Multi-objective simulation optimization on finite sets: Optimal allocation via scalarization

We consider the multi-objective simulation optimization problem on finite sets, where we seek the Pareto set corresponding to systems evaluated on multiple performance measures, using only Monte Carlo simulation observations from each system. We ask how a given simulation budget should be allocated across the systems, and a Pareto surface retrieved, so that the estimated Pareto set minimally deviates from the true Pareto set according to a rigorously defined metric. To answer this question, we suggest scalarization, where the performance measures associated with each system are projected using a carefully considered set of weights, and the Pareto set is estimated as the union of systems that dominate across the weight set. We show that the optimal simulation budget allocation under such scalarization is the solution to a bi-level optimization problem, for which the outer problem is concave, but some inner problems are non-convex. We comment on the development of tractable approximations for use when the number of systems is large.