Kallehauge, Brian5; Larsen, Jesper6; Madsen, Oli B.G.1
1 Logistics & ITS, Department of Transport, Technical University of Denmark2 Department of Transport, Technical University of Denmark3 Operations Research, Department of Informatics and Mathematical Modeling, Technical University of Denmark4 Department of Informatics and Mathematical Modeling, Technical University of Denmark5 Management Science, Department of Management Engineering, Technical University of Denmark6 Department of Management Engineering, Technical University of Denmark
This paper considers the vehicle routing problem with time windows, where the service of each customer must start within a specified time interval. We consider the Lagrangian relaxation of the constraint set requiring that each customer must be served by exactly one vehicle yielding a constrained shortest path subproblem. We present a stabilized cutting-plane algorithm within the framework of linear programming for solving the associated Lagrangian dual problem. This algorithm creates easier constrained shortest path subproblems because less negative cycles are introduced and it leads to faster multiplier convergence due to a stabilization of the dual variables. We have embedded the stabilized cutting-plane algorithm in a branch-and-bound search and introduce strong valid inequalities at the master problem level by Lagrangian relaxation. The result is a Lagrangian branch-and-cut-and-price (LBCP) algorithm for the VRPTW. Making use of this acceleration strategy at the master problem level gives a significant speed-up compared to algorithms in the literature based on traditional column generation. We have solved two test problems introduced in 2001 by Gehring and Homberger with 400 and 1000 customers respectively, which to date are the largest problems ever solved to optimality. We have implemented the LBCP algorithm using the ABACUS open-source framework for solving mixed-integer linear-programs by branch, cut, and price.
Computers and Operations Research, 2006, Vol 33, Issue 5, p. 1464-1487