An exact algorithm for the vehicle and crew scheduling problem

We present a model for the vehicle and crew scheduling problem in urban public transport Systems by combining models for vehicle and crew scheduling that cover a great variety of real world aspects, especially constraints for crews resulting from wage agreements and internal regulations. The main part of the model consists of a set partitioning formulation to cover the desired trips of the schedule. Because of the great number of columns, e.g. more than 5 million for a problem with 30 trips, a column generation algorithm is implemented to use all columns implicitly for the calculation of the continuous relaxation of the set partitioning problem. The column generation algorithm is embedded in a brauch and bound approach to generate an exact solution for the problem. To generate even better lower bound, polyhedral cuts basing on clique detection and a variant of the column generation algorithm that suits the cuts were tested.