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

Authors: Meng Zhao; Song-song Qin; Qi-wang Li; Fu-qiang Lu; Zhe Shen
Journal: Mathematical Problems in Engineering
 (2015)

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## 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.


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## Body

## 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 [0,1]. 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

LetX be a crisp set. Let Int⁡0,1 denote the set of all closed subintervals of 0,1. A mapping A: X→X0,1 is known as an IT2FS in X. For each x∈X, A(x)=A-x,A+x⊆0,1 represents the degree of membership of an element x to A. The type-1 fuzzy sets A-: X→[0,1] and A+: X→[0,1] are referred to as a lower fuzzy set of A and an upper fuzzy set of A, respectively. The values A-(x) and A+(x) represent the degrees of membership of x∈X to A- and A+, respectively, where 0≤A-(x)≤A+(x)≤1. If A-(x)=A+(x) for all x∈X, then the IT2FS A is fully determined by a single value from [0,1] for each x, and A reduces to a T1FS.Leta1-, a2-, a3-, a4-, a1+, a2+, a3+, and a4+ be nonnegative real values, where 0≤a1-≤a2-≤a3-≤a4-, 0≤a1+≤a2+≤a3+≤a4+, a1+≤a1-, and a4-≤a4+. Let hA- and hA+ denote the heights of A- and A+, respectively, where 0≤hA-≤hA+≤1. We consider that the lower and upper membership functions A-(x) and A+(x), respectively, of A are defined as follows:(1)A-x=hA-x-a1-a2--a1-a1-<x<a2-,hA-a2-≤x≤a3-,hA-a4--xa4--a3-a3-<x<a4-,0otherwise;A+x=hA+x-a1+a2+-a1+a1+<x<a2+,hA+a2+≤x≤a3+,hA+a4+-xa4+-a3+a3+<x<a4+,0otherwise.The lower and upper nonnegative trapezoidal fuzzy numbers with respect toA are denoted by A-=(a1-,a2-,a3-,a4-;hA-) and A+=(a1+,a2+,a3+,a4+;hA+), respectively, where A-⊆A+ (i.e., if and only if, ∀x∈X, A-(x)≤A+(x)). Additionally, A is a nonnegative IT2TrF number in X and is expressed as follows:(2)A=A-,A+=a1-,a2-,a3-,a4-;hA-,a1+,a2+,a3+,a4+;hA+.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 Aρ and Aβ be two nonnegative IT2TrF numbers in X, where (3)Aρ=a1ρ-,a2ρ-,a3ρ-,a4ρ-;hAρ-,a1ρ+,a2ρ+,a3ρ+,a4ρ+;hAρ+,Aβ=a1β-,a2β-,a3β-,a4β-;hAβ-,a1β+,a2β+,a3β+,a4β+;hAβ+.The addition operation between Aρ and Aβ is defined as follows:(4)Aρ⊕Aβ=a1ρ-+a1β-,a2ρ-+a2β-,a3ρ-+a3β-,a4ρ-+a4β-;min⁡hAρ-,hAβ-,a1ρ++a1β+,a2ρ++a2β+,a3ρ++a3β+,a4ρ++a4β+;min⁡hAρ+,hAβ+.The multiplication operation between a nonnegative real valueλ and Aρ is defined as follows: (5)λ·Aρ=λ·a1ρ-,λ·a2ρ-,λ·a3ρ-,λ·a4ρ-;hAρ-,λ·a1ρ+,λ·a2ρ+,λ·a3ρ+,λ·a4ρ+;hAρ+.


## 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 Sections4.1 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 Figure1, the lower limit and upper limit of IT2TrF A are A-=(a1-,a2-,a3-,a4-;hA-) and A+=(a1+,a2+,a3+,a4+;hA+), 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 A-, 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 A+, 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 A- and A+. Therefore, the value of risk preference of decision makers is also between 0 and 1. According to this principle, a risk preference coefficient θ of decision makers is introduced.Figure 1
Measurement of IT2TrF risk preferences.


#### 4.1.2. Measurement Method

According to the basic principle of risk preference measurement of decision makers and Figure1, we propose the following risk preference measurement formula:(1)
Whena1+<a2+<a1-<a2-<a3-<a4-<a3+<a4+,(6)θ=b4-b1+b3-b2hB-a4--a1-+a3--a2-hA-a4+-a1++a3+-a2+hA+-a4--a1-+a3--a2-hA-.(2)
When(a1+=a2+)∧(a1-=a2-)∧(a3-=a4-)<(a3+=a4+),(7)θ=0.(3)
Whena1+=a2+=a1-=a2-=a3-=a4-=a3+=a4+,(8)θ=0.According to formula (7)–(9), the range of risk preference coefficient is 0≤θ≤1. With increasing θ, 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 [30], and it has many excellent properties such as transmission and complementation. Particularly for interval type-2, Chen [25] 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 numbersAρ and Aβ in X, preference relation Aρ≥Aβ denotes that Aρ is better than or equal to Aβ. When preference relation Aρ≥Aβ holds, L(Aρ≥Aβ) is defined as the likelihood between trapezoidal fuzzy numbers.With respect to the two nonnegative trapezoidal fuzzy numbersAρ=(a1ρ,a2ρ,a3ρ,a4ρ;hAρ) and Aβ=(a1β,a2β,a3β,a4β;hAβ) in X, we assume that at least one of hAρ≠hAβ and aξρ-≠aξβ+(ξ=1,2,3,4) holds. Then, the likelihood L(Aρ≥Aβ) between trapezoidal fuzzy numbers is defined as follows:(9)LAρ≥Aβ=max⁡min⁡1-∑ξ=14max⁡aξβ-aξρ,0+a4β-a1ρ+2max⁡hAβ-hAρ,0∑ξ=14aξβ-aξρ+a4ρ-a1ρ+a4β-a1β+2hAβ-hAρ,1.The properties of likelihoodL(Aρ≥Aβ) are the following (Appendix B provides a demonstration):(1)
0≤L(Aρ≥Aβ)≤1;(2)
L(Aρ≥Aβ)+L(Aβ≥Aρ)=1;(3)
IfL(Aρ≥Aβ)=L(Aβ≥Aρ), L(Aρ≥Aβ)=L(Aβ≥Aρ)=0.5;(4)
L(Aρ≥Aρ)=0.5.


