A new approach for computing with fuzzy sets using interval analysis

We present a new approach for computing with fuzzy sets based on interval analysis techniques. Our proposed method is capable of managing multi-dimensional continuous membership functions of arbitrary form, such as, piecewise affine functions or non-linear expressions without any restrictions regarding convexity. We present a formal representation of fuzzy sets that allows to easily cast fuzzy problems into the set inversion framework. The SIVIA algorithm is presented as a convenient solution to solve this problem via interval analysis. It characterizes the resulting fuzzy set in an approximate (with the desired precision) but guaranteed way. Different combination operators were implemented using our method. We show that each operator is implemented following the same procedure and thus that, potentially, any fuzzy problem could be represented as a set inversion problem. Our approach is illustrated by examples at the end of the paper, where a discussion over the obtained results takes place.