AN EASY COMPUTATION OF MIN AND MAX OPERATIONS FOR FUZZY NUMBERS

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1 J. Appl. Math. & Computing Vol. 21(2006), No. 1-2, pp AN EASY COMPUTATION OF MIN AND MAX OPERATIONS FOR FUZZY NUMBERS DUG HUN HONG* AND KYUNG TAE KIM Abstract. Recently, Chiu and WangFuzzy sets and systems 126(2002) 273 introduced a method to implement the operations MIN and MAX for two continuous fuzzy numbers. In this paper, we propose a new method using α-cut representation for a simple computation of MIN and MAX operations for more than two upper semi-continuous fuzzy numbers. This method can be applied to non-continuous fuzzy numbers. Using this method, the computation of MIN and MAX operations for more than two fuzzy numbers can be done at once, where as using Chiu and Wang s method, it is done step by step. AMS Mathematics Subject Classification : 28E10. Key words and phrases : Fuzzy ordering; fuzzy minimum; fuzzy maximum 1. Introduction A fuzzy number is a convex normalized fuzzy set A on the real line R such that (i) there exists only one x 0 R, A(x 0 )=1(x 0 is called the mean value of A), and (ii) A(x) is piecewise continuous(not continuous in general), where A(x) denoted the membership function of the fuzzy number 4, 9. Min and MAX are the lattice operations to be used in the ordering of fuzzy numbers. There have been many papers talking about the lattice of fuzzy numbers recently 5, 6, They are defined for any two fuzzy numbers A and B on the universal set R as follows 9. MIN(A, B)(z) = sup mina(x),b(y), z=min(x,y) MAX(A, B)(z) = sup z=max(x,y) mina(x),b(y), Received September 10, Corresponding author. c 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 555

2 556 Dug Hun Hong and Kyung Tae Kim for all x X, y Y and z Z and X = Y = Z = R. In general, they are defined for any finite number of fuzzy numbers A i, i =1, 2,,n as follows: MIN(A 1,,A n )(z) = sup mina 1 (x 1 ),,A n (x n ), (1) z=min(x 1,,x n) MAX(A 1,,A n )(z) = sup z=max(x 1,,x n) mina 1 (x 1 ),,A n (x n ), (2) for x i R, i =1, 2,,n. Recently, Chiu and Wand3 proposed the following two results to show that the implement of MIN and MAX can be much easier. Theorem 1. For any two fuzzy numbers A and B as mentioned in Theorem 1, then the operation MAX can be implemented as { (A B)(z), as z<xm, MIN(A, B)(z) = (A B)(z), as z x m, where z Z = R, and and denote the standard fuzzy intersection and union, respectively. Theorem 2. For any two fuzzy numbers A and B as mentioned in Theorem 1, then the operation MAX can be implemented as { (A B)(z), as z<xm, MAX(A, B)(z) = (A B)(z), as z x m, But these results are very restrictive. Consider the following two cases. First, we consider two fuzzy numbers A and B as shown in Fig. 1. The membership function of fuzzy number A is not continuous, moreover, there does not exist x m such that A(x m )=B(x m ). So, we can not apply above two theorems to have MIN(A, B) and MAX(A, B). We next consider the following three fuzzy numbers A, B and C as shown in Fig.2. Figure 1. two fuzzy numbers(a is not continuous)

3 An easy computation of MIN and MAX operations for fuzzy numbers 557 We cannot have MIN(A, B, C) and MAX(A, B, C) at once by Theorem 1 and 2 above since the point x m of each two fuzzy numbers A, B and B, C are different. Figure 2. three fuzzy numbers We first find MIN(A, B) and next find MIN(A, B, C) =MIN(MIN(A, B),C) step by step. It will take more time. From these points of view, Chiu and Wang s 3 method is not efficient. Therefore, we need to look for an easier way to operate MIN and MAX. In this paper, we propose a new method using α-cut representation for a simple computation of MIN and MAX operations for more than two upper semi-continuous fuzzy numbers. This method can be applied to non-continuous fuzzy numbers. Using this method, the computation of MIN and MAX operations for more than two fuzzy numbers can be done at once where as using Chiu and Wang s method, the computation of MIN and MAX operations is done step by step. 2. Preliminaries We use a slight different of fuzzy number from now on. This definition can help us to identify a fuzzy number with the α-cut representation of fuzzy number. A fuzzy number is a fuzzy set A : R 0, 1 with the following properties; (1) A is normal, i.e., there exists x R such that A(x) =1. (2) A is upper semi-continuous. (3) suppa = cl{x R (x) > 0} is compact. (4) A is a convex fuzzy set, i.e., A(λx +(1 λ)y) min(a(x),a(y)) for x, y R and λ 0, 1. Let F (R) be the family of all fuzzy numbers. For a fuzzy set ũ, if we define { {x A(x) α}, 0 <α 1, A α = suppa, α =0. Then, A is a fuzzy number if and only if A 1 φ and A α is a closed bounded interval for each α 0, 1. If we use this characteristic of fuzzy number, a fuzzy