## 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 Figure1, the lower limit and upper limit of IT2TrF A are A-=(a1-,a2-,a3-,a4-;hA-) and A+=(a1+,a2+,a3+,a4+;hA+), 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 A-, 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 A+, 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 A- and A+. Therefore, the value of risk preference of decision makers is also between 0 and 1. According to this principle, a risk preference coefficient θ of decision makers is introduced.Figure 1
Measurement of IT2TrF risk preferences.


### 4.1.2. Measurement Method

According to the basic principle of risk preference measurement of decision makers and Figure1, we propose the following risk preference measurement formula:(1)
Whena1+<a2+<a1-<a2-<a3-<a4-<a3+<a4+,(6)θ=b4-b1+b3-b2hB-a4--a1-+a3--a2-hA-a4+-a1++a3+-a2+hA+-a4--a1-+a3--a2-hA-.(2)
When(a1+=a2+)∧(a1-=a2-)∧(a3-=a4-)<(a3+=a4+),(7)θ=0.(3)
Whena1+=a2+=a1-=a2-=a3-=a4-=a3+=a4+,(8)θ=0.According to formula (7)–(9), the range of risk preference coefficient is 0≤θ≤1. With increasing θ, 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.


## 4.1.1. Fundamental Principle

Each IT2TrF has a lower limit and upper limit simultaneously. For example, according to Figure1, the lower limit and upper limit of IT2TrF A are A-=(a1-,a2-,a3-,a4-;hA-) and A+=(a1+,a2+,a3+,a4+;hA+), 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 A-, 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 A+, 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 A- and A+. Therefore, the value of risk preference of decision makers is also between 0 and 1. According to this principle, a risk preference coefficient θ of decision makers is introduced.Figure 1
Measurement of IT2TrF risk preferences.


## 4.1.2. Measurement Method

According to the basic principle of risk preference measurement of decision makers and Figure1, we propose the following risk preference measurement formula:(1)
Whena1+<a2+<a1-<a2-<a3-<a4-<a3+<a4+,(6)θ=b4-b1+b3-b2hB-a4--a1-+a3--a2-hA-a4+-a1++a3+-a2+hA+-a4--a1-+a3--a2-hA-.(2)
When(a1+=a2+)∧(a1-=a2-)∧(a3-=a4-)<(a3+=a4+),(7)θ=0.(3)
Whena1+=a2+=a1-=a2-=a3-=a4-=a3+=a4+,(8)θ=0.According to formula (7)–(9), the range of risk preference coefficient is 0≤θ≤1. With increasing θ, 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 [30], and it has many excellent properties such as transmission and complementation. Particularly for interval type-2, Chen [25] 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 numbersAρ and Aβ in X, preference relation Aρ≥Aβ denotes that Aρ is better than or equal to Aβ. When preference relation Aρ≥Aβ holds, L(Aρ≥Aβ) is defined as the likelihood between trapezoidal fuzzy numbers.With respect to the two nonnegative trapezoidal fuzzy numbersAρ=(a1ρ,a2ρ,a3ρ,a4ρ;hAρ) and Aβ=(a1β,a2β,a3β,a4β;hAβ) in X, we assume that at least one of hAρ≠hAβ and aξρ-≠aξβ+(ξ=1,2,3,4) holds. Then, the likelihood L(Aρ≥Aβ) between trapezoidal fuzzy numbers is defined as follows:(9)LAρ≥Aβ=max⁡min⁡1-∑ξ=14max⁡aξβ-aξρ,0+a4β-a1ρ+2max⁡hAβ-hAρ,0∑ξ=14aξβ-aξρ+a4ρ-a1ρ+a4β-a1β+2hAβ-hAρ,1.The properties of likelihoodL(Aρ≥Aβ) are the following (Appendix B provides a demonstration):(1)
0≤L(Aρ≥Aβ)≤1;(2)
L(Aρ≥Aβ)+L(Aβ≥Aρ)=1;(3)
IfL(Aρ≥Aβ)=L(Aβ≥Aρ), L(Aρ≥Aβ)=L(Aβ≥Aρ)=0.5;(4)
L(Aρ≥Aρ)=0.5.


## 5. Model Algorithm

### 5.1. Problem

We consider an MCDA problem of IT2TrF in which the ratings of the alternative evaluations and criterion importance are expressed as IT2TrF numbers. We defineZ=z1,z2,…,zm as a set of decision alternatives, where m is the number of alternatives. We define C=c1,c2,…,cn as a criteria set that contains the criteria by which the alternative performances are measured and where n is the number of criteria. The set C can generally be divided into two sets, CI and CII(CI∩CII=∅,CI∪CII=C), where CI denotes a collection of benefit criteria (i.e., a larger value indicates greater preference) and CII denotes a collection of cost criteria (i.e., a smaller value indicates greater preference).Let a nonnegative IT2TrF numberAij denote the evaluative rating of an alternative zi∈Z with respect to a criterion cj∈C. Let Ai-(xj) and Ai+(xj) denote the lower and upper membership functions of Aij, respectively, where Aij is represented as follows:(10)Aij=Aij-,Aij+=a1ij-,a2ij-,a3ij-,a4ij-;hAj-,a1ij+,a2ij+,a3ij+,a4ij+;hAj+,where 0≤a1ij-≤a2ij-≤a3ij-≤a4ij-, 0≤a1ij+≤a2ij+≤a3ij+≤a4ij+, a1ij+≤a1ij-, a4ij-≤a4ij+, 0≤hAij-≤hAij+≤1, and Aij-⊆Aij+ (i.e., if and only if, ∀xj∈X, Ai-(xj)≤Ai+(xj)).Similarly, let a nonnegative IT2TrF numberWj denote the importance weight of a criterion cj∈C. Let W-(xj) and W+(xj) denote the lower and upper membership functions of Wj, respectively. Wj is expressed as follows:(11)Wj=Wj-,Wj+=W1j-,W2j-,W3j-,W4j-;hWj-,W1j+,W2j+,W3j+,W4j+;hWj+,where 0≤W1j-≤W2j-≤W3j-≤W4j-, 0≤W1j+≤W2j+≤W3j+≤W4j+, W1j+≤W1j-, W4j-≤W4j+, 0≤hWj-≤hWj+≤1, and Wj-⊆Wj+ (i.e., if and only if, ∀xj∈X, W-(xj)≤W+(xj)).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 coefficientθ. Thus, decision makers can select risk preference coefficient θ according to the environment to rank the alternatives (see Table 1).Table 1
Risk preference and corresponding risk preference coefficientθ.
Risk preference
Risk preference coefficientθ
Absolute risk aversion
0
Very strong risk aversion
0.1
Strong risk aversion
0.2
Medium risk aversion
0.3
Weak risk aversion
0.4
Risk neutral
0.5
Weak risk preference
0.6
Medium risk preference
0.7
Strong risk preference
0.8
Very strong risk preference
0.9
Absolute risk preference
1


