Computing optimal base-stock levels for an inventory system with imperfect supply

We study a single-item, single-site, periodic-review inventory system with negligible fixed ordering costs. The supplier to this system is not entirely reliable, such that each order is a Bernoulli trial, meaning that, with a given probability, the supplier delivers the current order and any accumulated backorders at the end of the current period, resulting in a Geometric distribution for the actual resupply lead time. We develop a recursive expression for the steady-state probability vector of a discrete-time Markov process (DTMP) model of this imperfect-supply inventory system. We use this recursive expression to prove the convexity of the inventory system objective function, and also to prove the optimality of our computational procedure for finding the optimal base-stock level. We present a service-constrained version of the problem and show how the computation of the optimal base-stock level using our DTMP method, incorporating the explicit distribution of demand over the lead time plus review (LTR) period, compares to approaches in the literature that approximate this distribution. We also show that the version of the problem employing an explicit penalty cost can be solved in closed-form for the optimal base-stock level for two specific period demand distributions, and we explore the behavior of the optimal base-stock level and the corresponding optimal service level under various values of the problem parameters.