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The family of such functions is denoted by where L, R ∈ Several selected examples of the discussed family of the membership functions p ≥ 0. The membership functions of the fuzzy numbers can be derived by putting together some of the above functions. For example, one has see also Fig. 4.1 that visualizes this fuzzy number for some parameters of the membership function.
4.3.2. Computing with fuzzy numbersThe calculations on fuzzy numbers rely on the extension principle (Mizumoto and Tanaka, 1976). Consider a function F transforming two fuzzy numbers A and B and producing the third number C such that The extension principle determines the membership function of C to be z ∈ R; f: R2 → R is a real function that is pointwise consistent with F meaning that F( {x}, {y}) = f(x, y). As usual, {x} describes a single-element set. Quite often, as implied by practice, it is enough to concentrate on a family of triangular fuzzy numbers. These are the simplest models of uncertain numerical quantities. The modal value represents an element that is the most representative to the concept. The lower and upper bound identify the region that is of interest to the discussed notion; all the elements situated beyond these boundaries are perceived as irrelevant to the linguistic concept. The analysis focusing on this class of membership functions helps reveal the most profound properties of fuzzy arithmetic. Consider two triangular numbers A = (x; a, m, b) and B = (x; c, n, d). More specifically, their membership functions are defined through the following piecewise linear relationships Each modal value (m and n, respectively) identifies a dominant (typical) value of the corresponding quantity whereas the lower and upper bounds (a or c and b or d) reflect the spread of the concept. Let us start with the addition operation. The sup-min composition applied to A and B yields The resulting fuzzy number is normal; that is C(z) = 1 for z = m+n. The computations of the spreads of C are dealt with separately:
The linearly increasing sections of the membership functions A and B, see Fig. 4.2, embrace the interval [a + c, m + n].
Based on that we derive along with x, y ∈ [a+c, m+n]. Expressing x and y as functions of ω one instantaneously derives and Furthermore, owing to the underlying addition operation, (z = x + y), we obtain We derive a similar relationship for the decreasing portions of the membership functions. The formulas below apply to x and y situated in [m + n, b + d] or equivalently Adding x and y gives Summarizing, the membership function of A ⊕ B equals Interestingly enough, C exhibits also a triangular membership function. To emphasize this fact, we use a concise notation In general, by adding triangular fuzzy numbers, the result becomes a fuzzy set of the same shape. More precisely, if Ai = (x; ai, mi, bi), i=1, 2, ..., n, then the sum of these numbers equals One can observe that the spreads of the result start accumulating. For instance, let mi = 1, ai = 0.95, bi =1.05 for all i = 1, 2, ... n. Iterating the addition operation, one derives a sequence of fuzzy numbers In virtue of the fundamental representation theorem, each fuzzy number can be regarded as a family of nested α-cuts. Subsequently, these α- cuts can be utilized to reconstruct the resulting fuzzy number. For instance, we get Generally, the use of α-cuts forms a sort of brute force method of carrying out computing with fuzzy quantities. Let us now perform a multiplication of two triangular fuzzy numbers. As before, we look first into the increasing parts of the membership functions expressed as Then their product becomes If ac ≤ z ≤ mn then the membership function of C is an inverse of F1 Similarly, consider the decreasing parts of A and B meaning that mn ≤ z ≤ bd As before, for any z in [mn, bd] we derive Evidently, the multiplication does not return a piecewise linear membership function. Instead, as clearly indicated by the results of the above computations, we produce a quadratic form of the resulting fuzzy number. The linearization of the membership functions can be treated as an approximation of the previously derived membership function. We require that this linearization consists of two linear functions coinciding with C at z = mn, ac, and bd. The quality of this linearization depends very much upon the spreads of the fuzzy numbers. To quantify the resulting linearization error, let us consider that m = n = 1 and The integral of error is of interest here while C∗ is a piecewise linear approximation of the original fuzzy number (C). Assuming that δ tends to zero, one can accept the linear approximation of the form
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