Linguistic multi-attribute decision making with considering decision makers’ risk preferences

Abstract

For qualitative decision making problems based on decision makers’ (DMs’) risk preferences and unbalanced linguistic term sets (LTSs), a new linguistic multi-attribute decision making (MADM) method is developed. Firstly, a concept of the generalized linguistic term set (GLTS) with triangular fuzzy semantic information is introduced. In order to capture and measure the DM’s risk preference, a triangular fuzzy membership function with risk preference parameters is constructed. Then, based on the expected semantic information of linguistic terms given by the DM and the distance between two triangular fuzzy numbers, a nonlinear programming model is established to obtain an optimal GLTS. An approach to linguistic MADM considering the DM’s risk preference is developed and its detailed steps are given. Finally, a numerical example and a sensitivity analysis of risk preference parameters are examined to illustrate the feasibility and effectiveness of the proposed models.

Keywords

Multi-attribute decision making (MADM)risk preference triangular fuzzy number generalized linguistic term set (GLTS)decision matrix

1 Introduction

Multi-attribute decision making (MADM) is an important research content of decision science, whose theory and methods have been widely applied in many fields [28, 35]. Because of the complexity of decision making environment and fuzziness of human cognition and judgment, it is difficult for decision makers (DMs) to use exact numbers to describe evaluation values of alternatives with respect to attributes in a large number of MADM problems. Thus, MADM has employed different forms to express evaluation values of alternatives with respect to attributes, such as interval numbers, triangular fuzzy numbers, trapezoid fuzzy numbers, intuitionistic fuzzy numbers, picture fuzzy numbers, linguistic terms, trapezoid fuzzy 2-tuples and so on [7 , 40]. In linguistic MADM, a DM uses linguistic terms to express evaluation values of alternatives with regard to attributes. Therefore, the distribution and semantic information of linguistic terms in a linguistic term set (LTS) are crucial to the rationality and validity of solving linguistic MADM problems.

LTSs can be divided into two broad categories [2, 29]: balanced LTSs and unbalanced LTSs. Herrera and Martínez [15] proposed a 2-tuple linguistic representation model to transform linguistic terms in a balanced LTS into linguistic 2-tuples and gave a weighted aggregation method based on 2-tuple linguistic information. The 2-tuple linguistic harmonic operators were developed by Jin et al. [18] and applied to solve MADM problems based upon a balanced LTS. Some aggregation operators were developed to solve MADM problems with hesitant fuzzy linguistic information in Wei [31, 33], Wei et al. [34] and Lu and Wei [24]. Herrera et al. [14] established a hierarchical linguistic structure model to solve MADM problems based on linguistic terms in unbalanced LTSs. Wang and Hao [30] offered a proportional 2-tuple linguistic representation model to compute with words in an unbalanced LTS. Based on this proportional 2-tuple model, Li and Dong [20] put forward unbalanced 2-tuple linguistic aggregation operators to solve MADM problems with risk preferences and completely unknown attribute weights. Li et al. [21] proposed a personalized individual semantics model and a computing with words (CW) framework to deal with linguistic group decision making (GDM) problems involving a consensus process. Dong et al. [10] presented a consensus-based linear programming model for GDM with multi-granular unbalanced 2-tuple linguistic preference relations. Dong and Liu [23] proposed a two-stage linear programming model to solve the GDM problem based on heterogeneous preference relations with self-confidence. Donget al. [8] proposed an optimization-based approach to improve the consistency level of unbalanced linguistic preference relations. Dong et al. [9 , 12] defined some CW methods to handle hesitant unbalanced fuzzy linguistic information and unbalanced 2-tuple linguistic information. Cabrerizo et al. [1, 3] studied on GDM problems with unbalanced linguistic information involving soft consensus measures and granular computing of linguistic information.

In practical linguistic MADM, DMs usually have risk preferences, such as the venture investment evaluation [20], the multi-attribute reverse auction problem [16] and so on. Risk preferences reflect DMs’ expectations and requirements on semantic values of linguistic terms in an LTS. Thus, it is an important content to study the semantic model of linguistic terms considering the DM’s risk preference on linguistic MADM. Zhou and Xu [41] extended the sigmoid function by introducing two parameters to reflect the DM’s risk preference. They constructed a generalized linguistic term set (GLTS) based on the extended sigmoid semantic function and applied it to qualitative MADM.

As the semantic value of a linguistic term based upon the extended sigmoid semantic function is in the form of exact numbers [41], it is difficult and inconvenient to describe the fuzziness and uncertainty of linguistic terms. Therefore, in this paper, a triangular fuzzy membership function is established to express semantic information of a linguistic term based on the DM’s risk preference. A nonlinear programming model is developed to construct a GLTS of the DM with the risk preference, and then a method is presented to deal with linguistic MADM involving DMs’ risk preferences.

The rest of this paper is laid out as follows. Basic concepts and properties of LTSs and triangular fuzzy numbers are introduced in Section 2. The definition of the GLTS based on triangular fuzzy numbers involving risk preference parameters is proposed and the method for solving risk preference parameters is designed in Section 3. Section 4 proposed an approach to solve linguistic MADM problems considering DMs’ risk preferences. In Section 5, a numerical example about contingency plans for emergency events and a sensitivity analysis of risk preference parameters are presented to explain that the proposed models are feasible and effective, and then their advantages are discussed. Finally, research conclusions and future works are given in Section 6.

2 Preliminaries

An LTS is made up of linguistic evaluation terms. Generally, a set of linguistic terms can be expressed as follows [15]: $S^{*} = {s_{i}^{*} | i = 0, 1, 2, \dots, g}$ (1) where g + 1 is a positive odd number and represents the granularity of a set of linguistic terms S^*. $s_{i}^{*}$ represents the label of the ith linguistic term.

For example, when g = 6, an LTS which is composed of seven linguistic evaluation terms is expressed as: $S^{*} = {\begin{matrix} s_{0}^{*} = extremely poor (EP), s_{1}^{*} = very poor (VP), \\ s_{2}^{*} = poor (P), s_{3}^{*} = medium (M), s_{4}^{*} = good (G), \\ s_{5}^{*} = very good (VG), s_{6}^{*} = extremely good (EG) \end{matrix}}$

The LTS S^* has the following characteristics [15]:

The order characteristic: when i > j, then $s_{i}^{*} > s_{j}^{*}$ and vice versa;

A negation operator is existed, $N e g (s_{i}^{*}) = s_{j}^{*}$ , in which j = g - i;

A maximization operator is existed: when $s_{i}^{*} \geq s_{j}^{*}$ , then $max (s_{i}^{*}, s_{j}^{*}) = s_{i}^{*}$ and vice versa;

A minimization operator is existed: when $s_{i}^{*} \leq s_{j}^{*}$ , then $min (s_{i}^{*}, s_{j}^{*}) = s_{i}^{*}$ and vice versa.

