A better adaptive importance sampling algorithm for estimating fuzzy reliability sensitivity of structures with multiple fuzzy failure modes in series

Aim: After reviewing the past algorithms^[3-7]and other relevant papers^{[1, 2]}, we believe we can work out a still better algorithm. Section 1 of the full paper briefly summarizes relevant parts of Ref. 8, whose first author is the second author of this paper. Section 2 briefly summarizes the relevant parts of Ref. 7, whose first author is also the second author of this paper. Section 3 derives eqs. (15) and (16) to estimate the fuzzy reliability sensitivities of series structures with the importance sampling obtained by simulated annealing optimization algorithm. Section 3 also gives eqs. (18) and (20) to calculate the variation coefficients of the reliability sensitivity. Section 4 presents three numerical examples respectively for three types of membership function of fuzzy failure mode, the three types being (1) linear distribution, (2) Cauchy's distribution and (3) semi-normal distribution. The calculated results, given in Tables 1 through 3 for the three examples respectively, show preliminarily that, with the same variation coefficients, the number of samples needed by our adaptive importance sampling algorithm is far fewer than that required by the Monte Carlo method, all the relative errors being within 5%.