Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control Models

Abstract

We study the classical multi-period capacitated stochastic inventory control problems in a data-driven setting. The objective is to match the on-hand inventory level with the random demand in each period, subject to supply constraints, while minimizing expected cost over a finite time horizon. Instead of assuming a full knowledge of the demand distributions, we assume that they can only be accessed through drawing random samples. Such data-driven models are ubiquitous in practice, where the cumulative distribution functions of the underlying random variables are either unavailable or too complicated to work with. We apply the Sample Average Approximation (SAA) method to the capacitated inventory control problem and establish an upper bound on the number of samples needed for the SAA method to achieve a near-optimal expected cost, under any level of required accuracy and prespecified confidence probability. The sample bound is polynomial in the number of time periods as wel as the confidence and accuracy parameters. Moreover the bound is independent of the underlying demand distributions. We crucially use an inequality of Massart, which is a stronger version of Chernoff inequality, to ensure a uniform approximation on the right derivatives.While the SAA method uses polynomially many samples, we show that the underlying SAA problem is in fact #P-hard. Thus the SAA method, which solves the SAA problem to optimality, is not a polynomial time algorithm in general, unless #P = P. Nevertheless, motivated by the SAA analysis, we propose a randomized polynomial time approximation scheme, which also uses polynomially many samples. The approximation scheme involves a sparsification procedure on the right derivatives of the cost-to-go functions, which maintains the tractability of the underlying dynamic program. Finally, we establish a lower bound on the number of samples required to solving the data-driven problem to near-optimality.