ELECTRE method for multiple attributes decision making problem with hesitant fuzzy information

Abstract

The ELECTRE method utilizes the concordance and discordance set. And then use concordance and discordance indices to construct concordance and discordance matrices. Both concordance and discordance indices have to be calculated for every pair of alternatives. Based on the aggregate matrix, a decision graph can be depicted to determine which alternative is preferable, incomparable or indifferent. It first uses concordance and discordance indices to analyze outranking relations among alternatives from real-world applications. In this paper, an extension of ELECTRE method, a multiple attributes hesitant fuzzy decision making technique, to a decision environment is investigated, where the individual decision value are provided by the hesitant fuzzy information. We first introduce some concepts of hesitant fuzzy set. Then, the hesitant fuzzy ELECTRE method is proposed in detail. Finally, an illustrative example for evaluating the performance of the industry zones is used to illustrate the developed methods.

Keywords

Multiple attributes decision making hesitant fuzzy set ELECTRE method industry zones

1 Introduction

Multiple attributes group decision making has been one of the fastest growing areas during the last two decades because of its usefulness and fascination [1 –7]. There are many techniques that have been developed to help experts rank alternatives according to many attributes. The existing decision making methods can be classified into three broad categories: (1) Value measurement models. AHP and Multiple Attribute Utility Theory (MAUT) are the best known method in this group. (2) Goal, aspiration and reference level models. Goal programming (GP) and TOPSIS are the most important methods that belong to the group. (3) Outranking models. ELECTRE and PROMETHEE are two main families of method in this group.

The ELECTRE (Elimination and Et Choice Translating Reality) method was first introduced by Roy [8, 9], which is the first outranking method thought the concordance and discordance matrices. The ELECTRE method utilizes the concordance and discordance set (the concordance set are combined to an overall index using criteria weights which are interpreted as a kind of “voting power” and the discordance indices are used to reduce the overall credibility of the hypothesis). And then use concordance and discordance indices to construct concordance and discordance matrices (the concordance matrix reflects weights of the concordance criteria and the discordance matrix reflect most relative differences according to the discordance criteria). Both concordance and discordance indices have to be calculated for every pair of alternatives. Based on the aggregate matrix, a decision graph can be depicted to determine which alternative is preferable, incomparable or indifferent. It first uses concordance and discordance indices to analyze outranking relations among alternatives from real-world applications.

Torra [10] proposed the hesitant fuzzy set which permitted the membership having a set of possible values and discussed the relationship between hesitant fuzzy set and intuitionistic fuzzy set. Xia and Xu [11] gave an intensive study on hesitant fuzzy information aggregation techniques and proposed a series of operators under various situations to solve the decision making problems with anonymity. Xu and Xia [12, 13] proposed a variety of distance measures for hesitant fuzzy sets, based on which the corresponding similarity measures are obtained, and defined the distance and correlation measures for hesitant fuzzy information and discussed their properties in detail. Xu et al. [14] developed several series of aggregation operators for hesitant fuzzy information with the aid of quasi-arithmetic means. Gu, et al. [15] investigated the evaluation model for risk investment with hesitant fuzzy information. They utilized the hesitant fuzzy weighted averaging (HFWA) operator to aggregate the hesitant fuzzy information corresponding to each alternative, and then ranked the alternatives according to the score function.

From above analysis, we can see that hesitant fuzzy set is a very useful tool to deal with uncertainty and fuzziness [16 –23]. Moreover, when giving the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error, or some possibility distribution on the possibility values, but because we have several possible values. For such cases, the hesitant fuzzy set is very useful in avoiding such issues in which each attribute can be described as a hesitant fuzzy element defined in terms of the opinions of experts. More and more multiple attribute decision making theories and methods under hesitant fuzzy environment have been developed. In this paper, we investigate multiple attributes group decision making problems, in which both the attribute weights information and the expert weights information take the form of hesitant fuzzy information, and attribute values also take the form of hesitant fuzzy information. To deal with the multiple attributes hesitant fuzzy group decision making problem, an extension of the ELECTRE method is proposed for solving the decision making problem with the goal of making effective decisions in this article. The paper is organized as follows. Section 2 introduces the basic definitions and notations of the measure of hesitant fuzzy sets. In Section 3, we present an extension of the ELECTRE method based on hesitant fuzzy information to cope with the MADM problems. And then, the proposed methods are illustrated with an example for evaluating the performance of the industry zones with hesitant fuzzy information in Section 4. Finally, some conclusions are drawn at the end of the paper.

2 Preliminaries

Here we give a brief review of some preliminaries.

Definition 1. [10] Let X be a fixed set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns a subset of [0,1]. To be easily understood, we express the HFS by a mathematical symbol: $E = {< x, h_{E} (x) > | x \in X}$ (1) where h _E (x) is a set of some values in [0,1], denoting the possible membership degrees of the element x to the set X. For convenience, we call h = h _E (x) a hesitant fuzzy element (HFE) and H the set of all HFEs.

