Ranking Fuzzy Numbers Using Targets

Transcription

1 Ranking Fuzzy Numbers Using Targets V.N. Huynh, Y. Nakamori School of Knowledge Science Japan Adv. Inst. of Sci. and Tech. {huynh, J. Lawry Dept. of Engineering Mathematics University of Bristol Abstract In this paper, we introduce a new fuzzy preference relation based on the alpha-cut representation of fuzzy numbers and comparison probabilities of interval values. Essentially, this approach is accompanied with a widely-accepted interpretation of fuzzy sets along with taking uncertain characteristic inherent in the representation of fuzzy numbers into consideration in the comparison process. The proposed fuzzy preference relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of targetbased evaluations. Several numerical examples are examined for illustration and comparison of the proposed approach. Keywords: Fuzzy numbers, Decision making, Ranking. Introduction The issue of comparison and ranking of fuzzy numbers has long been a concern since middle 97s and, it is mainly related to applications of fuzzy sets in decision analysis [5]. As we know, in practice evaluations for selection and for ranking among alternatives are two closely related and common facets of human decision making activities. Frequently, decision-makers are faced with a lack of precise information when assessing alternatives. In such situations, fuzzy numbers are extensively applied to represent the performance of alternatives and therefore, the ranking or selection of alternatives eventually leads to comparisons of the resulting fuzzy numbers. From reviewing the literature, we find that many methods for comparison and ranking of fuzzy numbers have been proposed. Most early ranking methods in the field have been reviewed and analyzed by Bortolan and Degani [4], and more recently by Chen and Hwang [5]. In particular, the collection of cases examined by Bortolan and Degani [4] has been widely used as benchmark examples for comparative studies of ranking methods. Though there has been a lot of methods for ranking fuzzy numbers presented in the last decades, none of them is a well accepted golden choice for all cases [4]. Main drawbacks found in most methods include: counter-intuitive, non-discriminating, inconsistency, using only local information or restricting the shape of fuzzy numbers to be ranked, difficult to understand [8, ]. Recently, Lee-Kwang and Lee [2] have proposed a new method for ranking fuzzy numbers based on the so-called satisfaction function (SF, for short) and a viewpoint-dependent evaluation method. Informally, the SF aims to represent the relation that a fuzzy number is less or greater than another one with some degree of possibility. The proposed ranking method is then based on evaluations of the SF function of every fuzzy number involved with a predefined viewpoint. Inspired from a target-based decision-making perspective [3],

2 it is very interesting here to observe that if fuzzy numbers involving in a ranking could be considered as the fuzzy performance assessments of alternatives, a predefined viewpoint in Lee-Kwang-Lee s method could be seen as the decision-maker s fuzzy target [9]. Then, according to the optimizing principle, the decision maker should choose an alternative that maximizes the possibility of meeting his target represented by the SF. This view could be considered as one of underlying motivations for ranking methods based on viewpointdependent evaluations. In the context of this paper, we will use the terms viewpoint and target interchangeably. Motivated by the above observation, we propose in this paper a new fuzzy preference relation, viewed as the SF in Lee-Kwang and Lee s work, based on a probabilistic approach. From this, the proposed fuzzy preference relation is then applied to the issue of ranking fuzzy numbers using fuzzy targets in terms of target-based evaluations. Our method of ranking fuzzy numbers works in a similar way to Lee-Kwang-Lee s method, i.e. consisting of two steps: evaluation and ordering, but with the new fuzzy preference relation interpreted as the probability of meeting the target. 2 Fuzzy Numbers and α-cut Based Interpretation A fuzzy number A is defined as a fuzzy subset with the membership function µ A (x) of the set R of all real numbers that satisfies the following properties: A is a normal fuzzy set, i.e. sup x R µ A (x) = ; A is a convex fuzzy set, i.e. µ A (λx +( λ)x 2 ) min(µ A (x ), µ A (x 2 )) for x, x 2 R and λ [, ]; the support of A, i.e. the set {x R µ A (x) > }, is bounded. As defined in [7], a fuzzy number A with the membership function µ A (x) can be represented as µ L A (x), a x b,, b x c, µ A (x) = µ R A (x), c x d,, otherwise. where µ L A (x) is a non-decreasing function and µ R A (x) is a non-increasing function. If µl A (x) and µ R A (x) are linear functions then A is called a trapezoidal fuzzy number and denoted by [a, b, c, d]. In particular, [a, b, c, d] becomes a triangular fuzzy number if we have b = c. For a fuzzy number A and α (, ], the α- cut A α of A is a crisp set defined as A α = {x R µ A (x) α} As A is a convex fuzzy set, A α is convex. Let us denote supp(a) = {x R µ A (x) } the support of A. It should be also noted that in fuzzy set theory, the concept of α-cuts essentially plays an important role in establishing the relationship between fuzzy sets and crisp sets. As is well known, each fuzzy set can be uniquely represented by the family of all of its α-cuts. Intuitively, each α-cut A α of a fuzzy set A can be viewed as a crisp approximation of A at the level α (, ]. Further, in computer applications, since the fuzzy number A possesses a continuous membership function, it can be usually approximated by sampling the membership function along the membership axis. That is, assuming uniform sampling and that the sample values are taken at membership grades α = > α 2 >... > α n > α n >, A can be approximated as the collection {A αi } n i=. We would also like to recall here that in the case where a fuzzy set A has a discrete membership function, i.e. A = {(x i, µ A (x i ))}, for x i R, i =,..., N with N being a finite positive integer, Dubois and Prade [6] pointed out that the family of its α-cuts forms a nested family of focal elements in terms of Dempster-Shafer theory [6]. In particular, assuming the range of the membership function µ A, denoted by

