Vehicle scheduling in public transit and Lagrangean pricing

This paper investigates the solution of the linear programming relaxation of the multi-commodity flow formulation of the multiple-depot vehicle scheduling problems arising in public mass transit. We develop a column generation technique that makes it possible to solve the huge linear programs that come up there. The technique, which we call Lagrangean pricing, is based on two different Lagrangean relaxations. We describe in detail the basic ingredients of our approach and give computational results for large-scale test data (with up to 70 million variables) from three German public transportation companies. Because of these results, we propose Lagrangean pricing as one of the basic ingredients of an effective method to solve multiple-depot vehicle scheduling problems to proven optimality.