Research Article | Open Access
The Likelihood Ranking Methods for Interval Type-2 Fuzzy Sets Considering Risk Preferences
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.
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  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  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 ). 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 . 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 .
In 1979, Kahneman and Tversky  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  studied the nature and operation of IT2FS, Zheng et al.  analyzed the similarity and acceptance of IT2FS, Zarinbal et al.  proposed a clustering analysis model of IT2FS with the method of relative entropy, Hwang et al.  proposed a similarity measurement method of interval type-2 fuzzy entropy, and Li et al.  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 , four-level , five-level , 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.  proposed IT2TFS MADM of controlling risk. Using IT2TFS decision problems, Ashtiani et al.  presented a feasible method to solve multiattribute group decision-making problems with the improved TOPSIS algorithm. Guo and Yin  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.  presented a conflict measurement model based on interval intuitionistic fuzzy number preference for large group decision-making problems. Zamri and Abdullah  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  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.
4.1.2. Measurement Method
According to the basic principle of risk preference measurement of decision makers and Figure 1, we propose the following risk preference measurement formula:)When ,(2) When ,(3)When ,
According to formula (7)–(9), the range of risk preference coefficient is . With increasing , the decision maker is inclined toward risk preference. The given calculation method of risk preference coefficient is based on the proportional relationship of the trapezoid area. The trapezoid area of the decision making is determined by subjective risk preference, in this sense that risk preference coefficient is computed subjectively.
4.2. Likelihood of Trapezoidal Fuzzy Number Preference Relations
The ranking method based on likelihood is one of the most widely used ranking methods for FMADM , and it has many excellent properties such as transmission and complementation. Particularly for interval type-2, Chen  proved through experimental studies that the ranking method based on likelihood is more convenient to calculate and more practical than other methods such as QUALIFLEX, ELECTRE, and TOPSIS. However, Chen did not consider the risk preferences of decision makers in his interval type-2 set ranking method based on likelihood. Considering that Chen’s study does not apply definition of likelihood, we provide the new definition of likelihood of trapezoidal fuzzy number preference relations.
Assuming two nonnegative trapezoidal fuzzy numbers and in , preference relation denotes that is better than or equal to . When preference relation holds, is defined as the likelihood between trapezoidal fuzzy numbers.
With respect to the two nonnegative trapezoidal fuzzy numbers and in , we assume that at least one of and ) holds. Then, the likelihood between trapezoidal fuzzy numbers is defined as follows:
The properties of likelihood are the following (Appendix B provides a demonstration):(1);(2);(3)If , ;(4).
5. Model Algorithm
We consider an MCDA problem of IT2TrF in which the ratings of the alternative evaluations and criterion importance are expressed as IT2TrF numbers. We define as a set of decision alternatives, where is the number of alternatives. We define as a criteria set that contains the criteria by which the alternative performances are measured and where is the number of criteria. The set can generally be divided into two sets, and , where denotes a collection of benefit criteria (i.e., a larger value indicates greater preference) and denotes a collection of cost criteria (i.e., a smaller value indicates greater preference).
Let a nonnegative IT2TrF number denote the evaluative rating of an alternative with respect to a criterion . Let and denote the lower and upper membership functions of , respectively, where is represented as follows:where , , , , , and (i.e., if and only if, , ).
Similarly, let a nonnegative IT2TrF number denote the importance weight of a criterion . Let and denote the lower and upper membership functions of , respectively. is expressed as follows:where , , , , , and (i.e., if and only if, , ).
Based on the approach of converting linguistic variables to IT2TrF numbers, where the importance weights of each attribute and the evaluative ratings of each alternative are converted to IT2TrF numbers, the following proposed algorithm is proposed to solve the problem of obtaining the different ranking results with different risk preference coefficient . Thus, decision makers can select risk preference coefficient according to the environment to rank the alternatives (see Table 1).
5.2. Proposed Algorithm
The proposed IT2TrF MADM considering risk preference is as follows.
(1) The Decision Matrix Is Initialized. An MCDA problem is formulated. The alternative set and the criterion set are specified. The decision matrix is established. denotes the evaluation value of alternative with respect to the criterion . Matrix is expressed as follows:
(2) The Risk Preference Coefficient Is Introduced. To differentiate the risk preferences of different decision makers, we introduce risk preference coefficient and translate IT2TrF into interval type-1 trapezoidal fuzzy number , shown in the following formula:where .
(3) Likelihood Computation Is Performed. Formula (9) is applied to compute the likelihoods for each criterion and each pair of alternatives and .
