Sampling-Based Approximation for Serial Multi-Echelon Inventory System

We consider inventory management of an infinite-horizon, serial system with unknown demand distribution. Each stage orders from its upstream stage and incurs inventory holding cost while the most downstream stage faces random demand and incurs inventory holding and demand backlogging cost. Lead times between stages are constants. The objective is to minimize the expected total discounted cost over the planning horizon. We apply the sample average approximation (SAA) method and study the performance of the resulting solution under the empirical distribution function constructed from a demand sample. We derive the sample size (i.e., distribution-free sample size) required to guarantee the performance of the SAA solution be close to the optimum with a given probability. This result is obtained by first deriving a decomposable and tight cost upper bound of the whole system that depends on the (given) echelon base-stock levels and then showing the cost difference between the SAA and optimal solutions can be measured by the distance between empirical and real demand distribution functions. Furthermore, when the one-period demand distribution function is absolutely continuous and has an increasing failure rate, we derive a smaller sample size (i.e., distribution-dependent sample size) that guarantees the same performance. A special case of this result for the newsvendor problem generalizes the existing result by Levi et al. (2015). In addition, both the distribution-free and the distribution-dependent sample sizes increase polynomially as the number of stages increases. Finally, our numerical study corroborates the theoretical results.}