Chapter 5 Fuzzy Number

5.1 Concept of Fuzzy Number 5.1.1 Interval interval A = [a1, a3] a1, a3  , a1 < a3 Expressing the interval as membership function a1 a3 A(x) x 1

5.1.2 Fuzzy Number Definition(Fuzzy number) convex fuzzy set normalized fuzzy set it’s membership function is piecewise continuous It is defined in the real number -cut interval of fuzzy number A = [a1(), a3()] ( < )  (a1()  a1(), a3()  a3())

5.1.2 Fuzzy Number a1 a3 A(x) x 1 a2 a1(0) A(x) x 1   a1() Fuzzy Number A = [a1, a2, a3] -cut of fuzzy number (’ < )  (A  A)

5.1.3 Operation of Interval operations of interval a1, a3, b1, b3  , A = [a1, a3], B = [b1, b3] Addition [a1, a3] (+) [b1, b3] = [a1 + b1, a3 + b3] Subtraction [a1, a3] (–) [b1, b3] = [a1 – b3, a3 – b1] Multiplication [a1, a3] () [b1, b3] = [a1  b1  a1  b3  a3  b1  a3  b3 , a1  b1  a1  b3  a3  b1  a3  b3]

5.1.3 Operation of Interval operations of interval Division [a1, a3] (/) [b1, b3] = [a1 / b1  a1 / b3  a3 / b1  a3 / b3 , a1 / b1  a1 / b3  a3 / b1  a3 / b3] excluding the case b1 = 0 or b3 = 0 Inverse interval [a1, a3]–1 = [1 / a1  1 / a3, 1 / a1  1 / a3] excluding the case a1 = 0 or a3 = 0

5.1.3 Operation of Interval Example 5.1 A = [3, 5], B = [–2, 7]

5.2 Operation of Fuzzy Number 5.2.1 Operation of -cut Interval -cut interval of fuzzy number A = [a1, a3] A = [a1(), a3()],   [0, 1], a1, a3, a1(), a3()   [a1(), a3()] (+) [b1(), b3()] = [a1() + b1(), a3() + b3()] [a1(), a3()] (–) [b1(), b3()] = [a1() – b3(), a3() – b1()]

5.2.2 Operation of Fuzzy Number Addition: A (+) B Subtraction: A (–) B Multiplication: A () B Division: A (/) B

5.2.2 Operation of Fuzzy Number Minimum: A () B Maximum: A () B multiply a scalar value to the interval a  a[b1, b3] = [a  b1  a  b3, a  b1  a  b3] multiply scalar value to -cut interval a[b1(), b3()] = [a  b1()  a  b3(), a  b1()  a  b3()]

5.2.3 Examples of Fuzzy Number Operation Example 5.3 : Addition A(+)B A = {(2, 1), (3, 0.5)}, B = {(3, 1), (4, 0.5)} for all x  A, y  B, z  A(+)B i. for z < 5, A(+)B(z) = 0 ii. z = 5 results from x + y = 2 + 3 A(2)  B(3) = 1  1 = 1 iii. z = 6 results from x + y = 3 + 3 or x + y = 2 + 4 A(3)  B(3) = 0.5  1 = 0.5 A(2)  B(4) = 1  0.5 = 0.5

5.2.3 Examples of Fuzzy Number Operation iv. z = 7 results from x + y = 3 + 4 A(3)  B(4) = 0.5  0.5 = 0.5 v. for z > 7 A(+)B(z) = 0 A(+)B = {(5, 1), (6, 0.5), (7, 0.5)} A(x) 1 3 0.5 2 A (+) B(x) B(x) 1 3 0.5 4 1 0.5 5 6 7 (a) Fuzzy set A (c) Fuzzy set A (+) B (b) Fuzzy number B

5.2.3 Examples of Fuzzy Number Operation Example 5.5 : Subtraction A()B A = {(2, 1), (3, 0.5)}, B = {(3, 1), (4, 0.5)} A()B = {(-2, 0.5), (-1, 1), (0, 0.5)} A () B(x) A(x) 1 3 0.5 2 B(x) 1 3 0.5 4 1 0.5 2 1

5.2.3 Examples of Fuzzy Number Operation Example 5.6 : Max operation A()B A = {(2, 1), (3, 0.5)}, B = {(3, 1), (4, 0.5)} A()B = {(3, 1), (4, 0.5)}

5.3 Triangular fuzzy number 5.3.1 Definition of Triangular fuzzy number A = (a1, a2, a3) membership functions a1 a3 A(x) x 1 a2 Triangular fuzzy number A = (a1, a2, a3)

5.3 Triangular fuzzy number -cut interval of triangular fuzzy number interval Aa from a1() = (a2 – a1) + a1 a3() = (a3  a2) + a3 thus A = [a1(), a3()] = [(a2  a1) + a1, (a3  a2) + a3]

5.3 Triangular fuzzy number Example 5.7 triangular fuzzy number A = (5, 1, 1) 6 5 4 3 2 1 1 2 0.5 A0.5  = 0.5 cut of triangular fuzzy number A = (5, 1, 1)

