Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ [email protected].

Extension PrincipleExtension Principle

Adriano Cruz ©2002NCE e IM/UFRJ

[email protected]

@2002 Adriano Cruz NCE e IM - UFRJ No. 2

Fuzzy NumbersFuzzy 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.


Fuzzy Numbers - ExampleFuzzy Numbers - Example

u(x)

x5 10 15

Fuzzy real number 10

u(x)

x5 10 15

Fuzzy integer number 10


Functions with Fuzzy ArgumentsFunctions 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.


Crisp MappingsCrisp Mappings

XXYYf(X)f(X)


Functions applied to intervalsFunctions applied to intervals

Compute the image of the interval.

An interval is a crisp set.

x

y

I

y=f(I)


MappingsMappings

XXYY

f(X)f(X)

Fuzzy argument?



Suppose that ff is a function from XX to YY and AA is a fuzzy set on XX defined as

A A = = µµAA((xx11)/)/xx1 1 + + µµAA((xx22)/)/xx2 2 + ... + + ... + µµAA((xxnn)/)/xxnn

The extension principle states that the image of fuzzy set AA under the mapping f(.)f(.) can be expressed as a fuzzy set BB.

B B = = ff((AA) = ) = µµAA((xx11)/)/yy1 1 + + µµAA((xx22)/)/yy2 2 + ... + + ... + µµAA((xxnn)/)/yynn

where yyii==ff((xxii))



If f(.) is a many-to-one mappingmany-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)(max)(

)(1xy A

yfxB


Monotonic Continuous FunctionsMonotonic Continuous Functions

For each point in the interval

– Compute the image of the interval.

– The membership degrees are carried through.

I


Monotonic Continuous FunctionsMonotonic Continuous Functions

x

y

x

y

u(x)u(y)


Monotonic Continuous Ex.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 + 0.3/8.2I


Monotonic Continuous Ex. Monotonic Continuous Ex.

f(x)

x5 10u(x)

x753

10

4

1 0.3

1

0.3

8.2

5.8


Nonmonotonic Continuous Nonmonotonic Continuous FunctionsFunctions

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.



x

y

x

y

u(x)u(y)


Nonmonotonic Continuous Ex.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



x

y

x

y

u(x)u(y)

2 3 4 5 6

0.30.61

0.30.61

1 0.3v


Function Example 1Function Example 1

Consider

Consider fuzzy set

Result

41)(

2xxfy

22|/||21~

xxxA

Y

B yyAfB /)()(~~



Result according to the principle

Y

A

Y

B xfxyyAfB )(/)(/)()(~~

|1|)(

||21)(

12

2

2

yx

xx

yx

A

A



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

0)(0

0)()]([)(

1

1

)(),,(),,( 111

yfif

yfifxy

iAiyfxxxxB

i

nn


Arithmetic OperationsArithmetic 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.


Fuzzy additionFuzzy addition

Using the extension principle fuzzy addition is defined as

zyx

yxBABA yxz

,

))()(()(


Fuzzy addition - ExamplesFuzzy 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)



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



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


Fuzzy additionFuzzy addition

A, x=3

B, y=11

0.3

0.60.5

C, x=14


Fuzzy ArithmeticFuzzy Arithmetic

Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as:

zyx

yxBAA

zyx

yxBABA

zyx

yxBABA

yxz

yxz

yxz

/

,/

*

,*

,

)()()(

)()()(

)()()(

Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ [email protected].

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Transcript of Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ [email protected].