Design optimization for robustness in multiple performance functions

In this paper, we propose a new approach to include robustness considerations in problems with multiple performance functions under uncertainty. Robustness of performance under uncertainty is typically ensured by minimizing the variation of the performance function, usually modeled by its variance or standard deviation. Such moment-based methods generally become cumbersome to solve as the number of performance functions increases, because of a large number of objective functions. The current paper proposes a new approach that tackles both robustness and optimality using the joint probability density function of all the performance functions. For a set of designer-specified thresholds, which can be viewed as upper bounds on the performance functions (for minimization), joint optimality is ensured by maximizing a single quantity irrespective of the number of performance functions: the joint probability that the design’s performance is bounded by the thresholds. To ensure joint robustness, we minimize a single scalar quantity irrespective of the number of performance functions: the determinant of the covariance matrix of the performance functions. Two examples are presented: an automobile side impact problem with two performance functions and a two-stage-to-orbit launch vehicle conceptual design problem with three performance functions.