms the linguistic expression ll, obtained by a context-free grammar G_H, into a HFLTS H_S: $E_{G_{H}} : ll \to H_{S}$ (1)

The linguistic expressions generated by production rules can be transformed into a HFLTS in different ways according to their meaning:

E_{G
_H} (s_i) ={ s_i|s_i ∈ S },

E_{G
_H} (lowerthan s_i) = {s_j|s_j ∈ S and s_j < s_i},

E_{G
_H} (greaterthan s_i) = {s_j|s_j ∈ S and s_j > s_i},

E_{G
_H} (atmost s_i) ={ s_j|s_j ∈ S and s_j ≤ s_i },

E_{G
_H} (atleast s_i) ={ s_j|s_j ∈ S and s_j ≥ s_i },

E_{G
_H} (between s_i and s_j) = {s_k|s_k ∈ S ands_i ≤ s_k ≤ s_j}.

3 A new possibility-degree formula for HFLTSs

3.1 Drawbacks of some existing comparison methods for HFLTSs

Let S ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, and $H_{S}^{1}$ and $H_{S}^{2}$ be two arbitrary HFLTSs on S. Rodríguez et al. [28] introduced the envelopes of HFLTSs and proposed a method to compare HFLTSs, shown as below:

$p_{R} (H_{S}^{1} \geq H_{S}^{2}) = \frac{max (0, I (H_{S^{+}}^{1}) - I (H_{S^{-}}^{2})) - max (0, I (H_{S^{-}}^{1}) - I (H_{S^{+}}^{2}))}{(I (H_{S^{+}}^{1}) - I (H_{S^{-}}^{1})) + (I (H_{S^{+}}^{2}) - I (H_{S^{-}}^{2}))}$ (2)

Lee and Chen [9] proposed the concept of the possibility-based comparison relation between two HFLTSs $H_{S}^{1}$ and $H_{S}^{2}$ as follows:

$p_{L} (H_{S}^{1} \geq H_{S}^{2}) = max {1 - max (\frac{I (H_{S^{+}}^{2}) - I (H_{S^{-}}^{1})}{(I (H_{S^{+}}^{1}) - I (H_{S^{-}}^{1})) + (I (H_{S^{+}}^{2}) - I (H_{S^{-}}^{2}))}, 0), 0}$ (3)

The following example shows that the aforementioned possibility-degree formulas (2) and (3) may be inappropriate for comparison among HFLTSs.

Example 3.1. Given two HFLTSs $H_{S}^{1} = {s_{3}, s_{4}}$ and $H_{S}^{2} = {s_{1}, s_{2}, s_{3}}$ , as per Equations (2 and 3), the possibility degree of $H_{S}^{1}$ over $H_{S}^{2}$ can be calculated as $p_{R} (H_{S}^{1} \geq H_{S}^{2}) = p_{L} (H_{S}^{1} \geq H_{S}^{2}) = 1$ . Thus, $H_{S}^{1}$ is absolutely superior to $H_{S}^{2}$ . However, this result may be inconsistent with common sense because it seems to be unreasonable to judge one HFLTS is absolutely superior to another if these two HFLTSs have some common elements. It is noted that s₃ is the possible linguistic term for $H_{S}^{1}$ and $H_{S}^{2}$ , so we could not say $H_{S}^{1}$ is absolutely superior to $H_{S}^{2}$ .

Additionally, if $H_{S}^{3} = {s_{6}}$ and $H_{S}^{4} = {s_{5}}$ , then $(I (H_{S^{+}}^{3}) - I (H_{S^{-}}^{3})) + (I (H_{S^{+}}^{4}) - I (H_{S^{-}}^{4})) = 0$ . Therefore, we fail to calculate the possibility degree of $H_{S}^{3}$ and $H_{S}^{4}$ via the Equations (2 and 3).

3.2 A new possibility-degree formula for HFLTSs

Definition 3.1. Let S ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, and $H_{S}^{1}$ and $H_{S}^{2}$ be two arbitrary HFLTSs on S, then the possibility degrees of $H_{S}^{1}$ being not less than $H_{S}^{2}$ is defined as

$p (H_{S}^{1} \geq H_{S}^{2}) = max {1 - max (\frac{I (H_{S^{+}}^{2}) - I (H_{S^{-}}^{1}) + 1}{l (H_{S}^{1}) + l (H_{S}^{2})}, 0), 0}$ (4)

Theorem 3.1. $p (H_{S}^{1} \geq H_{S}^{2}) + p (H_{S}^{2} \geq H_{S}^{1}) = 1$ ; especially, if $H_{S}^{1} = H_{S}^{2}$ , then $p (H_{S}^{1} \geq H_{S}^{2}) = p (H_{S}^{2} \geq H_{S}^{1}) = 0.5$ .

By the above theorem, we give the following definition.

Definition 3.2. If $p (H_{S}^{1} \geq H_{S}^{2}) > p (H_{S}^{2} \geq H_{S}^{1})$ , i.e., $p (H_{S}^{1} \geq H_{S}^{2}) > 0.5$ , then we say that $H_{S}^{1}$ is superior to $H_{S}^{2}$ with the degree of $p (H_{S}^{1} \geq H_{S}^{2})$ , denoted by $H_{S}^{1} ≻^{p (H_{S}^{1} \geq H_{S}^{2})} H_{S}^{2}$ . In this case, we also call that $H_{S}^{2}$ is inferior to $H_{S}^{1}$ with the degree of $p (H_{S}^{1} \geq H_{S}^{2})$ , denoted by $H_{S}^{2} ≺^{p (H_{S}^{1} \geq H_{S}^{2})} H_{S}^{1}$ .

If $p (H_{S}^{1} \geq H_{S}^{2}) = 1$ , then we call that $H_{S}^{1}$ is absolutely superior to $H_{S}^{2}$ , or $H_{S}^{2}$ is absolutely inferior to $H_{S}^{1}$ , denoted by $H_{S}^{1} ≻_{s} H_{S}^{2}$ or $H_{S}^{1} ≺_{s} H_{S}^{2}$ .