### 5.2. Proposed Algorithm

The proposed IT2TrF MADM considering risk preference is as follows.(1) The Decision Matrix
A
Is Initialized. An MCDA problem is formulated. The alternative set Z=z1,z2,…,zm and the criterion set C=c1,c2,…,cn are specified. The decision matrix A is established. Aij denotes the evaluation value of alternative zi with respect to the criterion cj. Matrix A is expressed as follows:(12)A=A11A12⋯A1nA21A22⋯A2n⋮⋮⋮Am1Am2⋯Amn.(2) The Risk Preference Coefficient
θ
Is Introduced. To differentiate the risk preferences of different decision makers, we introduce risk preference coefficient θ and translate IT2TrF A=[A-,A+] into interval type-1 trapezoidal fuzzy number B=[b1,b2,b3,b4;hb], shown in the following formula:(13)B=θA-+1-θA+B=B11B12⋯B1nB21B22⋯B2n⋮⋮⋮Bm1Bm2⋯Bmn,where Bij=(b1ij,a2ij,a3ij,a4ij;hij).(3) Likelihood Computation
L
(
B
i
j
≥
B
i
′
j
)
Is Performed. Formula (9) is applied to compute the likelihoods L(Bij≥Bi′j) for each criterion cj∈C and each pair of alternatives zi∈Z and zi′∈Z(i′=1,2,…,m,i′≠i).(4) The Likelihood-Based Performance Index
P
(
B
i
j
)
Is Calculated. The likelihood L(Bij≥Bi′j) represents the possibility of Bij≥Bi′j holding. If the evaluative rating Bij has a high possibility of being greater than or equal to the evaluative rating Bi′j, then the alternative zi∈Z results in improved performance for a benefit criterion cj∈CI compared to zi′∉Z. By contrast, if the evaluative rating Bi′j has a high possibility of being greater than or equal to the evaluative rating Bij, the alternative zi′∉Z results in improved performance for a benefit criterion cj∈CII compared to zi∈Z. P(Bij) is defined as follows:(14)PBij=∑i′=1,i′≠imLBij≥Bi′j,cj∈CI,∑i′=1,i′≠imLBi′j≥Bij,cj∈CII.(5) The Comprehensive Evaluation Value of Each Alternative
E
i
Is Calculated. In MADM problems, different attributes are usually given different weights. Considering the impact of attribute weights on the basis of the evaluation P(Bij), we introduce a comprehensive evaluation Ei of zi∈Z defined as follows:(15)Ei=⨁j=1nPBij·Wj=∑j=1nPBij·W1j-,∑j=1nPBij·W2j-,∑j=1nPBij·W3j-,∑j=1nPBij·W4j-;minnj=1⁡hWj-⁡,∑j=1nPBij·W1j+,∑j=1nPBij·W2j+,∑j=1nPBij·W3j+,∑j=1nPBij·W4j+;minnj=1⁡hWj+Ei=Ei-,Ei+=e1i-,e2i-,e3i-,e4i-;hEi-,e1i+,e2i+,e3i+,e4i+;hEi+,where 0≤e1i-≤e2i-≤e3i-≤e4i-, 0≤e1i+≤e2i+≤e3i+≤e4i+, e1i+≤e1i-, e4i-≤e4i+, and 0≤hEi-≤hEi+≤1, Ei-⊆Ei+.(6) Alternatives Are Ranked. Comprehensive evaluation Ei is IT2TrF, so Ei cannot be directly compared with each other. Therefore, to rank the comprehensive evaluation Ei(i=1,2,…,m), we define the signed distance [24] εi as follows:(16)εi=18e1i-+e2i-+e3i-+e4i-+4·e1i++2e2i++2e3i++4e4i++3e2i++e3i+-e1i+-e4i+hEi-hEi+,ε
i denotes the distance of comprehensive evaluation index Ei to [(0000;1),(0000;1)]. For zi∈Z, the greater the value εi is, the better the alternative is than others.


## 5.1. Problem

We consider an MCDA problem of IT2TrF in which the ratings of the alternative evaluations and criterion importance are expressed as IT2TrF numbers. We defineZ=z1,z2,…,zm as a set of decision alternatives, where m is the number of alternatives. We define C=c1,c2,…,cn as a criteria set that contains the criteria by which the alternative performances are measured and where n is the number of criteria. The set C can generally be divided into two sets, CI and CII(CI∩CII=∅,CI∪CII=C), where CI denotes a collection of benefit criteria (i.e., a larger value indicates greater preference) and CII denotes a collection of cost criteria (i.e., a smaller value indicates greater preference).Let a nonnegative IT2TrF numberAij denote the evaluative rating of an alternative zi∈Z with respect to a criterion cj∈C. Let Ai-(xj) and Ai+(xj) denote the lower and upper membership functions of Aij, respectively, where Aij is represented as follows:(10)Aij=Aij-,Aij+=a1ij-,a2ij-,a3ij-,a4ij-;hAj-,a1ij+,a2ij+,a3ij+,a4ij+;hAj+,where 0≤a1ij-≤a2ij-≤a3ij-≤a4ij-, 0≤a1ij+≤a2ij+≤a3ij+≤a4ij+, a1ij+≤a1ij-, a4ij-≤a4ij+, 0≤hAij-≤hAij+≤1, and Aij-⊆Aij+ (i.e., if and only if, ∀xj∈X, Ai-(xj)≤Ai+(xj)).Similarly, let a nonnegative IT2TrF numberWj denote the importance weight of a criterion cj∈C. Let W-(xj) and W+(xj) denote the lower and upper membership functions of Wj, respectively. Wj is expressed as follows:(11)Wj=Wj-,Wj+=W1j-,W2j-,W3j-,W4j-;hWj-,W1j+,W2j+,W3j+,W4j+;hWj+,where 0≤W1j-≤W2j-≤W3j-≤W4j-, 0≤W1j+≤W2j+≤W3j+≤W4j+, W1j+≤W1j-, W4j-≤W4j+, 0≤hWj-≤hWj+≤1, and Wj-⊆Wj+ (i.e., if and only if, ∀xj∈X, W-(xj)≤W+(xj)).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 coefficientθ. Thus, decision makers can select risk preference coefficient θ according to the environment to rank the alternatives (see Table 1).Table 1
Risk preference and corresponding risk preference coefficientθ.
Risk preference
Risk preference coefficientθ
Absolute risk aversion
0
Very strong risk aversion
0.1
Strong risk aversion
0.2
Medium risk aversion
0.3
Weak risk aversion
0.4
Risk neutral
0.5
Weak risk preference
0.6
Medium risk preference
0.7
Strong risk preference
0.8
Very strong risk preference
0.9
Absolute risk preference
1


