Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 680635, 12 pages

http://dx.doi.org/10.1155/2015/680635

## The Likelihood Ranking Methods for Interval Type-2 Fuzzy Sets Considering Risk Preferences

^{1}School of Business Administration, Northeastern University, Shenyang 110819, China^{2}Northeastern University at Qinhuangdao, Qinhuangdao 066004, China

Received 7 April 2015; Revised 10 August 2015; Accepted 19 August 2015

Academic Editor: Dan Simon

Copyright © 2015 Meng Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a ranking method that considers the risk preferences of decision makers for multiple-attribute decision-making problems in a multiple-interval type-2 trapezoidal fuzzy set environment. First, decision makers are classified according to the risk preferences and a measurement method of risk preferences is proposed. Second, a risk preference decision matrix is obtained and a new calculation formula of likelihood is defined. Finally, we obtain the ranking results of alternatives by calculating the signed distance. Our example analysis shows that the proposed method is scientific and reasonable, and different risk preferences influence the results of decision making. Comparison with previous methods shows that the proposed algorithm is more feasible; it is applicable for decision making on both risk preferences and risk conservation.

#### 1. Introduction

In an increasingly complex decision-making environment, decision-making information has become more uncertain and data with different attributes of options are difficult to determine. In 1965, Zadeh [1] proposed the fuzzy set (type-1 fuzzy set, T1FS) theory that since then has been widely used in multiple-attribute decision-making (MADM) problems. However, the concept of fuzzy sets could not solve the uncertainty of membership, so Zadeh [2] proposed type-2 fuzzy sets (T2FS). This type of fuzzy set is the generalization of the T1FS, describing membership with fuzzy sets in the interval of . Type-2 fuzzy number is portrayed by the primary and secondary membership. Therefore, T2FS has a stronger ability to deal with uncertain problems. To determine the T2FS, we need to provide an appropriate fuzzy set for membership of each element in the domain, which is difficult. To simplify the problem, we have to impose necessary restrictions on the form of T2FS. One approach is to limit the value of 0 or 1 and obtain the interval-value fuzzy sets (equivalent to intuitionistic fuzzy sets and vague set [3]). Another approach is to set the membership function to be the fuzzy number, namely, interval type-2 fuzzy set (IT2FS). IT2FS is a special case of T2FS. The value of the secondary membership is set to 1 and the value of the primary membership is set to a range, thus making it describe uncertainty better than type-1 fuzzy number. IT2FS has a greater application background, especially for MADM problems that require experts to judge the satisfaction degree of options with respect to different attributes. Human cognition has complexity, uncertainty, and other characteristics that cause difficulty for experts to provide a certain value and allows them to provide only linguistic variables that can be represented by fuzzy sets [4]. IT2FS has stronger language explanation ability than ordinary fuzzy sets [5, 6]. Thus, the MADM problem of IT2FS has become a highly valuable research topic [7].

In 1979, Kahneman and Tversky [8] proposed the significant prospect theory in the field of economics on the basis of a large number of experiments. Kahneman was awarded the Nobel Prize for Economics in 2002 in recognition of his important contribution to economics. Prospect theory states that when decision makers face several decision-making behaviors that have exactly the same amount of theoretical economic benefit, most of the decision makers choose small-risk decision behaviors. However, some decision makers still choose greater-risk decision behaviors. Thus, for uncertain (especially IT2FS) multiattribute group decision-making problems, different risk preferences directly influence the decision results. Therefore, we need to classify decision makers in accordance with their risk preference before entering the specific decision-making process. However, existing studies generally do not consider the risk preference or assume risk aversion, so no effective approach is available for IT2FS multiattribute decision problems.

Considering this situation, we propose the risk preference measurement method of IT2TrFS. First, we classify decision makers on the basis of different risk preferences and obtain a risk preference decision matrix for decision makers. Second, we propose a new calculation method of likelihood. Finally, we obtain the ranking results of options by calculating the signed distance. This method takes into account the risk preferences of decision makers and proposes a specific decision method for IT2TrFS MADM problems with attribute weights in the form of an exact value of IT2TrFS.

#### 2. Literature Review

At present, the theoretical study of interval type-2 fuzzy sets (IT2FS) mainly focuses on pure mathematics. For example, Chen [9] studied the nature and operation of IT2FS, Zheng et al. [10] analyzed the similarity and acceptance of IT2FS, Zarinbal et al. [11] proposed a clustering analysis model of IT2FS with the method of relative entropy, Hwang et al. [12] proposed a similarity measurement method of interval type-2 fuzzy entropy, and Li et al. [13] proposed the uncertainty measurement method of IT2FS. These theoretical studies serve as a good theoretical foundation for the research on IT2FS in the field of multiattribute decision making (MADM).

