An extended TODIM method for multiple attribute group decision making based on intuitionistic uncertain linguistic variables

Abstract

The intuitionistic uncertain linguistic variables are the good tools to express the fuzzy information, and the TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making) method can consider the bounded rationality of decision makers based on the prospect theory. However, the classical TODIM method can only process the multiple attribute decision making (MADM) problems where the attribute values take the form of crisp numbers. In this paper, we will extend the TODIM method to the multiple attribute group decision making (MAGDM) with intuitionistic uncertain linguistic information. Firstly, the definition, characteristics, expectation, comparison method and distance of intuitionistic uncertain linguistic variables are briefly introduced, and the steps of the classical TODIM method for MADM problems are presented. Then, on the basis of the classical TODIM method, the extended TODIM method is proposed to deal with MAGDM problems with intuitionistic uncertain linguistic variables, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Finally, an illustrative example is proposed to verify the developed approach.

Keywords

TODIM method intuitionistic uncertain linguistic information multiple attribute group decision making prospect theory

1 Introduction

Multiple attribute decision making (MADM) is an important research branch of decision theory, and it has been the wide applications in many fields. However, due to the fuzziness and uncertainty of real decision making problems, the attribute values in decision making problems are not always represented as real numbers, and they can be described as fuzzy numbers in more suitable occasions, such as interval numbers, triangular fuzzy numbers, trapezoid fuzzy numbers, linguistic variables or uncertain linguistic variables, and intuitionistic fuzzy numbers. Since Fuzzy set (FS), which is a very useful tool to process the fuzzy information, was firstly proposed by Zadeh [1], fuzzy multiple attribute decision making (FMADM) has got many research achievements. Atanassov [2] further proposed the intuitionistic fuzzy set (IFS) based on the FS. The different between IFS and FS is that IFS has a membership function and a non-membership function, and FS has only a membership function. Because IFS plays an important role in exquisitely describing the fuzzy information, so it is more suitable for decision makers to use in multi-attribute decision making and multi-attribute group decision making. However, because the values of membership function and non-membership in IFS was crisp numbers, some extensions of IFS were developed. Gargov and Atanassov [3], Atanassov [4] proposed the interval-valued intuitionistic fuzzy set (IVIFS), Liu and Zhang [5] gave the definition of the triangular intuitionistic fuzzy numbers (TIFNs), the relevant decision making methods based on TIFNs have been developed. Wang [6] defined intuitionistic trapezoidal fuzzy numbers (ITFNs) and interval intuitionistic trapezoidal fuzzy numbers (IITFNs), then some decision making methods based on them had been proposed[7, 8].

In real decision making, there exists a great deal of qualitative information which is more easily expressed by linguistic variables or uncertain linguistic variables than fuzzy numbers. Research on MADM methods for linguistic information has made many achievements [9–19 , 41]. However, when we use the linguistic variables or uncertain linguistic variables to express the qualitative information, it only means the membership degree belonged to a linguistic term is 1, and the non-membership degree or hesitation degree cannot be expressed. Similarly, IFS cannot express the membership degree and non-membership degree belonged to a linguistic term. Wang and Li [15] proposed the intuitionistic linguistic set by combined intuitionistic fuzzy set and linguistic variables, which can overcome the shortcomings of the intuitionistic fuzzy set and linguistic variables, for example, we can give an evaluation value by linguistic term “good” for the performance of a car, in general, for this result, we have not 100% sure, maybe we have the certainty degree of 80 percent and negation degree of 10 percent, in this case, we can more easily use the intuitionistic linguistic set to express the evaluation result. Liu [16] extended dependent aggregation operators to intuitionistic linguistic set, and developed intuitionistic linguistic generalized dependent hybrid weighted aggregation (ILGDHWA) operator, and proposed a MAGDM method. Liu and Wang [17] developed an intuitionistic linguistic power generalized weighted average (ILPGWA) operator and intuitionistic linguistic power generalized ordered weighted average (ILPGOWA) operator, and proposed two approaches for MAGDM problems with intuitionistic linguistic information. Liu and Jin [18] further proposed the concept of intuitionistic uncertain linguistic variables, and developed some intuitionistic uncertain linguistic aggregation operators, and discussed some properties of them; then proposed two MAGDM methods. Liu et al. [19] developed some intuitionistic uncertain linguistic Heronian mean operators, and developed a MADM method based on these operators.