If $p (H_{S}^{1} \geq H_{S}^{2}) = p (H_{S}^{2} \geq H_{S}^{1}) = 0.5$ , then we say that $H_{S}^{1}$ is indifferent with $H_{S}^{2}$ , denoted by $H_{S}^{1} \sim H_{S}^{2}$ .

Theorem 3.2.LetS ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, and $H_{S}^{1}$ and $H_{S}^{2}$ be two arbitrary HFLTSs onS. Then

$p (H_{S}^{1} \geq H_{S}^{2}) > 0.5$ if and only if $I (H_{S^{+}}^{1}) + I (H_{S^{-}}^{1}) > I (H_{S^{+}}^{2}) + I (H_{S^{-}}^{2})$ ;

$p (H_{S}^{1} \geq H_{S}^{2}) = 0.5$ if and only if $I (H_{S^{+}}^{1}) + I (H_{S^{-}}^{1}) = I (H_{S^{+}}^{2}) + I (H_{S^{-}}^{2})$ ;

$p (H_{S}^{1} \geq H_{S}^{2}) < 0.5$ if and only if $I (H_{S^{+}}^{1}) + I (H_{S^{-}}^{1}) < I (H_{S^{+}}^{2}) + I (H_{S^{-}}^{2})$ .

The following transitivity can be derived from Theorem 3.2.

Theorem 3.3.LetS ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, $H_{S}^{1}$ , $H_{S}^{2}$ and $H_{S}^{3}$ be three HFLTSs on S.

If $p (H_{S}^{1} \geq H_{S}^{2}) > 0.5$ and $p (H_{S}^{2} \geq H_{S}^{3}) \geq 0.5$ , or $p (H_{S}^{1} \geq H_{S}^{2}) \geq 0.5$ and $p (H_{S}^{2} \geq H_{S}^{3}) > 0.5$ , then $p (H_{S}^{1} \geq H_{S}^{3}) > 0.5$ .

If $p (H_{S}^{1} \geq H_{S}^{2}) = 0.5$ and $p (H_{S}^{2} \geq H_{S}^{3}) = 0.5$ , then $p (H_{S}^{1} \geq H_{S}^{3}) = 0.5$ .

Theorem 3.4.LetS ={ s₀, s₁, ⋯ , s_2τ } be a linguistic term set, $H_{S}^{1}$ , $H_{S}^{2}$ and $H_{S}^{3}$ be three HFLTSs onS. If $p (H_{S}^{1} \geq H_{S}^{2}) = 1$ , then $p (H_{S}^{1} \geq H_{S}^{3}) \geq p (H_{S}^{2} \geq H_{S}^{3})$ .

Example 3.2. (Continued with Example 3.1) From Equation (4), we can see that $H_{S}^{1}$ is absolutely superior to $H_{S}^{2}$ if and only if $I (H_{S^{+}}^{2}) < I (H_{S^{-}}^{1})$ . In Example 3.1, $H_{S}^{1} = {s_{3}, s_{4}}$ and $H_{S}^{2} = {s_{1}, s_{2}, s_{3}}$ . Then, by Equation (4), we can obtain $p (H_{S}^{1} \geq H_{S}^{2}) = \frac{4 - 1 + 1}{2 + 3} = 0.8$ . The comparison result implies $H_{S}^{1}$ is not absolutely superior to $H_{S}^{2}$ and consistent with our analysis in Example 3.1.

Additionally, in Example 3.1, $H_{S}^{3} = {s_{6}}$ and $H_{S}^{4} = {s_{5}}$ , then, by Equation (4), we have $p (H_{S}^{1} \geq H_{S}^{2}) = 1$ . This result is consistent with our intuition and overcomes the drawback of the possibility-degree formulas (2) and (3).

4 Possibility-based hesitant fuzzy linguistic TOPSIS method

In this section, we first describe a decision environment based on HFLTSs for MCDM problems involving the HFLTS importance weights Then, based on the possibility-based comparison approach with the positive and negative ideals, this section proposes a possibility-based hesitant fuzzy linguistic TOPSIS method to handle MCDM problems involving HFLTS importance weights.

4.1 Hesitant fuzzy linguistic decision context

Consider the following MCDM problem in the hesitant fuzzy linguistic context, in which the ratings of alternative evaluations and the importance weights of criteria are expressed as HFLTSs. Assume that there are m alternatives, denoted by Z ={ z₁, z₂, ⋯ , z_m }. Each alternative is assessed by means of n criteria, denoted by C ={ c₁, c₂, ⋯ , c_n }. In general, the criterion set C can be divided into two subsets C_I and C_II, where C_I denotes a collection of benefit criteria (i.e., a larger value of c_j indicates a greater preference), C_II denotes a collection of cost criteria (i.e., a smaller value of c_j indicates a greater preference), and where C_I∩ C_II = ∅ and C_I ∪ C_II = C.

The decision maker might feel much easier and are more willing to give their assessments by providing some linguistic expressions or sentences. Assume that there is a linguistic term set S ={ s₀, s₁, ⋯ , s_2τ }, together with a context-free grammar G_H that produces the linguistic expressions ll (z_i, c_j) based on comparative linguistic terms, is employed to assess the alternative z_i ∈ Z for the criterion c_j ∈ C. Such linguistic expressions can be transformed into HFLTSs by means of the transformation function E_{G
_H}. Hence, a judgment matrix with HFLTS information will be obtained as follows: $ℍ_{S} = [\begin{matrix} H_{S}^{11} & H_{S}^{12} & \dots & H_{S}^{1 n} \\ H_{S}^{21} & H_{S}^{22} & \dots & H_{S}^{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ H_{S}^{m 1} & H_{S}^{m 2} & \dots & H_{S}^{mn} \end{matrix}]$ (5) where $H_{S}^{ij}$ (i = 1, 2, ⋯ , m; j = 1, 2, ⋯ , n) is a HFLTS, denoting the hesitant fuzzy linguistic degree that the alternative z_i ∈ Z satisfies the criterionc_j ∈ C.

This paper explores MCDM problems involving HFLTS importance weights. Let a HFLTS $W_{S}^{j}$ denote the importance weight of criterion c_j ∈ C.

4.2 Proposed TOPSIS method involving HFLTS importance weights

Consider an MCDM problem in which both the evaluative ratings of alternatives and the importance weights of criteria are expressed by HFLTSs.

