On Integral Policies in Deterministic and Stochastic Distribution Systems

The ‘integrality’ question for dynamic optimization models of inventory control asks if there exists an integral optimal policy, given integral initial inventory levels, capacities, and demand realizations. One practical implication of this question lies in whether or not full‐truckload (FTL) shipping is optimal if customer demand is in integral number of truckloads. In this paper, we investigate the integrality question in single‐product, multiechelon distribution systems and show that integrality holds under deterministic demand but fails to hold under stochastic demand. In distribution systems with stochastic demand, less‐than‐truckload (LTL) shipping can be significantly cheaper than the cost of the optimal FTL shipping policy, even in the presence of economies of scale. For instance, this occurs in settings where shipping costs are expected to increase in the future and/or inventories are more expensive to hold upstream than downstream. In such situations, our results highlight the importance of strategically positioning inventory: LTL shipments can offer a more balanced allocation of inventory across the distribution network, leading to benefits that can exceed the savings from FTL shipments due to economies of scale. However, when the cost parameters are fairly constant across time and inventory holding costs are not significantly higher upstream than downstream, then the difference between the costs of optimal FTL and optimal LTL shipping is provably marginal. The online appendix is available at https://doi.org/10.1287/opre.2016.1586.