An epsilon-constraint method for integer-ordered bi-objective simulation optimization

Consider the context of integer-ordered bi-objective simulation optimization, in which the feasible region is a finite subset of the integer lattice. We propose a retrospective approximation (RA) framework to identify a local Pareto set that involves solving a sequence of sample-path bi-objective optimization problems at increasing sample sizes. We apply the epsilon-constraint method to each sample-path bi-objective optimization problem, thus solving a sequence of constrained single-objective problems in each RA iteration. We solve each constrained single-objective optimization problem using the SPLINE algorithm, thus exploiting gradient-based information. In early RA iterations, when sample sizes are small and standard errors are relatively large, we provide only a rough characterization of the Pareto set by making the number of epsilon-constraint problems a function of the standard error. As the RA algorithm progresses, the granularity of the characterization increases until we solve as many epsilon-constraint problems as there are points in the (finite) image of the local Pareto set. Our algorithm displays promising numerical performance.