Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets

doi:10.1016/j.eswa.2009.06.094

Expert Systems with Applications

Volume 37, Issue 1, January 2010, Pages 824-833

https://doi.org/10.1016/j.eswa.2009.06.094 Get rights and content

Abstract

In this paper, we present a new method to handle fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets, we present a new method to handle fuzzy multiple attributes group decision-making problems. The proposed method provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes.

Introduction

Many methods have been presented for handling fuzzy multiple attributes decision-making problems (Chen, 1988, Chen and Chen, 2003, Chen and Hwang, 1992, Fu, 2008, Hua et al., 2008, Lin et al., 2008, Hwang and Yoon, 1981, Wang and Chang, 2007, Yager and Xu, 2006). Chen and Hwang (1992) presented some methods for handling fuzzy multiple attributes decision-making problems. Chen (1988) presented a method for handling fuzzy multiple attributes decision-making problems based on the similarity measure between fuzzy sets. Fu (2008) presented a fuzzy optimization method for multi-criteria decision making. Hua et al. (2008) presented an approach for multi-attribute decision making problems with incomplete information. Hwang and Yoon (1981) presented the TOPSIS method for handling multiple attribute decision-making problems. Lin et al. (2008) presented a dynamic multi-attribute decision making model with grey number evaluations. Wang and Chang (2007) presented an application of TOPSIS in evaluating initial aircraft training under a fuzzy environment. Yager and Xu (2006) presented a continuous ordered weighted geometric operator and its application to decision-making. Moreover, some methods have been presented for handling fuzzy multiple attributes group decision-making problems (Chen, 2000; Chen, 2001; Lee & Chen, 2008; Lin & Wu, 2008; Tsabadze, 2006; Tsai & Wang, 2008). Chen (2000) extended the TOPSIS method to present a method to handle group decision-making problems in a fuzzy environment. Chen (2001) presented a method to evaluate the rate of aggregative risk in software development using the fuzzy set theory in the fuzzy group decision-making environment. Lin and Wu (2008) presented a causal analytical method for group decision-making in the fuzzy environment. Tsabadze (2006) presented a method for fuzzy aggregation based on group expert evaluations. Tsai and Wang (2008) presented a computing-coordination-based fuzzy group decision-making method for web service oriented architectures. However, the existing fuzzy multiple attributes group decision-making methods are based on traditional type-1 fuzzy sets (Zadeh, 1965). If we can use interval type-2 fuzzy sets (Mendel, John, & Liu, 2006) for handling fuzzy group decision-making problems, then there is room for more flexibility due to the fact that type-2 fuzzy sets provide more flexibility to represent uncertainties than traditional type-1 fuzzy sets (Zadeh, 1965).

In this paper, we present a new method for handling fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets, we present a new method to handle fuzzy multiple attributes group decision-making problems. The proposed method provide us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes.

The rest of this paper is organized as follows. In Section 2, we briefly review the definitions of interval type-2 fuzzy sets (Mendel et al., 2006). In Section 3, we present the arithmetic operations between trapezoidal interval type-2 fuzzy sets. In Section 4, we present a method for ranking interval type-2 fuzzy sets. In Section 5, we present a new method for handling fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. In Section 6, we use an example to illustrate the proposed method. The conclusions are discussed in Section 7.

Section snippets

Basic concepts of interval type-2 fuzzy sets

Let $\tilde{A}$ be a type-1 trapezoidal fuzzy set, $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4}; H_{1} (A), H_{2} (A))$ , as shown in Fig. 1, where $H_{1} (\tilde{A})$ denotes the membership value of the element $a_{2}, H_{2} (\tilde{A})$ denotes the membership value of the element $a_{3}, 0 ⩽ H_{1} (A) ⩽ 1$ and $0 ⩽ H_{2} (A) ⩽ 1$ . If $a_{2} = a_{3}$ , then the type-1 fuzzy set $\tilde{A}$ becomes a triangular type-1 fuzzy set.

In the following, we briefly review some definitions of type-2 fuzzy sets and interval type-2 fuzzy sets from (Mendel et al., 2006).

Definition 2.1

Mendel et al., 2006

A type-2 fuzzy set $\tilde{\tilde{A}}$ in the universe of discourse X can

Arithmetic operations between trapezoidal interval type-2 fuzzy sets

In this section, we briefly review the arithmetic operations between trapezoidal interval type-2 fuzzy sets we presented in (Lee & Chen, 2008).