Xu [37] proposed another representation model for an LTS: $S = {s_{i} | i = - g, \dots, - 1, 0, 1, \dots, g}$ (2) where g is a positive integer, and 2g + 1 is the granularity of the LTS S; s₀ denotes a neutral linguistic term, such as “medium”, “indifference”.

A set of linguistic terms S has the following two important properties:

The order property: when i > j, then s_i > s_j and vice versa;

There is a negation operator, Neg (s_i) = s_-i, where Neg (s₀) = s₀.

Although the representation of the subscript of linguistic term labels in the LTS S is different from the one in the LTS S^*, they have the same ability to express linguistic evaluation terms. For instance, the above LTS S^* with seven granularities can be equivalently represented as: $S = {\begin{matrix} s_{- 3} = extremely poor (EP), s_{- 2} = very poor (VP), \\ s_{- 1} = poor (P), s_{0} = medium (M), s_{1} = good (G), \\ s_{2} = very good (VG), s_{3} = extremely good (EG) \end{matrix}}$

In linguistic MADM analysis, a DM sometimes needs to adopt an unbalanced LTS (see Fig. 1) to express his/her evaluation information [9 , 41]. For the convenience of symbolic representation, this paper still employs S to indicate an unbalanced LTS. That is $S = {s_{- τ_{1}}, s_{- τ_{1} + 1}, \dots, s_{0}, \dots, s_{τ_{2} - 1}, s_{τ_{2}}}$ (3) where s₀ is a neutral linguistic term, τ₁ and τ₂ are positive integers, τ₁ + τ₂ + 1 is the granularity of the unbalanced LTS S.

Fig.1

A set of six asymmetrical and non-uniform linguistic terms.

Obviously, the granularity of an unbalanced LTS can be an even number (see Fig. 1). Thus, the inverse operation of linguistic terms may not always be found.

The semantic information of linguistic terms in an LTS can be described by fuzzy numbers, such as triangular fuzzy numbers, trapezoidal fuzzy numbers and so forth.

Let ${\tilde{v}}_{1} = (v_{1}^{L}, v_{1}^{M}, v_{1}^{U})$ , ${\tilde{v}}_{2} = (v_{2}^{L}, v_{2}^{M}, v_{2}^{U})$ be two non-negative triangular fuzzy numbers, and then the distance between them is defined as [6]:

$\begin{matrix} d ({\tilde{v}}_{1}, {\tilde{v}}_{2}) \\ = \sqrt{\frac{1}{3} ((v_{1}^{L} - v_{2}^{L})^{2} + (v_{1}^{M} - v_{2}^{M})^{2} + (v_{1}^{U} - v_{2}^{U})^{2})} \end{matrix}$ (4)

In order to compare and rank triangular fuzzy numbers, the following possibility degree formula was introduced by Xu [36]. $\begin{matrix} p ({\tilde{v}}_{1} \geq {\tilde{v}}_{2}) \\ = \frac{min {{len}^{-} ({\tilde{v}}_{1}) + {len}^{-} ({\tilde{v}}_{2}), max {v_{1}^{M} - v_{2}^{L}, 0}}}{2 {{len}^{-} ({\tilde{v}}_{1}) + {len}^{-} ({\tilde{v}}_{2})}} \\ + \frac{min {{len}^{+} ({\tilde{v}}_{1}) + {len}^{+} ({\tilde{v}}_{2}), max {v_{1}^{U} - v_{2}^{M}, 0}}}{2 {{len}^{+} ({\tilde{v}}_{1}) + {len}^{+} ({\tilde{v}}_{2})}} \end{matrix}$ (5)

where ${len}^{-} ({\tilde{v}}_{1}) = v_{1}^{M} - v_{1}^{L}$ , ${len}^{-} ({\tilde{v}}_{2}) = v_{2}^{M} - v_{2}^{L}$ , ${len}^{+} ({\tilde{v}}_{1}) = v_{1}^{U} - v_{1}^{M}$ , ${len}^{+} ({\tilde{v}}_{2}) = v_{2}^{U} - v_{2}^{M}$ .

A possibility degree matrix P = (p_ij) _n×n is obtained by Equation (5), where p_ij denotes a possibility degree of ${\tilde{v}}_{i} \geq {\tilde{v}}_{j}$ . Let $ω_{i} = \frac{\sum_{j = 1}^{m} p_{ij} + \frac{n}{2} - 1}{n (n - 1)}$ (6)

Then, a vector ω = (ω₁, …, ω_n) ^T can be received from the possibility degree matrix P. The corresponding ranking of alternatives is acquired by a descending order of ω_i (i = 1, 2, …, n).

3 The introduction of a new GLTS

In order to measure the uncertainty of linguistic terms and the influence of DMs’ risk preferences on semantic values of linguistic terms, this section proposes a concept of GLTSs with triangular fuzzy semantic information, and a nonlinear programming model is established to determine risk preference parameters in a GLTS.

3.1 The description of a GLTS based on triangular fuzzy numbers

Definition 1. Let S = {s_-τ₁, s_-τ₁+1, …, s₀, …, s_τ₂-1, s_{τ
₂}} be an LTS, then a GLTS with triangular fuzzy semantic information is defined as

$\begin{matrix} S_{(η)} & = & {〈 s_{t}, {\tilde{v}}_{t} 〉 | t = - τ_{1}, \dots, 0, \dots, τ_{2}, \\ {\tilde{v}}_{t} = (v_{t}^{L}, v_{t}^{M}, v_{t}^{U})} \end{matrix}$ (7) where η = τ₁ + τ₂ + 1 denotes the granularity of this LTS, and ${\tilde{v}}_{t}$ gives a triangular fuzzy semantic value of the linguistic term s_t.

For the sake of describing and measuring DMs’ risk preferences, the triangular fuzzy number ${\tilde{v}}_{t} = (v_{t}^{L}, v_{t}^{M}, v_{t}^{U})$ with two parameters is introduced as follows: $\begin{matrix} v_{t}^{L} & = & {\begin{matrix} {(1 + e^{- θ_{2} t})}^{- 1}, & t = - τ_{1} \\ {(1 + e^{- θ_{2} (t - 1)})}^{- 1}, & t = - τ_{1} + 1, \dots, - 1, 0 \\ {(1 + e^{- θ_{1} (t - 1)})}^{- 1}, & t = 1, 2, \dots, τ_{2} \end{matrix} \end{matrix}$ (8) $\begin{matrix} v_{t}^{M} & = & {\begin{matrix} {(1 + e^{- θ_{2} t})}^{- 1}, & t = - τ_{1}, \dots, - 1 \\ 0.5, & t = 0 \\ {(1 + e^{- θ_{1} t})}^{- 1}, & t = 1, 2, \dots, τ_{2} \end{matrix} \end{matrix}$ (9)

$\begin{matrix} v_{t}^{U} & = & {\begin{matrix} {(1 + e^{- θ_{2} (t + 1)})}^{- 1}, & t = - τ_{1}, \dots, - 1 \\ {(1 + e^{- θ_{1} (t + 1)})}^{- 1}, & t = 0, 1, \dots, τ_{2} - 1 \\ {(1 + e^{- θ_{1} t})}^{- 1}, & t = τ_{2} \end{matrix} \end{matrix}$ (10)

where θ₁ and θ₂ are the DM’s risk preference parameters, which satisfy θ₁, θ₂ > 0.

