Exact Membership Functions for the Fuzzy Weighted Average

The problem of computing the fuzzy weighted average, where both attributes and weights are fuzzy numbers, is well studied in the literature. Generally, the approach is to apply Zadeh's extension principle to compute alpha-cuts of the fuzzy weighted average from the alpha-cuts of the attributes and weights for fixed values of alpha epsilon [0..1]; this means that all values of the membership functions of the fuzzy weighted average are computed separately. In this paper, we generalise this approach in such a way that alpha is considered to be a parameter; this enables us to compute exact analytical membership functions for the fuzzy weighted average in case the attributes and weights are triangular or trapeizoidal fuzzy numbers. To illustrate the power of our algorithms, they are applied to the examples from the literature, providing exact membership functions in each case.