Let $H_{S}^{ij}$ and $H_{S}^{i^{'} j}$ be the evaluative ratings of the alternatives z_i and z_i′, respectively, with respect to the criterion c_j ∈ C. This paper uses the possibility $p (H_{S}^{ij} \geq H_{S}^{i^{'} j})$ of the HFLTS binary relationship $H_{S}^{ij} \geq H_{S}^{i^{'} j}$ to define the possibility-based outranking index of $H_{S}^{ij}$ . In general, alternatives are evaluated according to different criteria that must be maximized (i.e., benefit criteria) or minimized (i.e., cost criteria). In case of c_j ∈ C_I, $p (H_{S}^{ij} \geq H_{S}^{i^{'} j})$ indicates the possibility that $H_{S}^{ij}$ is better than or equal to $H_{S}^{i^{'} j}$ according to the principle of “the higher, the better.” In contrast, in the case of c_j ∈ C_II, $p (H_{S}^{ij} \geq H_{S}^{i^{'} j})$ represents the possibility that $H_{S}^{ij}$ is worse than or equal to $H_{S}^{i^{'} j}$ because of the principle of “the lower, the better.” As indicated in (iii) of Definition 3.1, it is known that $p (H_{S}^{i^{'} j} \geq H_{S}^{ij}) = 1 - p (H_{S}^{ij} \geq H_{S}^{i^{'} j})$ . Consequently, $p (H_{S}^{i^{'} j} \geq H_{S}^{ij})$ can be deemed the possibility that $H_{S}^{ij}$ is better than or equal to $H_{S}^{i^{'} j}$ with respect to c_j ∈ C_II. Following the discussion described, this paper presents the possibility-based outranking index $\bar{p} (H_{S}^{ij})$ of the HFLTS evaluative ratings $H_{S}^{ij}$ as shown below.

Definition 4.1. Let $p (H_{S}^{ij} \geq H_{S}^{i^{'} j})$ be the possibility of a HFLTS binary relationship $H_{S}^{ij} \geq H_{S}^{i^{'} j}$ , where $H_{S}^{ij}$ and $H_{S}^{i^{'} j}$ denote two HFLTS evaluative ratings of the alternatives z_i, z_i′ ∈ Z with respect to the criterion c_j ∈ C. The possibility-based outranking index $\bar{p} (H_{S}^{ij})$ of the HFLTS evaluative ratings $H_{S}^{ij}$ is defined as follows:

$\begin{matrix} \bar{p} (H_{S}^{ij}) \\ = {\begin{matrix} \frac{1}{m (m - 1)} (\sum_{i^{'} = 1}^{m} p (H_{S}^{ij} \geq H_{S}^{i^{'} j}) + \frac{m}{2} - 1) if c_{j} \in C_{I}, \\ \frac{1}{m (m - 1)} (\sum_{i^{'} = 1}^{m} p (H_{S}^{i^{'} j} \geq H_{S}^{ij}) + \frac{m}{2} - 1) if c_{j} \in C_{II} . \end{matrix} \end{matrix}$ (6)

Theorem 4.1.The possibility -based outranking index $\bar{p} (H_{S}^{ij})$ of a HFLTS evaluative rating $H_{S}^{ij}$ (forz_i ∈ Zandc_j ∈ C) satisfies the following properties:

$0 \leq \bar{p} (H_{S}^{ij}) \leq 1$ ;

$\sum_{i = 1}^{m} \bar{p} (H_{S}^{ij}) = 1$ .

Then, in regard to each criterion c_j ∈ C, the ranking order of all m alternatives can be subsequently obtained according to the descending order of the $\bar{p} (H_{S}^{ij})$ values.

The positive ideal and negative ideal solutions are regarded as the points of reference in the TOPSIS method. Denote $H_{S}^{+}$ as the hesitant fuzzy linguistic positive ideal solution concerning the MCDM problem based on HFLTSs, where

$H_{S}^{+} = {〈 c_{j}, H_{S}^{+ j} 〉 | c_{j} \in C}$ (7) where $H_{S}^{+ j} \in {H_{S}^{1 j}, H_{S}^{2 j}, \dots, H_{S}^{mj}}$ and $\bar{p} (H_{S}^{+ j}) = max_{i = 1, 2, \dots, m} {\bar{p} (H_{S}^{ij})}$ .

The hesitant fuzzy linguistic negative ideal solution, denoted as $H_{S}^{-}$ , is identified as follows:

$H_{S}^{-} = {〈 c_{j}, H_{S}^{- j} 〉 | c_{j} \in C}$ (8) where $H_{S}^{- j} \in {H_{S}^{1 j}, H_{S}^{2 j}, \dots, H_{S}^{mj}}$ and $\bar{p} (H_{S}^{- j}) = min_{i = 1, 2, \dots, m} {\bar{p} (H_{S}^{ij})}$ .

This paper establishes the concept of possibility-based comparison indices and then proposes a possibility-based closeness coefficient in the proposed hesitant fuzzy linguistic TOPSIS method. To do so, this paper first calculates the possibility of a hesitant fuzzy linguistic binary relation between the evaluative ratings of every alternative and the ideal solutions.

For each alternative z_i ∈ Z and the benefit criterion c_j ∈ C_I, we calculate the possibility $p (H_{S}^{+ j} \geq H_{S}^{ij})$ of $H_{S}^{ij}$ not being larger than $H_{S}^{+ j}$ and the possibility $p (H_{S}^{ij} \geq H_{S}^{- j})$ of $H_{S}^{ij}$ not being smaller than $H_{S}^{- j}$ . The alternative z_i has a good performance with respect to the benefit criterion c_j when the HFLTS evaluation $H_{S}^{ij}$ has a low possibility of being inferior to $H_{S}^{+ j}$ and a high possibility of being superior to $H_{S}^{- j}$ . For each alternative z_i ∈ Z and the cost criterion c_j ∈ C_II, we calculates $p (H_{S}^{ij} \geq H_{S}^{+ j})$ of $H_{S}^{ij}$ not being smaller than $H_{S}^{+ j}$ and $p (H_{S}^{- j} \geq H_{S}^{ij})$ of $H_{S}^{ij}$ not being larger than $H_{S}^{- j}$ . The alternative z_i performs well in terms of the cost criterion c_j, if the HFLTS evaluation $H_{S}^{ij}$ has a low possibility of being farther from $H_{S}^{+ j}$ and a high possibility of being farther from $H_{S}^{- j}$ .