4 558 Dug Hun Hong and Kyung Tae Kim number A is completely determined by the endpoints of the intervals A α = a 1 (α),a 2 (α). The following theorem(see Goetschel and Voxman7) implies that we can identify a fuzzy number A with the parameterized representation {(a 1 (α),a 2 (α)) 0 α 1}. Theorem 3. For A F (R), considering a 1 and a 2 as functions of α 0, 1. Then (1) a 1 is a bounded increasing function on 0, 1. (2) a 2 is a bounded decreasing function on 0, 1. (3) a 1 (1) a 2 (1). (4) a 1 and a 2 are left continuous on 0, 1 and right continuous at 0. (5) If b 1 and b 2 satisfy above (1)-(4), then there exists a unique B F (R) such that B α =b 1 (α),b 2 (α). 3. Main results We now propose a method to implement the operations MIN and MAX for fuzzy numbers using α-cut representation. Theorem 4. Let A i, i =1,,n, be fuzzy numbers with A i α =a 1 i (α),a2 i (α). Then the operation MIN and MAX can be implemented as MIN(A 1,,A n ) α = MAX(A 1,,A n ) α = min 1 i n a1 i (α), min 1 i n a2 i (α), (3) max 1 i n a1 i (α), max 1 i n a2 i (α), (4) Proof. We will prove (3) and can prove (4) similarly. z 0 MIN(A 1,,A n ) α for some i, ( sup x i=z 0,x j z 0,j =i ) min A 1 (x 1 ),,A i 1 (x i 1 ),A i (z 0 ),A i+1 (x i+1 ),, A n (x n ) α, for some i, A i (z 0 ) α and for all j i, there exists y i such that y j z 0 and

5 An easy computation of MIN and MAX operations for fuzzy numbers 559 A j (y j ) α for some i, z 0 a 1 i (α),a 2 i (α) and for all j i, z 0 a 2 j(α) n z 0 a 1 i (α),a 2 i (α) and z 0 min 1 j n a2 j(α) i=1 z 0 min 1 i n a1 i (α), min 1 i n a2 i (α) Example 1. For the aforementioned fuzzy numbers in the Fig. 1 and Fig.2 of Section 1, we can get MIN and MAX very quickly and easily (Fig.3, Fig.4) Figure 3. MIN and MAX Figure 4. MIN and MAX From above theorem, we can generalize the definitions of MIN and MAX for fuzzy numbers to INF and SUP as follows; Definition 1. Let A t, t T be a set of fuzzy numbers with A t α = INF(A t : t T ) α = inf t T a1 t (α), inf t T a2 t (α) SUP(A t : t T ) α = sup a 1 t (α), sup a 2 t (α). t T t T a 1 t (α),a 2 t (α). Example 2. Let A t, t 1 2, 1 be fuzzy numbers defined as A(x) =1 tx for x 1 t, 1 t and 0, otherwise. Then we can easily obtain ( ) INF A t : t 2, 1 2z for 2 z 0 (z) = 1 z for 0 z 1 0 otherwise SUP ( ) 1 A t : t 2, 1 (z) = 1+z for 1 z z for 0 z 2 0 otherwise. (see Fig. 5)

6 560 Dug Hun Hong and Kyung Tae Kim Figure 5. INF and SUP 4. Conclusion In this paper, we proposed an easy method to compute operations MIN and MAX of many fuzzy numbers at once using α-cut representations, where by Chiu and Wang s3 method, we can obtain the computation of them step by step, and hence it will take much time. We also considered SUP and INF as generalized concepts of MAX and MIN. References 1. R. Ambrosio, G.B. Martini, Maximum and minimum between fuzzy symbols in noninteractive and weakly non-interactive situations, Fuzzy Sets and Systems 12 (1984) J. C. Bezdek, Anaylsis of Fuzzy Information, CRC Press, Boca Raton, FL, C. H. Chiu, W. J. Wang, A simple computation of MIN and MAX operations for fuzzy numbers, Fuzzy Sets and Systems 126 (2002) D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, D. Dubois, H. Prade, in: J. Bezdek (Ed.), Fuzzy Numbers: An Overview, Analysis of Fuzzy Information, Vol. I, CRC Press, Boca Raton, FL, 1987, pp. 3-39(in Dubois, Prade, Yager(eds.), Readings in Fuzzy Sets for Intelligent Systems, Morgan & Kaufmann, San Francisco, 1993) 6. T. Geerts, A note on lattices of euclidean subspaces, Automatica 31 (2) (1995) R. Goetschel and W. Voxman, Elememtary fuzzy calculus, Fuzzy Sets and Systems 18 (1986), J. G. Kim, S. J. Cho, Structure of a lattice of fuzzy subgroups, Fuzzy Sets and Systems 89 (1997) G. J. Klir, S. J. Yuan, Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice-Hall PTR, NJ 07458, X. Liu, The least upper bound of content for realizable matrices on lattice 0, 1, Fuzzy Sets and Systems 80 (1996) M. Mares, Computation over fuzzy quantities, CRC Press, Boca Raton, FL, 1994.

7 An easy computation of MIN and MAX operations for fuzzy numbers K.L. Zhang, K. Hirota, On fuzzy number lattice ( R, ), Fuzzy Sets and Systems 92 (1997) Dug Hun Hong received the B.S., M.S. degrees in mathematics from Kyungpook National University, Taegu, Korea and Ph. D degree in mathematics from University of Minnesota, Twin City in 1981, 1983 and 1990, respectively. From 1991 to 2003, he worked with department of Statistics and School of Mechanical and Automotive Engineering, Catholic University of Daegu, Daegu, Korea, Since 2004, he has been a Professor in Department of Mathematics, Myongji University, Korea. His research interests include general fuzzy theory with application and probability theory. Department of Mathematics, Myongji University, Kyunggido , South Korea dhhong@mju.ac.kr Kyung Tae Kim received the B.S. degree in Electrical Engineering from Kyungpook National University, Taegu, Korea and M.E and Ph. D degrees in Electrical Engineering from Yonsei University, Seoul, Korea in 1978, 1980 and 1987, respectively. Since 1987, he has been a Professor in Department of Electronics and Electrical Information Engineering, Kyungwon University, Korea. His research interests include mobile communications and optical communications. Department of Electronics and Electrical Information Engineering, Kyungwon University, Sungnam Kyunggido, South Korea ktkim@kyungwon.ac.kr