A multi-criteria decision-making method for hesitant fuzzy linguistic term set based on the cloud model and evidence theory

Abstract

The hesitant fuzzy linguistic term set (HFLTS) is usually used in the uncertain decision situation. In order to solve the problem of multi-criteria decision-making (MCDM) with HFLTS, a new MCDM method based on the cloud model and evidence theory is proposed in this paper. A new envelope, whose representation is a synthetic cloud generated by the multiple linguistic term in the HFLTS, is presented for HFLTS to facilitate the computing processes and take both the randomness and fuzziness of linguistic variables into consideration. As to overcome the drawbacks of tradition aggregation operator, the criteria values in the form of the belief degrees are obtained from the synthetic clouds and aggregated using the evidential reasoning algorithm. An illustrative example, which is given to confirm the feasibility and validity, also shows that with the proposed method more reasonable and accurate ranking results can be obtained. Moreover, the belief degree of each linguistic term as well as the hesitant degree of the assessment can be obtained simultaneously.

Keywords

Hesitant fuzzy linguistic term set multi-criteria decision-making cloud model evidence theory uncertainty

1 Introduction

Multi-criteria decision-making (MCDM) problems are commonly specified in complex and uncertain situations [1]. In such cases, the decision makers (DMs) usually feel more confident of providing linguistic assessments than providing exact numerical values because of the inherent vagueness of human preferences. This decision-making case is defined as a linguistic MCDM problem, that is, the alternative assessments with regard to criteria are given with linguistic variables [2].

The research on the linguistic MCDM problem has obtained rich achievements [3 –6] since Zadeh [2] proposed the concept of linguistic variables. Herrera et al. [7] and Martínez et al. [8] survey that three main types of linguistic MCDM methods have been put forward: (1) The method on the basis of membership functions, which could convert the linguistic concept into some fuzzy numbers by membership functions [9 –12]. (2) The method on the basis of symbols, which could convert the linguistic concept into some real numbers [13 –16]. (3) The method on the basis of 2-tuple linguistic models, which could also transform the linguistic concept into some real numbers and introduce the uncertain decision-making into the precise domain [17 –20].

However, the methods mentioned above have some serious limitations because that each linguistic variable is indicated by only one linguistic term. There may be doubt in decision-making, especially when DMs participated in a decision situation hesitate when educe their preference rather than a single value due to the complexity, time pressure, and insufficiency of information among different possible values [21]. Therefore, a single linguistic term is often insufficient for DMs to express their preferences in the uncertain decision situation. Rodríguez et al. [22] put forward the hesitant fuzzy linguistic term set (HFLTS) which permits DMs to evaluate a linguistic variable by a few linguistic terms.

Recently, more and more studies have been devoted on the linguistic MCDM problem with HFLTS [22 –28]. Rodríguez et al. [22] use linguistic intervals generated by the envelope of HFLTS to facilitate the computing processes for HFLTS, but the results lose initial fuzzy representation. In Liu and Rodríguez’s work [1], the fuzziness of linguistic variables is considered, and a fuzzy representation for linguistic variables, in which the variables are represented by a fuzzy membership function generated from the multiple linguistic terms, is proposed based on a new fuzzy envelope for HFLTS. However, taking both the randomness and fuzziness into consideration simultaneously makes a representation of HFLTS more reasonable. As a new cognitive model of uncertainty, the cloud model is proposed by Li et al. [29] for the randomness of membership degree, which is founded on probability theory and fuzzy sets theory. The cloud model represents the linguistic concept with three numerical characteristics that reflect not only the average level of the linguistic concept but also the randomness and fuzziness. Some methods have successfully applied the cloud model to the linguistic MCDM problem, by transforming the linguistic concept into the cloud [30 –33]. However, there is no study on transferring HFLTS to the cloud until now. In this paper, considering both the randomness and fuzziness of linguistic variables, we propose a new representation of HFLTS based on the cloud model.

In the MCDM problem, aggregating different criteria values is crucial. Since Yager [34] first proposed the ordered weighted averaging (OWA) operator, many aggregation operators have been studied, e.g., the ordered weighted geometric averaging (OWGA) operator [35], the linguistic weighted ordered weighted averaging (LOWA) operator [36] and the hesitant trapezoidal fuzzy aggregation (HTFA) operator [37]. However, using traditional aggregation operators in the existing methods for HFLTS may bring some problems: (1) the defined discrete linguistic variables might be extended to the field of continuous variables. (2) the union of two HFLTSs might cause a set involving inconsecutive linguistic terms. (3) the results using aggregation operators might be contrary to common sense, as discussed by Wei et al. [26]. The evidence theory, firstly proposed by Dempster’s work [38] and further developed by Shafer [39], is a generalization of traditional probability, which allows us to get a better handle on the imprecise and incomplete uncertainty. It is particularly useful for dealing with uncertain subjective judgments when multiple pieces of evidence must be simultaneously considered. In 1990s, an Evidential Reasoning (ER) algorithm based on the decision theory and evidence theory was proposed by Yang et al [40, 41] to handle the uncertain MCDM problem. So far, the evidence theory and its extensions have been widely used in many fields such as pattern recognition [42], software requirements prioritization [43], multi-sensor data fusion [44], fault diagnosis [45], forex market analysis [46] and group decision analysis [47]. In this paper, we suppose that the HFLTS consists of the linguistic terms of which only the belief degree is different. Moreover, we consider the performance of each criterion is a reasonable source and can be fused to get a comprehensive evaluation for the uncertain MCDM problem. On the basis of the evidence theory, we aggregate all criteria values to solve the aforementioned problems of aggregation operators.

In this paper, we focus on how to solve the MCDM problem with HFLTS, and propose a method which integrates both the cloud model and evidence theory. We represent linguistic variables through the clouds generated from the multiple linguistic terms which form the HFLTS, and aggregate the criteria values in the form of the belief degrees using the evidence theory. The major contributions of this paper are as followings. Firstly, a new representation of HFLTS based on the cloud model is proposed. Secondly, the conversion from the cloud to the belief degrees is developed. Finally, use the evidential reasoning algorithm to aggregate all criteria values. Different from other MCDM methods, the proposed method can deal with the problems caused by traditional HFLTS representation and aggregation operator.

The rest of this paper is organized as follows: Section 2 introduces some relative definitions. Section 3 proposes the MCDM method for HFLTS based on the cloud model and evidential theory. To validate the proposed method, a numerical example is examined in Section 4. Section 5 concludes this paper.

2 Preliminaries

This section involves some basic concepts of HFLTS, cloud model and evidence theory.

Define a linguistic term set S = {s_i|i = 0, ⋯ g, g ∈ N}, where s_i is referred to as a possible value of a linguistic variable, and satisfies the following conditions:

The set is ordered: if α > β, then s_α > s_β;

A negation operator exists: neg (s_α) = s_g-α;

If s_α > s_β, then max {s_β, s_α} = s_α, min {s_β, s_α} = s_β.

2.1 Basics of HFLTS

Definition 1. [22] Define a linguistic term set S = {s₀, s₁ ⋯ s_g}, and then an HFLTS H_S is an ordered finite subset of the consecutive linguistic terms of S.

Example 1. Define a linguistic term set S = {s₀ = " Very Poor ", s₁ = " Poor ", s₂ = " Medium ", s₃ = " Good ", s₄ = " Very Good "}, and then an HFLTS might be H_S = {s₂, s₃, s₄}.

A context-free grammar G_H has been defined by Rodríguez et al. [22] to convert the comparative linguistic expressions into the HFLTS through the transformation function E_{G
_H}.