3 rng(µ A ), is rng(µ A ) = {α,..., α n }, where α i > α i+ >, for i =,..., n, then the body of evidence induced from A with the basic probability assignment, denoted by m A, is defined as m A (X) = { αi α i+, if X = A αi, otherwise with α n+ = by convention. In this case the normalization assumption of A insures the body of evidence does not contain the empty set. Interestingly, this view of fuzzy sets has been used by Baldwin in [] to introduce the so-called mass assignment of a fuzzy set with relaxing the assumption, and to provide a probabilistic based semantics for fuzzy sets. Under such a view of fuzzy sets, returning to the finite approximation of a fuzzy number A having a continuous-type membership function above, we can approximately represent A as a body of evidence (or, a random set) F A = {(A αi : α i α i+ )} n i= () In fact, this approximate representation of fuzzy numbers has been either implicitly or explicitly used by several authors previously, for instance in [8, 8]. In the following we also use this approximate representation of a fuzzy number to introduce a fuzzy preference relation on fuzzy numbers. 3 A Probability-Based Fuzzy Preference Relation 3. Intervals Case Let us consider two interval values denoted by X = [x, x 2 ] and Y = [y, y 2 ]. In [5], the authors proposed a ranking procedure for intervals based on the Hurwicz criterion as X ranks over Y if and only if δx + ( δ)x 2 > δy + ( δ)y 2 where δ [, ] is a parameter reflecting the strategy that is adopted by the decision maker. Roughly speaking, interval values are firstly mapped into real numbers taking the decision maker s attitude expressed by the Hurwicz criterion into account, and then a ranking is based on the natural order of resulted real numbers. Here we utilize an approach to comparing intervals motivated by a probabilistic view of the underlying uncertainty, instead. More formally, motivated by our later developments, we aim at defining a probability-based fuzzy preference relation over intervals. To this end, we consider intervals X and Y as uncertain values having uniform distributions f X (x) and f Y (y) over [x, x 2 ] and [y, y 2 ], respectively. Then, based on the probability theory, we can work out the probability of the ordering of uncertain values X and Y taking into account associated probability distributions f X (x) and f Y (y). Let us denote this probability P (Y X). As suggested in [7], the probability P (Y X) can be obtained equivalently by P (X Y ), where the probability distribution f Z (z) of uncertain value Z = X Y is defined as the convolution of f X (x) and f Y (y). Namely, f Z (z) = = Therefore, we have y 2 y f X (z + y)f Y (y)dy P (Y X) = P (Z ) = f X (z + y)dy (2) f Z (z)dz (3) Obviously, the result of computation for (3) depends on the relative position of x and x 2 with respect to y and y 2. In [7] the authors have studied all the particular cases and the result is briefly summarized in the following.. The disjoint cases. The two intervals X = [x, x 2 ] and Y = [y, y 2 ] are disjoint, i.e. x 2 y or x y 2. In these cases, we have {, if x2 y P (Y X) =,, if x y The overlapping cases. We have also two cases: x y x 2 y 2 ( ) or y x y 2 x 2 ( ), and {.5(x2 y ) 2 (x P (Y X) = 2 x )(y 2 y ), ( ),.5(y 2 x ) 2 (x 2 x )(y 2 y ), ( ).