5.3 Triangular fuzzy number Example 5.7(2) -cut interval from this fuzzy number A = [a1(), a3()] = [4  5, 2 + 1] If  = 0.5, substituting 0.5 for , we get A0.5 A0.5 = [a1(0.5), a3(0.5)] = [3, 0]

5.3.2 Operation of Triangular Fuzzy Number Properties of operations on triangular fuzzy number The results from addition or subtraction between triangular fuzzy numbers result also triangular fuzzy numbers. The results from multiplication or division are not triangular fuzzy numbers. Max or min operation does not give triangular fuzzy number

5.3.2 Operation of Triangular Fuzzy Number triangular fuzzy numbers A and B are defined A = (a1, a2, a3), B = (b1, b2, b3) Addition Subtraction Symmetric image (A) = (a3, a2, a1)

5.3.2 Operation of Triangular Fuzzy Number Example 5.8 A = (3, 2, 4), B = (1, 0, 6) A (+) B = (4, 2, 10) A () B = (9, 2, 5) 3 1 2 4 1 0.5 6 A B 4 2 1 0.5 10 A (+) B 9 2 1 0.5 5 A () B (a) Triangular fuzzy number A, B (b) A (+) B (c) A () B

5.3.2 Operation of Triangular Fuzzy Number Example 5.9 triangular fuzzy numbers A and B : A = (3, 2, 4), B = (1, 0, 6) -level intervals from -cut operation

5.3.2 Operation of Triangular Fuzzy Number Example 5.9 (cont’) A (+) B = [6  4, 8 + 10]  = 0 and  = 1, A0 (+) B0 = [4, 10] A1 (+) B1 = [2, 2] = 2 A () B = [11  9, 3 + 5]  = 0 and  = 1 A0 () B0 = [9, 5] A1 () B1 = [2, 2] = 2

5.3.3 Operation of general fuzzy numbers Example 5.10 Addition A () B A = (3, 2, 4), B = (1, 0, 6)

5.3.3 Operation of general fuzzy numbers Example 5.10 Addition A () B (cont’) think when z = 8. Addition to make z = 8 is possible for following cases 2 + 6, 3 + 5, 3.5 + 4.5, 

5.3.3 Operation of general fuzzy numbers Example 5.11 Multiplication A () B A = (1, 2, 4), B = (2, 4, 6)

5.3.3 Operation of general fuzzy numbers Example 5.11 Multiplication A () B (cont’) z = x  y = 8 is possible when z = 2  4 or z = 4  2 z = x  y = 12, 3  4, 4  3, 2.5  4.8, …

5.3.3 Operation of general fuzzy numbers Example 5.11 Multiplication A () B (cont’) From this kind of method membership function for all z  A () B A () B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0.5 A B Multiplication A () B of triangular fuzzy number

5.3.4 Approximation of Triangular Fuzzy Number Example 5.12 Approximation of multiplication A = (1, 2, 4), B=(2, 4, 6) When  = 0, A0()B0 = [2, 24] When  = 1, A0()B1 = [2+4+2, 4-20+24] = [8, 8] = 8 A () B  (2 , 8, 24)

5.3.4 Approximation of Triangular Fuzzy Number Example 5.13 Approximation of division When  = 0 When  = 1 approximated value of A () B

5.4 Other Types of Fuzzy Number 5.4.1 Trapezoidal Fuzzy Number Definition(Trapezoidal fuzzy number) A = (a1, a2, a3, a4) membership function a1 a4 A(x) x 1 a2 a3 Trapezoidal fuzzy number A = (a1, a2, a3, a4)

5.4.2 Operations of Trapezoidal Fuzzy Number 1)Addition and subtraction between fuzzy numbers become trapezoidal fuzzy number. 2)Multiplication, division, and inverse need not be trapezoidal fuzzy number. 3)Max and Min of fuzzy number is not always in the form of trapezoidal fuzzy number.

5.4.2 Operations of Trapezoidal Fuzzy Number Addition Subtraction

5.4.2 Operations of Trapezoidal Fuzzy Number Example 5.14 Multiplication A = (1, 5, 6, 9), B = (2, 3, 5, 8) A = [4 + 1, –3 + 9], B = [ + 2, –3 + 8] When  = 0 When  = 1 approximated value

5.4.2 Operations of Trapezoidal Fuzzy Number Example 5.14 Multiplication(Con’t) 10 20 30 40 50 60 70 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (1,5, 6, 9) = A (2,3, 5, 8) = B (1,5, 6, 9) () (2,3,5,8) = A () B (2,15, 30, 72) x Multiplication of trapezoidal fuzzy number A () B

5.4.2 Operations of Trapezoidal Fuzzy Number flat fuzzy number m1, m2  , m1 < m2 A(x) = 1, m1  x  m2 In this case, membership function in x < m1 and x < m2 need not be a line 1 m1 m2 Flat fuzzy number

5.4.3 Bell Shape Fuzzy Number Bell shape fuzzy number is often used in practical applications and its function is defined as follows is the mean of the function, is the standard deviation Bell shape fuzzy number