When θ₁ > θ₂, then the DM prefers to make a decision with risk seeking. When θ₁ = θ₂, then the DM prefers to make a decision with neutral risk. When θ₁ < θ₂, then the DM prefers to make a decision with risk aversion [41]. Clearly, $v_{- 1}^{U} = v_{0}^{M} = v_{1}^{L} = 0.5$ , and then S_(η) satisfies the order property: i > j if and only of $p ({\tilde{v}}_{i} \geq {\tilde{v}}_{j}) > 0.5$ .

When τ₁ = τ₂ > 1 and θ₁ ≠ θ₂, then $d ({\tilde{v}}_{- t}, {\tilde{v}}_{0}) \neq d ({\tilde{v}}_{t}, {\tilde{v}}_{0})$ (t = 1, 2, …, τ₂) can be obtained from Equation (4) and Equations (8– 10), and there also exists t ∈ {- τ₁ + 1, …, -1, 0, 1, …, τ₂ - 1} such that $d ({\tilde{v}}_{t}, {\tilde{v}}_{t + 1}) \neq d ({\tilde{v}}_{t - 1}, {\tilde{v}}_{t})$ . Then, S_(η) is an asymmetrical and non-uniform LTS.

If τ₁ = τ₂ > 1 and θ₁ = θ₂, then $d ({\tilde{v}}_{- t}, {\tilde{v}}_{0}) = d ({\tilde{v}}_{t}, {\tilde{v}}_{0})$ (t = 1, 2, …, τ₂). That is, S_(η) is a symmetrical LTS. In this case, if $d ({\tilde{v}}_{t}, {\tilde{v}}_{t + 1}) \approx d ({\tilde{v}}_{t - 1}, {\tilde{v}}_{t}) (t = - τ_{1} + 1, \dots, - 1, 0, 1, \dots, τ_{2} - 1)$ ,then S_(η) is an approximately uniform and symmetrical LTS; otherwise, it is a symmetrical and non-uniform LTS.

Based on Definition 1, for the set of seven linguistic terms S in Section 2, a GLTS considering risk preferences is denoted as: $S_{(7)} = {\begin{matrix} 〈 s_{- 3}, ({(1 + e^{3 θ_{2}})}^{- 1}, {(1 + e^{3 θ_{2}})}^{- 1}, {(1 + e^{2 θ_{2}})}^{- 1}) 〉, \\ 〈 s_{- 2}, ({(1 + e^{3 θ_{2}})}^{- 1}, {(1 + e^{2 θ_{2}})}^{- 1}, {(1 + e^{θ_{2}})}^{- 1}) 〉, \\ 〈 s_{- 1}, ({(1 + e^{2 θ_{2}})}^{- 1}, {(1 + e^{θ_{2}})}^{- 1}, 0.5) 〉, \\ 〈 s_{0}, ({(1 + e^{θ_{2}})}^{- 1}, 0.5, {(1 + e^{- θ_{1}})}^{- 1}) 〉, \\ 〈 s_{1}, (0.5, {(1 + e^{- θ_{1}})}^{- 1}, {(1 + e^{- 2 θ_{1}})}^{- 1}) 〉, \\ 〈 s_{2}, ({(1 + e^{- θ_{1}})}^{- 1}, {(1 + e^{- 2 θ_{1}})}^{- 1}, {(1 + e^{- 3 θ_{1}})}^{- 1}) 〉, \\ 〈 s_{3}, ({(1 + e^{- 2 θ_{1}})}^{- 1}, {(1 + e^{- 3 θ_{1}})}^{- 1}, {(1 + e^{- 3 θ_{1}})}^{- 1}) 〉 \end{matrix}}$

3.2 The determination of risk preference parameters

Suppose that ${\tilde{V}}^{*} = {{\tilde{v}}_{t_{α}}^{*}, {\tilde{v}}_{t_{β}}^{*}, \dots, {\tilde{v}}_{t_{γ}}^{*}}$ is a set of the DM’s expected or accepted triangular fuzzy semantic values, where t_α, t_β, …, t_γ ∈ {- τ₁, …, τ₂}. In order to obtain optimal values of risk preference parameters θ₁ and θ₂ in the GLTS S_(η), a nonlinear programming model is established as follows:

$\begin{matrix} min J = d ({\tilde{v}}_{t_{α}}, {\tilde{v}}_{t_{α}}^{*}) + d ({\tilde{v}}_{t_{β}}, {\tilde{v}}_{t_{β}}^{*}) + \dots + d ({\tilde{v}}_{t_{γ}}, {\tilde{v}}_{t_{γ}}^{*}) \\ s . t . θ_{1} > 0, θ_{2} > 0 . \end{matrix}$ (11)

Solving the above model (11), we obtain optimal values denoted by $θ_{1}^{*}, θ_{2}^{*}$ . By plugging $θ_{1}^{*}, θ_{2}^{*}$ into Equations (8– 10), a triangular fuzzy membership semantic function is determined. Thereby, an optimal GLTS comes into being.

Example 1. For the GLTS S₍₇₎ with risk preference parameters in Section 3.1, suppose that the DM’s expected triangular fuzzy semantic values are ${\tilde{V}}^{*} = {{\tilde{v}}_{- 2}^{*} = (0.03, 0.1, 0.2), {\tilde{v}}_{1}^{*} = (0.6, 0.7, 0.8),$ ${\tilde{v}}_{2}^{*} = (0.7, 0.8, 0.9)}$ .