4 For example, the case ( ) is graphically illustrated as in Fig. (left), where the area where Y is smaller than X is denoted by D y x and P (Y X) is the ratio of D y x to the whole rectangle, i.e. D y x + D x y. 3. The inclusion cases. Here we again have two cases: x y y 2 x 2 ( ) or y x x 2 y 2 ( ), and P (Y X) = { (x2.5(y +y 2 ) (x 2 x ), ( ),.5(x +x 2 ) y (y 2 y ), ( ). Similarly, the case ( ) is graphically illustrated as in Fig. (right). Note that in the case both intervals X and Y degenerate into scalar numbers, i.e. x = x 2 and y = y 2, we get, if x > y, P (Y X) = 2, if x = y,, if x < y. Furthermore, it is easily seen that the following holds Proposition We have. P (Y X) = P (X Y ), 2. If (x +x 2 ) = (y +y 2 ), P (Y X) = Fuzzy Numbers Case Now let us turn to the case of fuzzy numbers. Consider two fuzzy numbers A and B which are convex, normal and having bounded supports. In this case, according to (), A and B can be probabilistically approximately represented by F A = {(A αi : α i α i+ )} n i= (4) F B = {(B αi : α i α i+ )} n i= (5) respectively, with assuming a uniform sampling and that the sample values are taken at membership grades α = > α 2 >... > α n > α n > α n+ = Here, for all α i, A αi and B αi are intervals and denoted respectively by A αi = [a l (α i ), a r (α i )] B αi = [b l (α i ), b r (α i )] According to the probabilistic interpretation of (4) and (5), we can now approximately define the probability of the ordering of A and B, denoted by P (B A), in terms of probabilities of the ordering of their level-sets represented as intervals A αi and B αi as follows P (B A) n = α P (B αi A αi ) (6) i= where α is the separation between any two adjacent levels. It is of interest to note here that if α-cuts A α and B α can be intuitively viewed as crisp approximations of A and B respectively at the level α (, ], then P (B A) defined in (6) can be interpreted as expectation of the probability of the ordering of A and B. Clearly, the right side of the expression (6) is the Riemann sum of the function f(α) = P (B α A α ) over [, ] with respect to the partition α,..., α n+. Thus, we now generally define P (B A) as P (B A) = P (B α A α )dα (7) The approximation in (6) of the integral in (7) improves the finer the sample of membership grades. As a direct consequence of Proposition and (7), we obtain the following Proposition 2 For all fuzzy numbers A and B, we have. P (B A) = P (A B), 2. If (a l (α) + a r (α)) = (b l (α) + b r (α)), for any α (, ], P (B A) = Extension to Non-convex and Sub-normal Fuzzy Numbers Considering now two non-convex fuzzy numbers A and B, then for α (, ] we can express α-cuts A α and B α respectively as unions

