Partially Observed Inventory Systems: The Case of Rain Checks

In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to non-observation of demand, spoilage, misplacement, or theft of inventory. We study a discrete-time periodic-review inventory system where the unmet demand is backordered. When inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur and the inventory manager issues rain checks to customers. The shortages are fully observed via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.