Solving the nonlinear programming model (11) yields optimal risk preference parameters $θ_{1}^{*} = 0.7411$ and $θ_{2}^{*} = 1.2515$ . According to Equations (8– 10), the optimal GLTS is determined as $S_{(7)}^{1} = {\begin{matrix} 〈 s_{- 3}, (0.0229, 0.0229, 0.0756) 〉, \\ 〈 s_{- 2}, (0.0229, 0.0756, 0.2224) 〉, \\ 〈 s_{- 1}, (0.0756, 0.2224, 0.5000) 〉, \\ 〈 s_{0}, (0.2224, 0.5000, 0.6772) 〉, \\ 〈 s_{1}, (0.5000, 0.6772, 0.8149) 〉, \\ 〈 s_{2}, (0.6772, 0.8149, 0.9023) 〉, \\ 〈 s_{3}, (0.8149, 0.9023, 0.9023) 〉 \end{matrix}}$

It is easy to verify that $d ({\tilde{v}}_{- t}, {\tilde{v}}_{0}) \neq d ({\tilde{v}}_{t}, {\tilde{v}}_{0}) (t = 1, 2, 3)$ and $d ({\tilde{v}}_{t}, {\tilde{v}}_{t + 1}) \neq d ({\tilde{v}}_{t - 1}, {\tilde{v}}_{t}) (t = - 2, - 1, 0, 1, 2)$ . Thus, $S_{(7)}^{1}$ is an asymmetrical and non-uniform LTS, as shown in Fig. 2.

Fig.2

A set of asymmetrical and non-uniform linguistic terms $S_{(7)}^{1}$ .

Example 2. Suppose that ${\tilde{V}}^{*} = {{\tilde{v}}_{- 2}^{*} = (0.1, 0.2, 0.3), {\tilde{v}}_{1}^{*} = (0.6, 0.7, 0.8), {\tilde{v}}_{2}^{*} = (0.7, 0.8, 0.9)}$ is the set of the DM’s expected triangular fuzzy semantic values.

The optimal risk preference parameters are obtained as $θ_{1}^{*} = θ_{2}^{*} = 0.7411$ by solving the nonlinear programming model (11). Therefore, based on Equations (8– 10), the DM’s optimal GLTS is obtained as follows: $S_{(7)}^{2} = {\begin{matrix} 〈 s_{- 3}, (0.0977, 0.0977, 0.1851) 〉, \\ 〈 s_{- 2}, (0.0977, 0.1851, 0.3228) 〉, \\ 〈 s_{- 1}, (0.1851, 0.3228, 0.5000) 〉, \\ 〈 s_{0}, (0.3228, 0.5000, 0.6772) 〉, \\ 〈 s_{1}, (0.5000, 0.6772, 0.8149) 〉, \\ 〈 s_{2}, (0.6772, 0.8149, 0.9023) 〉, \\ 〈 s_{3}, (0.8149, 0.9023, 0.9023) 〉 \end{matrix}}$

It is easy to verify that $d ({\tilde{v}}_{- t}, {\tilde{v}}_{0}) = d ({\tilde{v}}_{t}, {\tilde{v}}_{0})$ (t = 1, 2, 3) and $d ({\tilde{v}}_{t}, {\tilde{v}}_{t + 1}) \neq d ({\tilde{v}}_{t - 1}, {\tilde{v}}_{t}) (t = - 2, - 1, 1, 2)$ . Thus, $S_{(7)}^{2}$ is a symmetrical and non-uniform LTS, as shown in Fig. 3.

Fig.3

A set of symmetrical and non-uniform linguistic terms $S_{(7)}^{2}$ .

Example 3. Suppose that ${\tilde{V}}^{*} = {\begin{matrix} {\tilde{v}}_{- 3}^{*} = (0.2, 0.2, 0.3), {\tilde{v}}_{- 2}^{*} = (0.2, 0.3, 0.4), \\ {\tilde{v}}_{- 1}^{*} = (0.3, 0.4, 0.5), {\tilde{v}}_{0}^{*} = (0.4, 0.5, 0.6), \\ {\tilde{v}}_{1}^{*} = (0.5, 0.6, 0.7), {\tilde{v}}_{2}^{*} = (0.6, 0.7, 0.8), \\ {\tilde{v}}_{3}^{*} = (0.7, 0.8, 0.8) \end{matrix}}$ is the set of the DM’s expected triangular fuzzy semantic values.

It is easy to verify that ${\tilde{V}}^{*}$ satisfies $d ({\tilde{v}}_{- t}^{*}, {\tilde{v}}_{0}^{*}) = d ({\tilde{v}}_{t}^{*}, {\tilde{v}}_{0}^{*}) (t = 1, 2, 3)$ and $d ({\tilde{v}}_{t}^{*}, {\tilde{v}}_{t + 1}^{*}) = d ({\tilde{v}}_{t - 1}^{*}, {\tilde{v}}_{t}^{*}) (t = - 1, 0, 1)$ , which indicates that the DM expects the LTS to be symmetrically and uniformly distributed.

By solving the nonlinear programming model (11), we obtain optimal risk preference parameters $θ_{1}^{*} = θ_{2}^{*} = 0.4383$ . Then, the DM’s optimal GLTS is determined as: $S_{(7)}^{3} = {\begin{matrix} 〈 s_{- 3}, (0.2117, 0.2117, 0.2939) 〉, \\ 〈 s_{- 2}, (0.2117, 0.2939, 0.3921) 〉, \\ 〈 s_{- 1}, (0.2939, 0.3921, 0.5000) 〉, \\ 〈 s_{0}, (0.3921, 0.5000, 0.6079 〉, \\ 〈 s_{1}, (0.5000, 0.6079, 0.7061) 〉, \\ 〈 s_{2}, (0.6079, 0.7061, 0.7883) 〉, \\ 〈 s_{3}, (0.7061, 0.7883, 0.7883) 〉 \end{matrix}}$

It is easy to verify that $d ({\tilde{v}}_{- t}, {\tilde{v}}_{0}) = d ({\tilde{v}}_{t}, {\tilde{v}}_{0}) (t = 1, 2, 3)$ and $d ({\tilde{v}}_{t}, {\tilde{v}}_{t + 1}) \approx d ({\tilde{v}}_{t - 1}, {\tilde{v}}_{t}) (t = - 2, - 1, 0, 1, 2)$ . Thus, $S_{(7)}^{3}$ is an approximately uniform and symmetrical LTS, as shown in Fig. 4.

Fig.4

A set of approximately uniform and symmetrical linguistic terms $S_{(7)}^{3}$ .

4 A method for linguistic MADM considering risk preferences

In linguistic MADM, let X ={ x₁, x₂, … , x_n } be a finite set of alternatives, A ={ a₁, a₂, …, a_m } be a finite set of attributes, where x_i (i = 1, 2, …, n) and a_j (j = 1, 2, …, m) denote the ith alternative and the jth attribute respectively. Suppose that w = (w₁, w₂, …, w_m) ^T is a weight vector with w_j ∈ [0, 1] and $\sum_{j = 1}^{m} w_{j} = 1$ for the attributes in A. A DM denotes his/her evaluation information of the alternative x_i (i = 1, 2, …, n) with respect to the attribute a_j (j = 1, 2, …, m) by r_ij in the form of linguistic terms in S. Thus, a decision matrix R = (r_ij) _n×m with linguistic terms is constructed, in which r_ij ∈ S. Then, the steps for linguistic MADM involving the DM’s risk preference are detailed as follows:

Step 1. Transform the decision matrix R = (r_ij) _n×m with linguistic terms into the one S° = (s_t°(i,j)) _n×m based on linguistic labels.

