An Extended Analytical Approach to Evaluating Monotonic Functions of Fuzzy Numbers

There is an increasing effort in the scientific community to provide suitable methods for the inclusion of uncertainties into mathematical models. One way to do so is to introduce parametric uncertainty by representing the uncertain model parameters as fuzzy numbers [1] and evaluating the model equations by means of Zadeh’s extension principle [2]. The evaluation of this classical formulation of the extension principle, however, turns out to be a highly complex task [3]. Fortunately, Buckley and Qu [4] provide an alternative formulation that operates on α-cuts and is applicable to continuous functions of independent fuzzy numbers. Powerful numerical techniques have been developed to implement this alternative formulation [5]. These techniques are particularly suitable for very complex simulation models [6]. In engineering design [7], however, the mathematical equations are usually less complex, and hence analytical methods might be more suitable for the inclusion of parameter uncertainties into the computations. For this purpose, a practical analytical approach to evaluating continuous, monotonic functions of independent fuzzy numbers was introduced by the authors [8], which is based on the alternative formulation of the extension principle. In this paper, we extend this approach in terms of computational efficiency depending on certain monotonicity conditions. An outline of this paper is as follows. In Section 2, we give a definition of fuzzy numbers and present two important types. In Section 3, we introduce the notion of a linguistic variable. In Section 4, we briefly recall Zadeh’s extension principle and introduce the alternative formulation based on α-cuts. In Section 5, we describe our extended analytical approach and give four illustrative examples. In Section 6, a practical engineering application is presented. Finally, in Section 7, some conclusions are drawn.