The multi-commodity network flow problem with soft transit time constraints: Application to liner shipping

The multi-commodity network flow problem (MCNF) consists in routing a set of commodities through capacitated at minimum cost and is relevant for containers liner shipping networks. As commodity transit times are often critical factor, the literature has introduced hard limits on times. In practical contexts, however, these may fail to provide sufficient flexibility since routes with even tiny delays would be discarded. Motivated by major operator, we study an MCNF generalization where time restrictions modeled as soft constraints, which discouraged using penalty functions time. Similarly, early arrivals can receive discount cost. We derive properties that distinguish this model from other variants adapt column generation procedure efficiently solve it. Extensive numerical experiments conducted realistic instances reveal explicit consideration lead significant reductions compared deadlines. Moreover, penalties used steer towards slower or faster configurations, resulting potential increase operational costs, generates trade-off quantify under varying functions.