SCORE Allocations for Bi-objective Ranking and Selection

The bi-objective ranking and selection (R&S) problem is a special case of the multi-objective simulation optimization problem in which two conflicting objectives are known only through dependent Monte Carlo estimators, the decision space or number of systems is finite, and each system can be sampled to some extent. The solution to the bi-objective R&S problem is a set of systems with non-dominated objective vectors, called the set of Pareto systems. We exploit the special structure of the bi-objective problem to characterize the asymptotically optimal simulation budget allocation, which accounts for dependence between the objectives and balances the probabilities associated with two types of misclassification error. Like much of the R&S literature, our focus is on the case in which the simulation observations are bivariate normal. Assuming normality, we then use a certain asymptotic limit to derive an easily-implementable Sampling Criteria for Optimization using Rate Estimators (SCORE) sampling framework that approximates the optimal allocation and accounts for correlation between the objectives. Perhaps surprisingly, the limiting SCORE allocation exclusively controls for misclassification-by-inclusion events, in which non-Pareto systems are falsely estimated as Pareto. We also provide an iterative algorithm for implementation. Our numerical experience with the resulting SCORE framework indicates that it is fast and accurate for problems having up to ten thousand systems.