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.

中文翻译：

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具有随机约束和嵌套采样的多级离散随机优化问题

我们考虑一个多阶段随机离散程序，其中任何阶段的约束都可能涉及无法轻松计算并通过模拟近似的期望。我们研究了一种使用嵌套抽样的样本平均近似 (SAA) 方法，在该方法中，在每个阶段检查多个场景，并为每个场景执行多个模拟复制以估计下一阶段的约束。这种方法提供了多级问题的近似解。为了建立 SAA 方法的一致性，我们首先考虑一个两阶段的问题，并表明在第二阶段的问题中，给定一个场景，当 SAA 的最优值和解以概率 1 收敛到真实问题时样本量趋于无穷大。对于第二阶段问题的所有可能场景，这些收敛结果并不统一。尽管如此，我们仍然能够证明，当两个阶段的样本量都增加到无穷大时，SAA 的最优值和解以概率 1 收敛到真实值。我们还证明了 SAA 最优值的大偏差概率、SAA 最优解的真实值以及 SAA 的任何最优解是真实问题的最优解的概率的指数收敛。所有这些结果都可以扩展到多阶段设置，我们将解释如何做到这一点。我们的框架和 SAA 结果涵盖了各种各样的资源分配问题，在第一个阶段之后的每个阶段，都有新的信息可用，并且可以重新调整分配，在涉及蒙特卡罗估计的期望的约束下。例如，我们将这种 SAA 方法应用于呼叫中心的人员配备问题，其目标是在服务质量 (QoS) 的某些约束下优化每种类型的座席数量。人员分配必须在不确定的到达率下决定，在第一阶段先分配，并且可以在稍后阶段获得关于到达率的更好信息时以一些额外的成本进行调整。