Multiplication Operation on Fuzzy Numbers

JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 331

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.

(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 ]

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

TABLE I.

z0 AND ITS MEMBERSHIP VALUE

z0 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9

vmax 0.0065 0.0110 0.0183 0.0300 0.0482 0.0759 0.1172 0.1767 0.2590 0.3679

z0 -8 -7 -6 -5 -4 -3 -2 -1 0 1

vmax 0.5034 0.6590 0.8171 0.9458 1.0000 0.9307 0.7095 0.3679 0.0183 0.0067

Figure 6 Triangular fuzzy number A = (a, b, c) ~ ~ ~ ~ ~ Suppose triangular fuzzy numbers M and N are Figure 5 Membership function of Q = M (×)N defined as, ~ ~ IV. ANALYTICAL METHOD M = (a1 ,b1 ,c1 ) , N = (a2 ,b2 ,c2 )

A. Decompose Theorem Same important properties of operations on triangular fuzzy number are summarized. It is not easy to solve nonlinear programming. We can use decompose theorem to calculate multiplication (i) The results from addition or subtraction operation. between triangular fuzzy numbers result also triangular fuzzy numbers. µ ~ ~ (z) = α[M (∗)N] M (∗)N ∫ α (ii) The results from multiplication or division are α∈[0,1] (5) not triangular fuzzy numbers. = α([m L ,m R ](∗)[n L ,n R ]) ∫ α α α α (iii) Max or min operation does not give triangular α∈[0,1] fuzzy number. B. Multiplication of Triangular Fuzzy Number ~ ~ Among the various shapes of fuzzy number, triangular M (+)N = (a1 + a2 ,b1 + b2 ,c1 + c2 ) fuzzy number (TFN) is the most popular one. It is a fuzzy ~ ~ M (−)N = (a1 − c2 ,b1 − b2 ,c1 − a2 ) number represented with three points as follows: ~ A = (a, b, c) − (N ) = (−c2 ,−b2 ,−a2 ) This representation is interpreted as membership L R function (Figure 6). M α = [mα , mα ] = [(b1 − a1 )α + a1 ,(b1 − c1 )α + c1 ] L R ⎧0 x ≤ a Nα = [nα , nα ] = [(b2 − a2 )α + a2 ,(b2 − c2 )α + c2 ] ⎪ ~ ~ ~ x − a α -cuts of Q = M (×)N is ⎪ a < x ≤ b L R L R L R ⎪b − a [mα , mα ]×[nα , nα ] = [qα , qα ] u A (x) = ⎨ (6) c − x Where L L L L R R L R R ⎪ b < x ≤ c qα = min{mα nα , mα nα , mα nα , mα nα } ⎪ R L L L R R L R R c − b and q = max m n , m n , m n , m n . ⎪ α { α α α α α α α α } ⎩0 x > c We suppose c1 ≥ b1 ≥ a1 and c2 ≥ b2 ≥ a2 firstly. L mα = (b1 − a1 )α + a1 ,

R R L L L R R L R R mα = (b1 − c1 )α + c1 , qα = max{mα nα ,mα nα ,mα nα ,mα nα } n L = (b − a )α + a , R R α 2 2 2 = mα nα n R = (b − c )α + c 2 α 2 2 2 = (b1 − c1 )(b2 − c2 )α + (c1b2 + c2b1 − 2c1c 2 )α q L = min{m L n L ,m L n R ,m R n L ,m R n R } α α α α α α α α α + c1c2 = m L n L L α α Substituting qα = z , we get 2 = (b1 − a1 )(b2 − a2 )α + (a1b2 + a2b1 − 2a1a 2 )α

+ a1a2 − (a b + a b − 2a a ) + (a b − a b )2 + 4(b − c )(b − a )z α = 1 2 2 1 1 2 1 2 2 1 1 1 2 2 （Omit α < 0 ） 2(b1 − a1 )(b2 − a2 ) R Substituting qα = z , we get − (c b + c b − 2c c ) − (c b − c b ) 2 + 4(b − c )(b − c )z α = 1 2 2 1 1 2 1 2 2 1 1 1 2 2 （Omit α > 1） 2(b1 − c1 )(b2 − c2 ) ~ ~ ~ Hence membership function of Q = M (×)N is ⎧ 2 − (a1b2 + a2b1 − 2a1a2 ) + (a1b2 − a2b1 ) + 4(b1 − c1 )(b2 − a2 )z ⎪ a1a2 ≤ z ≤ b1b2 ⎪ 2(b1 − a1 )(b2 − a2 ) ⎪ 2 ⎪− (c1b2 + c2b1 − 2c1c2 ) − (c1b2 − c2b1 ) + 4(b1 − c1 )(b2 − c2 )z ~ µQ (z) = ⎨ b1b2 ≤ z ≤ c1c2 ⎪ 2(b1 − c1 )(b2 − c2 ) （7） ⎪0 otherwise ⎪ ⎪ ⎩ ~ ~ Example 3 Suppose 3 = (2,3,5) , 5 = (3,5,6) , ~ ~ ~ calculate membership function of Q = 3 × 5 .

