Finite cut-based approximation of fuzzy sets and its evolutionary optimization

Given the representation theorem, it is well known that any fuzzy set can be represented by an infinite family of its α-cuts. While there have been a lot of theoretical investigations along this line, a surprisingly limited attention has been paid to the optimization of the representation (approximation) of fuzzy sets by some finite, usually quite limited, family of their α-cuts. In this study, we formulate a problem of the best approximation of a fuzzy set by a finite number of its α-cuts. Being concise, the task is formulated as follows: for a given fuzzy set and a prescribed finite number of a-cuts, optimize the values of the corresponding thresholds (a-cuts), so that the obtained finite representation as a nested set of intervals approximates the original fuzzy set to the highest possible extent. While for several (say, 2 or 3) threshold values detailed paper-and-pencil derivations could be accomplished thus leading to the construction of an analytic solution, in general, we need to resort to some optimization procedures. Considering the requirements of the resulting optimization problem formulated with this regard, we use here a certain biologically inspired optimization technique known as particle swarm optimization (PSO). In the paper, we elaborate on some categories of important and commonly encountered problems in which the capabilities of fuzzy sets are fully exploited, including decision-making and data analysis (supported by means of fuzzy clustering). The study includes a series of detailed numeric experiments that illustrate the performance of the PSO and demonstrate the effectiveness of the solutions developed through such optimization.