A Maritime Inventory Routing Problem: Discrete Time Formulations and Valid Inequalities

A single-product maritime inventory routing problem (MIRP) is studied in which the production and consumption rates vary over the planning horizon. The problem involves a heterogeneous fleet and multiple production and consumption ports with limited storage capacity. Two discrete time formulations are developed: an original model and a reformulated model that is a pure fixed charge network flow (FCNF) model with side constraints. Mixed integer sets arising from the decomposition of the formulations are identified. In particular, several lot-sizing relaxations are derived for the formulations and used to establish valid inequalities to strengthen the proposed formulations. Until now, the derivation of models and valid inequalities for MIRPs has mainly been inspired by the developments in the routing community. Here, we have developed a new model leading to new valid inequalities for MIRPs obtained by generalizing valid inequalities from the recent lot-sizing literature. Considering a set of instances based on real data, a computational study is conducted to test the formulations and the effectiveness of the valid inequalities. The FCNF formulation is generally much stronger than the original formulation. The developed valid inequalities reduce the integrality gap significantly for both formulations. By using a branch-and-bound scheme based on the strengthened FCNF formulation, most of our test instances are solved to optimality