The Multi-Item Newsvendor Problem with a Single Capacity Constraint

In many situations, retailers or manufacturers have a limited amount of capacity to stock, purchase, or manufacture goods where the time until obsolescence or spoilage may be relatively short. To address such situations, we consider a multi-item newsvendor problem with a single capacity constraint. We formulate a mathematical program for this model and discuss its solution. We show that the model falls into a special class of resource allocation models and that the optimal order quantities follow a stochastic version of the cut across the board rule. We also perform sensitivity analysis that provides the shadow price for additional capacity. Explicit expressions for the order quantities are developed for a broad range of demand distributions, including normal, uniform, beta, gamma, and Weibull. Finally, we provide an example of how this model could apply to a retailer faced with a capacity constraint. Additionally, we demonstrate that the results are quite robust to violations of the assumptions upon which the results are based