Lagrangian relaxation method for network flow modeled crew and vehicle rescheduling

We propose a method for solving the crew rescheduling problem (CRP) and the vehicle rescheduling problem (VRP) based on the Lagrangian relaxation method. The CRP/VRP is formulated as an integer programming problem on the basis of a network flow modeling approach from which a Lagrangian relaxation problem is constructed by relaxing the constraint that covers multiple resources. Using two procedures that generate the upper and lower bounds of the primal problem, both of which utilize an efficient shortest path algorithm for the directed acyclic graph (DAG), the proposed method gradually improves the gap between the upper and lower bounds while updating Lagrangian multipliers. Results of real-world vehicle rescheduling data from a Japanese railway line indicate that the proposed method generates a feasible solution within a practical amount of time, which is confirmed to be fairly close to the optimum according to the gap and a comparison with the heuristic solution method.