Optimizing stock levels for rental systems with a support warehouse and partial backordering

Various rental systems, such as public libraries and tool rental companies, have a support warehouse for storing rental products at low cost and carrying out expedited shipments in response to stock-outs at local warehouses. In case of stock-outs, rental systems commonly employ partial backordering, i.e., a limited number of demands is backordered while additional demand is lost. We include partial backordering in our models and demonstrate its relevance. We fully characterize costs and optimal base stock policies in cases with one support and one local warehouse and provide upper bounds for base stock levels in the general system. By enumerating base stock policies and evaluating exact costs numerically, we determine optimal base stock policies for small systems with up to six local warehouses. For larger systems we develop a greedy algorithm based on approximate cost evaluations. The resulting base stock policies have costs within 0.2% from optimality on average in experiments. Among other things, experiments indicate that existing methods with no backordering may lead to poor solutions with partial backordering and that partial backordering is an effective means to reduce lost demand even with limited partial backorder levels.