We consider a single-item, periodic review inventory control problem where 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, which might include costs associated with in-house rush production or outsourcing. We explicitly consider the case where expedited production is allowed to produce up to a positive inventory level. We also considered the case where expedited production beyond the shortfall 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 expediting policy and show that an (s, S) policy is optimal for regular production. In certain cases, we demonstrate that it may indeed be optimal to use expedited production to build up inventory. 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 expediting policy is generalized (s, S) or order-up-to, respectively. An explicit heuristic for policy calculation is given; a numerical study tests the heuristic and allows us to gain insight into when expediting above zero is costeffective. We find that, while excess expediting above zero is frequently optimal (particularly when the per-unit costs are close to those of regular production), the actual cost savings from the additional expediting are minimal.

We consider a two-stage supply chain under centralized control. The downstream facility faces discrete stochastic demand and passes supply requests to the upstream facility. The upstream facility always meets the supply requests from downstream. If the upstream facility cannot meet the supply requests from inventory on hand, the shortage must be filled by expediting, which will incur per unit and setup costs. Such expediting may take the form of overtime production, which occurs at the end of the period and incurs relatively high production costs, or premium freight shipments, which involves building products at the beginning of the period they are needed and shipping them very quickly with relatively high shipping costs. We consider the case where one method of filling shortages is available and determine novel optimal inventory policies under centralized control. At both stages, threshold policies that depend only on the current inventory in the system are optimal; for the total inventory in the system, a base-stock policy is optimal. Numerical analysis provides insight into the optimal policies and allows us to compare the supply chains under centralized and decentralized control.Expediting, Overtime, Supply Chain Coordination, Inventory, Markov Decision Processes

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