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Fuzzy functions

Fuzzy functions

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Fuzzy functions

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  1. Fuzzy functions ผศ.ดร.สุพจน์ นิตย์สุวัฒน์

  2. Topics • Kinds of Fuzzy Function, • Fuzzy Extrema of Function, • Integration and Differenciation of Fuzzy Function,

  3. 1. Kinds of Fuzzy Function

  4. Fuzzy function can be classified into following three groups according to which aspect of the crisp function the fuzzy concept was applied. • (1) Crisp function with fuzzy constraint. • (2) Crisp function which propagates the fuzziness of independent variable to dependent variable. • (3) Function that is itself fuzzy. This fuzzifying function blurs the image of a crisp independent variable.

  5. 1.1 Function with Fuzzy Constraint Definition (Function with fuzzy constraint) • Let X and Y be crisp sets, and f be a crisp function. • and are fuzzy sets defined on universal sets X and Y respectively. • Then the function satisfying the condition • is called a function with constraints on fuzzy domain and fuzzy range .

  6. Example 1. • There is a function y = f(x) • Assume that the function f has fuzzy constraint like this, • “If x is a member of A, then y is a member in B” • “The membership degree of x for A is less than that of y for B” or

  7. The previous fuzzy constraints denote the sufficient fuzzy condition for y to be a member of B. • “If membership degree of x for A is , then that of y for B would be no less than .

  8. Example 3 • We shall investigate a function with the following statement. “ A competent salesman gets higher income” • Let X and Y be sets of salesmen and of monthly income [0, ] respectively. • And A and B are fuzzy sets of “competent salesmen” and “high income”. • In this case, the functions f • satisfies the following for all and

  9. 1.2 Propagation of Fuzziness by Crisp Function Definition (Fuzzy extension function) • Fuzzy extension function propagatetes the ambiguity of independent variables to dependent variables, when f is a crisp function from X to Y, the fuzzy extension function f defines the image of fuzzy set .

  10. That is, the extension principle is applied. • where, is inverse image of y. • We use the sign ~ for the emphasis of fuzzy variable.

  11. Example 4 • There is a crisp function, • where its domain is A = {(0, 0.9), (1, 0.8), (2, 0.7), (3, 0.6), (4, 0.5)} and its range is B = [ 0, 20 ]

  12. The independent variables have ambiguity and the fuzziness is propagated to the crisp set B. • Then, we can obtain a fuzzy set B’ in B B’ = {(1, 0.9), (4, 0.8), (7, 0.7), (10, 0.6), (13, 0.5)}.

  13. 1.3 Fuzzifying Function of Crisp Variable • Fuzzifying function of crisp variable is a function which produces image of crisp domain in a fuzzy set. Definition (Single fuzzifying fuction) • Fuzzifying function from X to Y is the mapping of X in fuzzy power set .

  14. That is to say, the fuzzifying function is a mapping from domain to fuzzy set of range. • Fuzzifying function and the fuzzy relation coincides with each other in the mathematical manner. • So to speak, fuzzifying function can be interpreted as fuzzy relation R defined as following:

  15. Example 5 • Consider two crisp sets A = {2, 3, 4} and B = {2, 3, 4, 6, 8, 9, 12} • A fuzzifying function maps the elemets in A to power set in the following manner.

  16. B1 = {(2, 0.5), (4, 1), (6. 0.5)} • B2 = {(3, 0.5), (6, 1), (9, 0.5)} • B3 = {(4, 0.5), (8, 1), (12, 0.5)} • If we look at the mapping in detail, we can see the relationship as shown in (Fig 1)

  17. The function maps element 2 εA to element 2 εB1 with degree 0.5, to element 4 ε B1 with 0.1, and to element 6 ε B1 with 0.5. • Now we apply α-cut operation to the fuzzifying function.

  18. In the same manner

  19. We can see that the fuzzy function maps 2 to 2 with possibility 0.4 through f1, to 4 with 0.7 through f2 and to -1 with 0.5 through f3. • This result is represented by the above

  20. Example 7 • There is a fuzzy bunch of continuous functions (Fig 2).

  21. This fuzzy function maps 1.5 to 1.5 with possibility 0.4 through f1 , to 2.25 with 0.7 through f2 , and to 3.25 with 0.5 through f3 , that is

  22. 2. Fuzzy Extrema of Function

  23. 2.1 Maximizing and Minimizing Set Definition (Maximizing set) • Let f be the function having real values in X and the highest and the lowest value of f be sup(f) and inf(f) respectively. • At this time, the maximizing set M is defined as a fuzzy set.

  24. That is, the maximizing set M is a fuzzy set and defined by the possibility of x to make the maximum value sup(f). • The possibility of x to be in the range of M is defined from the relative normalized position in the interval [inf(f), sup(f)].

  25. Here the interval [inf(f), sup(f)] denotes the possible range of f(x) to have some values. • Minimizing set of f is defined as the maximizing set of –f.

  26. Example 8 • Let’s have a look at f(x) of (Fig 3.) • The interval of values is as follows : [inf(f), sup(f)] = [10, 20], • and when x = 5, f(x) = 15.

  27. Then the possibility of x = 5 to be in the maximizing set M is calculated as follows : • also if when x = 8, f(x) = 19, • denotes the possibility of x to make maximum value of f.

  28. Here, we might say that two independent variables x = 5 and x = 8 make the maximum value of f(x) =20 with the possibilities 0.5 and 0.9, respectively.

  29. Example 9 • Following example is to obtain the maximizing set M of (Fig. 4).

  30. 2.2 Maximum Value of Crisp Function (1) Crisp Domain • Assume x0 is the independent variable which makes function f be the maximum value in crisp domain D. • We might utilize the maximizing set M to find the value x0 . • That is, x0 shall be the element that enables to be the maximum value.

  31. is the membership function of maximizing set. • At this time, maximum value of f will be f(x0). • can be written as in the following, denoting domain D as a crisp set.

  32. Note that domain D is replaced by universal set X in the above formula. • We expressed the possibility of x to be in D as .

  33. Example 10 • There is a function (Fig. 5) and its domain.

  34. (2) Fuzzy Domain • Now getting the maximum value f(x0) when domain is expressed in fuzzy set. • To make f be the maximum value by x0, following two conditions should be met.

  35. Set as maximum • Set as maximum • For arbitrary element x1corresponding to the maximum f, it is necessary to satisfy the above two conditions on and . • The possibility of x1 to make the maxim value of f is determined by the minimum of and , that is,

  36. Therefore, the point x0 which enables the function f to be the maximum is defined as follows. • At this time, the maximum value is f(x0).

  37. Here is membership function of maximizing set and is that of fuzzy domain (Fig 6). • Comparing x0 with x1 in the figure, x1 enables f to be maximum rather than x0. • but since is very much smaller than , f(x0) is selected as the maximum value.