In this research, we investigate design optimization under uncertainties for problems with two objectives. Reliability-based design optimization (RBDO) that considers uncertainties as random variables and/or parameters and formulates constraints probabilistically has received extensive attention. However, research to date has focused primarily on single-objective problems only. We extend RBDO to problems for which multiple objectives are optimized simultaneously. Each constraint reliability value results in a Pareto set. The set of all Pareto frontiers at the various reliability values is denoted as the β-Pareto set. We study the relations between the deterministic Pareto set and the β-Pareto set and then develop a method to systematically determine the exact β-Pareto set of bi-objective linear programming problems. The method is also extended to predict the β-Pareto set of nonlinear problems using the sandwich technique. As a result, we are able to accurately predict the β-Pareto set in the objective space without solving multiple multi-objective optimization problems at various reliability levels. In the early stage of the product design process, the proposed approach can help decision-makers efficiently to determine how product performance varies with reliability level.

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