Under these considerations, we determine the possibility-based comparison index $p ({\tilde{h}}_{ij} ℝ {\tilde{h}}_{+ j})$ of $H_{S}^{ij}$ relative to $H_{S}^{+ j}$ by using the possibility $p (H_{S}^{+ j} \geq H_{S}^{ij})$ of a hesitant fuzzy linguistic binary relation of $H_{S}^{ij}$ not being larger than $H_{S}^{+ j}$ (i.e., $H_{S}^{ij} \leq H_{S}^{+ j}$ ) for c_j ∈ C_I and the possibility $p (H_{S}^{ij} \geq H_{S}^{+ j})$ of a hesitant fuzzy linguistic binary relation of $H_{S}^{ij}$ not being smaller than $H_{S}^{+ j}$ (i.e., $H_{S}^{ij} \geq H_{S}^{+ j}$ ) for c_j ∈ C_II, given in the following manner:

Definition 4.2. Let $H_{S}^{ij}$ and $H_{S}^{+ j}$ be the evaluative ratings of the alternative z_i ∈ Z and the positive ideal solution $H_{S}^{+}$ , respectively, with respect to criterion c_j ∈ C. Then the possibility-based comparison index $p (H_{S}^{ij} ℝ H_{S}^{+ j})$ of $H_{S}^{ij}$ relative to $H_{S}^{+ j}$ is defined as follows: $p (H_{S}^{ij} ℝ H_{S}^{+ j}) = {\begin{matrix} p (H_{S}^{+ j} \geq H_{S}^{ij}), & if c_{j} \in C_{I}, \\ p (H_{S}^{ij} \geq H_{S}^{+ j}), & if c_{j} \in C_{II} . \end{matrix}$ (9)

On the other hand, we define the possibility-based comparison index $L (H_{S}^{ij} ℝ H_{S}^{- j})$ of $H_{S}^{ij}$ relative to $H_{S}^{- j}$ via the possibility $L (H_{S}^{ij} \geq H_{S}^{- j})$ of a hesitant fuzzy linguistic binary relation of $H_{S}^{ij}$ not being smaller than $H_{S}^{- j}$ (i.e., $H_{S}^{ij} \geq H_{S}^{- j}$ ) for c_j ∈ C_I and the possibility $L (H_{S}^{- j} \geq H_{S}^{ij})$ of a hesitant fuzzy linguistic binary relation of $H_{S}^{ij}$ not being larger than $H_{S}^{- j}$ (i.e., $H_{S}^{ij} \leq H_{S}^{- j}$ ) for c_j ∈ C_II.

Definition 4.3. Let $H_{S}^{ij}$ and $H_{S}^{- j}$ be the evaluative ratings of the alternative z_i ∈ Z and the negative ideal solution $H_{S}^{-}$ , respectively, with respect to criterion c_j ∈ C. Then the possibility-based comparison index $L (H_{S}^{ij} ℝ H_{S}^{- j})$ of $H_{S}^{ij}$ relative to $H_{S}^{- j}$ is defined as follows: $p (H_{S}^{ij} ℝ H_{S}^{- j}) = {\begin{matrix} p (H_{S}^{ij} \geq H_{S}^{- j}), & if c_{j} \in C_{I}, \\ p (H_{S}^{- j} \geq H_{S}^{ij}), & if c_{j} \in C_{II} . \end{matrix}$ (10)

We utilize the concept of the possibility-based outranking index to transform the HFLTS importance weights into the exact importance weights.

Definition 4.4. Let $\bar{p} (W_{S}^{j})$ denotes the possibility-based outranking index of the HFLTS importance weight $W_{S}^{j}$ . $\bar{p} (W_{S}^{j})$ is defined by

$\begin{matrix} \bar{p} (W_{S}^{j}) \\ = \frac{1}{n (n - 1)} (\sum_{j^{'} = 1}^{n} p (W_{S}^{j} \geq W_{S}^{j^{'}}) + \frac{n}{2} - 1) \end{matrix}$ (11)

Theorem 4.2.The possibility -based outranking index $\bar{p} (W_{S}^{j})$ of the HFLTS importance weight $W_{S}^{j}$ satisfies the following properties:

$0 \leq \bar{p} (W_{S}^{j}) \leq 1$ ;

$\sum_{j = 1}^{n} \bar{p} (W_{S}^{j}) = 1$ .

Combining the exact importance weights $(\bar{p} (W_{S}^{1}), \bar{p} (W_{S}^{2}), \dots, \bar{p} (W_{S}^{n}))$ of the various criteria, this paper employs the indices $p (H_{S}^{ij} ℝ H_{S}^{+ j})$ and $p (H_{S}^{ij} ℝ H_{S}^{- j})$ to establish a possibility-based closeness coefficient for ranking all of the m alternatives. Let LCC_i denote the possibility-based closeness coefficient for the alternative z_i ∈ Z and it is given as follows:

${LCC}_{i} = \frac{\sum_{j = 1}^{n} [p (H_{S}^{ij} ℝ H_{S}^{- j}) \cdot \bar{p} (W_{S}^{j})]}{\sum_{j = 1}^{n} [p (H_{S}^{ij} ℝ H_{S}^{+ j}) \cdot \bar{p} (W_{S}^{j})] + \sum_{j = 1}^{n} [p (H_{S}^{ij} ℝ H_{S}^{- j}) \cdot \bar{p} (W_{S}^{j})]}$ (12) where 0 ≤ LCC_i ≤ 1. The higher the possibility-based closeness coefficient LCC_i, the better the alternative z_i. Therefore, the m alternatives can be ranked according to the decreasing order of LCC_i for all z_i ∈ Z. Then, we choose the best alternative with the maximum value among all of the LCC_i values.

To sum up all the above analysis, the proposed possibility-based TOPSIS method for solving an MCDM problem within the hesitant fuzzy linguistic decision environment involving the HFLTS importance weights can be summarized in the following sequence of steps.

