A PARAMETRIC REPRESENTATION OF FUZZY NUMBERS AND THEIR ARITHMETIC OPERATORS

Direct implementation of extended arithmetic operators on fuzzy number s is computationally complex. Implementation of the extension principl e is equivalent to solving a nonlinear programming problem, To overcom e this difficulty many applications limit the membership functions to certain shapes, usually either triangular fuzzy numbers (TFN) or trape zoidal fuzzy numbers (TrFN). Then calculation of the extended operator s can be performed on the parameters defining the fuzzy numbers, thus making the calculations trivial. Unfortunately the TFN shape is not cl osed under multiplication and division. The result of these operators is a polynomial membership function and the triangular shape only appr oximates the actual result. The linear approximation can be quite poor and may lead to incorrect results when used in engineering applicatio ns. We analyze this problem and propose six parameters which define pa rameterized fuzzy numbers (PFN), of which TFNs are a special case, We provide the methods for performing fuzzy arithmetic and show that the PFN representation is closed under the arithmetic operations. The new representation in conjunction with the arithmetic operators obeys many of the same arithmetic properties as TFNs. The new method has better accuracy and similar computational speed to using TFNs and appears to have benefits when used in engineering applications. (C) 1997 Elsevier Science B.V.