Definition 2. [22] Define a linguistic term set S = {s₀, s₁ ⋯ s_g}, as well as a function E_{G
_H} which transforms the linguistic expression ll generated from G_H into an HFLTS H_S: $E_{G_{H}} : ll \to H_{S}$ (1)

The comparative linguistic expressions obtained by G_H are transformed into the HFLTS through the following steps:

E_{G
_H} (s_i) = {s_i|s_i ∈ S};

E_{G
_H} (lessthan s_i) = {s_j|s_j ∈ S and s_j ≤ s_i};

E_{G
_H}(greater than 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 and s_k ≥ s_i and s_k ≤ s_j}.

2.2 Basics of cloud model

Define a qualitative concept T over a universe of discourse U = {u}. Then x ∈ U is defined as a random instantiation of T, and μ_T (x) ∈ [0, 1] is the certainty degree that x belongs to T, which corresponds to a random number with a steady trend. Define a cloud that represents the distribution of x in the universe U, where x is known as a cloud drop. ∀x ∈ U, the mapping μ_T (x), is essentially one-to-many mapping, which means the certainty degree that x belongs to T is a distribution of probability.

The cloud model can describe a concept with three numerical characteristics which are Expectation Ex, Entropy En and Hyper entropy He. In particularly, Ex is the mathematical expectation of the cloud drops that belongs to a concept in the universe. En is used to refer to the fuzziness measurement of a concept. He reflects the cloud drops dispersion. Suppose C is a cloud with three numerical characteristics Ex, En, and He, which can be represented as C (Ex, En, He).

The normal cloud model, which we discuss only in this paper, is founded on the normal distribution and normal membership function. The normal cloud model is generally applicable.

Definition 3. [48] Suppose U is the discourse universe and $\tilde{A}$ is a qualitative concept in U. If x ∈ U is a random instantiation of the concept $\tilde{A}$ , which follows x ∼ N (Ex, En^′2), En′ ∼ N (En, He²), and the certainty degree that x belongs to the concept $\tilde{A}$ follows $μ = e^{\frac{- (x - Ex)^{2}}{2 ({En}^{'})^{2}}}$ , then the x distribution in the universe U is called a normal cloud.

The cloud model is transformed from its qualitative representation to its quantitative representation by the forward cloud generator which generates cloud drops according to the three numerical characteristics, (Ex, En, He). Definition 4 details the generating procedure of the algorithm for the normal cloud as follows:

Definition 4. [48] Forward cloud generator FCG (Ex, En, He, N).

Input: Three parameters, Ex, En, He, and the number of cloud drops N.

Output: N cloud drops.

Steps:

Generate a random number $E n_{i}^{'}$ which follows normal distribution, with expectation En and variance He²;

Generate a random number x_i which follows normal distribution, with expectation Ex and variance $E n_{i}^{' 2}$ ;

Calculate $μ_{i} = e^{\frac{- (x_{i} - Ex)^{2}}{2 ({En}_{i}^{'})^{2}}}$ ;

x_i presented as drop (x_i, μ_i) is a cloud drop, whose certainty degree is μ_i;

Repeat steps 1 to 4 until N cloud drops have been generated.

2.3 Basics of evidence theory

The basic concepts of the evidence theory relevant to this paper are described briefly below:

Definition 5. Define Θ as a set of elements that are collectively exhaustive and mutually exclusive, where an element can be an object, a hypothesis, or in this paper a linguistic term. Θ is referred to as the frame of discernment; the set which consists of all the subsets of Θ is denoted by Ω (Θ) and seen as the power set of Θ.

Definition 6. [49] A basic probability assignment (BPA) is a function m : Ω (Θ) → [0, 1], which is called a mass function and satisfies the following equation: $m (Φ) = 0, \sum_{A \subseteq Ω (Θ)} m (A) = 1$ (2)

where Φ is empty, and A is any subset of Θ. The assigned probability m (A), which is also named probability mass, evaluates the belief assigned to A and indicates the support degree of the evidence to A accurately. It does not include the belief in any particular subset of A. Each subset A ⊆ Θ such that m (A) >0 is referred to as a focal element of m. When a mass value is committed to a subset that has more than one element, it explicitly states that there is not enough information for assigning this belief exactly to each individual element in the subset. Especially when there is no evidence about Θ at all, the total belief is assigned to the whole frame of discernment m (Θ) =1, where m (Θ) is called the ignorance degree.

3 MCDM method for HFLTS based on the cloud model and evidence theory

3.1 A new representation of HFLTS based on the cloud model

As a powerful transformation model of qualitative and quantitative, the cloud model makes it possible to address the potential randomness and fuzziness inherent in the linguistic concept. A proposal to get the envelope for HFLTS based on the cloud model is presented here. As using all the information included in the HFLTS is reasonable, every linguistic term in the HFLTS should be considered in the computing processes. At first, all the linguistic terms in the linguistic term set s_i ∈ S are transformed into the clouds.

Definition 7. Define the linguistic term set S = {s₀, s₁ ⋯ s_i ⋯ s_g}, and a valid non-negative universe [U_min, U_max] is provided. By the average method, S = {s₀, s₁ ⋯ s_i ⋯ s_g} = ${(B_{min}^{0}$ , $B_{max}^{0})$ , $(B_{min}^{1}$ , $B_{max}^{1})$ , ⋯ $(B_{min}^{i}$ , $B_{max}^{i})$ ⋯ $(B_{min}^{g}$ , $B_{max}^{g})}$ , where $Length (B_{min}^{i}$ , $B_{max}^{i})$ = (U_max - U_min)/N, $B_{min}^{0}$ = U_min, $B_{max}^{g}$ = U_max. Then, the cloud c_{s
_i} (Ex_i, En_i, He_i) representing s_i can be defined as:

${\begin{matrix} {Ex}_{i} = \frac{(B_{min}^{i} + B_{max}^{i})}{2} \\ {En}_{i} = \frac{(B_{max}^{i} - B_{min}^{i})}{6} \\ {He}_{i} = θ \end{matrix}$ (3)

The cloud model uses three numerical characteristics to describe a linguistic term. These characteristics realize the conversion of objective and its interchangeable between a qualitative linguistic concept and its quantitative value. The algorithm for computing the three numerical characteristics can be illustrated as shown below. Ex expresses the expectation of the linguistic concept. Thus, it is natural to use the median of the interval, where Interval [Ex-3En, Ex+3En] best describes the qualitative linguistic concept (99.74%, 6En rule). Therefore, 6En can be used to represent the fuzziness and bound of the linguistic concept. He is set to θ which can be adjusted according to actual conditions.

Example 2. Let the universe be [0, 1] and the linguistic term set be S = {s₀ = " Very Poor ", s₁ = " Poor ", s₂ = " Fair ", s₃ = " Good ", s₄ = " Very Good "}, then S = ${(B_{min}^{0}$ = 0, $B_{max}^{0}$ = 0.2), $(B_{min}^{1}$ = 0.2, $B_{max}^{1}$ = 0.4), $(B_{min}^{2}$ = 0.4, $B_{max}^{2}$ = 0.6), $(B_{min}^{3}$ = 0.6, $B_{max}^{3}$ = 0.8), $(B_{min}^{4}$ = 0.8, $B_{max}^{4}$ = 1)}. Given He = 0.01, the following five clouds can be obtained: C_S = {c_{s
₀} (0.1,0.033,0.01), c_{s
₁} (0.3, 0.033, 0.01), c_{s
₂} (0.5, 0.033, 0.01), c_{s
₃} (0.7, 0.033, 0.01), c_{s
₄} (0.9, 0.033, 0.01)}.

1000 cloud drops can be generated for each cloud using the forward cloud generator, and the predefined linguistic term set can be explicitly presented by clouds, as shown in Fig. 1.