Step 1. Formulate a MCDM problem. Specify the alternative set Z ={ z₁, z₂, ⋯ , z_m } and the criterion set C ={ c₁, c₂, ⋯ , c_n }, which is divided into two subsets C_I and C_II, where C_I∩ C_II = ∅, C_I ∪ C_II = C.

Step 2. Define the semantics and syntax of the linguistic term sets for each criterion. The expert then gives the linguistic evaluation values for the alternatives over the criteria and for importance weights of the criteria according to these semantics and syntax in expressions.

Step 3. Define the context free grammar G_H, then transform the initial linguistic expressions into the HFLTS evaluative rating $H_{S}^{ij}$ for the alternative z_i ∈ Z with respect to the criterion c_j ∈ C and the HFLTS importance weight $W_{S}^{j}$ for the criterion c_j ∈ C via the transformation function E_{G
_H}.

Step 4. Identify the hesitant fuzzy linguistic positive ideal and negative ideal solutions $H_{S}^{+ j}$ and $H_{S}^{- j}$ by using (7) and (8), respectively.

Step 5. Determine the possibility-based comparison index $p (H_{S}^{ij} ℝ H_{S}^{+ j})$ of $H_{S}^{ij}$ relative to $H_{S}^{+ j}$ by using (9) for each z_i ∈ Z with respect to c_j ∈ C.

Step 6. Determine the possibility-based comparison index $p (H_{S}^{ij} ℝ H_{S}^{- j})$ of $H_{S}^{ij}$ relative to $H_{S}^{- j}$ by using (10) for each z_i ∈ Z with respect to c_j ∈ C.

Step 7. Determine the exact importance weight $\bar{p} (W_{S}^{j})$ by using (11) for each c_j ∈ C.

Step 8. Determine the possibility-based closeness coefficient LCC_i using (12) for each alternativez_i ∈ Z.

Step 9. Rank the m alternatives in accordance with the LCC_i values. The alternative with the largest LCC_i value is the best choice.

5 Illustrative application of the proposed method

This section illustrates the proposed hesitant fuzzy linguistic TOPSIS method by applying it to a practical MCDM problem that concerns the car evaluation problem presented by Chen and Lee [4].

5.1 Decision context

Consider a car evaluation problem presented by Chen and Lee [4]. Assume that three decision makers used four criteria, including safety (c₁), price (c₂), appearance (c₃), and performance (c₄), to evaluate four cars z₁, z₂, z₃ and z₄. Since these criteria are all qualitative, it is convenient and only feasible for the decision makers to express their feelings by using linguistic terms rather than crisp judgment values on the criteria. As pointed out by Miller [24], most decision makers can’t handle more than 9 factors when making their decision. Hence, the decision-maker constructs a seven point linguistic scale to assess the evaluative ratings of cars, which is $\begin{matrix} S & = & {s_{0} = very low (VL), s_{1} = low (L), \\ s_{2} = medium low (ML), s_{3} = medium (M), \\ s_{4} = medium high (MH), s_{5} = high (H), \\ s_{6} = very high (VH)} \end{matrix}$

In addition, the decision-maker uses the following seven point linguistic scale to evaluate the importance weights of criteria. $\begin{matrix} S^{'} & = & {s_{0} = strongly less important (SLI), \\ s_{1} = moderately less important (MLI), \\ s_{2} = weakly less important (WLI), \\ s_{3} = equally important (EI), s_{4} = weakly \\ more important (WMI), s_{5} = moderately \\ more important (MMI), s_{6} = strongly more \\ important (SMI)} \end{matrix}$

As the car evaluation problem is very complicated, it is sometimes impossible for the decision maker to use just one linguistic term to express his/her opinion due to the fact that he may be hesitant when determining the values of each cars over the criteria.

With this linguistic term set and also the context-free grammar introduced in Subsection 2.3, the decision maker provides his/her assessments in linguistic expressions, shown in Table 1.

Table 1
Linguistic expressions provided by the decision-maker

c ₁ c ₂ c ₃ c ₄

Linguistic expressions of alternatives with respect to different criteria

z ₁ at least MH between M and H between M and MH L

z ₂ greater than MH at least MH between L and M lower than ML

z ₃ at least M between ML and M greater than MH at most ML

z ₄ between ML and MH between L and M between ML and H at least MH

Linguistic expressions of the importance weights of different criteria

between WLI and EI greater than EI at most WLI between WMI and MMI

	c ₁	c ₂	c ₃	c ₄
Linguistic expressions of alternatives with respect to different criteria
z ₁	at least MH	between M and H	between M and MH	L
z ₂	greater than MH	at least MH	between L and M	lower than ML
z ₃	at least M	between ML and M	greater than MH	at most ML
z ₄	between ML and MH	between L and M	between ML and H	at least MH
Linguistic expressions of the importance weights of different criteria
	between WLI and EI	greater than EI	at most WLI	between WMI and MMI

The linguistic expressions shown in Table 1 are similar to the human way of thinking and they can reflect the decision maker’s hesitant cognition intuitively. Using the transformation function E_{G
_H} given in Definition 2.5, we can generate the following HFLTS judgment matrix, shown in Table 2.

Table 2

Transformed hesitant fuzzy linguistic term sets of Table 1

	c ₁	c ₂	c ₃	c ₄
The HFLTS evaluative rating $H_{S}^{ij}$ of alternative z_i ∈ Z with respect to criterion c_j ∈ C
z ₁	{s₄, s₅, s₆}	{s₃, s₄, s₅}	{s₃, s₄}	{s₁}
z ₂	{s₅, s₆}	{s₄, s₅, s₆}	{s₁, s₂, s₃}	{s₀, s₁}
z ₃	{s₃, s₄, s₅, s₆}	{s₂, s₃}	{s₅, s₆}	{s₀, s₁, s₂}
z ₄	{s₂, s₃, s₄}	{s₁, s₂, s₃}	{s₂, s₃, s₄, s₅}	{s₄, s₅, s₆}
The HFLTS importance weights $W_{S}^{j}$ of criterion c_j ∈ C
	{s₂, s₃}	{s₄, s₅, s₆}	{s₀, s₁, s₂}	{s₄, s₅}

Step 8. The possibility-based closeness coefficient LCC_i of each alternative z_i ∈ Z is calculated as follows: $\begin{matrix} {LCC}_{1} = 0 . 4530, {LCC}_{2} = 0 . 3972, \\ {LCC}_{3} = 0 . 5281, {LCC}_{4} = 0 . 5732 . \end{matrix}$

Step 9. The four cars are ranked according to the increasing order of their LCC_i values. BecauseLCC₄ > LCC₃ > LCC₁ > LCC₂, the resulting ranking order is determined as z₄ ≻ z₃ ≻ z₁ ≻ z₂. Therefore, the best suitable car is z₄.