There are many decision making methods which are the good tools to solve the MADM or MAGDM problems, such as the TOPSIS [20, 21], the grey relational analysis [22, 23], the VIKOR [24 –26], the PROMETHEE [27], the ELECTRE [28, 29] and so on. They have also been extended to fit different types of attributes, such as fuzzy numbers, IFNs and linguistic information. However, there is a common shortcoming in these methods, i.e., they don’t consider the decision makers’ bounded rationality. In order to overcome this shortcoming, Gomes and Lima [30] proposed a TODIM (an acronym in Portuguese of Interactive and Multiple Attribute Decision Making) which is the first MCDM method which could consider the bounded rationality based on prospect theory. In the traditional TODIM method, according to Prospect Theory, the dominance degree of each alternative over the others is measured by the value function, which reflects DM’s risk attitude such as aversion of risk in face of gains and the propensity to risk in face of losses. Then, the ordering of alternatives can be got through calculating the overall value of each alternative. Recently, the TODIM method has received more attentions. Gomes et al. [31] used the TODIM method to deal with the problem of selecting the best option for the destination of the natural gas reserves in Brazil. Passos et al. [32] proposed a TODIM-FSE method for solving classification problems by combined TODIM and FSE (Fuzzy Synthetic Evaluation). Lourenzutti and Krohling [33] proposed the Fuzzy-TODIM method to deal with the MADM problems with intuitionistic fuzzy information and to be capable to consider underlying random vectors which affected the performance of the alternatives. The method could make use of some Bayesian ideas to provide an overall ranking of the alternatives. Krohling and de Souza [34] proposed a hybrid TODIM method by combining prospect theory and fuzzy numbers to deal with risk and uncertainty in MADM problems, and gave an application of this method in oil spill in the sea. Fan et al. [35] proposed an extended TODIM method to deal with the hybrid MADM problems in which the attribute values take the form of crisp numbers, interval numbers and fuzzy numbers. Krohling et al. [36] proposed an extended TODIM method in order to process the intuitionistic fuzzy information.

As introduced above, the intuitionistic uncertain linguistic variables can easily express the fuzzy information, and TODIM method can consider the decision makers’ bounded rationality based on prospect theory. Obviously, it is very important and meaningful to research the extended TODIM method for the MAGDM problems with intuitionistic uncertain linguistic information. In this paper,we combine the intuitionistic uncertain linguistic variables and the TODIM method to handle uncertain MADM problems in which the attribute weights and the expert weights are expressed in the form of crisp numbers, and attribute values take the form of intuitionistic uncertain linguisticvariables.

To achieve the above purposes, the remainder of this paper is organized as follows. In Section 1, we give a simple introduction of the research background. In Section 2, we briefly introduce some basic concepts of intuitionistic linguistic set, intuitionistic uncertain linguistic set, and TODIM method. In Section 3, we propose the extended TODIM method which combines the intuitionistic uncertain linguistic variables and the TODIM method. In Section 4, we give an illustrative example to show the method and its results. In Section 5, we conclude the paper.

2 Preliminaries

2.1 The linguistic set and uncertain linguistic variables

Suppose that S = (s ₀, s ₁, …, s _l-1) is the set of the finite and totally ordered linguistic elements, where l is an odd value. In general, l can be assigned to the values of 3,5, 7, 9, etc. For example, when l = 9, S = (s ₀, s ₁, s ₂, s ₃, s ₄, s ₅, s ₆, s ₇, s ₈)= (extremely poor, very poor, poor, slightly poor, fair, slightly good, good, very good, extremely good). Where, s _θ (θ = 0, 1, …, l - 1) can be called an linguistic variable.

Suppose s _i and s _j are any two elements in S, they must meet the following conditions [37, 38]:

If i > j, then si > sj (it means s _i is better than s _j);

There exists negative operator: neg (s _i) = s _j, where j = l - 1–i;

If s _i ≥ s _j (i.e., s _i is not worse than s _j), max(s _i, s _j) = s _i;

If s _i ≤ s _j (i.e., s _i is not better than s _j), min(s _i, s _j) = s _i.

The introduction about the operational rules and characteristics of linguistic variables can get from the references [37, 38].

For any linguistic set S = (s ₀, s ₁, …, s _l-1), the element s _i and its subscript i is strictly monotonically increasing [39]. In order to minimize the loss of linguistic information in operational process, the original discrete linguistic set S = (s ₀, s ₁, …, s _l-1) is expanded into continuous linguistic set $\bar{S} = {s_{α} | α \in R^{+}}$ which is also meet the strictly monotonically increasing. The operational laws are defined as follows: [37, 38] $(1) β s_{i} = s_{β \times i} β \geq 0$ (1) $(2) s_{i} \oplus s_{j} = s_{i + j}$ (2) $(3) s_{i} / s_{j} = s_{i / j}$ (3) $(4) (s_{i})^{n} = s_{i^{n}} n \geq 0$ (4)

Definition 1. [40] Suppose $\tilde{s} = [s_{a}, s_{b}]$ , $s_{a,} s_{b} \in \bar{S}$ with a ≤ b are the lower limit and the upper limit of $\tilde{s}$ , respectively, then $\tilde{s}$ is called an uncertain linguistic variable.