## 5.2. Proposed Algorithm

The proposed IT2TrF MADM considering risk preference is as follows.(1) The Decision Matrix
A
Is Initialized. An MCDA problem is formulated. The alternative set Z=z1,z2,…,zm and the criterion set C=c1,c2,…,cn are specified. The decision matrix A is established. Aij denotes the evaluation value of alternative zi with respect to the criterion cj. Matrix A is expressed as follows:(12)A=A11A12⋯A1nA21A22⋯A2n⋮⋮⋮Am1Am2⋯Amn.(2) The Risk Preference Coefficient
θ
Is Introduced. To differentiate the risk preferences of different decision makers, we introduce risk preference coefficient θ and translate IT2TrF A=[A-,A+] into interval type-1 trapezoidal fuzzy number B=[b1,b2,b3,b4;hb], shown in the following formula:(13)B=θA-+1-θA+B=B11B12⋯B1nB21B22⋯B2n⋮⋮⋮Bm1Bm2⋯Bmn,where Bij=(b1ij,a2ij,a3ij,a4ij;hij).(3) Likelihood Computation
L
(
B
i
j
≥
B
i
′
j
)
Is Performed. Formula (9) is applied to compute the likelihoods L(Bij≥Bi′j) for each criterion cj∈C and each pair of alternatives zi∈Z and zi′∈Z(i′=1,2,…,m,i′≠i).(4) The Likelihood-Based Performance Index
P
(
B
i
j
)
Is Calculated. The likelihood L(Bij≥Bi′j) represents the possibility of Bij≥Bi′j holding. If the evaluative rating Bij has a high possibility of being greater than or equal to the evaluative rating Bi′j, then the alternative zi∈Z results in improved performance for a benefit criterion cj∈CI compared to zi′∉Z. By contrast, if the evaluative rating Bi′j has a high possibility of being greater than or equal to the evaluative rating Bij, the alternative zi′∉Z results in improved performance for a benefit criterion cj∈CII compared to zi∈Z. P(Bij) is defined as follows:(14)PBij=∑i′=1,i′≠imLBij≥Bi′j,cj∈CI,∑i′=1,i′≠imLBi′j≥Bij,cj∈CII.(5) The Comprehensive Evaluation Value of Each Alternative
E
i
Is Calculated. In MADM problems, different attributes are usually given different weights. Considering the impact of attribute weights on the basis of the evaluation P(Bij), we introduce a comprehensive evaluation Ei of zi∈Z defined as follows:(15)Ei=⨁j=1nPBij·Wj=∑j=1nPBij·W1j-,∑j=1nPBij·W2j-,∑j=1nPBij·W3j-,∑j=1nPBij·W4j-;minnj=1⁡hWj-⁡,∑j=1nPBij·W1j+,∑j=1nPBij·W2j+,∑j=1nPBij·W3j+,∑j=1nPBij·W4j+;minnj=1⁡hWj+Ei=Ei-,Ei+=e1i-,e2i-,e3i-,e4i-;hEi-,e1i+,e2i+,e3i+,e4i+;hEi+,where 0≤e1i-≤e2i-≤e3i-≤e4i-, 0≤e1i+≤e2i+≤e3i+≤e4i+, e1i+≤e1i-, e4i-≤e4i+, and 0≤hEi-≤hEi+≤1, Ei-⊆Ei+.(6) Alternatives Are Ranked. Comprehensive evaluation Ei is IT2TrF, so Ei cannot be directly compared with each other. Therefore, to rank the comprehensive evaluation Ei(i=1,2,…,m), we define the signed distance [24] εi as follows:(16)εi=18e1i-+e2i-+e3i-+e4i-+4·e1i++2e2i++2e3i++4e4i++3e2i++e3i+-e1i+-e4i+hEi-hEi+,ε
i denotes the distance of comprehensive evaluation index Ei to [(0000;1),(0000;1)]. For zi∈Z, the greater the value εi 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 (c1), technological compatibility (c2), degree of complementarity (c3), win-win nature (c4), management experience (c5), collaborative degree (c6), credibility (c7), prospects (c8), and innovation (c9); namely, the attribute set C=(c1,c2,c3,…,c9). Chen et al. [7] 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 C can be considered as benefit criteria, and CI=C and CII=ϕ. Based on the approach of converting linguistic variables to IT2TrF numbers, the importance weight ωj of each cj and the evaluative ratings of each zi with respect to cj are established and presented in Tables 5 to 6.In the present study, we introduce risk preference coefficientθ 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.Table 2
The ranking results with differentθ.
ε
1
ε
2
ε
3
Ranking results
θ = 0
11.2168
11.1865
10.5695
z
1 ≻z2 ≻z3
θ = 0.1
11.2056
11.1770
10.5903
z
1
≻
z
2
≻
z
3
θ = 0.2
11.1935
11.1687
10.6107
z
1 ≻z2 ≻z3
θ = 0.3
11.1805
11.1617
10.6307
z
1 ≻z2 ≻z3
θ = 0.4
11.1666
11.1560
10.6502
z
1 ≻z2 ≻z3
θ = 0.5
11.1517
11.1519
10.6693
z
2 ≻z1 ≻z3
θ = 0.6
11.1374
11.1477
10.6878
z
2 ≻z1 ≻z3
θ = 0.7
11.1261
11.1387
10.7082
z
2 ≻z1 ≻z3
θ = 0.8
11.1394
11.0926
10.7409
z
2 ≻z1 ≻z3
θ = 0.9
11.1641
11.0679
10.7409
z
2 ≻z1 ≻z3
θ = 1.0
11.1891
11.0428
10.7409
z
2 ≻z1 ≻z3As shown in Table2, when decision makers are inclined to risk aversion, namely, θ∈[0,0.4], the best treatment option is Z1; and when they are inclined to risk preference and neutral, namely, θ∈[0.5,1.0], the best treatment option is Z2. 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 Z1.


