Consider the fuzzy functionfor triangular fuzzy numbers
16 CHAPTER 2. FUZZY SETS
Equation (2.23) defines the membership function of Z for any triangular (shaped) fuzzy number X in [a, b].
| z1(α) | = | min{ h(x) | x ∈ X[α] }, max{ h(x) | x ∈ X[α] }, | (2.24) | ||||
|---|---|---|---|---|---|---|---|
| = | (2.25) | ||||||
| z2(α) | |||||||
| Z(z) = sup |
X(x), Y (y) |
| h(x, y) = z | � | (2.26) | |||
| x,y | |||||||
| z1(α) | = | min{ h(x, y) | x ∈ X[α], y ∈ Y [α] }, max{ h(x, y) | x ∈ X[α], y ∈ Y [α] }, | (2.27) |
|---|---|---|---|
| = | (2.28) | ||
| z2(α) |
All the functions we usually use in engineering and science have a computer algorithm which, using a finite number of additions, subtractions, multipli-cations and divisions, can evaluate the function to required accuracy. Such functions can be extended, using α-cuts and interval arithmetic, to fuzzy func-tions. Let h : [a, b] → IR be such a function. Then its extension H(X) = Z, X in [a, b] is done, via interval arithmetic, in computing h(X[α]) = Z[α], α in [0, 1]. We input the interval X[α], perform the arithmetic operations needed to evaluate h on this interval, and obtain the interval Z[α]. Then put these α-cuts together to obtain the value Z. The extension to more independent variables is straightforward.
For example, consider the fuzzy function
| Z = H(X) =A X + B C X + D, | (2.29) |
|---|---|
Z = H(X) = 2X + 10 3X + 4, (2.31)
would be the extension of
| h(x) = 2x + 10 3x + 4. | (2.32) | ||
|---|---|---|---|
|
∗. But for basic fuzzy arithmetic | ||
in Section 2.3 the two methods give the same results. In the example below we show that for h(x) = x(1 − x), x in [0, 1], we can get Z X in [0, 1]. What is known ([3],[7]) is that for usual functions in science and∗ ̸= Z for some |
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Z = (1 − X) X, (2.33)
for X a triangular fuzzy number in [0, 1]. Let X[α] = [x1(α), x2(α)]. Using interval arithmetic we obtain
| z1(α) | = | (2.34) | ||||
|---|---|---|---|---|---|---|
| = | (2.35) | |||||
| z2(α) | ||||||
| Z | ∗(z) = sup | �X(x)|(1 − x)x = z, 0 ≤ x ≤ 1� | (2.36) | |||
| x | ||||||