For convenience, we can set $\tilde{S}$ is a set of all uncertain linguistic variables.

Let ${\tilde{s}}_{1} = [s_{a 1}, s_{b 1}]$ and ${\tilde{s}}_{2} = [s_{a 2}, s_{b 2}]$ be any two uncertain linguistic variables, the operational rules are defined as follows [40, 41]: $\begin{matrix} (1) {\tilde{s}}_{1} \oplus {\tilde{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] \oplus [s_{a_{2}}, s_{b_{2}}] \\ = [s_{a_{1} + a_{2}}, s_{b_{1} + b_{2}}] \end{matrix}$ (5) $\begin{matrix} (2) {\tilde{s}}_{1} \otimes {\tilde{s}}_{2} = [s_{a 1}, s_{b 1}] \otimes [s_{a 2}, s_{b 2}] \\ = [s_{a_{1} \times a_{2}}, s_{b_{1} \times b_{2}}] \end{matrix}$ (6) $\begin{matrix} (3) {\tilde{s}}_{1} / {\tilde{s}}_{2} = [s_{a_{1}}, s_{b_{1}}] / [s_{a_{2}}, s_{b_{2}}] \\ = [s_{a_{1} / b_{2}}, s_{b_{1} / a_{2}}] if a_{2} \neq 0, b_{2} \neq 0 \end{matrix}$ (7) $(4) λ {\tilde{s}}_{1} = λ [s_{a_{1}}, s_{b_{1}}] = [s_{λ * a_{1}}, s_{λ * b_{1}}]$ (8)

2.2 The intuitionistic linguistic set (ILS)

Definition 2. [15] Let $h_{θ (x)} \in \bar{S}$ , X be the given discourse domain, then $A = {< x [h_{θ (x)}, (u_{A} (x), v_{A} (x))] > | x \in X}$ (9) where u _A : X → [0, 1], v _A : X → [0, 1] and 0 ≤ u _A (x) + v _A (x) ≤1, ∀x ∈ X. The numbers u _A and v _A respectively represent the membership degree and non-membership degree of the element x to the linguistic index h _θ(x).

2.3 The intuitionistic uncertain linguistic set (IULS)

Definition 3. [18] Let $[s_{θ (x)}, s_{τ (x)}] \in \tilde{S}$ , and X be the given discourse domain, then $A = {< x | [s_{θ (x)}, s_{τ (x)}], (u_{A} (x), v_{A} (x))] > | x \in X}$ (10) is called an intuitionistic uncertain linguistic set (IULS) in which $s_{θ (x)}, s_{τ (x)} \in \bar{S}, u_{A} : X \to [0, 1]$ , and v _A : X → [0, 1] with the following condition 0 ≤ u _A (x) + v _A (x) ≤1, ∀x ∈ X. The u _A and v _A respectively express the membership degree and non-membership degree of the element x to the uncertain linguistic variable $[s_{θ (x)}, s_{τ (x)}] \in \tilde{S}$ .

For each IULS in X, if π (x) =1 - u _A (x) - v _A (x), ∀x ∈ X then π (x) is called the degree of uncertainty of x to the uncertain linguistic variable [s _θ(x), s _τ(x)]. Obviously, It can meet 0 ≤ π (x) ≤1, ∀x ∈ X.

