Methods for the fuzzy and random probability analysis problem based on Markov Chain and Saddle-point Approximation

Two novel methods are developed to solve the mixed probability analysis problem with both random and fuzzy uncertainties. The first one is Iterative Markov Chain Simulation First Order Saddle-point Approximation (IMCSFOSA) whose key idea is to get the upper limit and lower limit of the reliability for a given membership level with Markov Chain and First Order Saddle-point Approximation, and throughout the whole value domain of membership level by this process, the membership function of reliability can be obtained. Compared with the traditional methods, such as double-loop Monte Carlo simulation and iterative first order and second moment method, the IMCSFOSA is more efficient due to less simulation and more effective due to no transformation from the non-normal distribution to the normal one. The second method is Iterative Conditional Probability Markov Chain Simulation (ICPMCS), in which a nonlinear modification factor is introduced by Conditional Probability formula in solving the upper limit and lower limit of the reliability for a given membership level. The introduction of this factor highly improves the calculation accuracy for the highly nonlinear performance function. Several examples are introduced to illustrate the advantages of the presented methods.