JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 331 Multiplication Operation on Fuzzy Numbers Shang Gao and Zaiyue Zhang School of Computer Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China Email: [email protected] [email protected] Cungen Cao Institute of Computing Technology,The Chinese Academy of Sciences,Beijing 100080, China Email: [email protected] Abstract—A fuzzy number is simply an ordinary number programming, engineering (especially communications), whose precise value is somewhat uncertain. Fuzzy numbers and experimental science. S. M. Wang and J. Watada [6] are used in statistics, computer programming, engineering, discuss the laws of large numbers for T-independent L-R and experimental science. The arithmetic operators on fuzzy fuzzy variables based on continuous archimedean t-norm numbers are basic content in fuzzy mathematics. Operation and expected value of fuzzy variable. G. X. Wang, Q. of fuzzy number can be generalized from that of crisp interval. The operations of interval are discussed. Zhang and X. J. Cui [8] define a special sort discrete Multiplication operation on fuzzy numbers is defined by the fuzzy numbers - discrete fuzzy number on a fixed set extension principle. Based on extension principle, nonlinear with finite support set, and then obtain a representation programming method, analytical method, computer theorem of such discrete fuzzy numbers, study the drawing method and computer simulation method are used operations of scalar product, addition and multiplication, for solving multiplication operation of two fuzzy numbers. and obtain some results. H. M. Lee and L. Lin [9] The nonlinear programming method is a precise method weighted triangular fuzzy numbers to tackle the rate of also, but it only gets a membership value as given number aggregative risk in fuzzy circumstances during any phase and it is a difficult problem for solving nonlinear of the software development life cycle. X. W. Zhou, L. P. programming. The analytical method is most precise, but it is hard to α -cuts interval when the membership function is Wang and B. H. Zheng [10] calculated concentration of complicated. The computer drawing method is simple, but it pollution by means of triangle fuzzy number and need calculate the α -cuts interval. The computer established fuzzy risk assessment model of the potential simulation method is the most simple, and it has wide ecological risk index. D. Sanchez, M. Delgado and M. A. applicability, but the membership function is rough. Each Vila [11] define imprecise quantities on the basis of a method is illuminated by examples. new representation of imprecision introduced by the authors called RL-representation and show that the Index Terms—fuzzy number, membership function, imprecision of the quantities being operated can be extension principle, α -cuts; nonlinear programming increased, preserved or diminished. The concept takes into account the fact that all phenomena in the physical universe have a degree of I. INTRODUCTION inherent uncertainty. The arithmetic operators on fuzzy In most of cases in our life, the data obtained for numbers are basic content in fuzzy mathematics. decision making are only approximately known. In1965, Multiplication operation on fuzzy numbers is defined by Zadeh [1] introduced the concept of fuzzy set theory to the extension principle. The procedure of addition or meet those problems. In 1978, Dubois and Prade defined subtraction is simple, but the procedure of multiplication any of the fuzzy numbers as a fuzzy subset of the real or division is complex. The nonlinear programming, line [2]. Fuzzy numbers allow us to make the analytical method, computer drawing and computer mathematical model of linguistic variable or fuzzy simulation method are used for solving multiplication environment. A fuzzy number is a quantity whose value operation of two fuzzy numbers. The procedure of is imprecise, rather than exact as is the case with division is similar. "ordinary" (single-valued) numbers [3-6]. Any fuzzy number can be thought of as a function whose domain is II. CONCEPT OF FUZZY NUMBER a specified set. In many respects, fuzzy numbers depict the physical world more realistically than single-valued A. Fuzzy Number numbers. Fuzzy numbers are used in statistics, computer If a fuzzy set is convex and normalized, and its membership function is defined in R and piecewise continuous, it is called as fuzzy number. So fuzzy number Supported by the National Natural Science Foundation of China under Grant No.60773059. © 2009 ACADEMY PUBLISHER 332 JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 (fuzzy set) represents a real number interval whose (α'< α) ⇒ (a (α ') ≤ a (α ) ,a (α ') ≥ a (α ) ) boundary is fuzzy. 