Optimality of Four-Threshold Policies in Inventory Systems with Customer Returns and Borrowing/Storage Options

Consider a single commodity inventory system in which the demand is modelled by a sequence of i.i.d. random variables that can take negative values. Such problems have been studied in the literature under the name cash management and relate to the variations of the on-hand cash balances of financial institutions. The possibility of a negative demand also models product returns in inventory systems. This paper studies a model in which, in addition to standard ordering and scrapping decisions seen in the cash management models, the decision maker can borrow and store some inventory for one period of time. For problems with backorders, zero setup costs, and linear ordering, scrapping, borrowing, and storage costs, we show that an optimal policy has a simple four-threshold structure. These thresholds, in a nondecreasing order, are: order-up-to, borrow-up-to, store-down-to, and scrap-down-to levels. That is, if the inventory position is too low, an optimal policy is to order up to a certain level and then borrow up to a higher level. Analogously, if the inventory position is too high, the optimal decision is to reduce the inventory to a certain point after which one should store some of the inventory down to a lower threshold. This structure holds for the finite and infinite horizon discounted expected cost criteria and for the average cost per unit time criterion. We also provide sufficient conditions when the borrowing and storage options should not be used. In order to prove our results for average costs per unit time, we establish sufficient conditions when the optimality equations hold for a Markov decision process with an uncountable state space, noncompact action sets, and unbounded costs.