### 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 (c1), profit ratio of investment (c2), internal rate of return (c3), loan repayment rate (c4), and investment recovery period (c5), to evaluate the investment project. The set of all alternatives is denoted by z=(z1,z2,z3,z4). The set of evaluative attribute is denoted by C=(c1,c2,c3,c4,c5), where CI=(c1,c2,c3) and CII=(c4,c5). We use the GITFNWGA operator [32] 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 ωj of each cj and the evaluative ratings of each zi with respect to cj 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 Table3.Table 3
The ranking results with differentθ.
ε
1
ε
2
ε
3
ε
4
Ranking results
θ = 0
12.0645
12.7036
12.7192
12.5482
z
3 ≻z2 ≻z4 ≻z1
θ = 0.1
12.1504
12.6811
12.6274
12.5768
z
2 ≻z3 ≻z4 ≻z1
θ = 0.2
12.2387
12.6580
12.5329
12.6059
z
2 ≻z4 ≻z3 ≻z1
θ = 0.3
12.3235
12.6342
12.4818
12.5961
z
2 ≻z4 ≻z3 ≻z1
θ = 0.4
12.4058
12.6095
12.4348
12.5853
z
2 ≻z4 ≻z3 ≻z1
θ = 0.5
12.5129
12.5636
12.3874
12.5716
z
4 ≻z2 ≻z1 ≻z3
θ = 0.6
12.6462
12.4867
12.3483
12.5544
z
1 ≻z4 ≻z2 ≻z3
θ = 0.7
12.8194
12.3218
12.3736
12.5208
z
1 ≻z4 ≻z3 ≻z2
θ = 0.8
12.9711
12.1699
12.4113
12.4831
z
1 ≻z4 ≻z3 ≻z2
θ = 0.9
13.1231
12.0569
12.4122
12.4432
z
1 ≻z4 ≻z3 ≻z2
θ = 1.0
13.2799
11.9645
12.3986
12.3925
z
1 ≻z3 ≻z4 ≻z2According to this table, we can conclude that when a decision maker is absolutely risk averse, namely,θ=0, the best selection is program z3; when θ∈0.1,0.4, the best selection is program z2; when the decision maker is risk neutral, namely, θ=0.5, the best selection is program z4; when the decision maker is risk preferred, namely, θ∈0.6,1, the best selection is program z1. 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 z1.