Step 2. Suppose that the DM provides his/her expected or accepted triangular fuzzy semantic values denoted by a set ${\tilde{V}}^{*}$ . Then, solving the nonlinear programming model (11) yields optimal risk preferences parameters $θ_{1}^{*}$ and $θ_{2}^{*}$ .

Step 3. Put $θ_{1}^{*}, θ_{2}^{*}$ into Equations (8– 10) and obtain triangular fuzzy semantic values of linguistic terms, and a GLTS S_(η) is acquired by Equation (7). Then, convert the decision matrix S° = (s_t°(i,j)) _n×m based on linguistic labels into a triangular fuzzy decision matrix ${\tilde{V}}^{\circ} = {({\tilde{v}}_{t^{\circ} (i, j)})}_{n \times m}$ , where ${\tilde{v}}_{t^{\circ} (i, j)} = (v_{t^{\circ} (i, j)}^{L}, v_{t^{\circ} (i, j)}^{M}, v_{t^{\circ} (i, j)}^{U})$ .

Step 4. Utilize the triangular fuzzy weighted average operator to aggregate all the triangular fuzzy semantic values in the ith line of the triangular fuzzy decision matrix ${\tilde{V}}^{\circ}$ , and then derive a collective triangular fuzzy evaluation value ${\tilde{v}}_{t^{\circ} (i)}$ of the alternative x_i(i = 1, 2, …, n) as per Equation (12):

$\begin{matrix} {\tilde{v}}_{t^{\circ} (i)} \\ = (v_{t^{\circ} (i)}^{L}, v_{t^{\circ} (i)}^{M}, v_{t^{\circ} (i)}^{U}) \\ = (\sum_{j = 1}^{m} w_{j} v_{t^{\circ} (i, j)}^{L}, \sum_{j = 1}^{m} w_{j} v_{t^{\circ} (i, j)}^{M}, \sum_{j = 1}^{m} w_{j} v_{t^{\circ} (i, j)}^{U}) \end{matrix}$ (12)

Step 5. Based on Equation (5), construct a possibility degree matrix P = (p_ij) _n×n, where $p_{ij} = p ({\tilde{v}}_{t^{\circ} (i)} \geq {\tilde{v}}_{t^{\circ} (j)})$ represents a possibility degree of ${\tilde{v}}_{t^{\circ} (i)}$ being greater than ${\tilde{v}}_{t^{\circ} (j)}$ . Then, employ Equation (6) to get a ranking value ω_i for each alternative x_i (i = 1, 2, …, n), and all alternatives are ranked according to the values of ω_i (i = 1, 2, …, n) in descending order.

5 A numerical example and a sensitivity analysis

In this section, an example adapted from [26] is used to illustrate the feasibility and validity of the proposed method. A sensitivity analysis of risk preference parameters is presented to analyze the influence on the ranking order of alternatives.

5.1 The analysis of a numerical example

Example 4. Contingency plans for emergency events refer to the relevant plans or schemes proposed in advance in order to ensure the effective and ordered launching of contingency and rescue actions, and lower the damage in various possible emergency events. Choosing ideal contingency plans for emergency events can make a quick response to emergency events and then deal with them, in this way the expansion and escalation of emergency events can be avoided, furthermore, rehabilitation and reconstruction can also be conducted after emergency events happened.

Suppose that there are four contingency plans X ={ x₁, x₂, x₃, x₄ } for a company to choose after preliminary screening to deal with emergency events.

A DM evaluates the contingency plans in terms of the following four attributes:

Rationality a₁;

Feasibility a₂;

Quickness a₃;

Sufficiency a₄.

Assume that the values of attribute weights are the same, and the DM employs the set of seven linguistic terms S presented in Section 2 to provide a decision matrix with linguistic terms for these contingency plans, which is shown in Table 1.

Table 1
The decision matrix with linguistic terms given by a DM

a ₁ a ₂ a ₃ a ₄

x ₁ VP VG EG VG

x ₂ P VG VG EG

x ₃ P EG VG G

x ₄ M G VG VG

	a ₁	a ₂	a ₃	a ₄
x ₁	VP	VG	EG	VG
x ₂	P	VG	VG	EG
x ₃	P	EG	VG	G
x ₄	M	G	VG	VG

Based on the linguistic MADM method proposed in Section 4, the specific procedure is described as follows:

Step 1. Transform the decision matrix with linguistic terms given by the DM into the one S° = (s_t°(i,j)) _4×4 with linguistic labels: $S^{\circ} = (\begin{matrix} s_{- 2} & s_{2} & s_{3} & s_{2} \\ s_{- 1} & s_{2} & s_{2} & s_{3} \\ s_{- 1} & s_{3} & s_{2} & s_{1} \\ s_{0} & s_{1} & s_{2} & s_{2} \end{matrix})$

Step 2. In order to choose the best contingency plan, the DM gives accepted triangular fuzzy semantic values ${\tilde{V}}^{*} = {{\tilde{v}}_{- 2}^{*} = (0.03, 0.1, 0.2), {\tilde{v}}_{1}^{*} = (0.6, 0.7, 0.8), {\tilde{v}}_{2}^{*} = (0.7, 0.8, 0.9)}$ .

Solving the model (11) yields optimal risk preference parameters $θ_{1}^{*} = 0.7411, θ_{2}^{*} = 1.2515$ (see that in Example 1). This reveals that the DM is risk averse.

Step 3. Put $θ_{1}^{*}, θ_{2}^{*}$ into Equations (8– 10) and obtain the optimal GLTS $S_{(7)}^{1}$ (See Example 1). Then, the decision matrix S° with linguistic labels is converted into a triangular fuzzy decision matrix ${\tilde{V}}^{\circ} = {({\tilde{v}}_{t^{\circ} (i, j)})}_{4 \times 4}$ :

${\tilde{V}}^{\circ} = (\begin{matrix} (0.022 9, 0 . 0756, 0 . 2224) & (0.677 2, 0.814 9, 0.9023) & (0.814 9, 0.902 3, 0.902 3) & (0.6772, 0.8149, 0.9023) \\ (0.075 6, 0 . 2224, 0 . 5000) & (0.677 2, 0.814 9, 0.902 3) & (0.677 2, 0.814 9, 0.902 3) & (0.814 9, 0.902 3, 0.902 3) \\ (0.0756, 0.2224, 0.5000) & (0.814 9, 0.902 3, 0.902 3) & (0.6772, 0.8149, 0.9023) & (0.500 0, 0.677 2, 0.814 9) \\ (0.2224, 0.5000, 0.6772) & (0.5000, 0.6772, 0.8149) & (0.677 2, 0.814 9, 0.902 3) & (0.6772, 0.8149, 0.9023) \end{matrix})$