6 Comparative analysis and discussion

In this section, a comparative study with other hesitant fuzzy linguistic MCDM methods is conducted to clarify the advantages of the proposed possibility-based TOPSIS method.

6.1 Comparisons with the distance-based TOPSIS methods

Liao et al. [12] proposed a family of distance measures for HFLTSs. Based on the proposed distance measures, Liao et al. [16] further developed the distance measures-based TOPSIS method to solve the MCDM with HFLTSs. Liao and Xu [11] introduced a family of cosine distance measures for HFLTSs, and applied them to develop the cosine distance-based HFL-TOPSIS method for handling hesitant fuzzy linguistic MCDM problems. Beg and Rashid [1] proposed a TOPSIS for HFLTSs. By using the same example in Example 5.1, we carry out a side-by-side comparative analysis among Liao et al.’s distance-based TOPSIS method [12], Liao and Xu’s cosine distance-based HFL-TOPSIS method [11], Beg and Rashid’s TOPSIS method [1], and the proposed method. The comparison results are shown in Tables 5 and 6. From Table 5, we can see that the ranking orders of the alternatives derived by Liao et al.’s distance-based TOPSIS method [12], Liao and Xu’s cosine-distance-based HFL-TOPSIS method [11], Beg and Rashid’s TOPSIS method [1], and our algorithm are the same. This result explains the validity of our method. However, it should be noted that both Liao et al’s distance-based method and Liao and Xu’s distance-based method are based on the assumption that the input HFLTSs have equal length. This is not in accordance with real cases because it is impossible to make sure that all HFLTSs have equal length. If the lengths of different HFLTSs are different, it is necessary to add some artificial values to the HFLTSs until they are of equal length. It should be stated that filling some artificial values into a HFLTS would change the information of the original HFLTS and different methods of extension can produce different results. Thus, such an approach is less well justified theoretically and less reliable practically. But in the proposed method, the input HFLTSs are not assumed to have equal length, and thus the proposed method are much more convincing and applicable.

Table 3
Results of the mean possibilities $\bar{p} (H_{S}^{ij})$

$\bar{p} (H_{S}^{ij})$ c ₁ c ₂ c ₃ c ₄

z ₁ 0.2754 0.2111 0.2333 0.2222

z ₂ 0.3139 0.1528 0.1655 0.1861

z ₃ 0.2480 0.3083 0.3611 0.2167

z ₄ 0.1627 0.3278 0.2401 0.3750

$\bar{p} (H_{S}^{ij})$	c ₁	c ₂	c ₃	c ₄
z ₁	0.2754	0.2111	0.2333	0.2222
z ₂	0.3139	0.1528	0.1655	0.1861
z ₃	0.2480	0.3083	0.3611	0.2167
z ₄	0.1627	0.3278	0.2401	0.3750

Table 4

Results of the possibility-based comparison indices

	c ₁	c ₂	c ₃	c ₄
$p (H_{s}^{1 j} ℝ H_{s}^{+ j})$	0.6000	0.8333	1.0000	1.0000
$p (H_{s}^{1 j} ℝ H_{s}^{- j})$	0.8333	0.6667	0.8000	0.6667
$p (H_{s}^{2 j} ℝ H_{s}^{+ j})$	0.5000	1.0000	1.0000	1.0000
$p (H_{s}^{2 j} ℝ H_{s}^{- j})$	1.0000	0.5000	0.5000	0.5000
$p (H_{s}^{3 j} ℝ H_{s}^{+ j})$	0.6667	0.6000	0.5000	1.0000
$p (H_{s}^{3 j} ℝ H_{s}^{- j})$	0.7143	1.0000	1.0000	0.6000
$p (H_{s}^{4 j} ℝ H_{s}^{+ j})$	1.0000	0.5000	0.8333	0.5000
$p (H_{s}^{4 j} ℝ H_{s}^{+ j})$	0.5000	1.0000	0.7143	1.0000

Table 5

A comparison of the ranking orders of the alternatives for different distance-based TOPSIS methods

Methods	Closeness coefficients				Preference order
	z ₁	z ₂	z ₃	z ₄
Liao et al.’s Hamming-distance-based TOPSIS method [12]
Use the optimistic principle	0.3194	0.1476	0.4850	0.7588	z₄ ≻ z₃ ≻ z₁ ≻ z₂
Use the pessimistic principle	0.3159	0.1292	0.5541	0.8005	z₄ ≻ z₃ ≻ z₁ ≻ z₂
Liao et al.’s Euclidean-distance-based TOPSIS method [12]
Use the optimistic principle	0.3226	0.2617	0.4619	0.6741	z₄ ≻ z₃ ≻ z₁ ≻ z₂
Use the pessimistic principle	0.3141	0.2325	0.4856	0.7199	z₄ ≻ z₃ ≻ z₁ ≻ z₂
Liao and Xu’s cosine-distance-based TOPSIS method [11]
Use the optimistic principle	0.3351	0.1458	0.6932	0.8107	z₄ ≻ z₃ ≻ z₁ ≻ z₂
Use the pessimistic principle	0.3911	0.0903	0.6765	0.8785	z₄ ≻ z₃ ≻ z₁ ≻ z₂
Beg and Rashid’s TOPSIS method [1]	0.3704	0.1852	0.5926	0.6667	z₄ ≻ z₃ ≻ z₁ ≻ z₂
The proposed method	0.4530	0.3972	0.5281	0.5732	z₄ ≻ z₃ ≻ z₁ ≻ z₂

Table 6

Problem requirements of different MCDM methods

Methods	Problem with different characteristics
	Performance ratings	Criteria weights
Liao et al.’s Hamming-distance-based TOPSIS method [12]	HFLTSs	Crisp numbers
Liao et al.’s Euclidean-distance-based TOPSIS method [12]	HFLTSs	Crisp numbers
Liao and Xu’s cosine-distance-based TOPSIS method [11]	HFLTSs	Crisp numbers
Beg and Rashid’s TOPSIS method [1]	HFLTSs	N/A
The proposed method	HFLTSs	Crisp numbers or HFLTSs

5.2 Illustration of the proposed method

The proposed possibility-TOPSIS method is appropriate to handle MCDM problems in which the evaluative ratings of alternatives and the importance weights of criteria are expressed with HFLTSs. Accordingly, the car evaluation problem described in Subsection 5.1 is adopted for facilitating the illustration of the proposed method.