Definition 3.1

The addition operation between the trapezoidal interval type-2 fuzzy sets ${\tilde{\tilde{A}}}_{1} = ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) = ((a_{11}^{U}, a_{12}^{U}, a_{13}^{U}, a_{14}^{U}; H_{1} ({\tilde{A}}_{1}^{U}), H_{2} ({\tilde{A}}_{1}^{U})), (a_{11}^{L}, a_{12}^{L}, a_{13}^{L}, a_{14}^{L}; H_{1} ({\tilde{A}}_{1}^{L}), H_{2} ({\tilde{A}}_{1}^{L})))$ and ${\tilde{\tilde{A}}}_{2} = ({\tilde{A}}_{2}^{U}, {\tilde{A}}_{2}^{L}) = ((a_{21}^{U}, a_{22}^{U}, a_{23}^{U}, a_{24}^{U}; H_{1} ({\tilde{A}}_{2}^{U}), H_{2} ({\tilde{A}}_{2}^{U})), (a_{21}^{L}, a_{22}^{L}, a_{23}^{L}, a_{24}^{L}; H_{1} ({\tilde{A}}_{2}^{L}), H_{2} ({\tilde{A}}_{2}^{L})))$ is defined as follows: ${\tilde{\tilde{A}}}_{1} \oplus {\tilde{\tilde{A}}}_{2} = ({\tilde{A}}_{1}^{U}, {\tilde{A}}_{1}^{L}) \oplus ({\tilde{A}}_{2}^{U}, {\tilde{A}}_{2}^{L}) = ((a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} + a_{23}^{U}, a_{14}^{U} + a_{}))$

Ranking values of trapezoidal interval type-2 fuzzy sets

In this section, we present a new method for calculating the ranking values of trapezoidal interval type-2 fuzzy sets. Let ${\tilde{A}}_{s}^{U}$ and ${\tilde{A}}_{t}^{U}$ be upper trapezoidal membership functions of the interval type-2 fuzzy sets ${\tilde{A}}_{s}$ and ${\tilde{A}}_{t}$ , respectively, as shown in Fig. 4, where ${\tilde{A}}_{s}^{U} = (a_{s 1}^{U}, a_{s 2}^{U}, a_{s 3}^{U}, a_{s 4}^{U}; H_{1} ({\tilde{A}}_{s}^{U}), H_{2} ({\tilde{A}}_{s}^{U}))$ and ${\tilde{A}}_{t}^{U} = (a_{t 1}^{U}, a_{t 2}^{U}, a_{t 3}^{U}, a_{t 4}^{U}; H_{1} ({\tilde{A}}_{t}^{U}), H_{2} ({\tilde{A}}_{t}^{U}))$ . In order to define the likelihood $p ({\tilde{A}}_{s}^{U} ⩾ {\tilde{A}}_{t}^{U})$ of ${\tilde{A}}_{s}^{U} ⩾ {\tilde{A}}_{t}^{U}$ , we define the strength $E_{ts}$ of ${\tilde{A}}_{t}^{U}$ over ${\tilde{A}}_{s}^{U}$ by considering the difference

A new method for fuzzy multiple attributes group decision making

In this section, we present a new method for handling fuzzy multiple attributes group decision-making problems based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets. Assume that there is a set Z of alternatives and a set F of attributes, where $Z = {z_{1}, z_{2}, \dots, z_{n}}$ and $F = {f_{1}, f_{2}, \dots, f_{m}}$ . Assume that there are k decision-makers $D_{1}, D_{2}, \dots,$ and $D_{k}$ . The set F of attributes can be divided into two sets $F_{1}$ and $F_{2}$ , where $F_{1}$ denotes the set of benefit

Numerical example

In this section, we use an example to illustrate the proposed method. Table 2 shows the linguistic terms “Very Low” (VL), “Low” (L), “Medium Low” (ML), “Medium” (M), “Medium High” (MH), “High” (H), “Very High” (VH) and their corresponding interval type-2 fuzzy sets, respectively. Assume that there are three decision-makers $D_{1}, D_{2}$ and $D_{3}$ to evaluate cars, where there are three alternatives $z_{1}, z_{2}, z_{3}$ and four attributes (i.e., “Safety”, “Price”, “Appearance”, “Performance”), as shown in Table 3.

Conclusions

In this paper, we have presented a new method to handle fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the

Acknowledgement

This work was supported in part by the National Science Council, Republic of China, under Grant NSC 95-2221-E-011-116-MY2.

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Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets

Abstract

Introduction

Section snippets

Basic concepts of interval type-2 fuzzy sets

Mendel et al., 2006

Arithmetic operations between trapezoidal interval type-2 fuzzy sets

Ranking values of trapezoidal interval type-2 fuzzy sets

A new method for fuzzy multiple attributes group decision making

Numerical example

Conclusions

Acknowledgement

Automatica

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Expert Systems with Applications

Expert Systems with Applications

Computers and Mathematics with Applications

Expert Systems with Applications

Expert Systems with Applications

Fuzzy Sets and Systems

Expert Systems with Applications