Step 4. Utilize Equation (12) to aggregate all the triangular fuzzy semantic values in the ith line of the triangular fuzzy decision matrix ${\tilde{V}}^{\circ}$ to receive the collective triangular fuzzy evaluation value ${\tilde{v}}_{t^{\circ} (i)}$ of the alternative x_i(i = 1, 2, …, n). ${\tilde{v}}_{t^{\circ} (i)} = (\begin{matrix} (0.548 1, 0.651 9, 0.732 3) \\ (0.561 3, 0.688 6, 0.801 7) \\ (0.516 9, 0.654 2, 0.779 9) \\ (0.519 2, 0.701 8, 0.824 2) \end{matrix})$

Step 5. Utilize Equation (5) to calculate the possibility degree matrix $\begin{matrix} P & = & {(p_{ij})}_{4 \times 4} \\ = & (\begin{matrix} 0.5000 & 0.3090 & 0.469 4 & 0.307 1 \\ 0.6910 & 0.5000 & 0.6333 & 0.485 6 \\ 0.530 6 & 0.3667 & 0.5000 & 0.368 5 \\ 0.692 9 & 0.514 4 & 0.631 5 & 0.5000 \end{matrix}) \end{matrix}$

Employ Equation (6) to compute the ranking value vector ω = (0.2155, 0.2758, 0.2305, 0.2782) ^T based on the possibility degree matrix P. Because ω₄ > ω₂ > ω₃ > ω₁, the ranking order of the four alternatives is x₄ ≻ x₂ ≻ x₃ ≻ x₁. Thus, the best contingency plan is x₄.

5.2 A sensitivity analysis of risk preference parameters

Generally speaking, different DMs have different risk preferences. Thus, different ranking orders for all alternatives may be obtained based on the same decision matrix S° with linguistic labels by using the decision model in Section 4. For example, if the DM with neutral risk gives a set of accepted triangular fuzzy semantic values ${\tilde{V}}^{*}$ as shown in Example 3, the optimal GLTS will be determined as $S_{(7)}^{3}$ presented in Example 3. In this case, the decision matrix S° with linguistic labels in Example 4 can be transformed into the triangular fuzzy decision matrix ${\tilde{V}}^{\circ}$ as ${\tilde{V}}^{\circ} = (\begin{matrix} (0.2117, 0.2939, 0.3921) & (0.6079, 0.7061, 0.7883) & (0.7061, 0.7883, 0.7883) & (0.6079, 0.7061, 0.7883) \\ (0.2939, 0.3921, 0.5000) & (0.6079, 0.7061, 0.7883) & (0.6079, 0.7061, 0.7883) & (0.7061, 0.7883, 0.7883) \\ (0.2939, 0.3921, 0.5000) & (0.7061, 0.7883, 0.7883) & (0.6079, 0.7061, 0.7883) & (0.5000, 0.6079, 0.7061) \\ (0.3921, 0.5000, 0.6079) & (0.5000, 0.6079, 0.7061) & (0.6079, 0.7061, 0.7883) & (0.6079, 0.7061, 0.7883) \end{matrix})$

According to Step 4 and Step 5 given in Section 4.1, triangular fuzzy evaluation values of the four alternatives x_i (i = 1, 2, 3, 4) are obtained as follows: ${\tilde{v}}_{t^{\circ} (i)} = (\begin{matrix} (0.533 4, 0.623 6, 0.689 3) \\ (0.553 9, 0.648 2, 0.716 2) \\ (0.527 0, 0.623 6, 0.695 7) \\ (0.527 0, 0.630 0, 0.722 6) \end{matrix})$

Employ Equation (5) to get the following possibility degree matrix: $P = (\begin{matrix} 0.5000 & 0.342 5 & 0.496 9 & 0.437 2 \\ 0.657 5 & 0.5000 & 0.648 0 & 0.575 4 \\ 0.503 1 & 0.352 0 & 0.5000 & 0.441 3 \\ 0.562 8 & 0.424 6 & 0.558 7 & 0.5000 \end{matrix})$

Then utilize Equation (6) to derive the following ranking value vector $ω = {(0.231 4, 0.281 7, 0.233 0, 0 . 2538)}^{T}$

Because ω₂ > ω₄ > ω₃ > ω₁, the ranking order of the four alternatives is x₂ ≻ x₄ ≻ x₃ ≻ x₁. Accordingly, the best contingency plan is x₂.

From the above result we can see that the ranking order of alternatives and the best alternative may be changed by different risk preferences. So we should make a sensitivity analysis of risk preference parameters.

First of all, we make a detailed classification on the DM’s risk preference as per the risk types given in Section 2.1.

When θ₁ - θ₂ ≥ 1, the DM belongs to the type of strong risk seeking (SRS);

When 0 < θ₁ - θ₂ < 1, the DM belongs to the type of general risk seeking (GRS);

When θ₁ = θ₂, the DM belongs to the type of neutral risk (NR);

When θ₂ - θ₁ ≥ 1, the DM belongs to the type of strong risk aversion (SRA);

When 0 < θ₂ - θ₁ < 1, the DM belongs to the type of general risk aversion (GRA).

Based on the above five risk preference types, we have nine sets of risk preference parameters. The ranking results of the four alternatives under different risk preference types are obtained and shown in Table 2.