Definition 4. [18] Let A = {〈x| [s _θ(x), s _τ(x)] , (u _A (x), v _A (x))] 〉|x ∈ X} be IULS, the quaternion {[s _θ(x), s _τ(x)] , (u _A (x) , v _A (x))} is called an intuitionistic uncertain linguistic variable, and can also be regarded as the collection of the intuitionistic uncertain linguistic variables. So, the IULS can also be recorded as A = {〈 [s _θ(x), s _τ(x)] , (u _A (x) , v _A (x)) 〉|x ∈ X} Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}], (u (a_{1}), v (a_{1})) 〉$ and ${\tilde{a}}_{2} =$ 〈 [s _θ(a₂), s _τ(a₂)] , (u (a ₂) , v (a ₂)) 〉 be any two intuitionistic uncertain linguistic variables and λ ≥ 0 then the operational laws are defined as follows [18]: $\begin{matrix} (1) {\tilde{a}}_{1} + {\tilde{a}}_{2} = 〈 [s_{θ (a_{1}) + θ (a_{2})}, s_{τ (a_{1}) + τ (a_{2})}], \\ (1 - (1 - u (a_{1})) (1 - u (a_{2})), v (a_{1}) v (a_{2})) 〉 \end{matrix}$ (11) $\begin{matrix} (2) {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = 〈 [s_{θ (a_{1}) \times θ (a_{2})}, s_{τ (a_{1}) \times τ (a_{2})}], \\ (u (a_{1}) u (a_{2}), v (a_{1}) + v (a_{2}) - v (a_{1}) v (a_{2})) 〉 \end{matrix}$ (12) $\begin{matrix} (3) λ {\tilde{a}}_{1} = 〈 [s_{λ \times θ (a_{1})}, s_{λ \times τ (a_{1})}], \\ (1 - (1 - u (a_{1}))^{λ}, (v (a_{1}))^{λ}) 〉 \end{matrix}$ (13) $\begin{matrix} (4) {\tilde{a}}_{1}^{λ} = 〈 [s_{(θ (a_{1}))^{λ}}, s_{(τ (a_{1}))^{λ}}], \\ ((u (a_{1}))^{λ}, 1 - (1 - v (a_{1}))^{λ}) 〉 \end{matrix}$ (14)

Apparently, the operational results above still belong to intuitionistic uncertain linguistic variables.

Theorem 1. [18] Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}], (u (a_{1}),$ v (a ₁)) 〉 and ${\tilde{a}}_{2} = 〈 [s_{θ (a_{2})}, s_{τ (a_{2})}], (u (a_{2}), v (a_{2})) 〉$ be any two intuitionistic uncertain linguistic variables, it is easy to prove that the operational laws have the following properties: $(1) {\tilde{a}}_{1} + {\tilde{a}}_{2} = {\tilde{a}}_{2} + {\tilde{a}}_{1}$ (15) $(2) {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = {\tilde{a}}_{2} \otimes {\tilde{a}}_{1}$ (16) $(3) λ ({\tilde{a}}_{1} + {\tilde{a}}_{2}) = λ {\tilde{a}}_{1} + λ {\tilde{a}}_{2}, λ \geq 0$ (17) $(4) λ_{1} {\tilde{a}}_{1} + λ_{2} {\tilde{a}}_{1} = (λ_{1} + λ_{2}) {\tilde{a}}_{1}, λ_{1}, λ_{2} \geq 0$ (18) $(5) {\tilde{a}}_{1}^{λ_{1}} \otimes {\tilde{a}}_{1}^{λ_{2}} = ({\tilde{a}}_{1})^{λ_{1} + λ_{2}}, λ_{1}, λ_{2} \geq 0$ (19) $(6) {\tilde{a}}_{1}^{λ_{1}} \otimes {\tilde{a}}_{2}^{λ_{1}} = ({\tilde{a}}_{1} \otimes {\tilde{a}}_{2})^{λ_{1}}, λ_{1} \geq 0$ (20)

Definition 5. [18] Suppose ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}]$ , (u (a ₁), v (a ₁)) 〉 is an intuitionistic uncertain linguistic variable, then the expectation value $E ({\tilde{a}}_{1})$ of ${\tilde{a}}_{1}$ can be represented as

$\begin{matrix} E ({\tilde{a}}_{1}) & = & \frac{1}{2} \times (u (a_{1}) + 1 - v (a_{1})) \times s_{(θ (a_{1}) + τ (a_{1})) / 2} \\ = & s_{((θ (a_{1}) + τ (a_{1})) \times (u (a_{1}) + 1 - v (a_{1}))) / 4} \end{matrix}$ (21)

Definition 6. [18] Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}],$ (u (a ₁), v (a ₁)) 〉 be an intuitionistic uncertain linguistic variable, then the accuracy function $H ({\tilde{a}}_{1})$ of ${\tilde{a}}_{1}$ can be calculated as follows:

$\begin{matrix} H ({\tilde{a}}_{1}) & = & (u (a_{1}) + v (a_{1})) \times s_{(θ (a_{1}) + τ (a_{1})) / 2} \\ = & s_{((θ (a_{1}) + τ (a_{1})) \times (u (a_{1}) + v (a_{1}))) / 2} \end{matrix}$ (22)

Definition 7. [18] Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}]$ , (u (a ₁), v (a ₁)) 〉 and ${\tilde{a}}_{2} = 〈 [s_{θ (a_{2})}, s_{τ (a_{2})}], (u (a_{2}), v (a_{2})) 〉$ be any two intuitionistic uncertain linguistic variables, then

