1. Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ Adriano@nce.ufrj.br

2. @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.

3. @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

4. @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 A fuzzy arguments have membership degrees. n When computing a fuzzy mapping it is necessary to compute the image and its membership value.

5. @2002 Adriano Cruz NCE e IM - UFRJNo. 5 Functions applied to intervals n Compute the image of the interval. n An interval is a crisp set. x y I y=f(I)

6. @2002 Adriano Cruz NCE e IM - UFRJNo. 6 Monotonic Continuous Functions n For each point in the interval –Compute the image of the interval. –The membership degrees are carried through. I

7. @2002 Adriano Cruz NCE e IM - UFRJNo. 7 Monotonic Continuous Functions x y x y u(x) u(y)

8. @2002 Adriano Cruz NCE e IM - UFRJNo. 8 Monotonic Continuous Ex. n Function: y=f(x)=0.6*x+4 n Input: Fuzzy number - around-5 –Around-5 = 0.3/3 + 1.0/5 + 0.3/7 n 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

9. @2002 Adriano Cruz NCE e IM - UFRJNo. 9 Monotonic Continuous Ex. f(x) x 510 u(x) x 753 10 4 1 0.3 1 8.2 5.8

10. @2002 Adriano Cruz NCE e IM - UFRJNo. 10 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.

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

12. @2002 Adriano Cruz NCE e IM - UFRJNo. 12 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 + 1/3 + 0.6/6 + 0.3/11 I

13. @2002 Adriano Cruz NCE e IM - UFRJNo. 13 Nonmonotonic Continuous Functions x y x y u(x) u(y) 23 456 0.3 0.6 1 0.30.61

14. @2002 Adriano Cruz NCE e IM - UFRJNo. 14 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 V defined by

15. @2002 Adriano Cruz NCE e IM - UFRJNo. 15 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.

16. @2002 Adriano Cruz NCE e IM - UFRJNo. 16 Fuzzy addition n Using the extension principle fuzzy addition is defined as

17. @2002 Adriano Cruz NCE e IM - UFRJNo. 17 Fuzzy addition - Examples n A = 0.3/1 + 0.6/2 +1/3 + 0.6/4 + 0.3/5 n B = 0.5/10 + 1/11 + 0.5/12 n Getting the minimum of the membership values n 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 n Getting the maximum of the duplicates n A+B= 0.3/11 + 0.5/12 + 0.6/13 + 1/14 + 0.6/15 + 0.5/16 + 0.3/17

18. @2002 Adriano Cruz NCE e IM - UFRJNo. 18 Fuzzy addition A, x=3 B, y=11 0.3 0.6 0.5 C, x=14

19. @2002 Adriano Cruz NCE e IM - UFRJNo. 19 Fuzzy Arithmetic n Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as: