Transcript extensionv1

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.

A 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
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. 5
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. 6
Monotonic Continuous Functions
y
y
x
u(y)
u(x)
x
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 7
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) = 0.3/5.8 + 1.0/7 + 0.3/8.2
I
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 8
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. 9
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. 10
Nonmonotonic Continuous
Functions
y
y
x
u(y)
u(x)
x
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 11
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. 12
Nonmonotonic Continuous
Functions
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. 13
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 V
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. 14
Arithmetic Operations
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

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. 15
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. 16
Fuzzy addition - Examples
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


A = 0.3/1 + 0.6/2 +1/3 + 0.6/4 + 0.3/5
B = 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 + 1.0/14 + 0.6/15 +
0.3/16 + 0.3/13 + 0.5/14 + 0.5/15 + 0.5/16 +
0.3/17
Getting the maximum of the duplicates
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. 17
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. 18
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 )
x, y
x / yz
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 19