Substituting a1 = 2 , b1 = 3 , c1 = 5 , a2 = 3 , b2 = 5 , c2 = 6 into equation（7）, the result is the following expression ⎧1 (−7 + 1+ 8z ) 6 ≤ z ≤ 15 ⎪4 ⎪ ⎪1 µ ~ (z) = (17 − 49 + 8z ) 15 < z ≤ 30 ~ Q ⎨ Figure 7 Membership function of 3 = (2,3,5) ⎪4 (8) ⎪0 otherwise ⎪ ⎩ ~ The membership functions of 3 = (2,3,5) and ~ 5 = (3,5,6) are shown in Figure 7 and Figure 8. The ~ ~ ~ membership function of Q = 3 × 5 is shown in Figure 9.

~ Figure 8 Membership function of 5 = (3,5,6)

2 ⎪⎧e −(2− −z ) z ≤ 0 u ~ (z) = Q ⎨ −4−z ⎩⎪e z > 0 ~ ~ ~ The membership function of Q = 3 × 5 is shown in Figure 10.

~ ~ ~ Figure 9 Membership function of Q = 3 × 5 by analytical method The situation is complicated when it does not satisfy c1 ≥ b1 ≥ a1 and c2 ≥ b2 ≥ a2 .The exact analytic equations are hard to get. C. Multiplication of General Fuzzy Number As multiplication of fuzzy number, we can also multiply scalar value to α -cuts interval of fuzzy number. We demonstrate the method by example 2. ~ α -cuts of M is

L R ~ ~ ~ [mα , mα ] = [2 − − lnα ,2 + − lnα ] Figure 10 Membership function of Q = M (×)N by analytical ~ α -cuts of N is method L R [nα , nα ] = [−2 − − lnα ,−2 + − lnα ] V. COMPUTER DRAWING METHOD ~ ~ ~ α -cuts of Q = M (×)N is When it does not satisfy c1 ≥ b1 ≥ a1 and [m L , m R ]×[n L , n R ] = [q L , q R ] α α α α α α c2 ≥ b2 ≥ a2 , we can use Computer drawing method. Where The method based on decompose theorem is as follows: L L L L R R L R R Step 1 Set ; qα = min{}mα nα , mα nα , mα nα , mα nα α = 0.01 , Step 2 If α > 1, stop. The grey sector is the membership = −4 + lnα − 4 − lnα ~ R L L L R R L R R function of Q . Otherwise go to step 3; qα = max{mα nα , mα nα , mα nα , mα nα } ~ Step 3 Calculate α -cuts of M ([m L , m R ] ) and α -cuts −4 α α ⎪⎧− lnα − 4 0 ≤ α ≤ e . ~ = ⎨ of N ([n L , n R ]); ⎪ −4 α α ⎩− 4 + lnα + 4 − lnα e < α ≤ 1 Step 4 Calculate Substituting q L = z , we get L L L L R R L R R α qα = min{mα nα , mα nα , mα nα , mα nα } , R L L L R R L R R − lnα = −2 + − z (Omit − lnα < 0 ) qα = max{mα nα , mα nα , mα nα , mα nα }; Therefore ~ 2 Step 5 Set Q row axis and set membership function of α = e −(2− −z ) ~ Q column, then create a line from (q L ,α) to When 0 ≤ α ≤ e −4 , substituting R , we get α qα = z R (qα ,α) ; α = e−4−z −4 R Step 6 α ← a + 0.01, then go to step 2. When e < α ≤ 1, substituting qα = z , we get ~ −4 Example 4 Suppose − 1 = (−3,−1,2) (in Figure − lnα = 2 − − z (Omit α ≤ e ) ~ Therefore 11), 1 = (−2,1,4) (in Figure 12), then the membership 2 ~ ~ ~ ⎪⎧e −(2− −z ) z ≤ 0 function of Q = − 1 × 1 is shown in Figure 13. α = ⎨ −4−z ⎩⎪e z > 0 ~ ~ ~ Hence membership function of Q = M (×)N is

Step 4 Create a line from (zi ,0) to (zi ,v) , then go to step 2. ~ Example 5 Suppose 3 = (2,3,5) and ~ 5 = (3,5,6) , then the membership function of ~ ~ ~ Q = 3 × 5 is shown in Figure 14.

