Estimating the efficient frontier of a probabilistic bicriteria model

We consider a problem that trades off cost of system design with the risk of that design, where risk is measured by the probability of a bad event, such as system failure. Our interest lies in the problem class where we cannot evaluate this risk measure exactly, even for a given system design. We approach this problem via a bicriteria optimization model, replacing the risk measure by an Monte Carlo estimator and solving a parametric family of optimization models to produce an approximate efficient frontier. Optimizing system design with the risk estimator requires solution of a mixed integer program. We show that we can minimize risk over a range of cost thresholds or minimize cost over a range of risk thresholds and we examine associated asymptotics. The proximity of the approximate efficient frontier to the true efficient frontier is established via an asymptotically valid confidence interval with minimal additional work. Our approach is illustrated computationally using a facility-sizing problem.