1. Chapter 5 Fuzzy Number

2. 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

3. 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())

4. 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. 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]

6. 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

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

8. 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()]

9. 5.2.2 Operation of Fuzzy Number
Addition: A (+) B
Subtraction: A (–) B
Multiplication: A () B
Division: A (/) B

10. 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()]

11. 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 = or x + y = 2 + 4
A(3)  B(3) = 0.5  1 = 0.5
A(2)  B(4) = 1  0.5 = 0.5

12. 5.2.3 Examples of Fuzzy Number Operation
iv. z = 7
results from x + y = A(3)  B(4) = 0.5  0.5 = 0.5
v. for z > 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

13. 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

14. 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)}

15. 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)

16. 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]

17. 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)

18. 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]

19. 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

20. 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)

21. 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

22. 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

23. 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

24. 5.3.3 Operation of general fuzzy numbers
Example Addition A () B
A = (3, 2, 4), B = (1, 0, 6)

25. 5.3.3 Operation of general fuzzy numbers
Example Addition A () B (cont’)
think when z = 8.
Addition to make z = 8 is possible for following cases
2 + 6, 3 + 5, , 

26. 5.3.3 Operation of general fuzzy numbers
Example Multiplication A () B
A = (1, 2, 4), B = (2, 4, 6)

27. 5.3.3 Operation of general fuzzy numbers
Example 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, …

28. 5.3.3 Operation of general fuzzy numbers
Example 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

29. 5.3.4 Approximation of Triangular Fuzzy Number
Example Approximation of multiplication
A = (1, 2, 4), B=(2, 4, 6)
When  = 0, A0()B0 = [2, 24]
When  = 1, A0()B1 = [2+4+2, ] = [8, 8] = 8
A () B  (2 , 8, 24)

30. 5.3.4 Approximation of Triangular Fuzzy Number
Example Approximation of division
When  = 0
When  = 1
approximated value of A () B

31. 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)

32. 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.

33. 5.4.2 Operations of Trapezoidal Fuzzy Number
Addition
Subtraction

34. 5.4.2 Operations of Trapezoidal Fuzzy Number
Example 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

35. 5.4.2 Operations of Trapezoidal Fuzzy Number
Example 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

36. 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

37. 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