5 y = x y = x Dx y Dx y y2 y2 Dy x y y Dy x y x y2 x2 x y y2 x2 Case y x y2 x2 Case x y y2 x2 Figure : Comparison probability of two intervals of distinct intervals [8] A α = B α = n α i= m α [a i l (α), ai r(α)] (8) [b j l (α), bj r(α)] (9) j= Here we still assume that A and B are normal. Intuitively, recall that the probability P (Y X) of the ordering two intervals X and Y is defined by the ratio of the area where Y is smaller than X, i.e. D y x, to the whole area determined by the rectangle X Y (graphically, see for example Fig. ). Keeping this in mind, we can define P (B α A α ) as P (B α A α ) = n α m α i= j= P ( ) Bα j A i α M ij n α m α M ij i= j= () where A i α = [a i l (α), ai r(α)], Bα j = [b j l (α), bj r(α)], and M ij is the area determined by the rectangle [b j l (α), bj r(α)] [a i l (α), ai r(α)]. Note that in this case we also have P (A α B α ) = P (B α A α ) Further, once having defined P (B α A α ) by (), we also get P (B A) as defined in (7) above. Now let us consider the case of sub-normal fuzzy numbers A and B. Denote by hgt(a) and hgt(b) the heights of fuzzy sets A and B respectively. Assuming that A and B are non-empty, i.e. hgt(a) > and hgt(b) >. Let β = min(hgt(a), hgt(b)) Then the relation established in Proposition 2 suggests to define P (B A) as P (B A) = β β P (B α A α )dα () 4 An Application to Ranking Fuzzy Numbers 4. Ranking Procedure Given fuzzy numbers A and B, as discussed previously P (B A) could be interpreted as expected probability of the relation B A. From a perspective of decision making, assuming that A and B are considered as fuzzy performance assessments of two alternatives A and B respectively, then P (B A) can be also interpreted as the probability that A outperforms B. Under such an interpretation and motivated by the target-based approach to decision making [3], a procedure is proposed in the following, which ranks fuzzy numbers by the probability that they outperform some prespecified target or benchmark, which itself is also fuzzy. Assume that S = {A,..., A N } is a finite set of fuzzy numbers which need to be ranked. Then by a fuzzy target involving in the ranking problem we mean a fuzzy set T over R with the membership function µ T : R [, ] satisfying. µ T is a piecewise continuous function having a bounded support.

6 2. For any i, supp(a i ) supp(t ). 3. T is not empty, i.e. + µ T (x)dx >. Once having specified target T, the ranking procedure is simply carried out as follows neutral optimistic pessimistic. Evaluate v i = P (A i T ), i =,..., N. 2. Rank fuzzy numbers in S according to their evaluation values v i, i =,..., N. It would be noted that this ranking procedure is similar to that proposed by Lee-Kwang and Lee in [2], however, our motivation here is somehow different. Furthermore, their ranking procedure is based on the so-called satisfaction function defined as + y µ T (x)µ A (y)dxdy S(A > T ) = + + µ T (x)µ A (y)dxdy where S(A > T ) is interpreted as the possibility that A is greater than T. Example Let us consider the following example taken from [2]. Assume that we have four fuzzy numbers as depicted in Fig. 2. Let us consider three prototypical targets that are pessimist, optimist and neutral as depicted in Fig. 3. A A 2 A 3 A Figure 2: Fuzzy numbers. Then the probabilities that given fuzzy numbers meet various targets are shown in Table and the corresponding ranking result is A A 3 A 2 A Comparison with Previous Methods Now we examine the proposed ranking method in comparison with several previous methods. In particular, for the purpose of comparative study we select the Figure 3: Targets. Table : Results of the Example Targets A A 2 A 3 A 4 Optimistic Neutral Pessimistic following methods: Lee-Kwang-Lee s [2], Baldwin-Guild s [2], Liou-Wang s [3], Kim- Park s [], Peneva-Popchev s [4], all of which allow a change in the evaluation strategy which reflects decision attitude of the decision maker. A brief description of these methods is presented in [2]. Note here that targets pessimistic, neutral, and optimistic correspond to viewpoints V, V 2, and V 3 of Lee-Kwang-Lee s method. The comparative study is performed on eight cases, most of which are selected from the examples examined in [4] and more recently in [2]. The results are shown in Table 2 and Table 3. From these results we can see that in some cases the last four methods either are not discriminative (Baldwin-Guild s method in case (b), Liou-Wang s and Kim- Park s methods in case (e) with k = ) or provide counterintuitive results (Kim-Park s method in case (c) and Peneva-Popchev s method in case (e) with k = ). In contrast, in all cases our method shows good results, which are also consistent with those obtained by Lee-Kwang-Lee s method. Further, while Lee-Kwang-Lee s method cannot discriminate the second and the third in the case (f) with the neutral target, our method orders the third before the second which is closed to our intuitive observation.

