A heuristic moment-based framework for optimization design under uncertainty

To search an optimal design under uncertainty, this study proposes an effective framework that integrates the moment-based reliability analysis into a heuristic optimization algorithm. Integration of an equivalent single-variable performance function is an ideal concept to calculate the failure probability. However, such integration is often not available and is alternatively computed using the first four moments and a generalized moment-based reliability index is established, in which the Gaussian–Hermite integration and dimension reduction are implemented to enhance the effectiveness. To overcome the limited applicable range of moment-based approach, an adjustable optimization procedure is proposed, in which different reliability methods are performed depending on results of the constraint assessments. In addition, the ε level comparison is integrated into particle-swarm optimization to consider the constraint violation. Several literature studies are used to verify the accuracy of the proposed optimization framework including problems having linear, highly nonlinear, implicit probabilistic constraint functions with normal or non-normal variables and system-level reliability analysis. The effects of several parameters, such as the number of estimate point, the number of dimension, and the degree of uncertainty, are thoroughly investigated. Results indicating that tri-variate with seven points are able to provide a stable solution under a high degree of uncertainty.