Solving Vehicle and Crew Scheduling Problems Simultaneously

In the integrated vehicle and crew scheduling problem (VCSP) one has to simultaneously assign vehicles to timetabled trips and drivers to vehicles. In this paper two integer linear programming formulations are used to describe the integrated problem. On both formulations the multi-depot vehicle scheduling problem is modeled as a multicommodity flow problem while the crew scheduling problem is modeled as a set partitioning problem or as a set covering problem. Since, for real life applications, the set of feasible crew duties become too large to be considered explicitly, we solve the linear relaxations of the models by a column generation scheme. Whenever the resulting solution is not integer, branch-and-bound techniques are used over the set of feasible crew duties generated while solving the linear relaxation. We show that, under some conditions, just a subset of the decision variables is required to be integer and we propose different branching strategies depending on the subset of variables where branching is made. Computational experience, with random generated and real life data publicly available for benchmarking on the WEB, is reported and shows the effectiveness of the proposed approaches for the VCSP.