$\begin{matrix} (1) if E ({\tilde{a}}_{1}) > E ({\tilde{a}}_{2}), then {\tilde{a}}_{1} ≻ {\tilde{a}}_{2} . \\ (2) if E ({\tilde{a}}_{1}) = E ({\tilde{a}}_{2}), then : \\ if H ({\tilde{a}}_{1}) > H ({\tilde{a}}_{2}), then {\tilde{a}}_{1} ≻ {\tilde{a}}_{2} . \\ If H ({\tilde{a}}_{1}) = H ({\tilde{a}}_{2}), then {\tilde{a}}_{1} = {\tilde{a}}_{2} . \end{matrix}$

Definition 8. Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}], (u (a_{1}), v (a_{1})) 〉$ and ${\tilde{a}}_{2} = 〈 [s_{θ (a_{2})}, s_{τ (a_{2})}], (u (a_{2}), v (a_{2})) 〉$ be any two intuitionistic uncertain linguistic variables, then the normalized Hamming distance between ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ is expressed as

$\begin{matrix} d ({\tilde{a}}_{1}, {\tilde{a}}_{2}) = \frac{1}{4 (l - 1)} | (1 + u (a_{1}) - v (a_{1})) (θ (a_{1}) \\ + τ (a_{1})) - (1 + u (a_{2}) - v (a_{2})) (θ (a_{2}) + τ (a_{2})) | \end{matrix}$ (23)

Let ${\tilde{a}}_{1} = 〈 [s_{θ (a_{1})}, s_{τ (a_{1})}], (u (a_{1}), v (a_{1})) 〉, {\tilde{a}}_{2} = 〈 [s_{θ (a_{2})}, s_{τ (a_{2})}], (u (a_{2}), v (a_{2})) 〉$ and ${\tilde{a}}_{3} = 〈 [s_{θ (a_{3})}, s_{τ (a_{3})}], (u (a_{3}), v (a_{3})) 〉$ be any three intuitionistic uncertain linguistic variables, it is easy to prove that the distance $d ({\tilde{a}}_{1}, {\tilde{a}}_{2})$ between two intuitionistic uncertain linguistic variables ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ can meet the following conditions.

$\begin{matrix} (1) 0 \leq d ({\tilde{a}}_{1}, {\tilde{a}}_{2}) \leq 1; \\ (2) d ({\tilde{a}}_{1}, {\tilde{a}}_{1}) = 0; \\ (3) d ({\tilde{a}}_{1}, {\tilde{a}}_{2}) = d ({\tilde{a}}_{2}, {\tilde{a}}_{1}); \\ (4) d ({\tilde{a}}_{1}, {\tilde{a}}_{2}) + d ({\tilde{a}}_{2}, {\tilde{a}}_{3}) \geq d ({\tilde{a}}_{1}, {\tilde{a}}_{3}) . \end{matrix}$

2.4 The TODIM method

Let us consider the decision matrix A which is composed of alternatives and criteria described by

$A = \begin{matrix} A_{1} \\ ⋮ \\ A_{m} \end{matrix} \overset{\begin{matrix} C_{1} \dots C_{n} \end{matrix}}{[\begin{matrix} x_{11} & \dots & x_{1 n} \\ ⋮ & ⋱ & ⋮ \\ x_{m 1} & \dots & x_{mm} \end{matrix}]}$ where, A ₁, A ₂, …, A _m are the alternatives, and C ₁, C ₂, …, C _n are the criteria. x _ij is the rating of the ith alternative A _i under the jth criterion C _j, and W = (w ₁, w ₂, …, w _n) is the weight vector of the criteria (C ₁, C ₂, …, C _n) which satisfies $w_{j} \in [0, 1], \sum_{j = 1}^{n} w_{j} = 1$ . Because the source of the data of the decision matrix A is different, in order to compare with the different criteria, we should normalize the decision matrix A so that the matrix could be converted into a dimensionless matrix. Suppose the normalized decision matrix is R = [r _ij] _m×n (i = 1, 2, …, m, and j = 1, 2, …, n). The calculation of the partial dominance matrices and the final dominance matrix play a cardinal role in ranking the alternatives. Because TODIM method considered the decision makers’ bounded rationality, a reference criterion firstly needs to be determined. In general, we can set the criterion with the highest importance weight as a reference criterion. The steps of the TODIM method are described as follows.

Step 1: Calculate relative attribute weight w _cr for attribute C _c to reference attribute C _r, i.e., $w_{cr} = \frac{w_{c}}{w_{r}} (c = 1, 2, \dots n)$ (24) Where $w_{r} = max_{j} {w_{j}}$

Step 2: Calculate the dominance degree of alternative A _i over alternative A _j with respect to attribute C _c, i.e.,

$φ_{c} (A_{i}, A_{j}) = {\begin{matrix} \sqrt{\frac{w_{cr} (x_{ic} - x_{jc})}{\sum_{c = 1}^{n} w_{cr}}} if (x_{ic} - x_{jc}) > 0 \\ 0 if (x_{ic} - x_{jc}) = 0 \\ \frac{- 1}{θ} \sqrt{\frac{(\sum_{c = 1}^{n} w_{cr}) (x_{jc} - x_{ic})}{w_{cr}}} if (x_{ic} - x_{jc}) < 0 \end{matrix}$ (25) where θ is the attenuation factor of the losses.