Fig.1

Clouds of the linguistic terms.

For an HFLTS H_S, its fuzzy envelope env (H_S) can be defined as an synthetic cloud ${\tilde{c}}_{H_{S}}$ (Ex_{H
_S}, En_{H
_S}, He_{H
_S}). The synthetic cloud is generated from aggregation of the clouds of all the linguistic terms in the HFLTS. The synthetic cloud must meet the following requirements: 1) the synthetic cloud is required to cover all the linguistic terms in the HFLTS. 2) The synthetic cloud must be limited between the best and the worst linguistic terms and should not be out of the domain. We propose the cloud aggregation algorithm to carry this out.

Definition 8. Let H_S = {s_i, s_i+1, ⋯ s_k ⋯ , s_j} be an HFLTS, where s_k ∈ S and i ≤ k ≤ j. Hence c_{s
_k} (Ex_{s
_k}, En_{s
_k}, He_{s
_k}), i ≤ k ≤ j is a collection of clouds. The synthetic cloud ${\tilde{c}}_{H_{S}} ({Ex}_{H_{S}}, {En}_{H_{S}}, {He}_{H_{S}})$ can be defined as: ${\begin{matrix} {Ex}_{H_{S}} = (\sum_{k = i}^{j} {Ex}_{s_{k}}) / (j - i + 1) \\ {En}_{H_{S}} = (({Ex}_{s_{j}} + 3 {En}_{s_{j}}) - ({Ex}_{s_{i}} - 3 {En}_{s_{i}})) / 6 \\ {He}_{H_{S}} = \sum_{k = i}^{j} ({He}_{k}) / (j - i + 1) \end{matrix}$ (4)

Because of the hesitation among {s_i, s_i+1, ⋯ s_k ⋯, s_j}, such terms might have different importance. We believe the middle term might have more importance. Thus, Ex_{H
_S} is the average of all clouds. The definition domain of the synthetic cloud should be the same as the collection of clouds c_{s
_k} (Ex_{s
_k}, En_{s
_k}, He_{s
_k}), i ≤ k ≤ j. Thus, En_{H
_S} is greater than En_s_k of each individual cloud, and its value ensures that the synthetic cloud covers the domain of all original clouds without covering the domain of the irrelevant cloud.

Example 3. Given an HFLTS H_S = {s₁, s₂}, by aggregating two clouds c_{s
₁} (0.3, 0.033, 0.01) and c_{s
₂} (0.5, 0.033, 0.01), the synthetic cloud ${\tilde{c}}_{H_{S}}$ (Ex_{H
_S} = 0.4, En_{H
_S} = 0.066, He_{H
_S} = 0.01) can be obtained, as shown in in Fig. 2.

Fig.2

Synthetic cloud.

3.2 The conversion from the cloud to the belief degrees

We suppose that the HFLTS consists of the linguistic terms, just the belief degree of each linguistic term is different. A cloud is composed of cloud drops. Given a cloud drop (x, μ), its certainty degree that x belongs to the concept is μ. If $B_{min}^{i} \leq x \leq B_{max}^{i}$ , we consider that μ is the certainty degree that s_i belongs to the assessment. If enough cloud drops can be obtained as samples, the distribution of the belief degrees can be obtained.

Definition 9. Given a synthetic cloud and applied the forward cloud generator, n_all cloud drops can be generated. Suppose there are n₁ cloud drops {(x₁, μ₁), (x₂, μ₂)... (x_{t
₁}, μ_{t
₁}). . . (x_{n
₁}, μ_{n
₁})} between $[B_{min^{1}}$ , $B_{max^{1}}]$ , …, n_i cloud drops {(x₁, μ₁), (x₂, μ₂). . . (x_{t
_i}, μ_{t
_i}). . . (x_{n
_i}, μ_{n
_i})} between $[B_{min^{i}}$ , $B_{max^{i}}]$ , …, n_g cloud drops {(x₁, μ₁), (x₂, μ₂). . . (x_{t
_g}, μ_{t
_g}). . . (x_{n
_g}, μ_{n
_g})} between $[B_{min}^{g}$ , $B_{max}^{g}]$ , and $\sum_{i = 1}^{g}$ n_i = n_all. Then the distribution of the belief degrees can be calculated as follows: $β = {\begin{matrix} β (s_{1}) = \frac{\sum_{t_{1} = 1}^{n_{1}} μ_{t_{1}}}{\sum_{t = 1}^{n_{all}} μ_{t}} \forall x_{t_{1}} \in [B_{min}^{1}, B_{max}^{1}] \\ ⋮ \\ β (s_{i}) = \frac{\sum_{t_{i} = 1}^{n_{i}} μ_{t_{i}}}{\sum_{t = 1}^{n_{all}} μ_{t}} \forall x_{t_{i}} \in [B_{min}^{i}, B_{max}^{i}] \\ ⋮ \\ β (s_{g}) = \frac{\sum_{t_{g} = 1}^{n_{g}} μ_{t_{g}}}{\sum_{t = 1}^{n_{all}} μ_{t}} \forall x_{t_{g}} \in [B_{min}^{g}, B_{max}^{g}] \end{matrix}$ (5)

where β (s_i) denotes a belief degree of the assessment which is confirmed to s_i.

Example 4. Given a synthetic cloud ${\tilde{c}}_{H_{S}} ({Ex}_{H_{S}} = 0.4, {En}_{H_{S}} = 0.066, {He}_{H_{S}} = 0.01)$ , 1000 cloud drops can be generated through the cloud generator mentioned previously, as shown in Fig. 2. Then its distribution of the belief degrees can be calculated: β = {β (s₀) = 0, β (s₁) = 0.5, β (s₂) = 0.5, β (s₃) = 0, β (s₄) = 0}.

Consider the hesitation of the assessment, the discounting coefficient is used. The discounting coefficient can be regarded as a reliability coefficient. The more linguistic terms in the HFLTS, the less reliable the information source is. We use the discounting coefficient to modify the belief degrees.

Definition 10. Given the discounting coefficient α and a belief degree β (s_i), the discounted belief degree can be obtained as β′ (s_i) = αβ (s_i). Where α = (2g + 1 - n)/(2g), 0.5 ≤ α ≤ 1 and n is the number of the linguistic terms in the HFLTS. $β^{'} (Θ) = 1 - \sum_{i = 1}^{g} β^{'} (s_{i})$ is the uncertain degree caused by the hesitation. It can be considered as the hesitant degree.

Example 5. Suppose β = {β (s₀) =0, β (s₁) =0.5, β (s₂) =0.5, β (s₃) =0, β (s₄) =0}, through Definition 10, the following can be obtained: α = (2× 4 + 1 - 2)/(2 × 4) = 0.875 and β′ = {β′ (s₀) =0, β′ (s₁) =0.4375, β′ (s₂) =0.4375, β′ (s₃) =0, β′ (s₄) =0}.

3.3 The aggregation of the criteria values based on the evidence theory

Based on the evidence theory, an ER algorithm which we used to aggregate the criteria values has been developed for the uncertain MCDM problem [40 , 49]. The assessment on each criterion and the linguistic term set S are respectively treated as the evidence and the frame of discernment Θ. The assessments of each criterion can be fused to get the overall evaluation.

Firstly, the belief degrees are transformed into the BPA.

