Shipment consolidation with two demand classes: Rationing the dispatch capacity

We analyze the problem faced by a logistics provider that dispatches shipment orders (parcels or larger packages) of two order classes, viz. expedited and regular. Shipment orders arrive according to a compound Poisson process for each class. Upon an arrival, the logistics provider may continue consolidating arriving orders by paying a holding cost. Alternatively, the provider may dispatch, at a fixed cost, a vehicle containing (a portion of) the load consolidated so far. In addition, the provider must specify the composition of each dispatch by allocating (rationing) the volume of the vehicle between expedited and regular shipment orders. We model this problem as a continuous-time Markov Decision Process and minimize the expected discounted total cost. We prove the existence of quantity-based optimal threshold policies under particular conditions. We also structurally analyze the thresholds of these optimal policies. Based on these structural properties, we develop an efficient solution approach for large problem instances which are difficult to solve using the conventional policy-iteration method. For two real-life applications, we show that the quantity-based threshold policies derived using the proposed approach outperform the time policies used in practice.