A maritime inventory routing problem: Discrete time formulations and valid inequalities

4 A single product maritime inventory routing problem in which the production and con5 sumption rates vary over the planning horizon is studied. The problem includes a heteroge6 neous fleet and multiple production and consumption ports with limited storage capacity. 7 Two discrete time formulations are developed. One is a standard formulation and the other 8 is a fixed charge network flow reformulation. Mixed integer sets arising from the decomposition 9 of the formulations are identified. Several lot-sizing relaxations are derived for the network 10 formulation and used to establish valid inequalities to strengthen the proposed formulations. 11 Considering a set of instances based on real data, a computational study is conducted to 12 test the formulations and the effectiveness of the inclusion of valid inequalities. By using a 13 branch and bound scheme based on the strengthened network formulation a large majority of 14 the instances with up to sixty time periods are solved to optimality. 15