Definition 11. Through the following equations, the distribution of the discounted belief degrees for the criterion c_j can be transformed into the BPA via multiplying the given discounted degree of belief $β_{j}^{'} (s_{i})$ with the relative weight of the criterion w_j. $m_{j} = {\begin{matrix} m_{j} (s_{i}) = w_{j} β_{j}^{'} (s_{i}) \\ {\bar{m}}_{j} (Θ) = 1 - w_{j} \\ {\tilde{m}}_{j} (Θ) = w_{j} (1 - \sum_{i = 1}^{g} β_{j}^{'} (s_{i})) \end{matrix}$ (6)

where $m_{j} (Θ) = {\bar{m}}_{j} (Θ) + {\tilde{m}}_{j} (Θ)$ . As the ignorance degree, m_j (Θ) is divided into two parts: ${\bar{m}}_{j} (Θ)$ and ${\tilde{m}}_{j} (Θ)$ , where ${\bar{m}}_{j} (Θ)$ is determined by the relative importance of criterion c_j and ${\tilde{m}}_{j} (Θ)$ is determined by the uncertainness of the assessment with HFLTS.

Next, the BPAs on all criteria are aggregated into the combined probability assignment.

Definition 12. Suppose m_I(j) (s_i), ${\bar{m}}_{I (j)} (Θ)$ and ${\tilde{m}}_{I (j)} (Θ)$ are the combined probability masses obtained by aggregating the first j criteria. The following recursive formulas are then proposed to aggregate the first j criteria with the (j+1)th criterion. ${\begin{matrix} m_{I (j + 1)} (s_{i}) = K_{I (j + 1)} [m_{I (j)} (s_{i}) m_{j + 1} (s_{i}) + \\ m_{I (j)} (Θ) m_{j + 1} (s_{i}) + m_{I (j)} (s_{i}) m_{j + 1} (Θ)] \\ {\bar{m}}_{I (j + 1)} (Θ) = K_{I (j + 1)} [{\bar{m}}_{I (j)} (Θ) {\bar{m}}_{j + 1} (Θ)] \\ {\tilde{m}}_{I (j + 1)} (Θ) = K_{I (j + 1)} [{\tilde{m}}_{I (j)} (Θ) {\tilde{m}}_{j + 1} (Θ) + \\ {\bar{m}}_{I (j)} (Θ) {\tilde{m}}_{j + 1} (Θ) + {\tilde{m}}_{I (j)} (Θ) {\bar{m}}_{j + 1} (Θ)] \end{matrix}$ (7)

where $K_{I (j + 1)} = [1 - \sum_{i = 1}^{g} \sum_{\binom{i = 1}{t \neq i}}^{g} m_{I (j)} (s_{i}) m_{j + 1} (s_{t})]^{- 1}$ , $m_{I (j)} (Θ) = \bar{m} (Θ)_{I (j)} + {\tilde{m}}_{I (j)} (Θ)$ . m_I(j +1) (s_i) is a probability mass defined as the degree of support of the first (j+1) criteria to the hypothesis that the alternative is assessed to s_i.

Finally, normalize the combined probability assignment into the overall belief degrees.

Definition 13. Let β_all (s_i) and β_all (Θ) denote the overall belief degrees of the integration of all assessments, which are assigned to s_i and Θ respectively. Suppose there are m criteria, the distribution of the overall belief degrees can be calculated as follows: $β_{all} = {\begin{matrix} β_{all} (s_{i}) = \frac{m_{I (m)} (s_{i})}{1 - {\bar{m}}_{I (m)} (Θ)} \\ β_{all} (Θ) = \frac{{\tilde{m}}_{I (m)} (Θ)}{1 - {\bar{m}}_{I (m)} (Θ)} \end{matrix}$ (8)

β_all (s_i) denotes a belief degree to which the alternative is evaluated with s_I. β_all (Θ) is the hesitant degree of the assessment.

Example 6. Assume there is an alternative and two criteria c₁, c₂ with the weight vector {w₁ = 0.6, w₂ = 0.4}. Suppose the distribution of the discounted belief degrees on criterion c₁ and c₂ are $β_{1}^{'}$ = ${β_{1}^{'} (s_{0})$ = 0, $β_{1}^{'} (s_{1})$ = 0.4375, $β_{1}^{'} (s_{2}) = 0.4375$ , $β_{1}^{'} (s_{3}) = 0$ , $β_{1}^{'} (s_{4})$ = 0 }, $β_{2}^{'}$ = ${β_{2}^{'} (s_{0})$ = 0, $β_{2}^{'} (s_{1})$ = 0.4, $β_{2}^{'} (s_{2})$ = 0.3, $β_{2}^{'} (s_{3})$ = 0.2, $β_{2}^{'} (s_{4})$ = 0 }, respectively. Using Definition 11, the BPAs can be obtained: m₁ = {m₁ (s₀) = 0, m₁ (s₁) = 0.2625, m₁ (s₂) = 0.2625, m₁ (s₃) = 0, m₁ (s₄) = 0, ${\bar{m}}_{1} (Θ)$ = 0.4, ${\tilde{m}}_{1} (Θ)$ = 0.075}, m₂ = {m₂ (s₀) = 0, m₂ (s₁) = 0, m₂ (s₂) = 0.0264, m₂ (s₃) = 0.2472, m₂ (s₄) = 0.0264, ${\bar{m}}_{2} (Θ)$ = 0.6, ${\tilde{m}}_{2} (Θ)$ = 0.1}. Then, using Definition 12, the following can be obtained: $\begin{matrix} K_{I (2)} = [1 - m_{1} (s_{0}) (m_{2} (s_{1}) + m_{2} (s_{2}) + m_{2} (s_{3}) \\ + m_{2} (s_{4})) \\ - m_{1} (s_{1}) (m_{2} (s_{0}) + m_{2} (s_{2}) + m_{2} (s_{3}) + m_{2} (s_{4})) \\ - m_{1} (s_{2}) (m_{2} (s_{0}) + m_{2} (s_{1}) + m_{2} (s_{3}) + m_{2} (s_{4})) \\ - m_{1} (s_{3}) (m_{2} (s_{0}) + m_{2} (s_{1}) + m_{2} (s_{2}) + m_{2} (s_{4})) \\ - m_{1} (s_{4}) (m_{2} (s_{0}) + m_{2} (s_{1}) + m_{2} (s_{2}) + m_{2} (s_{3}))]^{- 1} \\ = 1.1773 \end{matrix}$ ${m_{I (2)} (s_{0}) = K_{I (2)} [m_{1} (s_{0}) m_{2} (s_{0}) + m_{1} (Θ) m_{2} (s_{0}) + m_{1} (s_{0}) m_{2} (Θ)] = 0 m_{I (2)} (s_{1}) = K_{I (2)} [m_{1} (s_{1}) m_{2} (s_{1}) + m_{1} (Θ) m_{2} (s_{1}) + m_{1} (s_{1}) m_{2} (Θ)] = 0.216 m_{I (2)} (s_{2}) = K_{I (2)} [m_{1} (s_{2}) m_{2} (s_{2}) + m_{1} (Θ) m_{2} (s_{2}) + m_{1} (s_{2}) m_{2} (Θ)] = 0.239 m_{I (2)} (s_{3}) = K_{I (2)} [m_{1} (s_{3}) m_{2} (s_{3}) + m_{1} (Θ) m_{2} (s_{3}) + m_{1} (s_{3}) m_{2} (Θ)] = 0.138 m_{I (2)} (s_{4}) = K_{I (2)} [m_{1} (s_{4}) m_{2} (s_{4}) + m_{1} (Θ) m_{2} (s_{4}) + m_{1} (s_{4}) m_{2} (Θ)] = 0.0148 {\bar{m}}_{I (2)} (Θ) = K_{I (2)} [{\bar{m}}_{1} (Θ) {\bar{m}}_{2} (Θ)] = 0.283 {\tilde{m}}_{I (2)} (Θ) = K_{I (2)} [{\tilde{m}}_{1} (Θ) {\tilde{m}}_{2} (Θ) + {\bar{m}}_{1} (Θ) {\tilde{m}}_{2} (Θ) + {\tilde{m}}_{1} (Θ) {\bar{m}}_{2} (Θ)] = 0.109$