Step 3: Calculate the overall dominance degree of each alternative A _i over each alternative A _j, and we can get $δ (A_{i}, A_{j}) = \sum_{c = 1}^{n} φ_{c} (A_{i}, A_{j})$ (26)

Step 4: Calculate the global value by normalizing the final matrix of dominance, and we can get $ξ_{i} = \frac{\sum_{j = 1}^{n} δ (A_{i}, A_{j}) - min_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j}))}{max_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j})) - min_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j}))}$ (27)

Step 5: Rank all the alternatives based on the global values of alternatives. The bigger ξ _i is, the better alternative A _i is.

3 An extended TODIM method for MAGDM problems based on the intuitionistic uncertain linguistic variables

The classical TODIM method can only process the MAGDM problems in which the attribute values take the form of crisp numbers. However, because of the complexity of the decision environment, sometimes, it is difficult to get the crisp numbers for the alternative under the different attributes because the pressure of time or cost. The intuitionistic uncertain linguistic variables can easily express the fuzzy information. In this section, we will extend the TODIM method to process the MAGDM problems with the intuitionistic uncertain linguistic information.

3.1 Description of the MAGDM problems

Consider the multiple attribute group decision making problems with intuitionistic uncertain linguistic information as follow. ${\tilde{R}}^{k} = \begin{matrix} A_{1} \\ ⋮ \\ A_{m} \end{matrix} \overset{\begin{matrix} C_{1} \dots C_{n} \end{matrix}}{[\begin{matrix} {\tilde{r}}_{11}^{k} & \dots & {\tilde{r}}_{1 n}^{k} \\ ⋮ & ⋱ & ⋮ \\ {\tilde{r}}_{m 1}^{k} & \dots & {\tilde{r}}_{mn}^{k} \end{matrix}]}$

Let A = {A ₁, A ₂, …, A _m} be a discrete set of alternatives, and C = {C ₁, C ₂, …, C _n} be a set of attributes. W = {w ₁, w ₂, …, w _n} is the weight vector of the attribute C _j (j = 1, 2, …, n), where w _j ≥ 0, j = 1, 2, …, n, $\sum_{j = 1}^{n} w_{j} = 1$ . Let D = {D ₁, D ₂, …, D _P} be the set of decision makers, and λ = (λ ₁, λ ₂, …, λ _P) ^T be the weight vector of decision makers D _k (k = 1, 2, …, p), where λ _k ≥ 0, $\sum_{k = 1}^{p} λ_{k} = 1$ . Suppose that ${\tilde{R}}^{k} = [{\tilde{r}}_{ij}^{k}]_{m \times n}$ is the decision matrix, where ${\tilde{r}}_{ij}^{k} = 〈 [s_{a_{ij}^{k}}, s_{b_{ij}^{k}}], (u_{ij}^{k}, v_{ij}^{k}) 〉$ takes the form of the intuitionistic uncertain linguistic variables given by the decision maker D _k for alternative A _i with respect to attribute A _j, and , $v_{ij}^{k} \geq 0$ , $0 \leq u_{ij}^{k} + v_{ij}^{k} \leq 1$ , $s_{a_{ij}^{k}}, s_{b_{ij}^{k}} \in S$ . Then, the ranking of alternatives is finally required.

3.2 The extended TODIM method for intuitionistic uncertain linguistic information

In the following, we will apply the TODIM to solve the multiple attribute group decision making problems with the intuitionistic uncertain linguistic information.

In order to solve the MAGDM problems with intuitionistic uncertain linguistic information, we need solve the following questions.

How to calculate the difference between two attribute values in (25), this is a key point to extend the TODIM method for intuitionistic uncertain linguistic information. In general, there are two methods, one is to convert the intuitionistic uncertain linguistic variables to crisp numbers by expectation value defined in (21), this process will cause the loss of information; another is to use the distance between two attribute values, this is reasonable. In this paper, we will adopt this method.

How to process the group decision making. In the previous studies, there are not the extended TODIM methods for group decision making. There are two methods to process the MAGDM problems, one is to aggregate personal evaluation information of the different decision makers to group information, then to extend TODIM methods, however, this method doesn’t consider the decision makers’ bounded rationality in aggregation process; another is to calculate the global dominance for each decision maker, then to aggregate to collective global dominance. This method can consider the bounded rationality of each decision maker, however, this method has a little more complicated in calculation process than the previous one.