Given three HFEs reprented by h, h ₁ and h ₂, Torra [10] defined some operations in them, which can be described as follows:

h ^c =∪ _γ∈h { 1 - γ } ;

h ₁∪ h ₂ = ∪ _{γ₁∈h₁, γ₂∈h₂} max { γ ₁, γ ₂ } ;

h ₁∩ h ₂ = ∪ _{γ₁∈h₁, γ₂∈h₂} min { γ ₁, γ ₂ }.

In the following, Torra [10] showed that the envelop of a HFE is an IFV.

Definition 2. Given a HFE h, we define the IFV A _env (h) as the envelope of h, where A _env (h) can be defined as (h ^-, 1 - h ⁺), with h ^- = min{ γ|γ ∈ h } and h ⁺ = max{ γ|γ ∈ h }.

Then, Torra [10] gave the further study of the relationship between HFEs and IFVs:

A _env (h ^c) = (A _env (h)) ^c ;

A _env (h ₁∪ h ₂) = A _env (h ₁) ∪ A _env (h ₂) ;

A _env (h ₁ ∩ h ₂) = A _env (h ₁) ∩ A _env (h ₂) .

Let h, h ₁ and h ₂ be three HFEs, then [11]

h ^λ =∪ _γ∈h { γ ^λ } ;

λh =∪ _γ∈h { 1 - (1 - γ) ^λ } ;

$h_{1} \cup h_{2} = \cup_{γ_{1} \in {\tilde{h}}_{1}, γ_{2} \in {\tilde{h}}_{2}} max {γ_{1}, γ_{2}}$

$h_{1} \cap h_{2} = \cup_{γ_{1} \in {\tilde{h}}_{1}, γ_{2} \in {\tilde{h}}_{2}} min {γ_{1}, γ_{2}}$

$h_{1} \oplus h_{2} = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} {γ_{1} + γ_{2} - γ_{1} γ_{2}};$

$h_{1} \otimes h_{2} = \cup_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} {γ_{1} γ_{2}}$ .

Definition 3. [12, 13] Let h ₁ and h ₂ be two HFEs on X ={ x ₁, x ₂, …, x _n }, then the hesitant normalized Euclidean distance measure between h ₁ and h ₂ is defined as: $∥ h_{1} - h_{2} ∥ = \sqrt{\frac{1}{l} \sum_{j = 1}^{l} | h_{1 σ (j)} - h_{2 σ (j)} | 2}$ (2) where l (h) is the number of the elements in the h, in most cases, l (h ₁) ≠ l (h ₂), and for convenience, let l = max{ l (h ₁) , l (h ₂) }. To operate correctly, we should extend the shorter one until both of them have the same length when we compare them.

3 ELECTRE method for multiple attributes decision making problem with hesitant fuzzy information

The ELECTRE (Elimination and Et Choice Translating Reality) method plays a prominent role in the group of outranking methods to solve the multiple attribute decision making problem, which is first idea concerning concordance, discordance and outranking concepts originate from real-world applications. Compared with other methods, the ELECTRE method is able to apply more complicated algorithm to deal with the complex and imprecise information from the decision problems and use these algorithms to rank the alternatives. The method uses concordance and discordance indexes to analyze the outranking relations among the alternatives by a decision graph. In general, many real world multiple attribute decision making problems take place in a complex environment and usually adhere to imprecise data and uncertainty. In this section, we extend the ELECTRE method for hesitant fuzzy multiple attribute decision making. Let A ={ A ₁, A ₂, …, A _m } be a discrete set of alternatives, and G ={ G ₁, G ₂, …, G _n } be the set of attributes, w = (w ₁, w ₂, …, w _n) is the weighting vector of the attribute G _j (j = 1, 2, …, n), where w _j∈ [0,1], $\sum_{j = 1}^{n} w_{j} = 1$ . If the decision makers provide several values for the alternative A _i under the attribute G _j with anonymity, these values can be considered as a hesitant fuzzy element h _ij. In the case where two decision makers provide the same value, then the value emerges only once in h _ij. Suppose that the decision matrix H = (h _ij) _m×n is the hesitant fuzzy decision matrix, where h _ij (i = 1, 2, …, m, j = 1, 2, …, n) are in the form of HFEs. The proposed method consists of the following steps:

Step 1. Determine the concordance and discordance sets using the concepts of score function. The concordance set is composed of all attribute which A _k is preferred to A _l, the concordance set $S_{kl}^{+}$ using the aforementioned concepts can be formulated as follows: $S_{kl}^{+} = {\begin{matrix} j | h_{kj} \geq h_{lj}; k, l = 1, 2, \dots, m, \\ j = 1, 2, \dots, n \end{matrix}}$ (3)