Finally, using Definition 13, the distribution of the overall belief degrees can be calculated: $\begin{matrix} β_{a l l} & = & {β_{a l l} (s_{0}) = \frac{m_{I (2)} (s_{0})}{1 - {\bar{m}}_{I (2)} (Θ)} = 0, β_{a l l} (s_{1}) = \frac{m_{I (2)} (s_{1})}{1 - {\bar{m}}_{I (2)} (Θ)} = 0.3015, \\ β_{a l l} (s_{2}) & = & \frac{m_{I (2)} (s_{2})}{1 - {\bar{m}}_{I (2)} (Θ)} = 0.3335, β_{a l l} (s_{3}) = \frac{m_{I (2)} (s_{3})}{1 - {\bar{m}}_{I (2)} (Θ)} = 0.1926, \\ β_{a l l} (s_{4}) & = & \frac{m_{I (2)} (s_{4})}{1 - {\bar{m}}_{I (2)} (Θ)} = 0.0206, β_{1} (Θ) = \frac{{\tilde{m}}_{I (2)} (Θ)}{1 - {\bar{m}}_{I (2)} (Θ)} = 0.1518} \end{matrix}$

3.4 The steps of the proposed method

With respect to a MCDM problem, suppose there are n alternatives A = {a₁, a₂ … a_t … a_n} and m criteria C = {c₁, c₂ … c_j … c_m} with the weight vector W = {w₁, w₂ … w_j … w_m} associated with C, where w_j ∈ [0, 1] and $\sum_{j = 1}^{m} w_{j} = 1$ . The DMs are more willing to provide their assessments by giving some linguistic expressions or sentences. Therefore, one linguistic term set S = {s₀, s₁ ⋯ s_g} is applied to assess the alternative a_t for the criterion c_j, together with a context-free grammar G_H which uses comparative linguistic terms to produce the linguistic expressions ll. Such linguistic expression can be converted into the HFLTS according to the transformation function E_{G
_H}. Hence, an evaluation matrix with HFLTS can be obtained as $H = {(h_{t j})}_{m \times n}$ , where $h_{t j}$ is the assessment for alternative a_t corresponding to criterion c_j and is in the form of HFLTS.

As a result of the discussion above, we are now in a position to describe a procedure for solving the MCDM problem with HFLTS. The procedure comprises the following steps:Step 1. Transform the HFLTS into the synthetic cloud as $(h_{t j})_{m \times n} \to ({\tilde{c}}_{tj} ({Ex}_{tj}, {En}_{tj}, {He}_{tj}))_{m \times n}$ .

After transforming all the linguistic terms in the linguistic term set into the corresponding clouds using Definition 7 in Section 3.1, the fuzzy envelopes of HFLTS $env (h_{t j})$ can be obtained according to Definition 8 in the same section: $env (h_{t j}) = {\tilde{c}}_{tj} ({Ex}_{tj}, {En}_{tj}, {He}_{tj})$ .

Step 2. Convert the synthetic cloud into the belief degrees as $({\tilde{c}}_{tj} ({Ex}_{tj}, {En}_{tj}, {He}_{tj}))_{m \times n} \to (β_{tj}^{'})_{m \times n}$ .

First, convert the synthetic cloud into the belief degrees using Definition 9 in Section 3.2, then use the discounting coefficient to modify the belief degrees according to Definition 10 in the same section. The distribution of the discounted belief degrees for alternative a_t on criterion c_j is denoted as $β_{tj}^{'}$ .

Step 3. Aggregate the criteria values of each alternative as ${aggreate}_{j = 1}^{m} β_{tj}^{'} \to β_{all} (a_{t})$ .

The criteria values are aggregated and the overall belief degrees of alternative a_t are acquired by the ER algorithm given in Definitions 11–13 in Section 3.3.

Step 4. Rank alternatives.

Rank the alternatives on the basis of β_all (a_t), and then, obtain the best one.

4 Illustrative example

This section demonstrates the implementation process of the proposed method with a numerical example. Let us consider the problem of evaluating three anti-air information warfare systems A = {a₁, a₂, a₃} (adapted from [25]). It is necessary to make a decision according to the following four criteria C = {c₁, c₂, c₃, c₄}: Information suppression c₁, Hard destruction c₂, Network resistance c₃, Psychological resistance c₄, with the associated weight vector W = {0.3, 0.2, 0.4, 0.1}. In addition, comparative linguistic expressions near the natural language are preferred by DMs. Therefore, the context-free grammar G_H and the linguistic term set S = {s₀ = " VeryPoor " , s₁ = " Poor " , s₂ = " MediumPoor " , s₃ = " Fair " , s₄ = " MediumGood " , s₅ = " Good " , s₆ = " VeryGood "} are used. The assessments given by the DMs in linguistic expressions for this problem are illustrated in Table 1. The linguistic expressions are transformed into the HFLTSs according to Definition 2, as shown in Table 2.

Table 1
Evaluation information

c₁ c₂ c₃ c₄

a ₁ Between s₂ and s₃ More than s₄ Less than s₂ Between s₄ and s₅

a ₂ s ₃ Between s₂ and s₄ Between s₂ and s₃ s ₆

a ₃ Between s₃ and s₄ Between s₃ and s₄ s ₄ Less than s₃

	c₁	c₂	c₃	c₄
a ₁	Between s₂ and s₃	More than s₄	Less than s₂	Between s₄ and s₅
a ₂	s ₃	Between s₂ and s₄	Between s₂ and s₃	s ₆
a ₃	Between s₃ and s₄	Between s₃ and s₄	s ₄	Less than s₃

Table 2

HFLTSs generated from the linguistic expressions

	c₁	c₂	c₃	c₄
a ₁	{s₂, s₃}	{s₄, s₅, s₆}	{s₀, s₁, s₂}	{s₄, s₅}
a ₂	{s₃}	{s₂, s₃, s₄}	{s₂, s₃}	{s₆}
a ₃	{s₃, s₄}	{s₃, s₄}	{s₄}	{s₀, s₁, s₂, s₃}

4.1 Procedure for MCDM

Step 1. Transform the HFLTS into the synthetic cloud.

Given the universe [U_min, U_max] = [0, 1] and He = 0.01, by Definition 7, the linguistic terms in the linguistic term set can be converted into 7 clouds as c_{s
₀} (0.071, 0.024, 0.01), c_{s
₁} (0.214, 0.024, 0.01), c_{s
₂} (0.357, 0.024, 0.01), c_{s
₃} (0.5, 0.024, 0.01), c_{s
₄} (0.643, 0.024, 0.01), c_{s
₅} (0.786, 0.024, 0.01), c_{s
₆} (0.929, 0.024, 0.01). Then, the fuzzy envelope of HFLTS env (H_S) = ${\tilde{c}}_{H_{S}}$ (Ex_{H
_S}, En_{H
_S}, He_{H
_S}) can be obtained through the cloud aggregation algorithm proposed in Definition 8, as shown in Table 3.

