Routing and fleet deployment in liner shipping with spot voyages

•Representation of contractual and spot voyages as nodes of a directed graph.•Definition of dominated nodes based on their attributes.•Development of a mixed integer formulation based on the directed graph.•Development of a test problem generator based on real world assumptions.•Insights on costs and revenues when contractual and spot voyages are combined.

The routing, scheduling and fleet deployment is an important integrated planning problem faced by liner shipping companies which also lift load from the spot market. This paper is concerned with coordinating the decisions of the assignment of ships to contractual and spot voyages, and the determination of ship routes and schedules in order to maximize profit. We propose a new model for representing voyages as nodes of a directed graph which is used to build a mixed integer programming formulation. Besides contractual and spot nodes, another type of node is put forward to represent a combination of a contractual voyage with one or more spot voyages. In addition, the concept of dominated nodes is introduced in order to discard them and reduce the effort of the search for an optimal solution. A set of test problems has been generated taking into account real world assumptions. The test problems are solved by an optimization software and computational results are reported. The results show the potential of the approach to solve test problems of moderate size.