Table 2
Ranking results based on different risk preference types

Numbering Risk preference Risk preference Ordering vector ω Ranking order

parameters types of alternatives

θ ₁ θ ₂

1 3 0.1 SRS ω = (0.2370, 0 . 2960, 0 . 1929, 0 . 2741) ^T x₂ ≻ x₄ ≻ x₁ ≻ x₃

2 3 0.4383 SRS ω = (0.1987, 0.2881, 0.2171, 0.2961) ^T x₄ ≻ x₂ ≻ x₃ ≻ x₁

3 0.8 0.2 GRS ω = (0.2612, 0.2867, 0.2202, 0.2319) ^T x₂ ≻ x₁ ≻ x₄ ≻ x₃

4 0.8 0.4383 GRS ω = (0.2403, 0.2841, 0.2263, 0.2493) ^T x₂ ≻ x₄ ≻ x₁ ≻ x₃

5 0.4383 0.4383 NR ω = (0.2314, 0.2817, 0.2330, 0.2538) ^T x₂ ≻ x₄ ≻ x₃ ≻ x₁

6 0.4383 0.8 GRA ω = (0.2128, 0.2770, 0.2355, 0.2747) ^T x₂ ≻ x₄ ≻ x₃ ≻ x₁

7 0.2 0.8 GRA ω = (0.1787, 0.2707, 0.2412, 0.3093) ^T x₄ ≻ x₂ ≻ x₃ ≻ x₁

8 0.4383 3 SRA ω = (0.2094, 0.2653, 0.2269, 0.2984) ^T x₄ ≻ x₂ ≻ x₃ ≻ x₁

9 0.1 3 SRA ω = (0.1760, 0.2510, 0.2299, 0.3432) ^T x₄ ≻ x₂ ≻ x₃ ≻ x₁

Numbering	Risk preference	Risk preference	Ordering vector ω	Ranking order
1	3	0.1	SRS	ω = (0.2370, 0 . 2960, 0 . 1929, 0 . 2741) ^T	x₂ ≻ x₄ ≻ x₁ ≻ x₃
2	3	0.4383	SRS	ω = (0.1987, 0.2881, 0.2171, 0.2961) ^T	x₄ ≻ x₂ ≻ x₃ ≻ x₁
3	0.8	0.2	GRS	ω = (0.2612, 0.2867, 0.2202, 0.2319) ^T	x₂ ≻ x₁ ≻ x₄ ≻ x₃
4	0.8	0.4383	GRS	ω = (0.2403, 0.2841, 0.2263, 0.2493) ^T	x₂ ≻ x₄ ≻ x₁ ≻ x₃
5	0.4383	0.4383	NR	ω = (0.2314, 0.2817, 0.2330, 0.2538) ^T	x₂ ≻ x₄ ≻ x₃ ≻ x₁
6	0.4383	0.8	GRA	ω = (0.2128, 0.2770, 0.2355, 0.2747) ^T	x₂ ≻ x₄ ≻ x₃ ≻ x₁
7	0.2	0.8	GRA	ω = (0.1787, 0.2707, 0.2412, 0.3093) ^T	x₄ ≻ x₂ ≻ x₃ ≻ x₁
8	0.4383	3	SRA	ω = (0.2094, 0.2653, 0.2269, 0.2984) ^T	x₄ ≻ x₂ ≻ x₃ ≻ x₁
9	0.1	3	SRA	ω = (0.1760, 0.2510, 0.2299, 0.3432) ^T	x₄ ≻ x₂ ≻ x₃ ≻ x₁

The following conclusions can be received from Table 2:

In the case of different values of risk preference parameters, decision results are basically consistent. The best alternative is x₂ or x₄. However, there are different ranking orders for the four alternatives. For the same risk preference type, different parameter values will lead to various ranking results of the four alternatives because of the different intensity of risk seeking or aversion. For instance, for the type of SRS, the intensity of the DM’s risk seeking listed in numbering #1 is stronger than the one listed in numbering #2. The bigger the value of θ₁ - θ₂, the stronger the intensity of the DM’s risk seeking. As can be seen from Table 2, the best contingency plan is x₂ under the situations of the risk preference parameters listed in numberings #1, #3, #4, #5 and #6, and the best contingency plan is x₄ under the situations of risk preference parameters listed in numberings #2, #7, #8 and #9. In addition, for the ranking value vectors listed in numberings #2 and #6, the difference between x₂ and x₄ is smaller, i.e., |ω₂ - ω₄| < 0.01. Thus, any of alternatives x₂ and x₄ can be chosen as the best contingency plan.

6 Conclusions

In this paper, we study the qualitative decision making problems based on an unbalanced LTS and considering the DM’s risk preference. Because different qualitative decision making problems may adopt different types of LTSs and the linguistic terms are uncertain, we propose the concept of a GLTS with triangular fuzzy semantic information. A triangular fuzzy membership semantic function with risk preference parameters is constructed. We establish a nonlinear programming model to obtain optimal risk preference parameters on the basis of accepted semantic values of linguistic terms. Afterwards, we present a linguistic MADM method. Finally, a numerical example and a sensitivity analysis about risk preference parameters are provided to show that the proposed models are feasible andeffective.

The advantages of the proposed models have two aspects. Firstly, in response to different qualitative decision making problems, the GLTS defined in Section 3 can simulate different types of LTSs and is capable of meeting the DM’s need better. Secondly, triangular fuzzy numbers with two risk preference parameters are introduced to characterize semantic information of linguistic terms. Based on the DM’s expected or accepted fuzzy semantic values, an optimal GLTS can be determined by solving the proposed nonlinear programming model.

In the future, we will study how to directly determine values of risk preference parameters and design a GLTS under the condition that the type of the DM’s risk preference is known. Another study is the application of the proposed models to real decision problems such as mobile decision support systems [25] and digital library evaluation [4, 5].

Footnotes

Acknowledgments

We are grateful to the Editor, Prof. Enrique Herrera-Viedma, and three anonymous reviewers for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China under Grant 71671160 and the Zhejiang Provincial Natural Science Foundation of China under Grant LY15G010004.

References

Cabrerizo

F.J.

, Al-Hmouz

, Morfeq

, Balamash

A.S.

, Martínez

M.A.

and Herrera-Viedma

, Soft consensus measures in group decision making using unbalanced fuzzy linguistic information, Soft Computing 21(11) (2017), 3037–3050.

Cabrerizo

F.J.

, Alonso

and Herrera-Viedma

, A consensus model for group decision making problems with unbalanced fuzzy linguistic information, International Journal of Information Technology and Decision Making 8(1) (2009), 109–131.

Cabrerizo

F.J.

, Herrera-Viedma

and Pedrycz

, A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts, European Journal of Operational Research 230(3) (2013), 624–633.

Cabrerizo

F.J.

, López-Gijón

, Martínez

M.A.

, Morente-Molinera

J.A.

and Herrera-Viedma

, A fuzzy linguistic extended libqual+model to assess service quality in academic libraries, International Journal of Information Technology and Decision Making 16(1) (2017), 225–244.

Cabrerizo

F.J.

, Morente-Molinera

J.A.

, Pérez

I.J.

, López-Gijón

and Herrera-Viedma

, A decision support system to develop a quality management in academic digital libraries, Information Sciences 323 (2015), 48–58.

Chen

C.T.

, Extension of the TOPSIS for group decision making under fuzzy environment, Fuzzy Sets and Systems 114(1) (2000), 1–9.

Cheng

C.B.

, Group opinion aggregation based on a grading process: A method for constructing triangular fuzzy numbers, Computers and Mathematics with Applications 48(10) (2004), 1619–1632.

Dong

Y.C.

, Li

C.C.

and Herrera

, An optimization-based approach to adjusting the unbalanced linguistic preference relations to obtain a required consistency level, Information Sciences 292(5) (2015), 27–38.

Dong

Y.C.

, Li

C.C.

and Herrera

, Connecting the linguistic hierarchy and the numerical scale for the 2-tuple linguistic model and its uses to deal with hesitant unbalanced linguistic information, Information Sciences 367-368 (2016), 259–278.

10.

Dong

Y.C.

, Li

C.C.