Step 1. The set of all alternatives is denoted by Z ={ z₁, z₂, z₃, z₄ } and the set of evaluative criteria is denoted by C ={ c₁, c₂, c₃, c₄ }, with C_I ={ c₁, c₃, c₄ } and C_II ={ c₂ }.

Steps 2 and 3 have been finished in Subsection 5.1.

Step 4. Calculate the possibility-based outranking index $\bar{p} (H_{S}^{ij})$ of the HFLTS evaluative ratings $H_{S}^{ij}$ of alternative z_i ∈ Z with respect to criterion c_j ∈ C. These computational results are presented in Table 3.

According to the HFLTS evaluations $H_{S}^{ij}$ in Table 2 and the possibility-based outranking index $\bar{p} (H_{S}^{ij})$ in Table 3, the hesitant fuzzy linguistic positive ideal solution $H_{S}^{+ j}$ was identified as follows: $\begin{matrix} H_{S}^{+ j} & = & {〈 c_{1}, {s_{5}, s_{6}} 〉, 〈 c_{2}, {s_{1}, s_{2}, s_{3}} 〉, \\ 〈 c_{3}, {s_{5}, s_{6}} 〉, 〈 c_{4}, {s_{4}, s_{5}, s_{6}} 〉} \end{matrix}$

Additionally, the hesitant fuzzy linguistic negative ideal solution $H_{S}^{- j}$ is determined in the following: $\begin{matrix} H_{S}^{- j} & = & {〈 c_{1}, {s_{2}, s_{3}, s_{4}} 〉, 〈 c_{2}, {s_{4}, s_{5}, s_{6}} 〉, \\ 〈 c_{3}, {s_{1}, s_{2}, s_{3}} 〉, 〈 c_{4}, {s_{0}, s_{1}} 〉} \end{matrix}$

Steps 5 and 6. Calculate the possibility-based comparison indices $p (H_{S}^{ij} ℝ H_{S}^{+ j})$ and $p (H_{S}^{ij} ℝ H_{S}^{- j})$ for each z_i ∈ Z and c_j ∈ C. These computational results are presented in Table 4.

In addition, as shown in Table 6, Liao and Xu’s distance-based HFL-TOPSIS method [11] and Liao et al.’s distance-based TOPSIS method [12] are applicable to the decision making problems where the performance ratings of alternatives on each criterion and the weights of criteria are expressed by HFLTSs and crisp numbers, respectively. Beg and Rashid’s TOPSIS method [1] is suitable for the problems where the performance ratings of alternatives on each criterion are represented by HFLTSs and the weights of criteria do not been taken into account. But these three methods fail to deal with the MCDM problems in which the weights of criteria take the form of HFLTSs. In contrast, the proposed algorithm is the preferred method for the MCDM problems where the weights of criteria are given in the form of HFLTSs or crisp numbers.

Step 7. The computation results of the exact importance weight $\bar{p} (W_{S}^{j})$ for each c_j ∈ C are shown as follows. $\begin{matrix} \bar{p} (W_{S}^{1}) & = & 0 . 1917, \bar{p} (W_{S}^{2}) = 0 . 3417, \\ \bar{p} (W_{S}^{3}) & = & 0 . 1417, \bar{p} (W_{S}^{4}) = 0 . 3250 . \end{matrix}$

6.2 Comparisons with the possibility-based methods

Lee and Chen [9] proposed a fuzzy decision making method based on the proposed possibility-degree formula (3) and the proposed HFLWA operator, the proposed HFLWG operator, the proposed HFLOWA operator, and the proposed HFLOWG operator of HFLTSs. Lee and Chen [8] presented a new fuzzy decision making method based on likelihood-based comparison relations of HFLTSs. Rodríguez et al. [28] presented a multicriteria linguistic decision making model in which experts provide their assessments by eliciting linguistic expressions. Wei et al. [38] used a hesitant fuzzy LWA operator, a hesitant fuzzy LOWA operator and the comparison methods to deal with MCDM problems in which the assessments of alternatives under criteria are represented by HFLTSs.

In the following, a numerical example is provided to illustrate that the proposed method can overcome the drawbacks of Rodríguez et al.’s method [28], Lee and Chen’s methods [8, 9], and Wei et al.’s method [38], which cannot distinguish the ranking order of alternatives in some situations.

Example 6.1. Assume that the linguistic expressions of the alternatives with respect to different criteria are shown in Table 7. Based on the transformation function E_{G
_H} shown in Definition 2.5, Table 7 can be transformed into the corresponding Table 8. Assume that the weights of the criteria c₁, c₂, c₃ and c₄ are 1/4, 1/4, 1/4 and 1/4, respectively.

