A differential evolution algorithm for fuzzy extension of functions

Abstract

The paper illustrates a differential evolution (DE) algorithm to calculate the level-cuts of the fuzzy extension of a multidimensional real valued function to fuzzy numbers. The method decomposes the fuzzy extension engine into a set of "nested" min and max box-constrained optimization problems and uses a form of the DE algorithm, based on multi populations which cooperate during the search phase and specialize, a part of the populations to find the the global min (corresponding to lower branch of the fuzzy extension) and a part of the populations to find the global max (corresponding to the upper branch), both gaining efficienty from the work done for a level-cut to the subsequent ones. A special version of the algorithm is designed to the case of differentiable functions, for which a representation of the fuzzy numbers is used to improve efficiency and quality of calculations. The included computational results indicate that the DE method is a promising tool as its computational complexity grows on average superlinearly (of degree less than 1.5) in the number of variables of the function to be extended. © 2007 Springer-Verlag Berlin Heidelberg.