### 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 [25]. A linear assignment method for multicriteria decision analysis with IT2FS [33], the extended QUALIFLEX method for multicriteria decision analysis is based on IT2FS and applications to medical decision making [7]. 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 z1≻z2≻z3; through the linear assignment method, we can obtain the final option as z2, but we cannot rank z1 and z3; through the extended QUALIFLEX method, we can obtain the final ranking of suppliers as z2≻z1≻z3. 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 z1≻z2≻z3, which is the same as the results of the method proposed in this study when decision makers are inclined to risk aversion, namely, θ∈[0,0.4]. The final ranking of the extended QUALIFLEX method is z2≻z1≻z3, which is the same as the results of our proposed method when decision makers are neutral or inclined to risk preference, namely, θ∈[0.5,1.0]. 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 z1. 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.Table 4
Comparison analysis of the obtained results.
Research source
Comparative methods
Ranking results
The supplier selection problem
The likelihood-based MCDA method [25]
z
1 ≻z2 ≻z3
The linear assignment method [33]
z
2 ≻z3, z2 ≻z1
The extended QUALIFLEX method [7]
z
2 ≻z1 ≻z3
The engineering investment project evaluation problems
The likelihood-based MCDA method [25]
z
4 ≻z2 ≻z1 ≻z3
The linear assignment method [33]
z
2 ≻z4 ≻z3 ≻z1
The extended QUALIFLEX method [7]
z
2 ≻z4 ≻z3 ≻z1Table 5
The corresponding weightsWj of criterion cj.
Corresponding weightWj of attribute cj
c
1
[(1.0000 1.0000 1.0000 1.0000; 0.8),(1.0000 1.0000 1.0000 1.0000; 1.0)];
c
2
[(0.0075 0.0075 0.0150 0.0525; 0.8),(0.0000 0.0000 0.0200 0.0700; 1.0)];
c
3
[(0.0875 0.1200 0.1600 0.1825; 0.8),(0.0400 0.1000 0.1800 0.2300; 1.0)];
c
4
[(1.0000 1.0000 1.0000 1.0000; 0.8),(1.0000 1.0000 1.0000 1.0000; 1.0)];
c
5
[(0.2325 0.2550 0.3250 0.3575; 0.8),(0.1700 0.2200 0.3600 0.4200; 1.0)];
c
6
[(0.4025 0.4525 0.5375 0.5675; 0.8),(0.3200 0.4100 0.5800 0.6500; 1.0)];
c
7
[(0.9475 0.9850 0.9925 0.9925; 0.8),(0.9300 0.9800 1.0000 1.0000; 1.0)];
c
8
[(0.6500 0.6725 0.7575 0.7900; 0.8),(0.5800 0.6300 0.8000 0.8600; 1.0)];
c
9
[(0.7825 0.8150 0.8850 0.9075; 0.8),(0.7200 0.7800 0.9200 0.9700; 1.0)]Table 6
The evaluative rating of the alternativezi with respect to the criterion cj.
Evaluation value of alternativez1 with respect to cj∈C
c
1
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
2
[(0.6500 0.6725 0.7575 0.7900; 0.8), (0.5800 0.6300 0.8000 0.8600; 1.0)];
c
3
[(0.2325 0.2550 0.3250 0.3575; 0.8), (0.1700 0.2200 0.3600 0.4200; 1.0)];
c
4
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
5
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
6
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
7
[(0.4025 0.4525 0.5375 0.5675; 0.8), (0.3200 0.4100 0.5800 0.6500; 1.0)];
c
8
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
9
[(0.2325 0.2550 0.3250 0.3575; 0.8), (0.1700 0.2200 0.3600 0.4200; 1.0)];
Evaluation value of alternativez2 with respect to cj∈C
c
1
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
2
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
3
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
4
[(0.0075 0.0075 0.0150 0.0525; 0.8), (0.0000 0.0000 0.0200 0.0700; 1.0)];
c
5
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
6
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
7
[(0.2325 0.2550 0.3250 0.3575; 0.8), (0.1700 0.2200 0.3600 0.4200; 1.0)];
c
8
[(1.0000 1.0000 1.0000 1.0000; 0.8), (1.0000 1.0000 1.0000 1.0000; 1.0)];
c
9
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
Evaluation value of alternativez3 with respect to cj∈C
c
1
[(0.4025 0.4525 0.5375 0.5675; 0.8), (0.3200 0.4100 0.5800 0.6500; 1.0)];
c
2
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
3
[(0.0000 0.0000 0.0000 0.0000; 0.8), (0.0000 0.0000 0.0000 0.0000; 1.0)];
c
4
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
5
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
6
[(0.6500 0.6725 0.7575 0.7900; 0.8), (0.5800 0.6300 0.8000 0.8600; 1.0)];
c
7
[(0.4025 0.4525 0.5375 0.5675; 0.8), (0.3200 0.4100 0.5800 0.6500; 1.0)];
c
8
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
9
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];Table 7
The collective IT2TrF data using the GITFNWGA operator in the engineering investment project evaluation problem.
The importance weights of eachωj of each cj
C1
[(0.90 0.97 0.97 0.98; 0.8), (0.83 0.97 0.97 1.00; 1.0)];
C2
[(0.90 0.97 0.97 0.98; 0.8), (0.83 0.97 0.97 1.00; 1.0)];
C3
[(0.53 0.63 0.63 0.73; 0.8), (0.43 0.63 0.63 0.83; 1.0)];
C4
[(0.85 0.93 0.93 0.97; 0.8), (0.77 0.93 0.93 1.00; 1.0)];
C5
[(0.68 0.70 0.75 0.80; 0.8), (0.56 0.70 0.78 0.90; 1.0)];
The evaluative ratings of eachzi with respect to cj
z
1
C1
[(0.73 0.78 0.90 0.94; 0.8), (0.73 0.78 0.90 0.94; 1.0)];
C2
[(0.56 0.63 0.77 0.82; 0.8), (0.56 0.63 0.77 0.82; 1.0)];
C3
[(0.78 0.84 0.95 0.98; 0.8), (0.78 0.84 0.95 0.98; 1.0)];
C4
[(0.88 0.94 0.99 0.99; 0.8), (0.85 0.91 0.97 1.00; 1.0)];
C5
[(0.42 0.50 0.60 0.75; 0.8), (0.40 0.48 0.50 0.75; 1.0)];
z
2
C1
[(0.67 0.73 0.88 0.93; 0.8), (0.67 0.73 0.88 0.93; 1.0)];
C2
[(0.55 0.75 0.93 0.95; 0.8), (0.76 0.80 0.93 0.95; 1.0)];
C3
[(0.85 0.91 0.97 0.99; 0.8), (0.85 0.91 0.97 0.99; 1.0)];
C4
[(0.79 0.85 0.93 0.95; 0.8), (0.79 0.85 0.93 0.95; 1.0)];
C5
[(0.32 0.48 0.84 0.90; 0.8), (0.22 0.60 0.84 0.90; 1.0)];
z
3
C1
[(0.85 0.91 0.97 0.99; 0.8), (0.85 0.91 0.97 0.99; 1.0)];
C2
[(0.56 0.63 0.77 0.82; 0.8), (0.56 0.65 0.77 0.82; 1.0)];
C3
[(0.42 0.55 0.70 0.80; 0.8), (0.53 0.60 0.78 0.90; 1.0)];
C4
[(0.79 0.85 0.93 0.95; 0.8), (0.79 0.85 0.93 0.95; 1.0)];
C5
[(0.77 0.82 0.88 1.00; 0.8), (0.84 0.96 0.98 1.0; 1.0)];
z
4
C1
[(0.73 0.78 0.90 0.94; 0.8), (0.73 0.78 0.90 0.94; 1.0)];
C2
[(0.46 0.55 0.70 0.75; 0.8), (0.46 0.55 0.70 0.75; 1.0)];
C3
[(0.70 0.73 0.88 0.98; 0.8), (0.67 0.73 0.88 0.94; 1.0)];
C4
[(0.85 0.91 0.97 0.99; 0.8), (0.85 0.91 0.97 0.99; 1.0)];
C5
[(0.93 0.98 1.00 1.00; 0.8), (0.93 0.98 1.00 1.00; 1.0)];For the engineering investment project evaluation problem, through the likelihood-based MCDA method, we can obtain the final ranking of the program asz4≻z2≻z1≻z3; through the linear assignment method, we can obtain the final ranking of the program as z2≻z4≻z3≻z1; through the extended QUALIFLEX method, we can obtain the final ranking of the program as z2≻z4≻z3≻z1. 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 z4≻z2≻z1≻z3, which is the same as the result of our proposed method when θ=0.5; the final ranking of the linear assignment method is z2≻z4≻z3≻z1, which is the same as the result of our proposed method when θ∈0.2,0.4; the final ranking of the extended QUALIFLEX method is z2≻z4≻z3≻z1, which is the same as the result of our proposed method when θ∈0.2,0.4. 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 z1. 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 (Section6.1); sometimes, with the change of θ, the ranking results also change many times (Section 6.2). However, overall, with the change of θ, 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 differentθ 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.


## 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 (c1), technological compatibility (c2), degree of complementarity (c3), win-win nature (c4), management experience (c5), collaborative degree (c6), credibility (c7), prospects (c8), and innovation (c9); namely, the attribute set C=(c1,c2,c3,…,c9). Chen et al. [7] 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 C can be considered as benefit criteria, and CI=C and CII=ϕ. Based on the approach of converting linguistic variables to IT2TrF numbers, the importance weight ωj of each cj and the evaluative ratings of each zi with respect to cj are established and presented in Tables 5 to 6.In the present study, we introduce risk preference coefficientθ 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.Table 2
The ranking results with differentθ.
ε
1
ε
2
ε
3
Ranking results
θ = 0
11.2168
11.1865
10.5695
z
1 ≻z2 ≻z3
θ = 0.1
11.2056
11.1770
10.5903
z
1
≻
z
2
≻
z
3
θ = 0.2
11.1935
11.1687
10.6107
z
1 ≻z2 ≻z3
θ = 0.3
11.1805
11.1617
10.6307
z
1 ≻z2 ≻z3
θ = 0.4
11.1666
11.1560
10.6502
z
1 ≻z2 ≻z3
θ = 0.5
11.1517
11.1519
10.6693
z
2 ≻z1 ≻z3
θ = 0.6
11.1374
11.1477
10.6878
z
2 ≻z1 ≻z3
θ = 0.7
11.1261
11.1387
10.7082
z
2 ≻z1 ≻z3
θ = 0.8
11.1394
11.0926
10.7409
z
2 ≻z1 ≻z3
θ = 0.9
11.1641
11.0679
10.7409
z
2 ≻z1 ≻z3
θ = 1.0
11.1891
11.0428
10.7409
z
2 ≻z1 ≻z3As shown in Table2, when decision makers are inclined to risk aversion, namely, θ∈[0,0.4], the best treatment option is Z1; and when they are inclined to risk preference and neutral, namely, θ∈[0.5,1.0], the best treatment option is Z2. 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 Z1.


