A parametric representation of fuzzy numbers and their arithmetic operators

Direct implementation of extended arithmetic operators on fuzzy numbers is computationally complex. Implementation of the extension principle is equivalent to solving a nonlinear programming problem. To overcome this difficulty many applications limit the membership functions to certain shapes, usually either triangular fuzzy numbers (TFN) or trapezoidal fuzzy numbers (TrFN). Then calculation of the extended operators can be performed on the parameters defining the fuzzy numbers, thus making the calculations trivial. Unfortunately the TFN shape is not closed under multiplication and division. The result of these operators is a polynomial membership function and the triangular shape only approximates the actual result. The linear approximation can be quite poor and may lead to incorrect results when used in engineering applications. We analyze this problem and propose six parameters which define parameterized 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.