Optimal Robust Sourcing with Volume Flexibility: Anticipatory Ordering Using the Shifting Operator

Abstract

We study an inventory model with volume flexibility. A firm may replenish its inventory by using its base capacity at regular sourcing cost, after which additional supply is available at a premium. The structure of the optimal replenishment policy is known to be a modified base-stock policy, yet one must resort to computational approaches to set the optimal base-stock levels. This is non-trivial in non-stationary and correlated environments, thus companies often rely on simplified, suboptimal heuristics. We employ robust optimization and leverage the central limit theorem to express the robustly optimal base-stock levels in closed form. A key novelty of our work is the emergence of a shifting operator that describes when one should source beyond the base capacity, postpone replenishment to avoid the sourcing premium, or temporarily increase orders beyond their demand to utilize the base capacity. From our robust formulation, we derive a robust heuristic that performs well when evaluated with respect to the expected average cost, especially when demand is non-stationary and when the base capacities fluctuate. Applying our model on data of a manufacturer reduces inventory and sourcing costs by 2.2% compared to the modified base-stock policy adopted by the manufacturer without a shifting operator. Our insights apply to a variety of practical settings such as production systems with overtime, sourcing from multiple sources, or transportation planning with a spot market.

Keywords: Robust Optimization, Inventory, Smoothing, Flexibility