## 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 (c1), profit ratio of investment (c2), internal rate of return (c3), loan repayment rate (c4), and investment recovery period (c5), to evaluate the investment project. The set of all alternatives is denoted by z=(z1,z2,z3,z4). The set of evaluative attribute is denoted by C=(c1,c2,c3,c4,c5), where CI=(c1,c2,c3) and CII=(c4,c5). We use the GITFNWGA operator [32] 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 ωj of each cj and the evaluative ratings of each zi with respect to cj 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 Table3.Table 3
The ranking results with differentθ.
ε
1
ε
2
ε
3
ε
4
Ranking results
θ = 0
12.0645
12.7036
12.7192
12.5482
z
3 ≻z2 ≻z4 ≻z1
θ = 0.1
12.1504
12.6811
12.6274
12.5768
z
2 ≻z3 ≻z4 ≻z1
θ = 0.2
12.2387
12.6580
12.5329
12.6059
z
2 ≻z4 ≻z3 ≻z1
θ = 0.3
12.3235
12.6342
12.4818
12.5961
z
2 ≻z4 ≻z3 ≻z1
θ = 0.4
12.4058
12.6095
12.4348
12.5853
z
2 ≻z4 ≻z3 ≻z1
θ = 0.5
12.5129
12.5636
12.3874
12.5716
z
4 ≻z2 ≻z1 ≻z3
θ = 0.6
12.6462
12.4867
12.3483
12.5544
z
1 ≻z4 ≻z2 ≻z3
θ = 0.7
12.8194
12.3218
12.3736
12.5208
z
1 ≻z4 ≻z3 ≻z2
θ = 0.8
12.9711
12.1699
12.4113
12.4831
z
1 ≻z4 ≻z3 ≻z2
θ = 0.9
13.1231
12.0569
12.4122
12.4432
z
1 ≻z4 ≻z3 ≻z2
θ = 1.0
13.2799
11.9645
12.3986
12.3925
z
1 ≻z3 ≻z4 ≻z2According to this table, we can conclude that when a decision maker is absolutely risk averse, namely,θ=0, the best selection is program z3; when θ∈0.1,0.4, the best selection is program z2; when the decision maker is risk neutral, namely, θ=0.5, the best selection is program z4; when the decision maker is risk preferred, namely, θ∈0.6,1, the best selection is program z1. 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 z1.


