Data-Driven Robust Resource Allocation with Monotonic Cost Functions

We consider two-stage planning problems (arising, e.g., in city logistics) in which a resource is first divided among a set of independent regions and then costs are incurred based on the allocation to each region. Costs are assumed to be decreasing in the quantity of the resource, but their precise values are unknown, for example, if they represent difficult expected values. We develop a new data-driven uncertainty model for monotonic cost functions, which can be used in conjunction with robust optimization to obtain tractable allocation decisions that significantly improve worst-case performance outcomes. Our model uses a novel uncertainty set construction that rigorously handles monotonic structure based on a statistical goodness-of-fit test with respect to a given sample of data. The practical value of this approach is demonstrated in three realistic case studies.