Redistributing stock in library systems with a depot

Public library organizations often utilize depots for carrying out shipments to libraries in case of stock-outs and for storing low demand rental items at low cost. Similar systems may be employed by rental companies for other rental products such as tools, DVDs, and jewelry. Since shipments deplete the depot's inventory, stock must be taken back from the libraries in order to deal with future shipment requests. These shipment and take-back operations are carried out periodically, e.g. daily or weekly. This work focuses on optimizing the decisions for shipments and take-backs. We model the system by means of a Markov decision process and investigate its optimal policy for various problem instances. For the take-back decision, we distinguish between so-called threshold, reactive, and preventive take-backs. We use the insights from the MDP to develop a three-phase take-back heuristic. In experiments, our heuristic performs within 1% on average from the optimal solution. For settings with a large number of libraries, it is shown that an acceptable performance can be achieved by setting a base-stock level at the depot and taking back sufficient stock from the libraries to achieve this level.