## 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 [25]. A linear assignment method for multicriteria decision analysis with IT2FS [33], the extended QUALIFLEX method for multicriteria decision analysis is based on IT2FS and applications to medical decision making [7]. 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 z1≻z2≻z3; through the linear assignment method, we can obtain the final option as z2, but we cannot rank z1 and z3; through the extended QUALIFLEX method, we can obtain the final ranking of suppliers as z2≻z1≻z3. 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 z1≻z2≻z3, which is the same as the results of the method proposed in this study when decision makers are inclined to risk aversion, namely, θ∈[0,0.4]. The final ranking of the extended QUALIFLEX method is z2≻z1≻z3, which is the same as the results of our proposed method when decision makers are neutral or inclined to risk preference, namely, θ∈[0.5,1.0]. 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 z1. 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.Table 4
Comparison analysis of the obtained results.
Research source
Comparative methods
Ranking results
The supplier selection problem
The likelihood-based MCDA method [25]
z
1 ≻z2 ≻z3
The linear assignment method [33]
z
2 ≻z3, z2 ≻z1
The extended QUALIFLEX method [7]
z
2 ≻z1 ≻z3
The engineering investment project evaluation problems
The likelihood-based MCDA method [25]
z
4 ≻z2 ≻z1 ≻z3
The linear assignment method [33]
z
2 ≻z4 ≻z3 ≻z1
The extended QUALIFLEX method [7]
z
2 ≻z4 ≻z3 ≻z1Table 5
The corresponding weightsWj of criterion cj.
Corresponding weightWj of attribute cj
c
1
[(1.0000 1.0000 1.0000 1.0000; 0.8),(1.0000 1.0000 1.0000 1.0000; 1.0)];
c
2
[(0.0075 0.0075 0.0150 0.0525; 0.8),(0.0000 0.0000 0.0200 0.0700; 1.0)];
c
3
[(0.0875 0.1200 0.1600 0.1825; 0.8),(0.0400 0.1000 0.1800 0.2300; 1.0)];
c
4
[(1.0000 1.0000 1.0000 1.0000; 0.8),(1.0000 1.0000 1.0000 1.0000; 1.0)];
c
5
[(0.2325 0.2550 0.3250 0.3575; 0.8),(0.1700 0.2200 0.3600 0.4200; 1.0)];
c
6
[(0.4025 0.4525 0.5375 0.5675; 0.8),(0.3200 0.4100 0.5800 0.6500; 1.0)];
c
7
[(0.9475 0.9850 0.9925 0.9925; 0.8),(0.9300 0.9800 1.0000 1.0000; 1.0)];
c
8
[(0.6500 0.6725 0.7575 0.7900; 0.8),(0.5800 0.6300 0.8000 0.8600; 1.0)];
c
9
[(0.7825 0.8150 0.8850 0.9075; 0.8),(0.7200 0.7800 0.9200 0.9700; 1.0)]Table 6
The evaluative rating of the alternativezi with respect to the criterion cj.
Evaluation value of alternativez1 with respect to cj∈C
c
1
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
2
[(0.6500 0.6725 0.7575 0.7900; 0.8), (0.5800 0.6300 0.8000 0.8600; 1.0)];
c
3
[(0.2325 0.2550 0.3250 0.3575; 0.8), (0.1700 0.2200 0.3600 0.4200; 1.0)];
c
4
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
5
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
6
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
7
[(0.4025 0.4525 0.5375 0.5675; 0.8), (0.3200 0.4100 0.5800 0.6500; 1.0)];
c
8
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
9
[(0.2325 0.2550 0.3250 0.3575; 0.8), (0.1700 0.2200 0.3600 0.4200; 1.0)];
Evaluation value of alternativez2 with respect to cj∈C
c
1
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
2
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
3
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
4
[(0.0075 0.0075 0.0150 0.0525; 0.8), (0.0000 0.0000 0.0200 0.0700; 1.0)];
c
5
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
6
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
7
[(0.2325 0.2550 0.3250 0.3575; 0.8), (0.1700 0.2200 0.3600 0.4200; 1.0)];
c
8
[(1.0000 1.0000 1.0000 1.0000; 0.8), (1.0000 1.0000 1.0000 1.0000; 1.0)];
c
9
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
Evaluation value of alternativez3 with respect to cj∈C
c
1
[(0.4025 0.4525 0.5375 0.5675; 0.8), (0.3200 0.4100 0.5800 0.6500; 1.0)];
c
2
[(0.0875 0.1200 0.1600 0.1825; 0.8), (0.0400 0.1000 0.1800 0.2300; 1.0)];
c
3
[(0.0000 0.0000 0.0000 0.0000; 0.8), (0.0000 0.0000 0.0000 0.0000; 1.0)];
c
4
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];
c
5
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
6
[(0.6500 0.6725 0.7575 0.7900; 0.8), (0.5800 0.6300 0.8000 0.8600; 1.0)];
c
7
[(0.4025 0.4525 0.5375 0.5675; 0.8), (0.3200 0.4100 0.5800 0.6500; 1.0)];
c
8
[(0.9475 0.9850 0.9925 0.9925; 0.8), (0.8300 0.8800 1.0000 1.0000; 1.0)];
c
9
[(0.7825 0.8150 0.8850 0.9075; 0.8), (0.7200 0.7800 0.9200 0.9700; 1.0)];Table 7
The collective IT2TrF data using the GITFNWGA operator in the engineering investment project evaluation problem.
The importance weights of eachωj of each cj
C1
[(0.90 0.97 0.97 0.98; 0.8), (0.83 0.97 0.97 1.00; 1.0)];
C2
[(0.90 0.97 0.97 0.98; 0.8), (0.83 0.97 0.97 1.00; 1.0)];
C3
[(0.53 0.63 0.63 0.73; 0.8), (0.43 0.63 0.63 0.83; 1.0)];
C4
[(0.85 0.93 0.93 0.97; 0.8), (0.77 0.93 0.93 1.00; 1.0)];
C5
[(0.68 0.70 0.75 0.80; 0.8), (0.56 0.70 0.78 0.90; 1.0)];
The evaluative ratings of eachzi with respect to cj
z
1
C1
[(0.73 0.78 0.90 0.94; 0.8), (0.73 0.78 0.90 0.94; 1.0)];
C2
[(0.56 0.63 0.77 0.82; 0.8), (0.56 0.63 0.77 0.82; 1.0)];
C3
[(0.78 0.84 0.95 0.98; 0.8), (0.78 0.84 0.95 0.98; 1.0)];
C4
[(0.88 0.94 0.99 0.99; 0.8), (0.85 0.91 0.97 1.00; 1.0)];
C5
[(0.42 0.50 0.60 0.75; 0.8), (0.40 0.48 0.50 0.75; 1.0)];
z
2
C1
[(0.67 0.73 0.88 0.93; 0.8), (0.67 0.73 0.88 0.93; 1.0)];
C2
[(0.55 0.75 0.93 0.95; 0.8), (0.76 0.80 0.93 0.95; 1.0)];
C3
[(0.85 0.91 0.97 0.99; 0.8), (0.85 0.91 0.97 0.99; 1.0)];
C4
[(0.79 0.85 0.93 0.95; 0.8), (0.79 0.85 0.93 0.95; 1.0)];
C5
[(0.32 0.48 0.84 0.90; 0.8), (0.22 0.60 0.84 0.90; 1.0)];
z
3
C1
[(0.85 0.91 0.97 0.99; 0.8), (0.85 0.91 0.97 0.99; 1.0)];
C2
[(0.56 0.63 0.77 0.82; 0.8), (0.56 0.65 0.77 0.82; 1.0)];
C3
[(0.42 0.55 0.70 0.80; 0.8), (0.53 0.60 0.78 0.90; 1.0)];
C4
[(0.79 0.85 0.93 0.95; 0.8), (0.79 0.85 0.93 0.95; 1.0)];
C5
[(0.77 0.82 0.88 1.00; 0.8), (0.84 0.96 0.98 1.0; 1.0)];
z
4
C1
[(0.73 0.78 0.90 0.94; 0.8), (0.73 0.78 0.90 0.94; 1.0)];
C2
[(0.46 0.55 0.70 0.75; 0.8), (0.46 0.55 0.70 0.75; 1.0)];
C3
[(0.70 0.73 0.88 0.98; 0.8), (0.67 0.73 0.88 0.94; 1.0)];
C4
[(0.85 0.91 0.97 0.99; 0.8), (0.85 0.91 0.97 0.99; 1.0)];
C5
[(0.93 0.98 1.00 1.00; 0.8), (0.93 0.98 1.00 1.00; 1.0)];For the engineering investment project evaluation problem, through the likelihood-based MCDA method, we can obtain the final ranking of the program asz4≻z2≻z1≻z3; through the linear assignment method, we can obtain the final ranking of the program as z2≻z4≻z3≻z1; through the extended QUALIFLEX method, we can obtain the final ranking of the program as z2≻z4≻z3≻z1. 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 z4≻z2≻z1≻z3, which is the same as the result of our proposed method when θ=0.5; the final ranking of the linear assignment method is z2≻z4≻z3≻z1, which is the same as the result of our proposed method when θ∈0.2,0.4; the final ranking of the extended QUALIFLEX method is z2≻z4≻z3≻z1, which is the same as the result of our proposed method when θ∈0.2,0.4. 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 z1. 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 (Section6.1); sometimes, with the change of θ, the ranking results also change many times (Section 6.2). However, overall, with the change of θ, 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 differentθ 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.


## 7. Conclusion

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 [34] and further examine the IT2TFS multiattribute group decision-making problems considering the risk preferences of decision makers.



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*Source: 680635-2015-09-14.xml*