Table 3
Synthetic clouds generated from HFLTSs ( ${\tilde{c}}_{tj} ({Ex}_{tj}, {En}_{tj}, {He}_{tj})$ )

c₁ c₂ c₃ c₄

a ₁ ${\tilde{c}}_{11} (0.429, 0.048, 0.01)$ ${\tilde{c}}_{12} (0.786, 0.072, 0.01)$ ${\tilde{c}}_{13} (0.214, 0.072, 0.01)$ ${\tilde{c}}_{14} (0.715, 0.048, 0.01)$

a ₂ ${\tilde{c}}_{21} (0.5, 0.024, 0.01)$ ${\tilde{c}}_{22} (0.5, 0.072, 0.01)$ ${\tilde{c}}_{23} (0.429, 0.048, 0.01)$ ${\tilde{c}}_{24} (0.929, 0.024, 0.01)$

a ₃ ${\tilde{c}}_{31} (0.572, 0.048, 0.01)$ ${\tilde{c}}_{32} (0.572, 0.048, 0.01)$ ${\tilde{c}}_{33} (0.643, 0.024, 0.01)$ ${\tilde{c}}_{34} (0.286, 0.096, 0.01)$

	c₁	c₂	c₃	c₄
a ₁	${\tilde{c}}_{11} (0.429, 0.048, 0.01)$	${\tilde{c}}_{12} (0.786, 0.072, 0.01)$	${\tilde{c}}_{13} (0.214, 0.072, 0.01)$	${\tilde{c}}_{14} (0.715, 0.048, 0.01)$
a ₂	${\tilde{c}}_{21} (0.5, 0.024, 0.01)$	${\tilde{c}}_{22} (0.5, 0.072, 0.01)$	${\tilde{c}}_{23} (0.429, 0.048, 0.01)$	${\tilde{c}}_{24} (0.929, 0.024, 0.01)$
a ₃	${\tilde{c}}_{31} (0.572, 0.048, 0.01)$	${\tilde{c}}_{32} (0.572, 0.048, 0.01)$	${\tilde{c}}_{33} (0.643, 0.024, 0.01)$	${\tilde{c}}_{34} (0.286, 0.096, 0.01)$

Step 2. Convert the synthetic cloud into the belief degrees.

For each synthetic cloud, use Definition 9 and Definition 10 to obtain the distribution of the discounted belief degrees. The result is shown in Table 4.

Table 4

Distribution of the discounted belief degrees ( $β_{tj}^{'} = {β_{tj}^{'} (s_{0}), β_{tj}^{'} (s_{1}), β_{tj}^{'} (s_{2}), β_{tj}^{'} (s_{3}), β_{tj}^{'} (s_{4}), β_{tj}^{'} (s_{5}), β_{tj}^{'} (s_{6})}$ )

	c₁	c₂	c₃	c₄
a ₁	{0, 0, 0 . 465, 0 . 451, 0, 0, 0}	{0, 0, 0, 0, 0 . 065, 0 . 7, 0 . 068}	{0 . 065, 0 . 7, 0 . 068, 0, 0, 0, 0}	{0, 0, 0, 0, 0 . 465, 0 . 451, 0}
a ₂	{0, 0, 0 . 002, 0 . 996, 0 . 002, 0, 0}	{0, 0, 0 . 065, 0 . 7, 0 . 068, 0, 0}	{0, 0, 0 . 465, 0 . 451, 0, 0, 0}	{0, 0, 0, 0, 0, 0 . 002, 0 . 998}
a ₃	{0, 0, 0, 0 . 465, 0 . 451, 0, 0}	{0, 0, 0, 0 . 465, 0 . 451, 0, 0}	{0, 0, 0, 0 . 002, 0 . 996, 0 . 002, 0}	{0 . 013, 0 . 364, 0 . 358, 0 . 016, 0, 0, 0}

Step 3. Aggregate the criteria values of each alternative.

Firstly, as shown in Table 5, the distribution of the discounted belief degrees of each criterion is transformed into the BPA by Definition 11. Next, the BPAs of all four criteria are aggregated into the combined probability assignment for each alternative by Definition 12. The result is shown in Table 6. Finally, by Definition 13, the combined probability assignment is normalized into the overall belief degrees, as described in Table 7.

Table 5

Basic probability assignment ( $m_{tj} = {m_{tj} (s_{0}), m_{tj} (s_{1}), m_{tj} (s_{2}), m_{tj} (s_{3}), m_{tj} (s_{4}), m_{tj} (s_{5}), m_{tj} (s_{6}), {\bar{m}}_{tj} (Θ), {\tilde{m}}_{tj} (Θ)}$ )

	c₁	c₂	c₃	c₄
a ₁	{0, 0, 0.14, 0.135, 0, 0,	{0, 0, 0, 0, 0.013, 0.14,	{0.026, 0.28, 0.027, 0,	{0, 0, 0, 0, 0.047, 0.045,
	0, 0.7, 0.025}	0.014, 0.8, 0.033}	0, 0, 0, 0.6, 0.067}	0, 0.9, 0.008}
a ₂	{0, 0, 0, 0.299, 0.001,	{0, 0, 0.013, 0.14, 0.014,	{0, 0, 0.186, 0.181, 0,	{0, 0, 0, 0, 0, 0,
	0, 0, 0.7, 0}	0, 0, 0.8, 0.033}	0, 0, 0.6, 0.033}	0.1, 0.9, 0}
a ₃	{0, 0, 0, 0.14, 0.135,	{0, 0, 0, 0.093, 0.09,	{0, 0, 0, 0.001, 0.398, 0.001,	{0.001, 0.036, 0.036, 0.002,
	0, 0, 0.7, 0.025}	0, 0, 0.8, 0.017}	0, 0.6, 0}	0, 0, 0, 0.9, 0.025}

Table 6

Combined probability assignment

	m_I(4) (s₀)	m_I(4) (s₁)	m_I(4) (s₂)	m_I(4) (s₃)	m_I(4) (s₄)	m_I(4) (s₅)	m_I(4) (s₆)	${\bar{m}}_{I (4)} (Θ)$	${\tilde{m}}_{I (4)} (Θ)$
a ₁	0.018	0.191	0.11	0.085	0.031	0.103	0.007	0.376	0.079
a ₂	0	0	0.122	0.442	0.007	0	0.042	0.352	0.035
a ₃	0.001	0.015	0.015	0.127	0.458	0.001	0	0.353	0.03

Table 7

Distribution of the overall belief degrees

	β_all (s₀)	β_all (s₁)	β_all (s₂)	β_all (s₃)	β_all (s₄)	β_all (s₅)	β_all (s₆)	β_all (Θ)
a ₁	0.029	0.306	0.176	0.136	0.049	0.165	0.012	0.127
a ₂	0	0	0.188	0.681	0.011	0	0.066	0.054
a ₃	0.001	0.023	0.023	0.197	0.708	0.001	0	0.047

Step 4. Rank the alternatives.

The distributed assessment and their hesitation for three alternatives can be shown graphically as in Figs. 3 and 4. From Fig. 3, the differences among three alternatives can be identified and be used to rank them. For example, a₃ is apparently preferred to a₂ as the former is assessed to be Medium Good at a high degree and assessed to be Fair at a certain degree while the latter is assessed to be Fair at a high degree and assessed to be Medium Poor at a certain degree. It can be seen from Fig. 3 that among three alternatives only a₁ is to a high degree assessed to Poor and also it is to a certain extent assessed to Medium Poor. Furthermore, it can be seen from Figs. 3 and 4 that the evaluation of a₁ is more distributed and hesitant. Therefore, a₁ should be the worst in performance. We can obtain the following intuitive ranking order based on the above observations: a₃ > a₂ > a₁.

Fig.3

Distributed assessment for three alternatives.

Fig.4

Hesitation for three alternatives.

