Pii: S0898-1221(01)00124-9

-This study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance. The proposed approach only requires a few left and right spreads at some a-levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other. The total dominance is then determined by combining the left and right dominance based on a decision maker's optimistic perspectives. Such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired. The approach proposed herein is relatively simple i in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers. By using a few left and right spreads, two groups of examples demonstrate the accuracy and applicability of the proposed approach. © 2001 Elsevier Science Ltd. All rights reserved. Keywords--Ranking fuzzy numbers, Dominance, Decision making. 1. I N T R O D U C T I O N Decision makers are normally faced with the lack of precise information to assess a set of alternatives in an uncertain environment. Imprecise evaluations may be attributed to (1) unquantifiable information, (2) incomplete information, (3) nonobtainable information, and (4) partial ignorance [1]. To resolve this problem, the fuzzy set theory pioneered by Zadeh [2] has been extensively used. Fuzzy numbers or fuzzy subsets of the real line ~ are applied to represent the imprecise numerical measurements of different alternatives. Therefore, comparing the different alternatives is actually comparing the resulting fuzzy numbers. More than twenty fuzzy ranking indices have been proposed since 1976 [1,3]. Various techniques are applied to compare the fuzzy numbers. Some investigations [3-15] defined a ranking function to map a fuzzy number to a real number and, then, used natural orderings. Other investigators [16-25] defined a comparison function that maps two fuzzy numbers to a real number when determining the degree to which one dominates the other. Most approaches are biased on the possibility concept and/or the probability measure of fuzzy events concept [3]. Several 0898-1221/01/$ see front matter (~) 2001 Elsevier Science Ltd. All rights reserved. Typeset by J4A4S-TEX PII: S0898-1221(01)00124-9 1590 L.-H. CHEN AND H.-W. LU authors [3,9,26,27] have reviewed and compared some of them using the same set of examples, as provided by Bortolan and Degani [28]. Chen and Hwang [1] thoroughly reviewed the existing approaches, pointing out some illogical conditions that arise among them. Some of the existing approaches are difficult to understand and have suffered from different plights, e.g., the lack of discrimination, producing counterintuitive orderings, and ultimately resulting in inconsistent orderings if a new fuzzy number is added; high complexity and cumbersome computational efforts are also characteristic [29,30]. Nearly all approaches should acquire membership functions of fuzzy numbers before the ranking is performed; however, this may be infeasible in real applications. Furthermore, accuracy and efficiency should be of priority concern in the ranking process if ranking a large amount of fuzzy numbers. In light of the above discussion, this study presents an approximate ranking approach based on the left and right dominance, which follows the concept of area measurement. Many ranking methods have already been developed to rank fuzzy numbers based on area measurement. Some of those methods compute the Hamming distance measurements between each fuzzy number and fuzzy maximum (or minimum) as the ranking basis, such as in the investigations of Yager [14], Kerre [8], Nakamura [22], and Kolodziejczyk [10]. However, these methods are illogical owing to the neglect of the fuzzy number's relative locations on the X-axis [1]. Tseng and Klein [23] proposed a ranking algorithm based on the difference concept. A preference relation is developed for the ranking process using the dominance and indifference, which adheres to the concept of difference between two fuzzy numbers. Yager [15] proposed a ranking index, F3, by measuring the area from the membership axis to the average of the left and right membership functions. This index has consistent comparisons in Bortolan and Degani's examples [27,28]. Liou and Wang [11] ranked fuzzy numbers using the total integral values according to a decision maker's attitude of risk. Using their method, a neutral decision maker, who specifies the value (~/) of the index of optimum to be 0.5, will obtain the total integral value equivalent to F3. Fortemps and Roubens [27] recently proposed a ranking method based on the concepts of area compensation, which corresponds to Yager's F3 and the total integral value. Fortemps and Roubens's method produces equivalent outcomes as Yager's F3 and Liou and Wang's total integral value with 7 = 0.5, if the fuzzy numbers are normal and convex. The above investigations require membership functions when computing the area compensation and the total integral value. Chen and Klein [29,30] employed several existing concepts to develop a ranking method based on area measurement without membership functions. Two crisp maximizing and minimizing barriers, proposed by Choobineh and Li [26], are first defined to construct a referential rectangle. The ranking index is then determined by the difference between the fuzzy number and the referential rectangle. However, the index is affected by the choice of the two barriers. For circumventing the above-mentioned problems, the proposed approximate approach only uses a-cuts and performs simple arithmetic operations for the ranking purpose. Initially, the left (right) dominance is determined by summing the difference of the left (right) spreads at each a-level to denote the degree to which one fuzzy number dominates the other at the left(right)hand side. According to our results, the left (right) dominance approximates the area difference of two fuzzy numbers from the membership axis to the left (right) membership function, when the number of a-cuts approaches the infinity. Moreover, to reflect the decision maker's optimistic or pessimistic perspectives, a convex combination of the left and right dominance using an index of optimism is employed to rank the fuzzy numbers [9]. The proposed approach corresponds to the F3 index, the total integral value with V = 0.5, and Fortemps and Roubens' area compensation approximately, when the index of optimism is assigned 0.5. An example is used to compare two fuzzy numbers, while considering different values of the index of optimism. Particularly, the proposed approach can also be applied to rank the combination case of some fuzzy numbers and crisp numbers and the case of discrete fuzzy numbers. Also described herein are some properties which are useful in ranking a large quantity of fuzzy numbers. Ranking Fuzzy Numbers 1591 Comparing the proposed approximate approach with the existing methods using both Bortolan and Degani's examples [28] and Tseng and Klein's examples [23] reveals that the former is more simple, efficient, and consistent. The rest of this paper is organized as follows• Section 2 introduces the ranking approach. Section 3 describes some useful properties. Next, Section 4 presents some comparative examples which demonstrate the accuracy and efficiency of the proposed approach over the existing methods. Concluding remarks are finally made in Section 5. 2. T H E R A N K I N G M E T H O D A real fuzzy number can be defined as a fuzzy subset of the real line ~, which is convex and normal [31,32]. That is, for a fuzzy number A of ~ defined by the membership function #A(X), x E ~, the following relations exist: maxlzA(X) = 1, (la) ~A[/~XI "[(1 )~)x2] _ min[/~A(Xi), ~A(X2)], ( lb) where x l , x2 E ~, V)~ E [0, 1]. A fuzzy number A with the membership function #A(X), x E ~, can be defined as [33] #L(x), a < x < b , 1, b < x < c , #A(X) = #~(x), c < x < d, (2) 0, otherwise, where #L(x) is the left membership function that is an increasing function and # L : [a, b] ---, [0, 1]. Meanwhile, #Aa(X) is the right membership function that is a decreasing function and pA R : [c, d] -* [0, 1]. In addition, a trapezoidal fuzzy number is denoted by [a, b, c, a~. In particular, [a, b, c, d] can also signify a triangular fuzzy number if b = c. Assume that every fuzzy number is bounded; i.e., o o < a, d < or. For a fuzzy number A, the a-cuts (level sets) As = {x E ~ I #A(X) >_ a}, a E [0, 1], are convex subsets of ~. The lower and upper limits of the k th oL-cut for the fuzzy number A~ are defined as li,k = inf {XI#A(X) > otk}, (3a) xE~ ri,k = sup{z i #A(Z) >_ ak }, (3b) xE~ respectively, where li,k and ri,k are left and right spreads, respectively [29]. While comparing two fuzzy numbers Ai and Aj, Figure 1 illustrates their corresponding left and right spreads at the ak level•