(4) The Likelihood-Based Performance Index Is Calculated. The likelihood represents the possibility of holding. If the evaluative rating has a high possibility of being greater than or equal to the evaluative rating , then the alternative results in improved performance for a benefit criterion compared to . By contrast, if the evaluative rating has a high possibility of being greater than or equal to the evaluative rating , the alternative results in improved performance for a benefit criterion compared to . is defined as follows:
(5) The Comprehensive Evaluation Value of Each Alternative Is Calculated. In MADM problems, different attributes are usually given different weights. Considering the impact of attribute weights on the basis of the evaluation , we introduce a comprehensive evaluation of defined as follows:where , , , , and , .
(6) Alternatives Are Ranked. Comprehensive evaluation is IT2TrF, so cannot be directly compared with each other. Therefore, to rank the comprehensive evaluation , we define the signed distance  as follows:
denotes the distance of comprehensive evaluation index to . For , the greater the value is, the better the alternative is than others.
6. Applications and Comparative Discussions
6.1. Application to Supplier Selection Problem
An aviation project needs to select one supplier from four candidate enterprises. The supplier charges the production of an aircraft wing-body fairing. Nine attributes can be based on facilities assurance (), technological compatibility (), degree of complementarity (), win-win nature (), management experience (), collaborative degree (), credibility (), prospects (), and innovation (); namely, the attribute set . Chen et al.  used a nine-point linguistic rating scale (i.e., absolutely poor, very poor, poor, medium poor, fair, medium good, good, very good, and absolutely good) to evaluate the options. Higher linguistic rating values indicate greater preference; thus, all of the criteria in can be considered as benefit criteria, and and . Based on the approach of converting linguistic variables to IT2TrF numbers, the importance weight of each and the evaluative ratings of each with respect to are established and presented in Tables 5 to 6.
In the present study, we introduce risk preference coefficient for different risk preferences. Using the proposed method for IT2TrFS MCDA, we obtain different ranking results of alternatives with different risk preference coefficients, which are presented in Table 2.
As shown in Table 2, when decision makers are inclined to risk aversion, namely, , the best treatment option is ; and when they are inclined to risk preference and neutral, namely, , the best treatment option is . In the case of the aviation project, the decision makers are risk averse and the company is on a steady growth path; thus, it needs to make sound decisions. The risk preference coefficient of decision makers is 0.2, so the best selection is program .
6.2. Application to Engineering Investment Project Evaluation Problem
To solve the engineering investment project evaluation problem, three decision makers use the following five attributes: total investment amount (), profit ratio of investment (), internal rate of return (), loan repayment rate (), and investment recovery period (), to evaluate the investment project. The set of all alternatives is denoted by . The set of evaluative attribute is denoted by , where and . We use the GITFNWGA operator  to combine the opinions of the three decision makers to obtain an average weighting matrix and an average decision matrix. Based on the two average matrices, the importance weight of each and the evaluative ratings of each with respect to are shown in Table 7.
Using the proposed method for IT2TrFS MCDA, we obtain different ranking results of alternatives with different risk preference coefficients, which are presented in Table 3.
According to this table, we can conclude that when a decision maker is absolutely risk averse, namely, , the best selection is program ; when , the best selection is program ; when the decision maker is risk neutral, namely, , the best selection is program ; when the decision maker is risk preferred, namely, , the best selection is program . According to the actual situation, the company is in its infancy and decision makers need to make risk decisions to promote the rapid growth of the company. Thus, decision makers tend toward risk preference, and the risk preference coefficient is 0.8, which means that the final option is .
6.3. Algorithm Comparison Analysis
As discussed, the supplier selection for the aviation project and the engineering investment project evaluation problem verify the validity and feasibility of the proposed method. In this section, we further illustrate the superiority of the proposed method through comparison analysis.
We have chosen several classical algorithms of the existing IT2FS MADM including an interval type-2 fuzzy PROMETHEE method using a likelihood-based outranking comparison approach . A linear assignment method for multicriteria decision analysis with IT2FS , the extended QUALIFLEX method for multicriteria decision analysis is based on IT2FS and applications to medical decision making . As shown in Table 4, in the aviation supplier selection problem, through the likelihood-based MCDA method, we can obtain the final ranking of suppliers as ; through the linear assignment method, we can obtain the final option as , but we cannot rank and ; through the extended QUALIFLEX method, we can obtain the final ranking of suppliers as . Given the preceding results, we can conclude that the likelihood-based MCDA method and the extended QUALIFLEX method are superior to the linear assignment method because these two programs can provide not only the best solutions but also the ranking of programs. However, neither of these two methods take into account the risk preferences; the results show only an extreme case of the proposed method. For example, the final ranking of the likelihood-based MCDA method is , which is the same as the results of the method proposed in this study when decision makers are inclined to risk aversion, namely, . The final ranking of the extended QUALIFLEX method is , which is the same as the results of our proposed method when decision makers are neutral or inclined to risk preference, namely, . In reality, according to risk appetite and enterprise development, the airline decision makers select the risk preference coefficient 0.2 and then the final option is . Therefore, the proposed method not only can obtain a complete sort of result, but also can fully reflect the flexibility of the decision process and the subjective views of decision makers.
