Capacitated Spatial Newsvendor

Abstract

We revisit the classical newsvendor problem by introducing capacitated supply and spatial transportation in a setting where a firm needs to position available supplies across a service region to meet random demand over the space. Leveraging optimal transport theory, we derive optimality conditions that minimize the sum of aggregate newsvendor loss and transportation expenses in a general framework, accounting for demand over continuous space or discrete locations or a combination of both. The solution structure retains the simplicity of the classical newsvendor solution but also differs due to the interplay of demand, transportation, and supply constraints. The demand effect drives supply placement to high-demand areas, the transportation effect prioritizes nearby demand, and the supply constraint restricts the ability to fully meet all demand, especially when the total supply is insufficient. The solution is explicitly characterized using Laguerre cell division in a specific semi-discrete setting, where initial supplies are stored at discrete facilities and demands are distributed over a continuous space. This structure reveals how transportation costs adjust the optimal demand quantile and how limited local supply influences this quantile through Laguerre cells' weight parameters. The derived optimality conditions further inform the development of numerical algorithms in both semi-discrete and fully discrete settings.