Base-Stock Models for Lost Sales: A Markovian Approach

We consider the lost sales model with discrete demand that is filled only from inventory on-hand. The inventory is reviewed every T periods and an order is placed to bring the inventory position back to a target base-stock level R. The order is received after a lead time of L periods. Based on the outstanding orders in the pipeline, we represent the state of the system as a Markov chain for a given base-stock level. We develop structural properties of the best stationary base-stock policy. These properties include a stopping rule and a method for finding strictly improving lower-bounds that facilitate the development of an optimization algorithm. We also show that the structure of the transition probability matrix is recursive in R and L and that the matrix is sparse with special structure. This special structure is used to facilitate computation of the stationary distribution. In turn this distribution is used to compute the long-run average cost for a given base-stock from which a search yields the best, minimum cost, base-stock value. Analytical results complemented by numerical examples reveal that neither the best stationary base-stock nor its average cost is monotone in L for a given T.