Inventory Control with a Cash Register : Sales Recorded but Not Demand or Shrinkage

Inventory inaccuracy is common at retailers. At many retailers, a cash register records incoming orders and outgoing sales, but not the demand or the shrinkage. The shrinkage refers to spoilage or pilferage of inventory. The demand differs from the sales in the periodic-review lost-sales inventory model presented in this paper. When the inventory exceeds the demand, the remaining inventory is subject to a shrinkage unobserved by the inventory manager. Otherwise, the unmet demand is lost and unobserved. Our objective is to minimize the expected discounted cost over an infinite horizon. We use dynamic programming along with the concept of unnormalized probability, and prove the existence of an optimal feedback policy and that the value function is the solution of the dynamic programming equation. We obtain a theoretical lower bound for the cost via the formulation of a fictitious inventory problem. For discrete-valued inventories, we provide the analogous formulation, develop an iterative algorithm, and compare its solution to a myopic solution and a lower bound. We identify many cases of parameter values for which the solution of the iterative algorithm performs significantly better than the myopic solution. Moreover, the achieved cost is close to the lower bound