For the engineering investment project evaluation problem, through the likelihood-based MCDA method, we can obtain the final ranking of the program as ; through the linear assignment method, we can obtain the final ranking of the program as ; through the extended QUALIFLEX method, we can obtain the final ranking of the program as . As the preceding findings show, all of the ranking results of the three methods are an extreme case of this proposed method. For example, the final ranking of the likelihood-based MCDA method is , which is the same as the result of our proposed method when ; the final ranking of the linear assignment method is , which is the same as the result of our proposed method when ; the final ranking of the extended QUALIFLEX method is , which is the same as the result of our proposed method when . According to the actual situation, the company is at the initial stage and needs bold decisions to facilitate its rapid growth, so the decision makers choose risk preference coefficient 0.8 and the final choice of investment program is . This example further confirms the advantages of the proposed method. When the comparison methods are used, one can go against the wishes of policymakers and make an inappropriate choice of programs.
Of course, because of the different research questions and initial program data, not every interval type-2 fuzzy decision problem presents the results of these two examples. Sometimes, different risk preference coefficients can obtain results similar to the decision-making results; sometimes, from risk aversion to risk neutrality and risk preference, one change is observed in the ranking results (Section 6.1); sometimes, with the change of , the ranking results also change many times (Section 6.2). However, overall, with the change of , different results have existing larger likelihood, so decision makers need to select the appropriate results based on actual risk preference.
The given two examples can fully reflect the advantages of the proposed method.
First, the proposed algorithm fully reflects the flexibility of the decision-making process, which means that decision makers can select different to obtain different decision results according to their own risk preference and the actual decision environment. Compared with current methods, the proposed algorithm is more reasonable.
Second, the proposed method can be used widely, which means that, for the actual decision problems, regardless of the risk preference of decision makers and the decision environment, all decision makers can obtain reasonable decision results based on the risk preference coefficient. However, the current methods can obtain decision results based on only one situation.
Finally, the proposed method is easy to implement on a computer and has a good potential to solve practical problems. At present, the method has been successfully applied to practical engineering problems with different risk preferences, such as supplier selection in aviation projects and engineering investment project evaluation.
IT2TFS have more advantages in describing uncertain information than the general fuzzy sets. IT2TrF can effectively describe qualitative indices [7, 24, 25, 33]. In this study, considering the background of the MADM problem, we conclude the following three research tasks.
(1) We propose the measurement method of risk preference for IT2TrF decision makers. Based on different attitudes of decision makers that have different risk preference toward the uncertainty range of IT2TrF, the method measures the risk preference coefficient of decision makers. This method applies to both IT2TFS and IT2FS.
(2) We propose the calculation method of preference likelihood of trapezoidal fuzzy numbers. This method is simple to calculate and easy to implement on a computer. Using the formula of likelihood, we establish the ranking method of preference relations of attributes, which applies not only to decision problems that have given attribute weights and IT2TrF but also to decision problems that have given attribute weights and exact value.
(3) From the subjective point of view, considering the different risk preferences of decision makers, we introduce the risk preference coefficient and propose a likelihood decision-making algorithm considering the risk preferences of decision makers. Through example analysis and algorithm comparison, the algorithm is proven reasonable.
In this study, the proposed algorithm is mainly applied to IT2TrF. However, the basic idea of this algorithm is also applicable to all IT2TFS. The proposed algorithm can be further extended to the interval Gaussian type-2 fuzzy set  and further examine the IT2TFS multiattribute group decision-making problems considering the risk preferences of decision makers.
A. The Initial Data of Application Research
B. Demonstration of the Properties of Likelihood
Appendix B provides a demonstration of the properties of likelihood .
Property B.1. If at least one of , , , and holds, we can obtainNamely, .
Property B.2. Given(1) if(2) ifwhen(3) if.
Property B.3. According to Property B.2,