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Extension Principle
Adriano Cruz ©2002
NCE e IM/UFRJ
[email protected]
Fuzzy Numbers

A fuzzy number is fuzzy subset of the
universe of a numerical number.
– A fuzzy real number is a fuzzy subset of
the domain of real numbers.
– A fuzzy integer number is a fuzzy subset of
the domain of integers.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 2
Fuzzy Numbers - Example
u(x)
Fuzzy real number 10
5
10
15
x
u(x)
Fuzzy integer number 10
5
@2002 Adriano Cruz
10
15
NCE e IM - UFRJ
x
No. 3
Functions with Fuzzy Arguments

A crisp function maps its crisp input
argument to its image.

Fuzzy arguments have membership
degrees.

When computing a fuzzy mapping it is
necessary to compute the image and
its membership value.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 4
Crisp Mappings
X
@2002 Adriano Cruz
f(X)
NCE e IM - UFRJ
Y
No. 5
Functions applied to intervals

Compute the image of the interval.

An interval is a crisp set.
y
y=f(I)
I
@2002 Adriano Cruz
x
NCE e IM - UFRJ
No. 6
Mappings
f(X)
Y
X
Fuzzy
argument?
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 7
Extension Principle

Suppose that f is a function from X to Y
and A is a fuzzy set on X defined as
A = µA(x1)/x1 + µA(x2)/x2 + ... + µA(xn)/xn

The extension principle states that the
image of fuzzy set A under the mapping
f(.) can be expressed as a fuzzy set B.
B = f(A) = µA(x1)/y1 + µA(x2)/y2 + ... + µA(xn)/yn
where yi=f(xi)
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 8
Extension Principle

If f(.) is a many-to-one mapping, then
there exist x1, x2 X, x1  x2, such that
f(x1)=f(x2)=y*, y*Y.

The membership grade at y=y* is the
maximum of the membership grades at x1
and x2

more generally, we have
 B ( y )  max  A ( x)
x  f 1 ( y )
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 9
Monotonic Continuous Functions

For each point in the interval
– Compute the image of the interval.
– The membership degrees are carried
through.
I
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 10
Monotonic Continuous Functions
y
y
x
u(y)
u(x)
x
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 11
Monotonic Continuous Ex.

Function: y=f(x)=0.6*x+4

Input: Fuzzy number - around-5
– Around-5 = 0.3 / 3 + 1.0 / 5 + 0.3 / 7

f(around-5) = 0.3/f(3) + 1/f(5) + 0.3/f(7)

f(around-5) = 0.3/0.6*3+4 + 1/ 0.6*5+4 + 0.3/ 0.6*7+4

f(around-5) = 0.3/5.8 + 1.0/7 + 0.3/8.2
I
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 12
Monotonic Continuous Ex.
1
0.3
8.2
f(x)
10
5.8
4
5
10
x
u(x)
1
0.3
3
@2002 Adriano Cruz
NCE e IM - UFRJ
5
7
x
No. 13
Nonmonotonic Continuous
Functions

For each point in the interval
– Compute the image of the interval.
– The membership degrees are carried
through.
– When different inputs map to the same
value, combine the membership degrees.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 14
Nonmonotonic Continuous
Functions
y
y
x
u(y)
u(x)
x
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 15
Nonmonotonic Continuous Ex.


Function: y=f(x)=x2-6x+11
Input: Fuzzy number - around-4
Around-4 = 0.3/2+0.6/3+1/4+0.6/5+0.3/6
y = 0.3/f(2)+0.6/f(3)+1/f(4)+0.6/f(5)+0.3/f(6)
y = 0.3/3+0.6/2+1/3+0.6/6+0.3/11
y = 0.6/2+(0.3 v 1)/3+0.6/6+0.3/11
y = 0.6/2 + 1/3 + 0.6/6 + 0.3/11
I
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 16
Nonmonotonic Continuous
Functions
1 v 0.3
y
y
x
u(y) 1 0.6
0.3
u(x)
1
0.6
0.3
x
2 3 4 5 6
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 17
Function Example 1

