Modeling Inventory Systems with Imperfect Supply

Abstract

We study inventory systems operating under an infinite-horizon, periodic-review base-stock control policy with stochastic demand and imperfect (i.e., less than 100% reliable) supply. We model demand using a general discrete distribution and replenishment lead time using a geometric distribution, resulting from a Bernoulli trial-based model of supply uncertainty. We develop a computational approach using a discrete time Markov process (DTMP) model to minimize the total system cost and obtain the optimal base-stock level when the backorder penalty is given. We develop a general, recursive solution for the steady state probability of each inventory level and use this to find the optimal base-stock level in this setting. Moreover, for specific demand distributions we are able to develop closed-form solutions for these outcomes. The lead-time demand (LTD) distribution can also be obtained from these recursive equations to determine the base-stock level when a target customer service level is specified in lieu of a backorder penalty cost. We conduct extensive computational experiments to observe the robustness of various approximate solutions under two scenarios for the lead-time distribution. The first scenario assumes a geometric lead time. The second scenario considers a general lead-time distribution. We conduct computational experiments to observe the conditions in which the DTMP model performs well, including situations where the demand and the lead-time distributions are specified separately, and where the LTD distribution is given and follows either a Beta distribution or a bimodal distribution. Finally, for a two-location inventory system consisting of a single retailer supplied by a single distributor, whose supply ultimately comes from an unreliable supplier upstream, we propose a computational approach to determine optimal or near-optimal base-stock levels at the retailer and distributor. We develop two decomposition-based approximation methods, solving the separate single-site inventory problems (distributor, retailer) sequentially, but with different methods to compute the implied backorder penalty at the distributor that induces near-optimal base-stock levels at both locations.