## Presentation on theme: "Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ"— Presentation transcript:

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

Fuzzy Numbers - Example
u(x) Fuzzy real number 10 5 10 15 x u(x) Fuzzy integer number 10 5 10 15 x @2002 Adriano Cruz NCE e IM - UFRJ

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

Crisp Mappings Y f(X) X @2002 Adriano Cruz NCE e IM - UFRJ

Functions applied to intervals
Compute the image of the interval. An interval is a crisp set. y y=f(I) I x @2002 Adriano Cruz NCE e IM - UFRJ

Mappings f(X) Y X Fuzzy argument? @2002 Adriano Cruz NCE e IM - UFRJ

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)/x 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)/y µA(xn)/yn where yi=f(xi) @2002 Adriano Cruz NCE e IM - UFRJ

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 @2002 Adriano Cruz NCE e IM - UFRJ

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

Monotonic Continuous Functions
y y x u(y) u(x) x @2002 Adriano Cruz NCE e IM - UFRJ

Monotonic Continuous Ex.
Function: y=f(x)=0.6*x+4 Input: Fuzzy number - around-5 Around-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* / 0.6* / 0.6*7+4 f(around-5) = 0.3/ / /8.2 I @2002 Adriano Cruz NCE e IM - UFRJ

Monotonic Continuous Ex.
f(x) 8.2 10 5.8 4 x 5 10 1 0.3 u(x) 1 0.3 x 3 5 7 @2002 Adriano Cruz NCE e IM - UFRJ

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

Nonmonotonic Continuous Functions
y y x u(y) u(x) x @2002 Adriano Cruz NCE e IM - UFRJ

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/ / /11 I @2002 Adriano Cruz NCE e IM - UFRJ

Nonmonotonic Continuous Functions
y y 1 v 0.3 x 1 0.6 0.3 u(y) u(x) 1 0.6 0.3 x 2 3 4 5 6 @2002 Adriano Cruz NCE e IM - UFRJ

Function Example 1 Consider Consider fuzzy set Result
@2002 Adriano Cruz NCE e IM - UFRJ

Function Example 2 Result according to the principle
@2002 Adriano Cruz NCE e IM - UFRJ

Function Example 3 @2002 Adriano Cruz NCE e IM - UFRJ

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 @2002 Adriano Cruz NCE e IM - UFRJ

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

Fuzzy addition Using the extension principle fuzzy addition is defined as @2002 Adriano Cruz NCE e IM - UFRJ

B =(11~)= 0.5/10 + 1/ /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

B =(11~)= 0.5/10 + 1/ /12 Getting the minimum of the membership values A+B=0.3/ / / / / / /13 + 1/ / / / / / / /17 @2002 Adriano Cruz NCE e IM - UFRJ

B =(11~)= 0.5/10 + 1/ /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)/ /17 A+B=0.3 / / / / / / / 17 @2002 Adriano Cruz NCE e IM - UFRJ

Fuzzy addition B, y=11 A, x=3 C, x=14 0.6 0.5 0.3 @2002 Adriano Cruz
NCE e IM - UFRJ

Fuzzy Arithmetic Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as: @2002 Adriano Cruz NCE e IM - UFRJ