Inventory model with incomplete information: sales and zero-balance signals

Inventory problems with incomplete inventory information arise frequently in practice because demand or invisible demand is not observed directly but both reduce the inventory level. In this paper, we develop a periodic-review lost-sales inventory model, where the sales is always observed while the inventory level is observed only when it reaches zero. Our objective is to minimize the expected discounted cost over an infinite horizon, and we use dynamic programming along with the concept of unnormalized probability to solve the problem. The interaction between the sales and zero-balance walk signal simplifies the updating process of the inventory level distribution. Interestingly, the evolution of inventory distribution is independent of the demand. We also find a mean-based approximation has the customary dynamic program of the completely observed problem giving rise to a lower bound on the optimal cost of the original problem. Furthermore, incorporating the variance of inventory level improves the bound.