Sampling-Based Approximation for Serial Multi-Echelon Inventory System
70 Pages Posted: 4 Jun 2021 Last revised: 23 Dec 2022
Date Written: June 25, 2022
Abstract
We study inventory management of an infinite-horizon, series system with multiple stages. Each stage orders from its upstream stage and the most upstream stage orders from an external supplier. Random demand with unknown distribution occurs at the most downstream stage. Each stage incurs inventory holding cost while the most downstream stage also incurs demand backlogging cost when it has inventory shortage. The objective is to minimize the expected total discounted cost over the planning horizon. We apply the sample average approximation (SAA) method to obtain a heuristic policy (SAA policy) using the empirical distribution function constructed from a demand sample (of the underlying demand distribution). We derive an upper bound of sample size (i.e., distribution-free bound) that guarantees the performance of the SAA policy be arbitrarily close (i.e., with arbitrarily small relative error) to the optimal policy under known demand distribution with high probability. This result is obtained by first deriving a separable and tight cost upper bound of the whole system that depends on (given) echelon base-stock levels and then showing the cost difference between the SAA and optimal solutions can be measured by the distance between the empirical and the underlying demand distribution functions. We also provide a lower bound of sample size that matches the upper bound. Furthermore , when the one-period demand distribution is absolutely continuous and has an increasing failure rate (IFR), we derive a tighter sample size upper bound (i.e., distribution-dependent bound). This distribution-dependent bound for the newsvendor problem generalizes the existing result. In addition, we show that both the distribution-free and the distribution-dependent bounds increase polynomially as the number of stages increases. The performance of SAA policy and the sample size bounds are illustrated numerically. Finally, we extend the results to finite-horizon series systems.
Keywords: Clark-Scarf model, echelon base-stock policy, SAA, sample size, log-concave demand
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