The Maximum Flow Problem with Minimum Lot Sizes

In many transportation systems, the shipment quantities are subject to minimum lot sizes in addition to regular capacity constraints. That is, either the quantity must be zero, or it must be between the two bounds. In this work, we consider a directed graph, where a minimum lot size and a flow capacity are defined for each arc, and study the problem of maximizing the flow from a given source to a given terminal. We prove that the problem is NP-hard. Based on a straightforward mixed integer programming formulation, we develop a Lagrangean relaxation technique, and demonstrate how this can provide strong bounds on the maximum flow. For fast computation of near-optimal solutions, we develop a heuristic that departs from the zero solution and gradually augments the set of flow-carrying (open) arcs. The set of open arcs does not necessarily constitute a feasible solution. We point out how feasibility can be checked quickly by solving regular maximum flow problems in an extended network, and how the solutions to these subproblems can be productive in augmenting the set of open arcs. Finally, we present results from preliminary computational experiments with the construction heuristic.