Decomposition and Intersection of Two Fuzzy Numbers for Fuzzy Preference Relations

: In fuzzy decision problems, the ordering of fuzzy numbers is the basic problem. The fuzzy preference relation is the reasonable representation of preference relations by a fuzzy membership function. This paper studies Nakamura’s and Kołodziejczyk’s preference relations. Eight cases, each representing different levels of overlap between two triangular fuzzy numbers are considered. We analyze the ranking behaviors of all possible combinations of the decomposition and intersection of two fuzzy numbers through eight extensive test cases. The results indicate that decomposition and intersection can affect the fuzzy preference relations, and thereby the ﬁnal ranking of fuzzy numbers


Introduction
For solving decision-making problems in a fuzzy environment, the overall utilities of a set of alternatives are represented by fuzzy sets or fuzzy numbers.A fundamental problem of a decision-making procedure involves ranking a set of fuzzy sets or fuzzy numbers.Ranking functions, reference sets and preference relations are three categories with which to rank a set of fuzzy numbers.For a detailed discussion, we refer the reader to surveys by Chen and Hwang [1] and Wang and Kerre [2,3].For ranking a set of fuzzy numbers, this paper concentrates on those fuzzy preference relations that are able to represent preference relations in linguistic or fuzzy terms and to make pairwise comparisons.To propose the fuzzy preference relation, Nakamura [4] employed a fuzzy minimum operation followed by the Hamming distance.Kołodziejczyk [5] considered the common part of two membership functions and used the fuzzy maximum and Hamming distance.Yuan [6] compared the fuzzy subtraction of two fuzzy numbers with real number zero and indicated that the desirable properties of a fuzzy ranking method are the fuzzy preference presentation, rationality of fuzzy ordering, distinguishability and robustness.Li [7] included the influence of levels of possibility of dominance.Lee [8] presented a counterexample to Li's method [7] and proposed an additional comparable property.The methods of Wang et al. [9] and Asady [10] were based on deviation degree.Zhang et al. [11] presented a fuzzy probabilistic preference relation.Zhu et al. [12] proposed hesitant fuzzy preference relations.Wang [13] adopted the relative preference degrees of the fuzzy numbers over average.This paper evaluates and compares two fundamental fuzzy preference relations-one is proposed by Nakamura [4] and the other by Kołodziejczyk [5].The intersection of two membership functions and the decomposition of two fuzzy numbers are main differences between these two preference relations.Since the desirable criteria cannot easily be represented in mathematical forms, their performance measures are often tested by using test examples and judged intuitively.To this end, we consider eight complex cases that represent all the possible cases the way two fuzzy numbers can overlap with each other.For Nakamura's and Kołodziejczyk's fuzzy preference relations, this paper analyzes and compares the ordering behaviors of the decomposition and intersection through a group of extensive cases.
The organization of this paper is as follows-Section 2 briefly reviews the fuzzy sets and fuzzy preference relations and presents the eight test cases.Section 3 analyzes Nakamura's fuzzy preference relation and presents an algorithm.Section 4 presents the behaviors of Kołodziejczyk's fuzzy preference relation.Section 5 analyzes the effect of the decomposition and intersection on fuzzy preference relations.Finally, some concluding remarks and suggestions for future research are presented.

Fuzzy Sets and Test Problems
We first review the basic notations of fuzzy sets and fuzzy preference relations.Consider a fuzzy set A defined by a universal set of real numbers R by the membership function A(x), where The set of all triangular fuzzy numbers on R is denoted by TF( ).
Definition 4. For a fuzzy number A, the upper boundary set A of A and the lower boundary set A of A are respectively defined as: A(x) = sup y≥x A(y) and: A(x) = sup y≤x A(y).
Definition 5.The Hamming distance between two fuzzy numbers A and B is defined by:

Definition 6.
Let A and B be two fuzzy numbers and × be an operation on R , such as +, -, *, ÷ . . . .By extension principle, the extended operation ⊗ on fuzzy numbers can be defined by: For simplicity, we denote R (A, B) for the degree of preference of fuzzy number B over fuzzy number A.
The evaluation criteria for the comparison of two fuzzy numbers cannot easily be represented in mathematical forms therefore it is often tested on a group of selected examples.The membership functions of two fuzzy numbers can be overlapping/nonovelapping, convex/nonconvex, and normal/non-normal.All the approaches proposed in the literature seem to suffer from some questionable examples, especially for the portion of overlap between two membership functions.
Let A(a 1 , b 1 , c 1 ) and B(a 2 , b 2 , c 2 ) be two triangular fuzzy numbers.Figure 1 displays eight test cases of representing all the possible cases the way two fuzzy numbers A and B can overlap with each other.Table 1 shows the area Q i of i-th region in each case.More precisely, the eight extensive test cases are as follows: For simplicity, we denote for the degree of preference of fuzzy number B over fuzzy number A.
The evaluation criteria for the comparison of two fuzzy numbers cannot easily be represented in mathematical forms therefore it is often tested on a group of selected examples.The membership functions of two fuzzy numbers can be overlapping/nonovelapping, convex/nonconvex, and normal/non-normal.All the approaches proposed in the literature seem to suffer from some questionable examples, especially for the portion of overlap between two membership functions.
Let ( 1 ,  1 ,  1 ) and ( 2 ,  2 ,  2 ) be two triangular fuzzy numbers.Figure 1 displays eight test cases of representing all the possible cases the way two fuzzy numbers A and B can overlap with each other.Table 1 shows the area   of i-th region in each case.More precisely, the eight extensive test cases are as follows:     Table 1.The area of i-th region for eight cases.
Case Area Table 1.The area Q i of i-th region for eight cases.
Case Area

Case Area
3

Nakamura's Fuzzy Preference Relation
Using fuzzy minimum, fuzzy maximum, and Hamming distance, Nakamura's fuzzy preference relations [4] are defined as follows: Definition 8.For two fuzzy numbers A and B, Nakamura [4] defines N(A, B) and N (A, B) as fuzzy preference relations by the following membership functions: respectively.Yuan [6] showed that N(A, B) is reciprocal and transitive, but not robust.Wang and Kerre [3] derived that: It follows that: For two triangular fuzzy numbers A(a 1 , b 1 , c 1 ) and B(a 2 , b 2 , c 2 ), then: Define: The steps for implementing the Nakamura's fuzzy preference relation N(A, B) are as in Algorithm 1: Table 2. N(A, B) and N (A, B) for eight cases.

Kołodziejczyk's Fuzzy Preference Relation
By considering the common part of two membership functions, Kołodziejczyk's method [5] is based on fuzzy maximum and Hamming distance to propose the following fuzzy preference relations: Definition 9.For two fuzzy numbers A and B, Kołodziejczyk [5] defines K1 (A, B) and K2 (A, B) as fuzzy preference relations by the following membership functions: and: respectively.K1 (A, B) is reciprocal, transitive and robust [3,5].Since: the results of K2 (A, B) can be obtained from those of N(A, B).
For two triangular fuzzy numbers A(a 1 , b 1 , c 1 ) and B(a 2 , b 2 , c 2 ), then: Define: and: Then: In Table 3, we display the values of K1 (A, B) and K2 (A, B) for each test case.An examination of the table reveals that: and: If b 1 = b 2 , we have: and: and: .
It follows that: and:

