An Optimization of ac-cuts of Fuzzy Sets Through Particle Swarm Optimization

Given the representation theorem, it is well-known that any fuzzy set can be represented by its infinite family of alpha-cuts. While there have been a lot of theoretical investigations along this line, a surprisingly limited attention was paid to the optimization of the representation (approximation) of fuzzy sets by some finite, usually quite limited, family of their alpha-cuts. In this study, we formulate a problem of the best approximation of a fuzzy set by a finite number of its alpha-cuts. Being concise, the task is formulated as follows: for a given fuzzy set A and a certain finite number of "n" alpha-cuts, optimize the values of these cuts (thresholds), alpha₁ < alpha₂ < ... < alpha_n where alpha_i isin (0,1] so that this finite alpha-cut representation of A approximates the original fuzzy set A to the highest possible extent. While for several (say, 2 or 3) threshold values detailed paper-and-pencil derivations could be easily completed 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