Inventory Control with Generalized Expediting

We consider a single-item, periodic-review inventory control problem in which discrete stochastic demand must be satisfied. When shortages occur, the unmet demand must be filled by some form of expediting; we allow a very general form for the cost structure of expediting. We explicitly consider the case where expedited production is allowed to produce up to a positive inventory level. We also consider the case where expedited production beyond the deficit is not permitted; an alternate application for this model is an inventory system with general lost sales costs. For the infinite-horizon discounted problem, we characterize the structure of the optimal stationary expediting policy and show that an (s, S) policy is optimal for regular production. For the special cases where the expediting cost function is concave or consists of a fixed and linear per-unit cost, we show that the optimal stationary expediting policy is generalized (s, S) or order-up-to, respectively. A numerical study allows us to gain insight into when expediting above and beyond the deficit is cost-effective.