An Incomplete Information Inventory Model with Presence of Inventories or Backorders as Only Observations

In many real-life contexts, inventory levels are only incompletely observed due to nonobservation of demand, discrepancies in transmitting sales data, transaction errors, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system where the demand is not observed and the unmet demand is backordered. As a result, the inventory manager cannot tell the exact quantities of inventories or backorders. However, by looking at the shelf, he knows whether the inventory is positive or non-positive. Only with this information, the inventory manager must determine the order quantity in each period that would minimize the expected total discounted cost over an infinite-horizon. The dynamic programming formulation of this problem has an infinitedimensional state space. We use the concept of unnormalized probability to establish the existence of an optimal feedback policy and the uniqueness of the solution of the dynamic programming equation when the periodic cost has linear growth