Consider
2
x
y  f ( x)  1 
4

Consider fuzzy set
~
A   1 2 | x | / x |  2  x  2

Result
@2002 Adriano Cruz
~
~
B  f ( A)    B ( y ) / y
NCE e IM - UFRJ
Y
No. 18
Function Example 2

Result according to the principle
~
~
B  f ( A)    B ( y ) / y    A ( x) / f ( x)
x  2 1 y
Y
2
Y
 A ( x)  1 2 | x |
 A ( x) | 1  y |
2
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 19
Function Example 3
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 20
Extension Principle



Let f be a function with n arguments that
maps a point in X1xX2x...xXn to a point in Y
such that y=f(x1,…,xn).
Let A1x…xAn be fuzzy subsets of
X1xX2x...xXn
The image of A under f is a subset of Y
defined by
1

[


(
x
)]
if
f
( y)  0
i Ai
i

 B ( y )  ( x1 ,xn ),( x1 ,, xn ) f 1 ( y )

0
if f 1 ( y )  0
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 21
Arithmetic Operations



Applying the extension principle to
arithmetic operations it is possible to
define fuzzy arithmetic operations
Let x and y be the operands, z the
result.
Let A and B denote the fuzzy sets that
represent the operands x and y
respectively.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 22
Fuzzy addition

Using the extension principle fuzzy
addition is defined as
 A B ( z )   (  A ( x)   B ( y ))
x, y
x y z
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 23
Fuzzy addition - Examples

A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5

B =(11~)= 0.5/10 + 1/11 + 0.5/12

A+B=(0.3^0.5)/(1+10) + (0.6^0.5)/(2+10) +
(1^0.5)/(3+10) + (0.6^0.5)/(4+10) + (0.3^0.5)/(5+10)
+ (0.3^1)/(1+11) + (0.6^1)/(2+11) + (1^1)/(3+11) +
(0.6^1)/(4+11) + (0.3^1)/(5+11)
+( 0.3^0.5)/(1+12) + (0.6^0.5)/(2+12) +
(1^0.5)/(3+12) + (0.6^0.5)/(4+12) + (0.3^0.5)/(5+12)
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 24
Fuzzy addition - Examples




A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5
B =(11~)= 0.5/10 + 1/11 + 0.5/12
Getting the minimum of the
membership values
A+B=0.3/11 + 0.5/12 + 0.5/13 + 0.5/14 + 0.3/15 +
0.3/12 + 0.6/13 + 1/14 + 0.6/15 + 0.3/16 + 0.3/13 +
0.5/14 + 0.5/15 + 0.5/16 + 0.3/17
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 25
Fuzzy addition - Examples

A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5

B =(11~)= 0.5/10 + 1/11 + 0.5/12

Getting the maximum of the duplicated
values

A+B=0.3/11 + (0.5 V 0.3)/12 + (0.5 V 0.6 V 0.3)/13 +
(0.5 V 1 V 0.5)/14 + (0.3 V 0.6 V 0.5)/15 + (0.3 V
0.5)/16 + 0.3/17

A+B=0.3 / 11 + 0.5 / 12 + 0.6 / 13 + 1 / 14 + 0.6 / 15
+ 0.5 / 16 + 0.3 / 17
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 26
Fuzzy addition
A,
x=3
B,
y=11
C,
x=14
0.6
0.5
0.3
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 27
Fuzzy Arithmetic

Using the extension principle the remaining
fuzzy arithmetic fuzzy operations are defined
as:  A B ( z )    A ( x )   B ( y )
x, y
x y z
 A*B ( z )    A ( x)   B ( y )
x, y
x* y  z
 A / ( z )    A ( x)   B ( y )
@2002 Adriano Cruz
x, y
x / yz
NCE e IM - UFRJ
No. 28