A Novel Modeling Approach for Express Package Carrier Planning

Express package carrier networks have large numbers of heavily-interconnected and tightly-constrained resources, making the planning process difficult. A decision made in one area of the network can impact virtually any other area as well. Mathematical programming therefore seems like a logical approach to solving such problems, taking into account all of these interactions. The tight time windows and non-linear cost functions of these systems, however, often make traditional approaches such as multicommodity flow formulations intractable. This is due to both the large number of constraints and the weakness of the linear programming (LP) relaxations arising in these formulations. To overcome these obstacles, we begin by proposing an alternative formulation to the load matching and routing problem that incorporates much of the problem complexity implicitly within the variable definition, rather than explicitly within the constraints. This formulation enables us to linearize the cost structure, strengthen the LP relaxation of the formulation, and drastically reduce the number of constraints. We then use templates, rather than traditional column generation, to identify promising candidate vairables, ensuring tractability while also enabling the inclusion of additional operational considerations that would be difficult if not impossible to capture in a traditional approach. Finally, we extend our approach to also include two additional planning problems — trailer assignment and empty balancing — rather than solving these three closely related problems sequentially, as is typically done in practice. Computational results are presented using data from a major international package carrier.