The Convergence Modes in Random Fuzzy Theory

Yan-Kui Liu, Xiao-Dong Dai

Abstract

Random fuzzy optimization problems include uncertain parameters defined only through probability and possibility distributions, they are inherently infinite-dimensional optimization problems that can rarely be solved directly. Thus, algorithms to solve such optimization problems must rely on intelligent computing and approximation schemes. This fact motivates us to discuss the modes of convergence in random fuzzy theory. Several new convergence concepts such as convergence almost uniform, and convergence in chance for sequences of random fuzzy variables were presented. Then the criteria for convergence almost sure, convergence almost uniform, and convergence in chance are established. Finally, the interconnections between convergence almost uniform and convergence almost sure, convergence almost uniform and convergence in chance, and convergence in chance and convergence almost sure are discussed. All these results can be regarded as the theoretical foundation of the intelligent computing and approximation schemes for random fuzzy optimization problems.