By calculating the utilities, ranking order which is more precise can be obtained. Suppose the utilities of linguistic terms are U = {u (s₀) = 0, u (s₁) = 0.167, u (s₂) = 0.333, u (s₃) =0.5, u (s₄) = 0.667, u (s₅) = 0.833, u (s₆) = 1}. Then we can calculate the maximum, minimum and average utilities of the alternative as follows [49]: ${\begin{matrix} u_{max} = \sum_{i = 1}^{g - 1} β_{all} (s_{i}) u (s_{i}) + (β_{all} (s_{g}) + β_{all} (Θ)) u (s_{g}) \\ u_{min} = \sum_{i = 2}^{g} β_{all} (s_{i}) u (s_{i}) + (β_{all} (s_{1}) + β_{all} (Θ)) u (s_{1}) \\ u_{avg} = \frac{u_{max} + u_{min}}{2} \end{matrix}$ (9)

We can rank the alternatives according to their average utilities, which are calculated as u_avg (a₁) = 0.423, u_avg (a₂) = 0.503, u_avg (a₃) = 0.606. It can be known that a₃ > a₂ > a₁, which is the same as the ranking result got previously by intuitive analysis.

4.2 Comparative analysis

For validation of the feasibility of the proposed method, a comparative analysis is conducted by another four linguistic MCDM methods. This analysis is based on the same numerical example mentioned above. A comparison of four methods is shown in Table 8.

Table 8
Comparison with different methods

Methods Ranking results

The multi-criteria linguistic decision-making model [22] a₂ > a₃ > a₁

The MCDM method based on the HLWA operator [26] {a₂, a₃} > a₁

The MCDM outranking approach based on HFLTS [25] a₃ > a₂ > a₁

The MCDM method based on the fuzzy envelope [1] a₃ > a₂ > a₁

The proposed method a₃ > a₂ > a₁

Methods	Ranking results
The multi-criteria linguistic decision-making model [22]	a₂ > a₃ > a₁
The MCDM method based on the HLWA operator [26]	{a₂, a₃} > a₁
The MCDM outranking approach based on HFLTS [25]	a₃ > a₂ > a₁
The MCDM method based on the fuzzy envelope [1]	a₃ > a₂ > a₁
The proposed method	a₃ > a₂ > a₁

As Table 8 shows, the ranking of alternatives obtained by the proposed method is identical to the results of the methods in [1] and [25], but is different from the results of the methods in [22] and [26]. These results indicate: (1) The multi-criteria linguistic decision-making model [22] considers the hesitance among the linguistic variables to produce a range directly, but distort the fuzziness of the information. Therefore, it’s not reliable to conclude that a₂ > a₃. (2) A total order can not be provided by the MCDM method based on the HLWA operator [26], so a₂ and a₃ can not be ranked. (3) The result obtained by the MCDM outranking approach based on HFLTS [25] is consistent with the proposed method, but the basis is not similar. The outranking approach involves systematic comparisons of the assessments of alternatives for each criterion, and the calculation process is complex. (4) The result obtained by the MCDM method based on the fuzzy envelope [1] is also the same as the proposed method. The fuzzy envelope of an HFLTS is expressed by a fuzzy membership function and then integrated with the fuzzy TOPSIS model. However, the randomness of linguistic variables is not considered.

In contrast, considering both the randomness and fuzziness of linguistic variables, the proposed method transforms the HFLTS to the synthetic cloud. Considering that the HFLTS consists of the linguistic terms each of which only the belief degree varies, the proposed method converts the synthetic cloud into the belief degrees. Considering the hesitation of the assessment, the proposed method aggregates the criteria values with the evidential reasoning algorithm. Furthermore, compared to the considered methods, the result of the proposed method shows not only the ranking order but also the belief degree of each linguistic term and the hesitant degree of the assessment using percentage values.

5 Conclusion

HFLTS is more appropriate for the linguistic MCDM problem than traditional linguistic set. In this paper, a new MCDM method for HFLTS is proposed, which takes advantages of both the cloud model and evidence theory. To facilitate the computing processes and take both the randomness and fuzziness of linguistic variables into consideration, the envelope for HFLTS is obtained based on the cloud model. To overcome the drawbacks of tradition aggregation operators, the criteria values are aggregated based on the evidence theory. The contributions of this paper are threefold. First, a new envelope for HFLTS is proposed, in which a synthetic cloud is generated by aggregating the clouds of the linguistic terms in the HFLTS. Second, a method for converting the synthetic cloud into the belief degrees is developed, according to the distribution of cloud drops and the discounting coefficient. Last but not the least, we use the ER algorithm to aggregate different criteria values and rank the alternatives based on the belief degrees. A numerical example together with the corresponding comparison analysis with other methods is used to demonstrate the feasibility and validity of the proposed method. The results show that more reasonable and accurate ranking outcomes can be obtained by the new method. Moreover, the belief degree of each linguistic term and the hesitant degree of the assessment can be obtained simultaneously. Therefore, the new method has much application potential. In future research, we will apply our method in resolution of the problems that may include different types of evaluation information such as numerical, linguistic and interval-valued information.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 71301081, 61373139, 61572261), Natural Science Foundation of Jiangsu Province (No. BK20130877, BK20150868), Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (No.17KJB520027), Natural Science Foundation of Nanjing University of Posts and Telecommunications (No. NY218073).

References

Liu

H.B.

and Rodriguez

R.M.

, A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multi-criteria decision making, Inform Sci 258 (2014),220–238.

Zadeh

L.A.

, The concept of a linguistic variable and its application approximate reasoning, Inform Sci 9(1) (1976), 43–58.

S.M.

, Zhang

H.Y.

and Wang

J.Q.

, Hesitant fuzzy linguistic Maclaurin symmetric mean operators and their applications to multi-criteria decision-making problem, Int J Intell Syst (2017). doi:.

Herrera-Viedma

, Lopez-Herrera

A.G.

, Luque

and Porcel

, A fuzzy linguistic IRS model based on a 2-tuple fuzzy linguistic approach, Int J Unc Fuzz Knowl Based Syst 15(2) (2007), 225–250.

Zhang

X.Y.

and Wang

J.Q.

, Consensus-based framework to MCGDM under multi-granular uncertain linguistic environment, J Intell Fuzzy Syst (2017), doi:.

Y.J.

, Ma

, Xu

W.J.

and Wang

H.M.

, An incomplete multi-granular linguistic model and its application in emergency decision of unconventional outburst incidents, J Intell Fuzzy Syst 29 (2015), 619–633.

Herrera

, Alonso

, Chiclana

and Herrera-Viedma

, Computing with words in decision making: Foundations, trends and prospects, Fuzzy Optim Decis Making 8 (2009), 337–364.

Martínez

, Ruan

and Herrera

, Computing with words in decision support systems: An overview on models and applications, Int J Comput Intell Syst 3(4) (2010), 382–395.

Delgado

, Verdegay

J.L.

and Vila

M.A.

, Linguistic decision making models, Int J Intell Syst 7(4) (1992), 479–492.

10.

Wang

, Wang

J.Q.

and Zhang

H.Y.

, A likelihood-based TODIM approach based on multi-hesitant fuzzy linguistic information for evaluation in logistics outsourcing, Comput Ind Eng 99 (2016), 287–299.

11.

S.M.

, Wang

and Wang

J.Q.

, An interval type-2 fuzzy likelihood-based MABAC approach and its application in selecting hotels on the tourism website, Int J Fuzzy Syst 19(1) (2017), 47–61.

12.

S.M.

, Wang

and Wang

J.Q.

, An extended TODIM approach with intuitionistic linguistic numbers, Int Trans Oper Res (2016). doi:.

13.

Bordogna

, Fedrizzi

and Passi

, A linguistic modeling of consensus in group decision making based on OWA operators, IEEE Trans Syst Man Cybern 27(1) (1997), 126–132.

14.

Z.S.

, A method based on fuzzy linguistic assessments and linguistic ordered weighted averaging operator for multi-attribute group decision making problems, Syst Eng 20(5) (2002), 79–82.

15.

J.P.

