On fuzzy simulations for expected values of functions of fuzzy numbers and intervals

Abstract

Based on existing fuzzy simulation algorithms, this article presents two innovative techniques for approximating the expected values of fuzzy numbers' monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of Liu and Liu (2002) is enhanced by updating the discretization procedure for the simulation of the membership function and the calculation formula for the expected values. This is achieved through initiating a novel uniform sampling process and employing a formula for discrete fuzzy numbers, respectively, as the generated membership function in the stochastic discretization algorithm would adversely affect its accuracy to some extent. What is more, considering that the bisection procedure involved in the numerical integration algorithm of Li (2015) is time-consuming and also, not necessary for the specified types of fuzzy numbers, a special numerical integration algorithm is proposed, which can simplify the simulation procedure by adopting the analytical expressions of α-optimistic values. Subsequently, concerning the extensive applications of regular fuzzy intervals, several theorems are introduced and proved as an extended effort to apply the improved stochastic discretization algorithm and the special numerical integration algorithm to the issues of fuzzy intervals. Throughout this article, a series of numerical experiments are conducted from which the superiority of both the two novel techniques over others are conspicuously displayed in aspects of accuracy, stability, and efficiency.