| In many real-life contexts, inventory levels are only partially (i.e. not fully observed). This is may be due to non-observation of demand, spoilage, or misplacement. This dissertation examines different aspects of an inventory system where the demand is not observed. We provide below a brief description of specific problems considered in this study.;In Chapter 2, we present a periodic-review inventory system where the unmet demand is backordered. When inventory level is nonnegative, the inventory level is unknown and represented by a conditional distribution. Otherwise, inventory shortages occur and the inventory manager issues rain checks to customers. The shortages are fully observable via the rain checks. In Chapter 3, we extend the model to the case of partially observed backorders. That is, neither the inventory levels or backordered quantities are observable to the IM. However, by looking at the shelf, he knows whether the inventory level is positive or nonpositive.;Based on all these partial information, the inventory manager determines the order quantity in each period to minimize the expected total discounted cost over an infinite-horizon. The Dynamic Programming formulation of these problems has an infinite-dimensional state space. The methodology of unnormalized probability is adopted to establish the existence of an optimal feedback policy and the uniqueness of the solution to the Dynamic Programming equation when the operational cost each period has linear growth.;In Chapter 4, we approximate the inventory distribution discussed in Chapter 2 by Chebyshev polynomials to compute the optimal order quantity/cost for the system. Moreover, we use Fast Fourier Transforms to speed up the computations. We also propose a heuristic termed a base mean-stock policy. The order quantity for the heuristic policy is computed by regarding the mean of the inventory level as the inventory level in a fully observed inventory system, and then using a base stock policy. We show numerically that the optimal order quantity is very close to the base mean-stock order quantity, when the variance of the inventory distribution is small. When the mean of the inventory distribution is large, the optimal order quantity is more than the base mean-stock quantity, it is the other way around when the mean is small or there are backorders. |
