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Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ

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@2002 Adriano Cruz NCE e IM - UFRJNo. 2 Fuzzy Numbers n 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.

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@2002 Adriano Cruz NCE e IM - UFRJNo. 3 Fuzzy Numbers - Example u(x) x51015 Fuzzy real number 10 u(x) x51015 Fuzzy integer number 10

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@2002 Adriano Cruz NCE e IM - UFRJNo. 4 Functions with Fuzzy Arguments n A crisp function maps its crisp input argument to its image. n Fuzzy arguments have membership degrees. n When computing a fuzzy mapping it is necessary to compute the image and its membership value.

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@2002 Adriano Cruz NCE e IM - UFRJNo. 5 Crisp Mappings X Y f(X)

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@2002 Adriano Cruz NCE e IM - UFRJNo. 6 Functions applied to intervals n Compute the image of the interval. n An interval is a crisp set. x y I y=f(I)

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@2002 Adriano Cruz NCE e IM - UFRJNo. 7 Mappings X Y f(X) Fuzzy argument?

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@2002 Adriano Cruz NCE e IM - UFRJNo. 8 Extension Principle fXY AX n Suppose that f is a function from X to Y and A is a fuzzy set on X defined as A = µ A (x 1 )/x 1 + µ A (x 2 )/x µ A (x n )/x n A f(.)B n 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) = µ A (x 1 )/y 1 + µ A (x 2 )/y µ A (x n )/y n y i =f(x i ) where y i =f(x i )

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@2002 Adriano Cruz NCE e IM - UFRJNo. 9 Extension Principle many-to-one mapping n If f(.) is a many-to-one mapping, then there exist x 1, x 2 X, x 1 x 2, such that f(x1)=f(x 2 )=y*, y* Y. n The membership grade at y=y* is the maximum of the membership grades at x 1 and x 2 n more generally, we have

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@2002 Adriano Cruz NCE e IM - UFRJNo. 10 Monotonic Continuous Functions n For each point in the interval –Compute the image of the interval. –The membership degrees are carried through. I

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@2002 Adriano Cruz NCE e IM - UFRJNo. 11 Monotonic Continuous Functions x y x y u(x) u(y)

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@2002 Adriano Cruz NCE e IM - UFRJNo. 12 Monotonic Continuous Ex. n Function: y=f(x)=0.6*x+4 n Input: Fuzzy number - around-5 –Around-5 = 0.3 / / / 7 n f(around-5) = 0.3/f(3) + 1/f(5) + 0.3/f(7) n f(around-5) = 0.3/0.6* / 0.6* / 0.6*7+4 n f(around-5) = 0.3/ / /8.2 I

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@2002 Adriano Cruz NCE e IM - UFRJNo. 13 Monotonic Continuous Ex. f(x) x 510 u(x) x

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@2002 Adriano Cruz NCE e IM - UFRJNo. 14 Nonmonotonic Continuous Functions n 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.

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@2002 Adriano Cruz NCE e IM - UFRJNo. 15 Nonmonotonic Continuous Functions x y x y u(x) u(y)

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@2002 Adriano Cruz NCE e IM - UFRJNo. 16 Nonmonotonic Continuous Ex. n Function: y=f(x)=x 2 -6x+11 n 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

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@2002 Adriano Cruz NCE e IM - UFRJNo. 17 Nonmonotonic Continuous Functions x y x y u(x) u(y) v

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@2002 Adriano Cruz NCE e IM - UFRJNo. 18 Function Example 1 n Consider n Consider fuzzy set n Result

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@2002 Adriano Cruz NCE e IM - UFRJNo. 19 Function Example 2 n Result according to the principle

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@2002 Adriano Cruz NCE e IM - UFRJNo. 20 Function Example 3

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@2002 Adriano Cruz NCE e IM - UFRJNo. 21 Extension Principle n Let f be a function with n arguments that maps a point in X 1 xX 2 x...xX n to a point in Y such that y=f(x 1,…,x n ). n Let A 1 x…xA n be fuzzy subsets of X 1 xX 2 x...xX n n The image of A under f is a subset of Y defined by

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@2002 Adriano Cruz NCE e IM - UFRJNo. 22 Arithmetic Operations n Applying the extension principle to arithmetic operations it is possible to define fuzzy arithmetic operations n Let x and y be the operands, z the result. n Let A and B denote the fuzzy sets that represent the operands x and y respectively.

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@2002 Adriano Cruz NCE e IM - UFRJNo. 23 Fuzzy addition n Using the extension principle fuzzy addition is defined as

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@2002 Adriano Cruz NCE e IM - UFRJNo. 24 Fuzzy addition - Examples n A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5 n B =(11~)= 0.5/10 + 1/ /12 n 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)

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@2002 Adriano Cruz NCE e IM - UFRJNo. 25 Fuzzy addition - Examples n A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5 n B =(11~)= 0.5/10 + 1/ /12 n Getting the minimum of the membership values n A+B=0.3/ / / / / / /13 + 1/ / / / / / / /17

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@2002 Adriano Cruz NCE e IM - UFRJNo. 26 Fuzzy addition - Examples n A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5 n B =(11~)= 0.5/10 + 1/ /12 n Getting the maximum of the duplicated values n 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 n A+B=0.3 / / / / / / / 17

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@2002 Adriano Cruz NCE e IM - UFRJNo. 27 Fuzzy addition A, x=3 B, y= C, x=14

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@2002 Adriano Cruz NCE e IM - UFRJNo. 28 Fuzzy Arithmetic n Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as:

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