The method involves the following steps:

Step 1: Normalize decision matrix

Generally, the criteria have two types, i.e., benefit type and cost type. If the criteria are cost type, the criteria should be converted to benefit type. The intuitionistic uncertain linguistic variables decision matrix ${\tilde{R}}^{k} = [{\tilde{r}}_{ij}^{k}]_{m \times n}$ (i = 1, 2, …, m; j = 1, 2, …, n and k = 1, 2, …, p) is normalized to form the matrix ${\tilde{X}}^{k} = {[{\tilde{x}}_{ij}^{k}]}_{m \times n}$ $({\tilde{x}}_{ij}^{k} = 〈 [s_{{\dot{a}}_{ij}^{k}}, s_{{\dot{b}}_{ij}^{k}}], ({\dot{u}}_{ij}^{k}, {\dot{v}}_{ij}^{k}) 〉)$ by the following formula. ${\begin{matrix} {\tilde{x}}_{ij}^{k} = 〈 [s_{{\dot{a}}_{ij}^{k}}, s_{{\dot{b}}_{ij}^{k}}], ({\dot{u}}_{ij}^{k}, {\dot{v}}_{ij}^{k}) 〉 \\ = 〈 [s_{a_{ij}^{k}}, s_{b_{ij}^{k}}], (u_{ij}^{k}, v_{ij}^{k}) com>rangle benefit type \\ {\tilde{x}}_{ij}^{k} = 〈 [s_{{\dot{a}}_{ij}^{k}}, s_{{\dot{b}}_{ij}^{k}}], ({\dot{u}}_{ij}^{k}, {\dot{v}}_{ij}^{k}) 〉 \\ = 〈 neg (s_{b_{ij}^{k}}), neg (s_{a_{ij}^{k}}) (u_{ij}^{k}, v_{ij}^{k}) com>rangle cost type \end{matrix}$ (28)

Step 2: Calculate the relative weight w _cr of criterion C _c to reference criterion C _r, $w_{cr} = \frac{w_{c}}{w_{r}}$ (29) where w _r = max {w _c|c = 1, 2, …, n}.

Step 3: Calculate the dominance of each alternative A _i over each alternative A _j according to the kth decision maker under the criterion C _c by the followingexpression:

$φ_{c}^{k} (A_{i}, A_{j}) = {\begin{matrix} \sqrt{\frac{w_{cr}}{\sum_{c = 1}^{n} w_{cr}} d ({\tilde{x}}_{ic}^{k}, {\tilde{x}}_{jc}^{k})} if ({\tilde{x}}_{ic}^{k} - {\tilde{x}}_{jc}^{k}) > 0 \\ 0 if ({\tilde{x}}_{ic}^{k} - {\tilde{x}}_{jc}^{k}) = 0 \\ \frac{- 1}{θ} \sqrt{\frac{(\sum_{c = 1}^{n} w_{cr})}{w_{cr}} d ({\tilde{x}}_{ic}^{k}, {\tilde{x}}_{jc}^{k})} \\ if ({\tilde{x}}_{ic}^{k} - {\tilde{x}}_{jc}^{k}) < 0 \end{matrix}$ (30)

Step 4: Calculate the global dominance of alternative A _i over each alternative A _j according the kth decision maker by the following expression: $δ^{k} (A_{i}, A_{j}) = \sum_{c = 1}^{n} φ_{c}^{k} (A_{i}, A_{j})$ (31)

Step 5: Calculate the collective overall dominance of alternative A _i over each alternative A _j by the following expression. $δ (A_{i}, A_{j}) = \sum_{K = 1}^{p} λ_{k} δ^{k} (A_{i}, A_{j})$ (32)

Step 6: Calculate the overall value of the alternative A _i by normalizing the collective overall dominance of alternative A _i by the following expression: $ξ_{i} = \frac{\sum_{j = 1}^{n} δ (A_{i}, A_{j}) - min_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j}))}{max_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j})) - min_{i} (\sum_{j = 1}^{n} δ (A_{i}, A_{j}))}$ (33)

Step 7: Rank all the alternatives and select the best one by the overall value of the alternative A _i. The bigger ξ _i is, the better alternative A _i is.