However, to solve IT2FS MADM problems, we need to address the translation problem between IT2FS and linguistics. In recent years, research on the translation between IT2FS and linguistics mainly includes three-level [14], four-level [15], five-level [16], seven-level [17, 18], and nine-level [19, 20] languages. Using these language-level systems, decision makers can easily convert linguistic variables into IT2FS to make decisions.

IT2FS mainly includes IT2IFS, interval type-2 triangular fuzzy sets (IT2TFS), interval type-2 trapezoidal fuzzy sets (IT2TrFS), and others. The IT2IFS domain is a discrete set and can only roughly indicate whether an attribute is the member of an option or not. Therefore, conducting research based on IT2TFS and IT2TrFS is more meaningful than research on IT2IFS. Previous studies on IT2TFS MADM are as follows. Mokhtarian et al. [21] proposed IT2TFS MADM of controlling risk. Using IT2TFS decision problems, Ashtiani et al. [22] presented a feasible method to solve multiattribute group decision-making problems with the improved TOPSIS algorithm. Guo and Yin [23] proposed type-2 intuitionistic fuzzy information MADM methods. Compared to IT2TrFS, IT2TFS can describe the qualitative index [7, 24, 25] more effectively. Studies on IT2TrFS have started only recently. For example, Chen et al. [7, 24, 25] improved the classic QUALIFLEX, ELECTRE, and PROMETHEE algorithms and proposed a specific MADM algorithm. Xu et al. [26] presented a conflict measurement model based on interval intuitionistic fuzzy number preference for large group decision-making problems. Zamri and Abdullah [27] proposed IT2TrFS entropy MADM methods. These studies provided specific ideas and laid a good foundation for our study. However, existing studies do not consider the impact of risk preferences of decision makers on decision results. For fuzzy MADM problems, decision makers have to be classified in accordance with their risk preferences. As IT2FS is used to portray ambiguity and uncertainty, introducing the risk preferences of decision makers in making decisions is a practical approach [28–30].

#### 3. Preliminary Knowledge

Let be a crisp set. Let denote the set of all closed subintervals of . A mapping : is known as an IT2FS in . For each , represents the degree of membership of an element to . The type-1 fuzzy sets : and : are referred to as a lower fuzzy set of and an upper fuzzy set of , respectively. The values and represent the degrees of membership of to and , respectively, where . If for all , then the IT2FS is fully determined by a single value from for each , and reduces to a T1FS.

Let , , , , , , , and be nonnegative real values, where , , , and . Let and denote the heights of and , respectively, where . We consider that the lower and upper membership functions and , respectively, of are defined as follows:

The lower and upper nonnegative trapezoidal fuzzy numbers with respect to are denoted by and , respectively, where (i.e., if and only if, , ). Additionally, is a nonnegative IT2TrF number in and is expressed as follows:

By applying the extension principle proposed by Zadeh [31] to the IT2TrF environment, the operations of addition and multiplication by a nonnegative ordinary number defined on nonnegative IT2TrF numbers produce another nonnegative IT2TrF number. Let and be two nonnegative IT2TrF numbers in , where The addition operation between and is defined as follows:

The multiplication operation between a nonnegative real value and is defined as follows:

#### 4. Theoretical Basis

In this paper, the proposed IT2IFS MADM method considering risk preference mainly refers to two core contents. One aspect is measuring the risk appetite of IT2IFS makers. Another is choosing the suitable ranking method. The two aspects are explained in Sections 4.1 and 4.2.

##### 4.1. Measurement of Risk Preferences of Decision Makers

###### 4.1.1. Fundamental Principle

Each IT2TrF has a lower limit and upper limit simultaneously. For example, according to Figure 1, the lower limit and upper limit of IT2TrF are and , respectively. These lower and upper limits are type-1 trapezoidal fuzzy numbers. The area between the lower and upper limits is the fuzzy zone of an IT2TrF; namely, IT2TrF is the set of all type-1 trapezoidal fuzzy numbers in a fuzzy field. The core of risk preference measurement of decision makers lies in different attitudes toward the fuzzy zone. When the decision that a decision maker makes is closer to , the risk preference is close to 0 and the decision maker belongs to the type of complete risk avoidance. When the decision is closer to , the risk preference is close to 1 and the decision maker belongs to the type of complete risk preference. Based on the different risk preferences of decision makers, the value of decision making is between and . Therefore, the value of risk preference of decision makers is also between 0 and 1. According to this principle, a risk preference coefficient of decision makers is introduced.