The discordance set consists of all attribute which A _k is not preferred to A _l. The discordance set $S_{kl}^{-}$ using the aforementioned concepts can be formulated as follows: $S_{kl}^{-} = {\begin{matrix} j | h_{kj} < h_{lj}; k, l = 1, 2, \dots, m, \\ j = 1, 2, \dots, n \end{matrix}}$ (4)

Step 2. Utilizing the concept of concordance and discordance sets to calculate concordance and discordance matrix:

The concordance matrix for each pair-wise comparison of the alternatives is defined as follows: $U = [\begin{matrix} - & u_{12} & \dots & \dots & u_{1 m} \\ u_{21} & - & u_{23} & \dots & u_{2 m} \\ \dots & \dots & - & \dots & \dots \\ u_{(m - 1) 1} & \dots & \dots & - & u_{(m - 1) m} \\ u_{m 1} & u_{m 2} & \dots & u_{m (m - 1)} & - \end{matrix}]$ (5) where $u_{kl} = \sum_{j \in S_{kl}^{+}} w_{j}$ , and a higher value of u _kl indicates that A _k is preferred to A _l.

The discordance matrix is defined as follows: $V = [\begin{matrix} - & v_{12} & \dots & \dots & v_{1 m} \\ v_{21} & - & v_{23} & \dots & v_{2 m} \\ \dots & \dots & - & \dots & \dots \\ v_{(m - 1) 1} & \dots & \dots & - & v_{(m - 1) m} \\ v_{m 1} & v_{m 2} & \dots & v_{m (m - 1)} & - \end{matrix}]$ (6) where $v_{kl} = \frac{max_{j \in S_{kl}^{-}} ∥ h_{kj} - h_{lj} ∥}{max_{j} ∥ h_{kj} - h_{lj} ∥}$ , and a higher value of r _kl indicates that A _k is less favorable than A _l.

Step 3. Computing the Boolean matrix X and Y with the minimum concordance level and minimumdiscordance level.

The Boolean matrix X is computed in accordance with the minimum concordance level $\bar{U}$ , $X = [\begin{matrix} - & x_{12} & \dots & \dots & x_{1 m} \\ x_{21} & - & x_{23} & \dots & x_{2 m} \\ \dots & \dots & - & \dots & \dots \\ x_{(m - 1) 1} & \dots & \dots & - & x_{(m - 1) m} \\ x_{m 1} & x_{m 2} & \dots & x_{m (m - 1)} & - \end{matrix}]$ (7) where $u_{kl} \geq \bar{U} \Rightarrow x_{kl} = 1, u_{kl} < \bar{U} \Rightarrow x_{kl} = 0$ , $\bar{U} = \sum_{k = 1}^{m} \sum_{l = 1}^{m} u_{kl} / m (m - 1)$ , can be defined as the average values of all the elements in the matrix U.

The Boolean matrix Y is computed by a minimum discordance level as $Y = [\begin{matrix} - & y_{12} & \dots & \dots & y_{1 m} \\ y_{21} & - & y_{23} & \dots & y_{2 m} \\ \dots & \dots & - & \dots & \dots \\ y_{(m - 1) 1} & \dots & \dots & - & y_{(m - 1) m} \\ y_{m 1} & y_{m 2} & \dots & y_{m (m - 1)} & - \end{matrix}]$ (8) where $v_{kl} < \bar{V} \Rightarrow y_{kl} = 1, v_{kl} \geq \bar{V} \Rightarrow y_{kl} = 0$ , and $\bar{V} = \sum_{k = 1}^{m} \sum_{l = 1}^{m} v_{kl} / m (m - 1)$ is defined as the average value of all the elements in the matrix V.

Step 4. Computing the matrix Z by the multiplication of the elements of the matrix X and Y as follows: $Z = X \otimes Y$ (9) where each element in the matrix Z is computed by z _kl = x _kl y _kl.

Step 5. Ranking all the order in accordance with z _kl and select the best alternative.

4 Numerical example

The problem of evaluating the performance of the industry zones with hesitant fuzzy information is the multiple attribute decision making problems. In this section, we shall present a numerical example to evaluate the performance evaluation of the industry zones with hesitant fuzzy information to illustrate the method proposed in this paper. There are five industry zones A _i (i = 1, 2, 3, 4, 5) for four attributes G _j (j = 1, 2, 3, 4). The four attributes include the gross value of industrial output (G ₁), the industrial added value (G ₂), the realization of tax (G ₃) and the actual utilization of foreign capital (G ₄). In order to avoid influence each other, the decision makers are required to evaluate the performance of the five possible industry zones A _i (i = 1, 2, …, 5) under the above four attributes in anonymity and the decision matrix H = (h _ij) _m×n is presented in Table 1, where h _ij (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4) are in the form of HFEs.

