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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 numeri...
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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, 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

µ B ( y ) = max µ 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 5

@2002 Adriano Cruz

NCE e IM - UFRJ

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) = ∫ µ B ( y ) / y NCE e IM - UFRJ

Y

No. 18

Function Example 2 

Result according to the principle ~

~

B = f ( A) = ∫ µ B ( y ) / y = ∫ µ A ( x) / f ( x) x = 2 1− y

Y 2

Y

µ A ( x) = 1 2 | x | µ 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  [ ⊗ µ ( x )] if f ( y) ≠ 0 i Ai i ⊕ µ B ( y ) = ( x1 ,Kxn ),( x1 ,K, 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

µ A+ B ( z ) = ∨ ( µ A ( x) ∧ µ 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: µ A− B ( z ) = ∨ µ A ( x ) ∧ µ B ( y ) x, y

x− y= z

µ A*B ( z ) = ∨ µ A ( x) ∧ µ B ( y ) x, y

x* y = z

µ A / ( z ) = ∨ µ A ( x) ∧ µ B ( y ) @2002 Adriano Cruz

x, y x / y= z NCE e IM - UFRJ

No. 28