4 An illustrative example

In the section, we will provide an example to illustrate the extended TODIM method based on intuitionistic uncertain linguistic variables (Cited from [18]). Suppose that an investment company wants to invest an amount of money to a company. There are four candidate companies A _i (i = 1, 2, 3, 4) evaluated by three decision makers {D ₁, D ₂, D ₃}. The weight vector of the decision makers is γ = (0.4, 0.32, 0.28) ^T, and the attributes are considered include: C ₁ (the risk index), C ₂ (the growth index), C ₃ (the social–political impact index), and C ₄ (the environmental impact index). Suppose the attribute weight vector is W = (0.32, 0.26, 0.18, 0.24) ^T. The three decision makers {D ₁, D ₂, D ₃} evaluate the four companies A _i (i = 1, 2, 3, 4) with respect to the attributes C _j (j = 1, 2, 3, 4) by using the intuitionistic uncertain linguistic variables. Suppose that the decision makers use linguistic term set S = (s ₀, s ₁, s ₂, s ₃, s ₄, s ₅, s ₆) to express their evaluation results. For C ₁, C ₃ and C ₄, S = (s ₀, s ₁, s ₂, s ₃, s ₄, s ₅, s ₆)= (very big, big, slightly big, general, slightly little, little,very little), and for C ₂, S = (s ₀, s ₁, s ₂, s ₃, s ₄, s ₅, s ₆)= (very little, little, slightly little, general, slightly big, big, very big). and construct three following decision matrices ${\tilde{R}}^{k} = {[{\tilde{r}}_{ij}^{k}]}_{4 \times 4} (k = 1, 2, 3)$ as listed in Tables 1–3.

4.1 The evaluation steps

Since the all criteria are benefit type and the all evaluation values are described as intuitionistic uncertain linguistic variables, we don’t need to normalize the decision matrix.

Calculate the relative weight w _cr. According to (29), we can get w _cr = (1, 0 . 8125, 0 . 5625, 0 . 75) ^T.

Calculate the dominance of each alternative A _i over each alternative A _j according to the kth decision maker under the criterion C _c. By (30) (suppose the attenuation coefficient θ is 1), we get the following results listed in Tables 4–15.

Calculate the global dominance of alternative A _i over each alternative A _j according the kth decision maker. By (31), we can get the following results listed in Tables 16–18.

Calculate the collective overall dominance of alternative A _i over each alternative A _j. By (32), we can get the following result listed in Table 19.

Calculate the overall value of the alternativec. By (33), we can get the following result listed in Table 20.

Rank all the alternatives A _i (i = 1,2,3,4) in accordance with the value ξ _i, we can get the ranking of investments A ₂ ≻ A ₄ ≻ A ₁ ≻ A ₃.

4.2 Analysis on the effect of the factor of attenuation of losses

The θ is attenuation factor of the losses, Kahneman and Tversky [42] proposed the θ = 2.25. Tversky and Kahneman [43] proposed that the θ can get the value between 1.0 and 2.5. In order to illustrate the effects on decision making about attenuation factor of the losses θ, we gave the ranking results on different values of θ shown as Table 21.

Form the Table 21, we can find that the ordering of the alternatives does not make any change by increasing the value of attenuation of losses from θ = 1 to θ = 2.5.

4.3 Analysis on the effectiveness of the extended TODIM method

In order to analyze the effectiveness of the extended TODIM method, we can compare it with that proposed by Liu and Jin [18]. Firstly, we can get same ranking results for these two methods. However, the method proposed in this paper is based on Prospect Theory, and it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Of course, the method proposed by Liu and Jin [18] can give the comprehensive evaluation values of all alternatives and the ranking results by generalized aggregation operators, and the method in this paper can only provide the ranking results. Obviously, this is the two ideas to solve MAGDM problems.

5 Conclusion

In the classical TODIM method, according to Prospect Theory, the dominance degree of each alternative over the others is measured by building a multi-attribute value function, which reflects DM’s aversion of risk in face of gains and the propensity to risk in face of losses. However, the classical TODIM method can only process the MADM problems in which the attribute values take the form of crisp numbers, in this paper, we extended TODIM method to deal with the intuitionistic uncertain linguistic information which can more easily express fuzzy information than uncertain linguistic variables or intuitionistic fuzzy set, and the steps of decision making for MAGDM problems were given. Further, we analyze the effects on decision making about attenuation factor of the losses θ in this method, and its effectiveness. In the further research, studying the applications of the new method is necessary and meaningful because the bounded rationality is the actual decision behavior of the human beings, such as selection of supplier, science-technology assessment, the performance evaluation and so on. In addition, we should further expand TODIM method to adapt to other fuzzy information.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), Shandong Provincial Social Science Planning Project (No. 13BGLJ10), the Natural Science Foundation of Shandong Province (No. ZR2011FM036), Graduate education innovation projects in Shandong Province (SDYY12065) and National Soft Science Project (2014GXQ4D192).

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