1 1 3 3 Fuzzy number is expressed as a fuzzy set defining a The convex condition may also be written as, fuzzy interval in the real number R. Since the boundary of this interval is ambiguous, the interval is also a fuzzy (α'< α) ⇒ (A ⊂ A ) set. Generally a fuzzy interval is represented by two end α α ' Operation of fuzzy number can be generalized from points a1 and a3 and a peak point a2 as [a1, a2, a3 ] (Figure 1). The α -cut operation can be also applied to the fuzzy that of crisp interval. Let’s have a look at the operations of interval. number. If we denote α -cut interval for fuzzy number A as Aα , the obtained interval Aα is defined as ∀ ∈ a1, a3, b1, b3 R (α ) (α ) Aα = [a1 ,a3 ] A = [a1, a3], B = [b1, b3] We can also know that it is an ordinary crisp interval Assuming A and B as numbers expressed as interval, (Figure 2). main operations of interval are i) Addition [a1, a3] (+) [b1, b3] = [a1 + b1, a3 + b3] ii) Subtraction [a1, a3] (-) [b1, b3] = [a1 - b3, a3- b1] iii) Multiplication ∧ ∧ ∧ [a1, a3] (•) [b1, b3] = [a1 • b1 a1 • b3 a3 • b1 a3 • b3, a • b ∨ a • b ∨ a • b ∨ a • b ] Figure 1 Fuzzy Number A = [a1 ,a2 ,a3 ] 1 1 1 3 3 1 3 3 Fuzzy number should be normalized and convex. Here iv) Division the condition of normalization implies that maximum membership value is 1. ∧ ∧ ∧ [a1, a3] (/) [b1, b3] = [a1 / b1 a1 / b3 a3 / b1 a3 / b3, ∃x ∈ R , µ ~ (x ) = 1 ∨ ∨ ∨ 0 A 0 a1 / b1 a1 / b3 a3 / b1 a3 / b3] B. Operation of α -cut Interval excluding the case b1 = 0 or b3 = 0 v) Inverse interval -1 ∧ ∨ [a1, a3] = [1 / a1 1 / a3, 1 / a1 1 / a3] excluding the case a1 = 0 or a3 = 0 When previous sets A and B is defined in the positive real number R+, the operations of multiplication, division, and inverse interval are written as, iii′) Multiplication [a , a ] (•) [b , b ] = [a • b , a • b ] 1 3 1 3 1 1 3 3 Figure 2 α -cut of fuzzy number iv′) Division (α'< α) ⇒ (Aα ⊂ Aα ' ) [a , a ] (/) [b , b ] = [a / b , a / b ] The convex condition is that the line by α -cut is 1 3 1 3 1 3 3 1 continuous and α -cut interval satisfies the following v) Inverse Interval relation. -1 (α ) (α ) [a1, a3] = [1 / a3, 1 / a1] Aα = [a1 ,a3 ] © 2009 ACADEMY PUBLISHER JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 333 vi) Minimum Therefore multiplication operation on fuzzy numbers is expressed as [a , a ] (∧) [b , b ] = [a ∧ b , a ∧ b ] 1 3 1 3 1 1 3 3 ~ ~ ~ ~ µ M (×)N (z) = sup min{µ M (x),µ N (y)} (2) z=x×y vii) Maximum The procedure of addition or subtraction is simple, ∨ ∨ ∨ but the procedure of multiplication or division is complex. [a1, a3] ( ) [b1, b3] = [a1 b1, a3 b3] Example 1 There are two intervals A and B, III. NONLINEAR PROGRAMMING METHOD Based on multiplication operation on fuzzy numbers, A = [3, 5], B = [-2, 7] multiplication operation problem is formulated as a nonlinear programming. Then following operation might be set. max v A(+)B=[3-2, 5+7]=[1, 12] ~ s.t. µ M (x) ≥ v (3) A(-)B=[3-7, 5-(-2)]=[-4, 7] ~ µ N (y) ≥ v ∧ ∧ ∧ A(•)B=[3•(-2) 3•7 5•(-2) 5•7, xy = z0 3•(-2) ∨3•7∨5•(-2) ∨5•7] Given z0 , we can get the maximum vmax of nonlinear programming (1). vmax is membership value of z0 . =[-10, 35] ~ ~ Example 2 Suppose that the membership of M , N A(/)B=[3/(-2) ∧3/7∧5/(-2) ∧5/7, −(x−2)2 are ~ (Figure 3) and µ M (x) = e 3/(-2) ∨3/7∨5/(-2) ∨5/7] −( y+2)2 ~ ~ ~ ~ (Figure 4). is µ N (y) = e Q = M (×)N =[-2.5, 5/7] formulated as following: -1 -1 max v A =[3,5] = [1/5, 1/3] 2 s.t. e −(x−2) ≥ v -1 -1 ∧ ∨ 2 (4) B =[-2,7] = [1/(-2) 1/7, 1/(-2) 1/7] e −( y+2) ≥ v =[-1/2, 1/7] xy = z0 A(∧)B=[3∧(-2),5∧7] When z0 =-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1 =[-2, 5] respectively, we can get the vmax correspondingly. The membership value of z is shown in table 1 and A(∨)B=[3∨(-2),5∨7] 0 membership function is shown in Figure 5. =[3, 7] C. Operation of fuzzy numbers Based on the extension principle, arithmetic operations on fuzzy numbers are defined by following: ~ ~ If M and N are fuzzy numbers, membership of ~ ~ M (∗)N is defined as follow: µ ~ ~ (z) = sup min µ ~ (x),µ ~ (y) (1) ~ M (∗)N { M N } Figure 3 Membership function of z=x∗y M Where * stands for any of the four arithmetic operations. ~ ~ ~ ~ µ M (+)N (z) = sup min{µ M (x),µ N (y)} z=x+ y ~ ~ ~ ~ µ M (−)N (z) = sup min{µ M (x),µ N (y)} z=x− y ~ ~ ~ ~ µ M (/)N (z) = sup min{µ M (x),µ N (y)} z=x / y ~ Figure 4 Membership function of N © 2009 ACADEMY PUBLISHER 334 JOURNAL OF SOFTWARE, VOL.