Download A(+)B

Chapter 5
Fuzzy Number
1
5.1 Concept of Fuzzy Number
5.1.1 Interval


interval A = [a1, a3] a1, a3   , a1 < a3
Expressing the interval as membership function
A(x)
0,

 A ( x)  1,
0,

x  a1
a1  x  a3
x  a3
1
a1
a3
x
2
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())
3
5.1.2 Fuzzy Number
A(x)
A(x)
1

1
A

a1(0)
a1() a1()
a3() a3()
a3(0)
A = [a1(), a3()]
a1
a2
a3
Fuzzy Number A = [a1, a2, a3]
x
A = [a1(), a3()]
-cut of fuzzy number (’ < )  (A  A)
4
x
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
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
6
5.1.3 Operation of Interval
Example 5.1

A = [3, 5], B = [–2, 7]
A() B  [3  2, 5  7]  [1, 12]
A() B  [3  7, 5  (2)]  [ 4, 7]
A() B  [3  (2)  3  7  5  (2)  5  7, 3  (2)  ]
 [ 10, 35]
A(/) B  [3 /( 2)  3 / 7  5 /( 2)  5 / 7, 3 /( 2)  ]
 [ 2.5, 5 / 7]
 1
1
1
1  1 1
B  [2,7]  
 ,
    ,

 (2) 7 (2) 7   2 7 
1
1
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()]
8
5.2.2 Operation of Fuzzy Number
Addition:
A (+) B
 A( ) B ( z)   ( A ( x)   B ( y))
z  x y
Subtraction:
A (–) B
 A( ) B ( z)   ( A ( x)   B ( y))
z  x y
Multiplication:
A () B
 A() B ( z)   ( A ( x)   B ( y))
z  x y
Division:
A (/) B
 A(/)B ( z)   ( A ( x)   B ( y))
zx / y
9
5.2.2 Operation of Fuzzy Number
Minimum: A () B
 A( ) B ( z)   ( A ( x)   B ( y))
z  x y
Maximum: A () B
 A( ) B ( z)   ( A ( x)   B ( y))
z  x y
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()]
10
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
 A(  ) B (5)   (1)  1
5 2  3
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
 A(  ) B (6)  633 (0.5,0.5)  0.5
6 2 4
11
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
 A(  ) B (7)   (0.5)  0.5
7  3 4
v. for z > 7
A(+)B(z) = 0
A(+)B = {(5, 1), (6, 0.5), (7, 0.5)}
A(x)
A (+) B(x)
B(x)
1
1
1
0.
5
0.
5
0.
5
2 3
(a) Fuzzy set A
3 4
(b) Fuzzy number B
5 6 7
(c) Fuzzy set A (+) B 12
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)
B(x)
1
1
1
0.
5
0.5
0.
5
2 3
3 4
2 1 0
13
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)}
14
5.3 Triangular fuzzy number
5.3.1 Definition of Triangular fuzzy number
A = (a1, a2, a3)
membership functions
0,


 x  a1 ,

 a2  a1
 ( A) ( x)  
 a x
 3
,
 a3  a2


0,
 A ( x)
x  a1
1
a1  x  a2
a2  x  a3
x  a3
a1
a2
a3
x
Triangular fuzzy number A = (a1, a2, a3)
15
5.3 Triangular fuzzy number
-cut interval of triangular fuzzy number
interval Aa
from
a1( )  a1

a2  a1
a3  a3( )

a3  a 2
a1() = (a2 – a1) + a1
a3() = (a3  a2) + a3
thus
A = [a1(), a3()]
= [(a2  a1) + a1, (a3  a2) + a3]
16
5.3 Triangular fuzzy number
Example 5.7
triangular fuzzy number A = (5, 1, 1)
1
0.5
6 5 4 3 2 1
0
1
2
0,

x  5

,
 4
 ( A) ( x)  
1  x

,
 2

0,
x  5
 5  x  1
1  x  1
x 1
A0.5
 = 0.5 cut of triangular fuzzy number A = (5, 1, 1)
17
5.3 Triangular fuzzy number
Example 5.7(2)

-cut interval from this fuzzy number
x5

4
1 x

2

x  4  5

x  2  1
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]
18
5.3.2 Operation of Triangular Fuzzy
Number
Properties of operations on triangular
fuzzy number
1.
2.
3.
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
19
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
A() B  (a1 , a2 , a3 )( )(b1 , b2 , b3 )
 (a1  b1 , a2  b2 , a3  b3 )

Subtraction
A() B  (a1 , a2 , a3 )( )(b1 , b2 , b3 )
 (a1  b3 , a2  b2 , a3  b1 )

Symmetric image
(A) = (a3, a2, a1)
20
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)
A
1
1
1
B
0.5

3

1
0
0.5
2
4
6
(a) Triangular fuzzy number A, B

4
0
2
(b) A (+) B
A (+) B
10
0.5
9
0
2
A () B
5
(c) A () B
21
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
A
 [a1( ) , a 3( ) ]
B
 [b1( ) , b3( ) ]




