Marginal improvement procedures for top-m selection

Given a fixed simulation budget, the problem of selecting the best and top-$m$ alternatives among a finite set of alternatives have been studied separately in simulation optimization literature, because the existing sampling procedures are often dedicated to one problem. Under a Bayesian framework, we formulate the top-$m$ selection into a stochastic dynamic program, and characterize the optimal sampling policy via Bellman equations. To determine sequential sampling decisions, we measure the expected marginal improvement from obtaining one additional simulation observation based on predictive distributions, and then develop two cheaply computational approximations to the improvement, thereby yielding two generic sampling procedures that are efficient in selecting top-$m$ alternative(s). The two procedures are proved to be consistent, in a sense that the best and top-$m$ alternatives can be correctly identified as the simulation budget goes to infinity. Numerical experiments on synthetic problems and a coronavirus transmission control application are conducted to demonstrate the efficiency and generality of our procedures.