The time constrained multi-commodity network flow problem and its application to liner shipping network design

• Show that it is essential to include transit time constraints in LSND.
• Present a tailored algorithm for the solution of the MCF problem.
• Show through computational experiments that solution time does not increase.
• Provide new models for accurately modeling transshipment operations in a graph.
• Describe a contraction scheme that can reduce the LSN graph size.

The multi-commodity network flow problem is an important sub-problem in several heuristics and exact methods for designing route networks for container ships. The sub-problem decides how cargoes should be transported through the network provided by shipping routes. This paper studies the multi-commodity network flow problem with transit time constraints which puts limits on the duration of the transit of the commodities through the network. It is shown that for the particular application it does not increase the solution time to include the transit time constraints and that including the transit time is essential to offer customers a competitive product.