[( a 2  a1 )  a1 ,(a 3  a 2 )  a 3 ]
[5  3,2  4]
[(b2  b1 )  b1 ,(b3 b 2 )  b3 ]
[  1,6  6]
22
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
23
5.3.3 Operation of general fuzzy
numbers
Example 5.10 Addition A () B
A = (3, 2, 4), B = (1, 0, 6)
0,


x  3,
2  3
 ( A) ( x)  
4  x

,
4  2

0,
x  3
3 x  2
2 x4
x4
0,


 y 1,
0 1
 ( B) ( y)  
6  y

,
6

0


0,
y  1
1  y  0
0 y6
y6
24
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, 
 A(  ) B 

8 x  y
[  A (2)   B (6),  A (3)   B (5),  A (3.5)   B (4.5), ]
 
[1  0, 0.5  1 / 6, 0.25  0.25,]
 
[0,1 / 6, 0.25,]
0,


z  4,
 6
 A(  ) B ( z )  
10  z

,
 8

0,
z  4
4 z  2
2  z  10
z  10
25
5.3.3 Operation of general fuzzy
numbers
Example 5.11 Multiplication A () B
A = (1, 2, 4), B = (2, 4, 6)
0,

 x  1,

 ( A) ( x)  
 1 x  2,
 2

0,
x 1
1 x  2
2 x4
x4
0,


 1 y  1,
2
 ( B) ( y)  
 1
 y  3,
 2

0,
y2
2 y4
4 y6
y6
26
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
 A() B   [  A (2)   B (4),  A (4)   B (2), ]
x y 8
  [1  1, 0  0, ]
 1

z = x  y = 12, 3  4, 4  3, 2.5  4.8, …
 A(  ) B 
 [  A (3)   B (4),  A (4)   B (3),  A (2.5)   B (4.8), ]
x y 12
  [0.5  1, 0  0.5, 0.75  0.6, ]
  [0.5, 0, 0.6, ]
 0.6
27
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  (2, 8, 24)
1
0.5
A B
A () B
0
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
Multiplication A () B of triangular fuzzy number
28
5.3.4 Approximation of Triangular Fuzzy
Number
Example 5.12 Approximation of multiplication
A = (1, 2, 4), B=(2, 4, 6)
A  [(2  1)  1,  (4  2)  4]  [  1,  2  4]
B  [(4  2)  2,  (6  4)  6]  [2  2,  2  6]
A () B
 [  1,
 2  4]()[ 2  2,
 2  6]
 [(  1)( 2  2), (2  4)( 2  6)]
 [2 2  4  2, 4 2  20  24]
When  = 0, A0()B0 = [2, 24]
When  = 1, A0()B1 = [2+4+2, 4-20+24] = [8, 8] = 8
A () B  (2 , 8, 24)
29
5.3.4 Approximation of Triangular Fuzzy
Number
Example 5.13 Approximation of division
A (/) B  [(  1) /( 2  6), (2  6) /( 2  2)]
When  = 0
A0 (/) B0
When  = 1
A1 (/) B1  [(1  1) /( 2  6), (2  4) /( 2  2)]
 [1 / 6, 4 / 2]
 [0.17, 2]
 [2 / 4, 2 / 4]
 0.5
approximated value of A () B
A(/) B

(0.17,
0.5,
2)
30
5.4 Other Types of Fuzzy Number
5.4.1 Trapezoidal Fuzzy Number
Definition(Trapezoidal fuzzy number)
A = (a1, a2, a3, a4)
membership function
 A ( x)
0,
 x  a1
,

a

a
 2 1
 A ( x)  1,
 a4  x
,

a

a
 4 3
0,
x  a1
a1  x  a2
a2  x  a3
a3  x  a4
x  a4
1
a1
a2
a3
a4 x
Trapezoidal fuzzy number A = (a1, a2, a3, a4)
31
5.4.2 Operations of Trapezoidal Fuzzy
Number
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.
32
5.4.2 Operations of Trapezoidal Fuzzy
Number
Addition
A() B  (a1 , a2 , a3 , a4 )( )(b1 , b2 , b3 , b4 )
 (a1  b1 , a2  b2 , a3  b3 , a4  b4 )
Subtraction
A() B  (a1  b4 , a2  b3 , a3  b2 , a4  b1 )
33
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]
A () B


[( 4  1)(  2), ( 3  9)( 3  8)]
[ 4 2  9  2, 9 2  51  72]
When  = 0
When  = 1
A0 () B0
A1 () B1
 [2,
72]
 [4  9  2,
 [15, 30]
9  51  72]
approximated value
A() B

[2,
15,
30,
72]
34
5.4.2 Operations of Trapezoidal Fuzzy
Number
Example 5.14 Multiplication(Con’t)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(1,5,
6,
9)
()
(2,3,5,8) = A () B
(2,15, 30, 72)
(1,5, 6, 9)
=A
(2,3, 5, 8)
=B
10
20
x
30
40
50
60
70
Multiplication of trapezoidal fuzzy number A () B
35
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
36
5.4.3 Bell Shape Fuzzy Number
Bell shape fuzzy number is often used in practical
applications and its function is defined as follows
2



(
x

m
)


f
 f ( x)  exp 

2
2



f


 f is the mean of the function,  f is the standard deviation
Bell shape fuzzy number
37