~ Figure11 Membership function of − 1 = (−3,−1,2)

~ ~ ~ Figure 14 Membership function of Q = 3 × 5 by computer simulation method

VII. CONCLUSIONS Four methods for solving multiplication operation of ~ two fuzzy numbers are given. Each method has Figure 12 Membership function of 1 = (−2,1,4) advantages and disadvantages. The nonlinear programming method is a precise method also, but it only gets a membership value as given number and it is a difficult problem for solving nonlinear programming. The analytical method is most precise, but it is hard to α - cuts interval when the membership function is complicated. The computer drawing method is simple also, but it need calculate the α -cuts interval. The computer simulation method is the most simple, and it has wide applicability, but the membership function is rough.

ACKNOWLEDGMENT ~ ~ ~ Figure 13 Membership function of Q = − 1 × 1 by computer This work was partially supported by National Basic drawing method Research Program of Jiangsu Province University (08KJB520003) and the Open Project Program of the VI. COMPUTER SIMULATION METHOD State Key Lab of CAD&CG. In computer drawing method, it need calculate α -cuts REFERENCES interval. It is hard to calculate α -cuts interval when the membership function is complicate. The computer [1] L. A. Zadeh, “Fuzzy set”, Information and Control, simulation method needn’t calculate α -cuts interval. The vol.8,no.3, pp.338-353, 1965. [2] D. Dubois and H. Prade, “Operations on fuzzy numbers”. computer simulation method is as follows: International Journal of Systems Science, vol.9, Step 1 set i = 1 and simulation times N ; no.6,pp.613-626,1978. Step 2 If i > N , stop. The grey sector is the membership [3] S. Heilpern, “Representation and application of fuzzy ~ numbers”, Fuzzy sets and Systems, vol.91, no.2, pp.259- function of Q . Otherwise go to step 3; 268, 1997. [4] G. J. Klir, Fuzzy Sets: An Overview of Fundamentals, Step 3 Generate two random number xi and yi on the Applications, and Personal views. Beijing Normal University Press, 2000, pp.44-49. interval and interval . Calculate [a1 ,c1 ] [a2 ,c2 ] [5] R. Fuller and P. Majlender, “On weighted possibilistic z = x y , v = min(µ ~ (x ), µ ~ (y )) ; mean and variance of fuzzy numbers”, Fuzzy Sets and i i i M i N i Systems, vol.136, pp.363-374,2003. [6] G. J. Klir, “Fuzzy arithmetic with requisite constraints”. Fuzzy Sets System, vol. 91, pp. 165–175,1997.

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[8] G. X. Wang, Q. Zhang and X. J. Cui, “The discrete fuzzy numbers on a fixed set with finite support set”, IEEE International Conference on Cybernetics and Intelligent Systems, 2008, pp.812-817. Zaiyue Zhang was born in 1961, and received his M.S. [9] H. M. Lee and L. Lin, “Using weighted triangular fuzzy degree in mathematics in 1991 from the Department of numbers to evaluate the rate of aggregative risk in software Mathematics, Yangzhou Teaching College, and the Ph.D. development”, Proceedings of the 7th International degree in mathematics in 1995 from the Institute of Software, Conference on Machine Learning and Cybernetics, the Chinese Academy of Sciences. Now he is a professor of the ICMLC, vol. 7, 2008, pp.3762-3767. School of Computer Science and Technology, Jiangsu [10] X. W. Zhou, L. P. Wang and B. H. Zheng, “Ecological risk University of Science and Technology. His current research assessment of sediment pollution based on triangular fuzzy areas are recursion theory, knowledge representation and number”, Huanjing Kexue, vol. 29, no.11, 2008, pp.3206- knowledge reasoning. 3212(in Chinese).

[11] D. Sanchez, M. Delgado and M. A. Vila, “RL-numbers: An alternative to fuzzy numbers for the representation of imprecise quantities”, 2008 IEEE International Conference on Fuzzy Systems, 2008, pp.2058-2065. Cungen Cao was born in 1964, and received his M.S. degree in 1989 and Ph.D. degree in 1993 both in mathematics from the Institute of Mathematics, the Chinese Academy of Sciences. Now he is a professor of the Institute of Computing Technology, the Chinese Academy of Sciences. His research area is large scale knowledge processing.

Shang Gao was born in 1972, and received his M.S. degree in 1996 and Ph.D degree in 2006. He now works in school of