Time and quantity dependent waiting costs in a newsvendor problem with backlogged shortages

Upon demand realization in the newsvendor problem, it is often assumed that shortages result in lost sales penalties. However, in some practical situations, shortages are backlogged and the inventory manager is penalized based on the magnitude and duration of the shortage. In this paper, we investigate a variation of the newsvendor problem in which all shortages are backlogged and replenished through an emergency procurement process. Costs incurred during emergency procurement include a variable emergency ordering cost and a non-linear customer waiting cost. We derive closed form expressions for the optimal order quantity and emergency procurement rate for two special cases, and we propose a heuristic approximation for the general case. The effectiveness of the approximate solution is assessed through numerical experiments and sensitivity analysis.