On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

Abstract

We consider a multistage stochastic discrete program in which constraints on any stage might involve expectations that cannot be computed easily and are approximated by simulation. We study a sample average approximation (SAA) approach that uses nested sampling, in which at each stage, a number of scenarios are examined and a number of simulation replications are performed for each scenario to estimate the next-stage constraints. This approach provides an approximate solution to the multistage problem. To establish the consistency of the SAA approach, we first consider a two-stage problem and show that in the second-stage problem, given a scenario, the optimal values and solutions of the SAA converge to those of the true problem with probability one when the sample sizes go to infinity. These convergence results do not hold uniformly over all possible scenarios for the second stage problem. We are nevertheless able to prove that the optimal values and solutions of the SAA converge to the true ones with probability one when the sample sizes at both stages increase to infinity. We also prove exponential convergence of the probability of a large deviation for the optimal value of the SAA, the true value of an optimal solution of the SAA, and the probability that any optimal solution to the SAA is an optimal solution of the true problem. All of these results can be extended to a multistage setting and we explain how to do it. Our framework and SAA results cover a large variety of resource allocation problems for which at each stage after the first one, new information becomes available and the allocation can be readjusted, under constraints that involve expectations estimated by Monte Carlo. As an illustration, we apply this SAA method to a staffing problem in a call center, in which the goal is to optimize the numbers of agents of each type under some constraints on the quality of service (QoS). The staffing allocation has to be decided under an uncertain arrival rate with a prior distribution in the first stage, and can be adjusted at some additional cost when better information on the arrival rate becomes available in later stages.