7 Table 2: Comparative Examples A B C (a) (b) (c) (d) (a) (b) (c) (d) Methods A B A B C A B A B Pessimistic New Neutral Optimistic Pessimistic Lee-Kwang-Lee Neutral Optimistic k = Baldwin-Guild k = k = k = Liou-Wang k = k = k = Kim-Park k = k = k = Paneva-Popchev k = k = Conclusion We have introduced in the paper a new fuzzy preference relation based on the alpha-cut representation of fuzzy numbers and comparison probabilities of interval values. The proposed fuzzy preference relation was then applied to the issue of ranking fuzzy numbers in terms of target-based evaluations. For further work, we are planning to develop a targetbased method for solving the problem of fuzzy decision making under uncertainty based on the proposed fuzzy preference relation. Acknowledgment Support in part from the JAIST International Joint Research Grant is gratefully acknowledged. References [] J. F. Baldwin, The management of fuzzy and probabilistic uncertainties for knowledge based systems, in: S. A. Shapiro (Ed.), The Encyclopaedia of AI, New York: Wiley, 992, pp [2] J. F. Baldwin, N. C. F. Guild, Comparison of fuzzy sets on the same decision space, Fuzzy Sets Syst. 2 (979) [3] R. Bordley, M. LiCalzi, Decision analysis using targets instead of utility functions, Decisions in Economics and Finance 23 () (2) [4] G. Bortolan, R. Degani, A review of some methods for ranking fuzzy subsets, Fuzzy Sets Syst. 5 (985) 9. [5] S. J. Chen, C. L. Hwang, Fuzzy Multiple Attribute Decision Making, New York: Springer, 992. [6] D. Dubois, H. Prade, Properties of measures of information in evidence and possibility theories, Fuzzy Sets Syst. 24 (987) 6 82.

8 Table 3: Comparative Examples 2 A B C (e) (f) (g) (h) (e) (f) (g) (h) Methods A B C A B C A B A B Pessimistic New Neutral Optimistic Pessimistic Lee-Kwang-Lee Neutral Optimistic k = Baldwin-Guild k = k = k = Liou-Wang k = k = k = Kim-Park k = k = k = Paneva-Popchev k = k = [7] D. Dubois, H. Prade, Operations on fuzzy numbers, Int. J. Syst. Sci. 9 (6) (978) [8] P. Fortemps, M. Roubens, Ranking and defuzzification methods based on area compensation, Fuzzy Sets Syst. 82 (996) [9] V. N. Huynh, Y. Nakamori, M. Ryoke, T. B. Ho, Decision making under uncertainty with fuzzy targets, submitted. [] K. Kim, K. S. Park, Ranking fuzzy numbers with index of optimism, Fuzzy Sets Syst. 35 (99) [] K. M. Lee, C. H. Cho, H. Lee-Kwang, Ranking fuzzy values with satisfaction function, Fuzzy Sets Syst. 64 (994) [2] H. Lee-Kwang, J.-H. Lee, A method for ranking fuzzy numbers and its application to decision-making, IEEE Trans. Fuzzy Syst. 7 (6) (999) [3] T. S. Liou, M. J. Wang, Ranking fuzzy numbers with integral value, Fuzzy Sets Syst. 5 (992) [4] V. Peneva, I. Popchev, Comparison of clusters from fuzzy numbers, Fuzzy Sets Syst. 97 (998) [5] J. Saade, H. Schwarzlander, Ordering fuzzy sets over the real line: an approach based on decision making under uncertainty, Fuzzy Sets Syst. 5 (992) [6] G. Shafer, A Mathematical Theory of Evidence. Princeton: Princeton University Press, 976. [7] R. R. Yager, M. Detyniecki, B. Bouchon- Meunier, A context-dependent method for ordering fuzzy numbers using probabilities, Inf. Sci. 38 (2) [8] R. R. Yager, D. Filev, On ranking fuzzy numbers using valuations, Int. J. Intell. Syst. 4 (999)