, Wu

Z.B.

and Zhang

, A consensus based method for multi-criteria group decision making under uncertain linguistic setting, Group Decision Negotiation 23 (2014), 127–148.

16.

Z.B.

and Xu

J.P.

, Possibility distribution-based approach for MAGDM with hesitant fuzzy linguistic information, IEEE Trans Cybern 46 (2016), 694–705.

17.

Martínez

and Herrera

, An overview on the 2-tuple linguistic model for computing with words in decision making: Extensions, applications and challenges, Inform Sci 207 (2012), 1–18.

18.

Liu

H.C.

, Qin

J.T.

, Mao

L.X.

and Zhang

Z.Y.

, Personnel selection using interval 2-tuple linguistic VIKOR method, Hum Factors Ergon Manuf Service Ind 25 (2015), 370–384.

19.

Wang

, Wang

J.Q.

, Zhang

H.Y.

and Chen

X.H.

, Multi-criteria group decision making approach based on 2-tuple linguistic aggregation operators with multi-hesitant fuzzy linguistic information, Int J Fuzzy Syst 18 (2016), 81–97.

20.

Dong

Y.C.

, Li

C.C.

and Herrera

, Connecting the linguistic hierarchy and the numerical scale for the 2-tuple linguistic model and its use to deal with hesitant unbalanced linguistic information, Inform Sci 367 (2016), 259–278.

21.

Rodrígueza

R.M.

, Bedregalb

, Bustincec

, ., A position and perspective analysis of hesitant fuzzy sets on information fusion in decision making. Towards high quality progress, Inform Fus 29 (2016), 89–97.

22.

Rodriguez

R.M.

, Martinez

and Herrera

, Hesitant fuzzy linguistic term sets for decision making, IEEE Trans Fuzzy Syst 20(1) (2012), 109–119.

23.

Jordi

M.A.

, Nuria

, Monica

, ., Consensus, dissension and precision in group decision making by means of an algebraic extension of hesitant fuzzy linguistic term sets, Inform Fus 42 (2018), 1–11.

24.

Zhang

B.W.

, Liang

H.M.

and Zhang

G.Q.

, Reaching a consensus with minimum adjustment in MAGDM with hesitant fuzzy linguistic term sets, Inform Fus 42 (2018), 12–23.

25.

Wang

J.Q.

, Wang

, Chen

Q.H.

, Zhang

H.Y.

and Chen

X.H.

, An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets, Inform Sci 280 (2014), 338–351.

26.

Wei

C.P.

, Zhao

and Tang

X.J.

, Operators and comparisons of hesitant fuzzy linguistic term sets, IEEE Trans Fuzzy Syst 22 (2014), 575–585.

27.

Y.Z.

, Li

C.C.

, Chen

, ., Group decision making based on linguistic distributions and hesitant assessments: Maximizing the support degree with an accuracy constraint, Inform Fus 41 (2018), 151–160.

28.

C.C.

, Rodriguez

R.M.

, Martinez

, ., Personalized individual semantics based on consistency in hesitant linguistic group decision making with comparative linguistic expressions, Knowl Based Syst 145 (2018), 156–165.

29.

D.Y.

, Meng

H.J.

and Shi

X.M.

, Membership clouds and membership cloud generators, Int J Intell Syst 32 (1995), 16–21.

30.

Wang

J.Q.

, Peng

J.J.

, Zhang

H.Y.

, ., An uncertain linguistic multi-criteria group decision-making method based on a cloud model, Group Decision Negotiation 24 (2015), 171–192.

31.

Wang

J.Q.

, Peng

, Zhang

H.Y.

and Chen

X.H.

, Method of multi-criteria group decision-making based on cloud aggregation operators with linguistic information, Inform Sci 274 (2014), 177–191.

32.

Wang

J.Q.

, Wang

, Zhang

H.Y.

and Chen

X.H.

, Atanassov’s interval-valued intuitionistic linguistic multi-criteria group decision-making method based on the trapezium cloud model, IEEE Trans Fuzzy Syst 23(3) (2015), 542–554.

33.

Yang

, Yan

and Zeng

, How to handle uncertainties in AHP: The Cloud Delphi hierarchical analysis, Inform Sci 222 (2013), 384–404.

34.

Yager

R.R.

, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Trans Syst Man Cybern 18 (1988), 183–190.

35.

Z.S.

and Da

Q.L.

, The ordered weighted geometric averaging operators, Int J Intell Syst 17 (2002), 709–716.

36.

Herrera

and Martínez

, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans Fuzzy Syst 8 (2000), 746–752.

37.

Peng

X.D.

, Hesitant trapezoidal fuzzy aggregation operators based on Archimedean t-norm and t-conorm and their applica-tion in MADM with completely unknown weight information, Int J Uncertainty Quantification (2017). doi:.

38.

Dempster

A.P.

, A generalization of Bayesian inference, J R Statist Soc Ser B 30(2) (1968), 205–247.

39.

Shafer

, A Mathematical Theory of Evidence, Princeton University Press, 1976.

40.

Yang

J.B.

and Singh

M.G.

, An evidential reasoning approach for multiple attribute decision making with uncertainty, IEEE Trans Syst Man Cybern 24(1) (1994), 1–18.

41.

Yang

J.B.

and Sen

, A general multi-level evaluation process for hybrid MADM with uncertainty, IEEE Trans Syst Man Cybern 24(10) (1994), 1458–1473.

42.

Liu

Z.G.

, Pan

, Dezert

and Martin

, Adaptive imputation of missing values for incomplete pattern classification, Pattern Recogn 52 (2016), 85–95.

43.

Voola

and Vinaya

B.A.

, Study of aggregation algorithms for aggregating imprecise software requirement, Eur J Oper Res 259 (2017), 1191–1199.

44.

Zhou

, Liu

L.F.

, Guo

and Sun

L.J.

, Multi-sensor data fusion for water quality evaluation using Dempster-Shafer evidence theory, Int J Distrib Sensor Networks 12 (2013), 1–6.

45.

Yuan

, Xiao

, Fei

, ., Modeling sensor reliability in fault diagnosis based on evidence theory, Sensors 16 (2016), 1–13.

46.

Dymova

, Sevastjanov

and Kaczmarek

, A forex trading expert system based on a new approach to the rule base evidential reasoning, Expert Syst Appl 51 (2016), 1–13.

47.

, Huhns

and Yang

, A consensus framework for multiple attribute group decision analysis in an evidential reasoning context, Inform Sci 17 (2014), 22–35.

48.

D.Y.

, Liu

C.Y.

and Gan

W.Y.

, A new cognitive model: Cloud model, Int J Intelligent Syst 24 (2009), 357–375.

49.

Yang

J.B.

, Wang

Y.M.

, Xu

D.L.

and Chin

K.S.

, The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties, Eur J Oper Res 171 (2006), 309–343.

A multi-criteria decision-making method for hesitant fuzzy linguistic term set based on the cloud model and evidence theory

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Basics of HFLTS

2.3 Basics of evidence theory

3.1 A new representation of HFLTS based on the cloud model

4 Illustrative example

Table 1 Evaluation information c1 c2 c3 c4 a 1 Between s2 and s3 More than s4 Less than s2 Between s4 and s5 a 2 s 3 Between s2 and s4 Between s2 and s3 s 6 a 3 Between s3 and s4 Between s3 and s4 s 4 Less than s3

Footnotes

Acknowledgments

References

Table 1
Evaluation information

c₁ c₂ c₃ c₄

a ₁ Between s₂ and s₃ More than s₄ Less than s₂ Between s₄ and s₅

a ₂ s ₃ Between s₂ and s₄ Between s₂ and s₃ s ₆

a ₃ Between s₃ and s₄ Between s₃ and s₄ s ₄ Less than s₃