Two Comparative Studies of Decomposition and Intersection of Two Fuzzy Numbers
If the fuzzy number A is less than the fuzzy number B, then the Hamming distance between A and max(A, B) is large.Two representations are adopted.One is d(A, max(A, B)).The other is d(A, max(A, B)) + d A, max A, B which decomposes A into A and A. To analyze the effect of decomposition, we consider the following preference relations without decomposition: and: which are the counterparts of the Kołodziejczyk's preference relations K1 (A, B) and K2 (A, B).Therefore, the preference relations K1 (A, B) and K2 (A, B) consider the decomposition of fuzzy numbers, while T1 (A, B) and T2 (A, B) do not.The preference relations K1 (A, B) and T1 (A, B) consider the intersection of two membership functions, while K2 (A, B) and T2 (A, B) do not.For completeness, Table 4 displays the values of N(A, B), N (A, B), K1 (A, B), K2 (A, B), T1 (A, B) and T2 (A, B) of each test case in terms of the values of Q i .The K1 (A, B) considers both decomposition and intersection of two fuzzy numbers, while T2 (A, B) do not.From K1 (A, B) to T2 (A, B), two representations are: and: Table 4. N(A, B) , N (A, B), K1 (A, B), K2 (A, B), T1 (A, B) and T2 (A, B) for eight cases.
The first feature of Table 4 is that the differences between K1 (A, B) and T1 (A, B) and between K2 (A, B) and T2 (A, B) are Q 3 .More precisely, the numerators and denominators of both K1 (A, B) and K2 (A, B) include 2Q 3 for cases 1, 3, 5 and 7, the denominators of both K1 (A, B) and K2 (A, B) include 2Q 3 for cases 2, 4, 6 and 8. Therefore, 2Q 3 represents the effect of the decomposition of fuzzy numbers.The differences between K1 (A, B) and K2 (A, B) and between T1 (A, B) and T2 (A, B) are Q 6 .More precisely, the numerators and denominators of both K1 (A, B) and T1 (A, B) include Q 6 and 2Q 6 , respectively.Therefore, Q 6 represents the effect of the intersection of two membership functions.After some computations, the characteristics of K1 (A, B), K2 (A, B), T1 (A, B) and T2 (A, B) are described as follows: For each test case of two triangular fuzzy numbers A(a 1 , b 1 , c 1 ) and B(a 2 , b 2 , c 2 ), we analyze the behaviors of K1 (A, B), K2 (A, B), T1 (A, B) and T2 (A, B) by applying Theorem 1 as follows: Firstly, for b 1 ≤ b 2 , we have: for case 1.For cases 3, 5 and 7, we have the following results.
(    It follows that: and: For this simple case, all the preference relations give the same degree of preference of B over A. We have: From the viewpoint of probability, the fuzzy numbers A and B have the same mean, but B has a smaller standard deviation.The results indicate that the differences between the decomposition and intersection of A and B cannot affect the degree of preference for B over A. Case (c) A(a, a + b, a + 2b) and B(a + α, a + b + α + β, a + α + 2b + 2β).
For this case, the fuzzy number B is a right shift of A. Therefore, B should have a higher ranking than A based on the intuition criterion.We obtain: and: All methods prefer B, but T1 (A, B) is indecisive.More precisely, If 2b + β < α, then T1 (A, B) < 1/2, so A > B If 2b + β = α, then T1 (A, B) = 1/2, so A = B If 2b + β > α, then T1 (A, B) > 1/2, so A < B.
Hence, a conflicting ranking order of T1 (A, B) exists in this case.This case is more complex for the partial overlap of A and B. The membership function of B has the right peak, B expands to the left of A for the left membership function, and A expands to the right of B for the right membership function.We have: and: It follows that:
Case (b) A(c − a, c, c + a) and B(c − b, c, c + b).
Case (d) A(a, a, a + b) and B(c, c + b, c + b) with a ≥ c.
Definition 2. A normal and convex fuzzy set whose membership function is piecewise continuous is called a fuzzy number.Definition 3. A triangular fuzzy number A, denoted A = (a, b, c), is a fuzzy number with membership function given by: R is a fuzzy total ordering if, and only if, R is both reciprocal and transitive.4. R is robust if, and only if, for any given fuzzy numbers ,  and ε > 0, there exists δ > 0 for which |(, ) − ( ′ , )| <  , for all fuzzy number ′ and max