Table 7
Linguistic expressions of the alternatives with respect to different criteria

c ₁ c ₂ c ₃ c ₄

z ₁ at least MH VH at most M VL

z ₂ VH at least MH VL lower than ML

z ₃ at least M VL greater than MH VH

z ₄ VH between L and M VL at least MH

	c ₁	c ₂	c ₃	c ₄
z ₁	at least MH	VH	at most M	VL
z ₂	VH	at least MH	VL	lower than ML
z ₃	at least M	VL	greater than MH	VH
z ₄	VH	between L and M	VL	at least MH

Table 8

Transformed hesitant fuzzy linguistic term sets of Table 7

	c ₁	c ₂	c ₃	c ₄
z ₁	{s₄, s₅, s₆}	{s₆}	{s₀, s₁, s₂, s₃}	{s₀}
z ₂	{s₆}	{s₄, s₅, s₆}	{s₀}	{s₀, s₁}
z ₃	{s₀}	{s₀}	{s₅, s₆}	{s₆}
z ₄	{s₆}	{s₁, s₂, s₃}	{s₀}	{s₄, s₅, s₆}

Table 9 shows the ranking orders of the alternatives z₁, z₂, z₃ and z₄ for different possibility-based methods. From Table 9, we can see that Lee and Chen’s method [8] cannot distinguish the preference order between the alternatives z₁ and z₄; Wei et al.’s pessimistic and optimistic methods [38], Lee and Chen’s method [9] and Rodríguez et al.’s method [28] cannot distinguish the preference order among the alternatives z₁, z₂, z₃ and z₄. We also can see that the proposed algorithm A can distinguish the preference order among the alternatives z₁, z₂, z₃ and z₄. By comparing the experimental results shown in Table 9, we can see that the proposed method can overcome the drawbacks of Rodríguez et al.’s method [28], Lee and Chen’s methods [8, 9], and Wei et al.’s method [38] due to the fact the proposed method can distinguish the preference order among the alternatives z₁, z₂, z₃ and z₄, whereas Rodríguez et al.’s method [28], Lee and Chen’s methods [8, 9], and Wei et al.’s method [38] cannot distinguish the preference order among the alternatives z₁, z₂, z₃ and z₄.

Table 9

A comparison of the ranking orders of the alternatives for different possibility-based methods for Example 6.1

Methods	Preference values				Preference orders
	z ₁	z ₂	z ₃	z ₄
Lee and Chen’s method [9]
Use the HFLWA operator	1.0000	1.0000	1.0000	1.0000	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Use the HFLWG operator	0.0000	0.0000	0.0000	0.0000	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Use the HFLOWA operator	1.0000	1.0000	1.0000	1.0000	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Use the HFLOWG operator	0.0000	0.0000	0.0000	0.0000	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Wei et al.’s method [38]
Optimistic	{s₆}	{s₆}	{s₆}	{s₆}	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Neutral	{s₃}	{s₃}	{s₃}	{s₃}	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Pessimistic	{s₀}	{s₀}	{s₀}	{s₀}	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Rodriguez et al.’s method [28]	1.0000	1.0000	1.0000	1.0000	z₁ ∼ z₂ ∼ z₃ ∼ z₄
Lee and Chen’s method [8]	0.5167	0.4792	0.4732	0.5167	z₁ ∼ z₄ ≻ z₂ ≻ z₃
The proposed method	0.4275	0.4545	0.5833	0.5185	z₃ ≻ z₄ ≻ z₂ ≻ z₁

7 Conclusions

Based on the possibility-based comparison approach with the approximate ideals, this paper developed a hesitant fuzzy linguistic TOPSIS method for addressing MCDM problems in the HFLTS framework. First, this paper introduced the concept of possibilities of HFLTS binary relations and studied its properties. Second, this paper defined the possibility-based outranking index of the HFLTS evaluative ratings, which was used to determine the positive-ideal and negative-ideal solutions. Then, this paper defined the possibility-based comparison indices of the evaluative ratings between a specific alternative and the ideal, based on which the possibility-based closeness coefficient of each alternative relative to the ideals was proposed. Third, a possibility-based hesitant fuzzy linguistic TOPSIS method was developed to address the MCDM problems involving HFLTS importance weights. Finally, this paper furnished a numerical example concerning the car evaluation problem to illustrate the feasibility and the applicability of the proposed method. Moreover, the advantages of the proposed method have been validated via the comparisons with other hesitant fuzzy linguistic decision-making methods. The comparative results have demonstrated that the proposed method is appropriate and effective for handling MCDM problems in the HFLTS context and has the potential for use in practical applications.

Footnotes

Acknowledgments

The author thanks the anonymous referees for their valuable suggestions in improving this paper. This work was supported by the National Natural Science Foundation of China (Nos. 61375075, 61672205, and 11626079), the Natural Science Foundation of Hebei Province of China (No. F2015402033), the Science and Technology Research Project of Colleges and Universities of Hebei Province (No. BJ2017031), the Scientific Research Project of Department of Education of Hebei Province of China (Nos. QN2015026 and QN2016235), and the Natural Science Foundation of Hebei University (Nos. 799207217073 and 799207217108).

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Hesitant fuzzy linguistic TOPSIS method using a possibility-based comparison approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Linguistic term sets

2.2 Hesitant fuzzy linguistic term sets

2.3 Transforming linguistic expressions into HFLTSs

3.1 Drawbacks of some existing comparison methods for HFLTSs

4.1 Hesitant fuzzy linguistic decision context

5.1 Decision context

6.1 Comparisons with the distance-based TOPSIS methods

Table 3 Results of the mean possibilities p ¯ ( H S ij ) p ¯ ( H S ij ) c 1 c 2 c 3 c 4 z 1 0.2754 0.2111 0.2333 0.2222 z 2 0.3139 0.1528 0.1655 0.1861 z 3 0.2480 0.3083 0.3611 0.2167 z 4 0.1627 0.3278 0.2401 0.3750

6.2 Comparisons with the possibility-based methods

Table 7 Linguistic expressions of the alternatives with respect to different criteria c 1 c 2 c 3 c 4 z 1 at least MH VH at most M VL z 2 VH at least MH VL lower than ML z 3 at least M VL greater than MH VH z 4 VH between L and M VL at least MH

Footnotes

Acknowledgments

References

Table 3
Results of the mean possibilities $\bar{p} (H_{S}^{ij})$

$\bar{p} (H_{S}^{ij})$ c ₁ c ₂ c ₃ c ₄

z ₁ 0.2754 0.2111 0.2333 0.2222

z ₂ 0.3139 0.1528 0.1655 0.1861

z ₃ 0.2480 0.3083 0.3611 0.2167

z ₄ 0.1627 0.3278 0.2401 0.3750

Table 7
Linguistic expressions of the alternatives with respect to different criteria

c ₁ c ₂ c ₃ c ₄

z ₁ at least MH VH at most M VL

z ₂ VH at least MH VL lower than ML

z ₃ at least M VL greater than MH VH

z ₄ VH between L and M VL at least MH