Optimal Policies and Bounds for Stochastic Inventory Systems with Lost Sales

This paper studies the classical discrete-time, single-location inventory model with stochastic demand, lost sales, and positive lead time. We transform the problem into an equivalent problem of the Markovian demand inventory model with zero lead time and zero initial inventory, which provides a better understanding to the original problem. Based on this transformation, we introduce a key concept—effectual demand, which determines both the performance in the current period and the evolution into future periods. In this way, we provide a bridge of two research streams in the literature: Markovian demand inventory model and lost sales inventory model. We believe this Markovian approach is more straightforward to work with various applications. In this way, we easily show the existence of optimal policies in discounted and average cost, and finite and infinite horizon cases, when cost functions are of polynomial growth. The polynomial growth cost functions virtually cover all practical scenarios in real business settings. We then present simpler proofs and examples for the linear order cost case. The analytical solutions are first. We also derive bounds analytically on the optimal policy; these bounds are equivalent to those of the myopic policy but tighter than the popular bound in the literature.