, Xu

Y.F.

and Gu

, Consensus-based group decision making under multi-granular unbalanced 2-tuple linguistic preference relations, Group Decision and Negotiation 24(2) (2015), 217–242.

11.

Dong

Y.C.

, Xu

Y.F.

and Yu

, Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model, IEEE Transactions on Fuzzy Systems 17(6) (2009), 1366–1378.

12.

Dong

Y.C.

, Zhang

G.Q.

, Hong

W.C.

and Yu

, Linguistic computational model based on 2-tuples and intervals, IEEE Transactions on Fuzzy Systems 21(6) (2013), 1006–1018.

13.

Fan

Z.P.

and Liu

, A method for group decision-making based on multi-granularity uncertain linguistic information, Expert Systems with Applications 37(5) (2010), 4000–4008.

14.

Herrera

, Herrera-Viedma

and Martínez

, A fuzzy linguistic methodology to deal with unbalanced linguistic term sets, IEEE Transactions on Fuzzy Systems 16(2) (2008), 354–370.

15.

Herrera

and Martínez

, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8(6) (2000), 746–752.

16.

Huang

, Qian

X.H.

, Fang

S.C.

and Wang

X.W.

, Winner determination for risk aversion buyers in multi-attribute reverse auction, Omega 59 (2015), 184–200.

17.

Jiang

X.P.

and Wei

G.W.

, Some Bonferroni mean operators with 2-tuple linguistic information and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 27(5) (2014), 2153–2162.

18.

Jin

H.P.

, Park

J.M.

and Kwun

Y.C.

, 2-Tuple linguistic harmonic operators and their applications in group decision making, Knowledge-Based Systems 44(1) (2013), 10–19.

19.

Y.B.

, Liu

X.Y.

and Yang

S.H.

, Trapezoid fuzzy 2-tuple linguistic aggregation operators and their applications to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems 27(3) (2014), 1219–1232.

20.

C.C.

and Dong

Y.C.

, Unbalanced linguistic approach for venture investment evaluation with risk attitudes, Progress in Artificial Intelligence 3(1) (2014), 1–13.

21.

C.C.

, Dong

Y.C.

, Herrera

, Herrera-Viedma

and Martínez

, Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching, Information Fusion 33(C) (2017), 29–40.

22.

Liu

D.N.

, Model for evaluating the electrical power system safety with hesitant fuzzy linguistic information, Journal of Intelligent and Fuzzy Systems 29(2) (2015), 725–730.

23.

Liu

, Dong

Y.C.

, Chiclana

, Cabrerizo

F.J.

and Herrera-Viedma

, Group decision-making based on heterogeneous preference relations with self-confidence, Fuzzy Optimization and Decision Making (2017). article in press, 10.1007/s10700-016-9254-8.

24.

and Wei

G.W.

, Models for multiple attribute decision making with dual hesitant fuzzy uncertain linguistic information, International Journal of Knowledge-based and Intelligent Engineering Systems 20(4) (2016), 217–227.

25.

Pérez

I.J.

, Cabrerizo

F.J.

and Herrera-Viedma

, A mobile decision support system for dynamic group decision making problems, IEEE Transactions on Systems, Man and Cybernetics - Part A: Systems and Humans 40(6) (2010), 1244–1256.

26.

Ruan

C.Y.

and Yang

J.H.

, Hesitant fuzzy information aggregation operators considering confidence level and priority level, Systems Engineering and Electronics 38 (2016), 2324–2330.

27.

Tang

, Wen

L.L.

and Wei

G.W.

, Approaches to multiple attribute group decision making based on the generalized Dice similarity measures with intuitionistic fuzzy information, International Journal of Knowledge-based and Intelligent Engineering Systems 21(2) (2017), 85–95.

28.

Wan

S.P.

, 2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making, Knowledge-Based Systems 45(3) (2013), 31–40.

29.

Wang

B.L.

, Liang

J.Y.

and Qian

Y.H.

, A normalized numerical scaling method for the unbalanced multi-granular linguistic sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23(2) (2015), 221–243.

30.

Wang

J.H.

and Hao

J.Y.

, A new version of 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 14(3) (2006), 435–445.

31.

Wei

G.W.

, Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making, International Journal of Machine Learning and Cybernetics 7(6) (2016), 1093–1114.

32.

Wei

G.W.

, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management 17(4) (2016), 491–502.

33.

Wei

G.W.

, Alsaadi

F.E.

, Hayat

and Alsaedi

, Hesitant fuzzy linguistic arithmetic aggregation operators in multiple attribute decision making, Iranian Journal of Fuzzy Systems 13(4) (2016), 1–16.

34.

Wei

G.W.

, Xu

X.R.

and Deng

D.X.

, Interval-valued dual hesitant fuzzy linguistic geometric aggregation operators in multiple attribute decision making, International Journal of Knowledge-based and Intelligent Engineering Systems 20(4) (2016), 189–196.

35.

, Wan

S.P.

and Dong

J.Y.

, Aggregating decision information into Atanassov’s intuitionistic fuzzy numbers for heterogeneous multi-attribute group decision making, Applied Soft Computing 41(C) (2016), 331–351.

36.

Z.S.

, Study on method for triangular fuzzy number-based multi-attribute decision making with preference information on alternatives, Systems Engineering and Electronics 24(8) (2002), 9–12.

37.

Z.S.

, Deviation measures of linguistic preference relations in group decision making, Omega 33(3) (2005), 249–254.

38.

Z.S.

, An interactive approach to multiple attribute group decision making with multigranular uncertain linguistic information, Group Decision and Negotiation 18(2) (2009), 119–145.

39.

Yue

Z.L.

, An extended TOPSIS for determining weights of decision makers with interval numbers, Knowledge-Based Systems 24 (2011), 146–153.

40.

Zhang

X.Y.

and Wang

Z.J.

, Approaches to multiple attribute group decision making with multi-granularity linguistic term sets, International Journal of Innovative Computing, Information and Control 12(1) (2016), 275–291.

41.

Zhou

and Xu

Z.S.

, Generalized asymmetric linguistic term set and its application to qualitative decision making involving risk appetites, European Journal of Operational Research 254(2) (2016), 610–621.

Linguistic multi-attribute decision making with considering decision makers’ risk preferences

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 The description of a GLTS based on triangular fuzzy numbers

5.1 The analysis of a numerical example

Table 1 The decision matrix with linguistic terms given by a DM a 1 a 2 a 3 a 4 x 1 VP VG EG VG x 2 P VG VG EG x 3 P EG VG G x 4 M G VG VG

Footnotes

Acknowledgments

References

Table 1
The decision matrix with linguistic terms given by a DM

a ₁ a ₂ a ₃ a ₄

x ₁ VP VG EG VG

x ₂ P VG VG EG

x ₃ P EG VG G

x ₄ M G VG VG