Gaussian Partially Observable Inventory Control

In the classic single-product periodic-review stochastic inventory control problem with backorders and setup costs, it is well-known that there exist optimal (s t , S t ) policies under the finite-horizon, infinite-horizon discounted, and average cost criteria. We consider a partially observable version of this classic problem. We show that when both the per-period demands and additive observation disturbances are Gaussian, there exist optimal (s t , S t ) policies under the aforementioned criteria, where the time-dependent reorder points and order-up-to points are relative to the posterior mean of the true inventory level. This is proved by reducing the original problem to a fully observable inventory control problem with non-stationary demand.