The proposed method consists of the following steps:

Step 1. Computing the Concordance set S ⁺ and discordance set S ^- by utilizing the score function of the hesitant fuzzy values, $\begin{matrix} S^{+} & = & [\begin{matrix} - & 1, 3, 4 & 2 & 3 & 2 \\ 1, 4 & - & 4 & 1, 3 & - \\ 2, 4 & 1, 3 & - & 1, 2, 3 & 2, 3 \\ 3, 4 & 1, 4 & 4 & - & 2, 4 \\ 1, 2, 4 & 3, 4 & 2, 3 & 3, 4 & - \end{matrix}] \\ S^{-} & = & [\begin{matrix} - & 2 & 1, 3, 4 & 1, 3 & 2, 4 \\ 1, 4 & - & 1, 3, 4 & 2, 4 & 4 \\ 3 & - & - & 3 & 3, 4 \\ 1 & 2, 4 & 1, 3 & - & 1, 3 \\ - & - & 1, 4 & 3, 4 & - \end{matrix}] \end{matrix}$

Step 2. Utilizing the sets of concordance and discordance to calculate concordance matrix U and discordance matrix V. $U = [\begin{matrix} - & 0 . 700 & 0 . 380 & 0 . 350 & 0 . 300 \\ 0 . 390 & - & 0 . 400 & 0 . 300 & 0 . 000 \\ 0 . 450 & 1 . 000 & - & 0 . 550 & 0 . 490 \\ 0 . 600 & 0 . 600 & 0 . 350 & - & 0 . 600 \\ 1 . 000 & 1 . 000 & 0 . 550 & 0 . 600 & - \end{matrix}]$ $V = [\begin{matrix} - & 1.000 & 1.000 & 0.000 & 1.000 \\ 0.636 & - & 1.000 & 0.500 & 0.600 \\ 0.135 & 0.000 & - & 0.367 & 0.280 \\ 0.334 & 0.400 & 0.800 & - & 0.000 \\ 0.000 & 0.000 & 1.000 & 0.500 & - \end{matrix}]$

Step 3. Achieving the Boolean matrices X and Y. $\begin{matrix} X & = & [\begin{matrix} - & 0 & 1 & 0 & 0 \\ 0 & - & 1 & 0 & 0 \\ 0 & 1 & - & 0 & 0 \\ 1 & 0 & 1 & - & 0 \\ 1 & 0 & 1 & 0 & - \end{matrix}] \\ Y & = & [\begin{matrix} - & 0 & 1 & 0 & 0 \\ 0 & - & 1 & 0 & 0 \\ 1 & 0 & - & 0 & 1 \\ 0 & 1 & 1 & - & 0 \\ 0 & 1 & 1 & 0 & - \end{matrix}] \end{matrix}$

Step 4. We construct the matrix Z as follows: $\begin{matrix} Z & = & X \otimes Y \\ = & [\begin{matrix} - & 0 & 1 & 0 & 0 \\ 0 & - & 1 & 0 & 0 \\ 0 & 1 & - & 0 & 0 \\ 1 & 0 & 1 & - & 0 \\ 1 & 0 & 1 & 0 & - \end{matrix}] \otimes [\begin{matrix} - & 0 & 1 & 0 & 0 \\ 0 & - & 1 & 0 & 0 \\ 1 & 0 & - & 0 & 1 \\ 0 & 1 & 1 & - & 0 \\ 0 & 1 & 1 & 0 & - \end{matrix}] \\ = & [\begin{matrix} - & 0 & 0 & 0 & 0 \\ 0 & - & 1 & 0 & 0 \\ 1 & 1 & - & 1 & 0 \\ 0 & 0 & 1 & - & 0 \\ 0 & 1 & 1 & 0 & - \end{matrix}] \end{matrix}$

Step 5. Rank the alternatives according to the decision graph, and thus the most desirable industry zone is A ₃.

5 Conclusion

Many real-world decision problems take place in a complex environment and involve conflicting systems of attributes, uncertainty and imprecise information. However, when giving the membership degree of an element, the difficulty of establishing the membership degree is not because we have a margin of error, or some possibility distribution on the possibility values, but because we have several possible values. For such cases, Torra [10] proposed the hesitant fuzzy set. The hesitant fuzzy set is adequate for dealing with the vagueness of experts’ judgments over alternatives with respect to attributes, which can be accurately and perfectly described in terms of the opinions of experts. This study proposes a novel hesitant fuzzy ELECTRE method to solve the hesitant fuzzy multiple attributes group decision making problem. Finally, a practical application for evaluating the performance of the industry zones with hesitant fuzzy information is demonstrated. In the future, we shall continue working in the application of the hesitant fuzzy multiple attribute decision-making to other domains [24 –40].

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