Chapter 6

FUZZY FUNCTION

Chi-Yuan Yeh

Kinds of fuzzy function

• Crisp function with fuzzy constraint

• Fuzzy extension function which propagates the fuzziness of independent variable to dependent variable

• Fuzzifying function which blurs the image of a crisp independent variable

2

Function with fuzzy constraint

3

Fuzzy extension function

4

tes

Fuzzy extension function

5

x~

Single fuzzifying function

6

Single fuzzifying function

7

Single fuzzifying function

8

Fuzzy bunch of functions

9

Fuzzy bunch of functions

10

1

Fuzzy bunch of functions

11

Fuzzy bunch of functions

12

Fuzzy bunch of functions

13

Maximizing set

14

Minimizing set

Minimizing set of f is defined as the maximizing set of -f

15

Xx

xxxm

inf(f)sup(f)

)f(sup(f)

f)inf(f)sup(

f)inf())f(()(f

Maximizing set

16

Maximizing set

17

Maximizing set

18

Maximum value in crisp domain

19

Maximum value in crisp domain

20

Maximum value in crisp domain

21

Maximum value in fuzzy domain

22

Maximum value in fuzzy domain

23

Maximum value in fuzzy domain

24

Maximum value in fuzzy domain

25

Maximum value in fuzzy domain

26

Integration of fuzzying function

27

Integration of fuzzying function

28

Integration of fuzzying function

29

Integration of fuzzying function

30

Integration in fuzzy interval

31

y

Integration in fuzzy interval

32

33

Differentiation of crisp function in fuzzy interval

34

Differentiation of crisp function in fuzzy interval

35

36

0 0.5 x

Differentiation of fuzzifying function

function

37

Thanks for your attention!