Joint Probability Formulation for Multiobjective Optimization Under Uncertainty

Abstract

This paper presents a new approach to solve multiobjective optimization problems under uncertainty. Unlike the existing state-of-the-art, where means/variances of the objectives are used to ensure optimality, we employ a distributional formulation. The proposed formulations are based on joint probability, i.e., probability that all objectives are simultaneously bound by certain design thresholds under uncertainty. For minimization problems, these thresholds can be viewed as the desired upper bounds on the individual objectives. The tradeoffs are illustrated using the so-called decision surface, which is the surface of maximized joint probabilities for a set of design thresholds. Two optimization formulations to generate the decision surface are proposed, which provide the designer with the distinguishing capability that is not available in the existing literature, namely, decision making under uncertainty, while ensuring joint objective performance: (1) Maximum probability design: Given a set of thresholds (preferences within each objective), find a design that maximizes the joint probability while using a probabilistic aggregation as against an ambiguous weight-based method. (2) Optimum threshold design: Given a designer-specified joint probability, find a set of thresholds that satisfy the joint probability specification while allowing for a specification of preferences among the objectives.