The vehicle routing problem with time windows (VRPTW) consists of finding least-cost vehicle routes to satisfy the demands of customers that can be visited within specific time windows. We introduce two enhancements for the exact solution of the VRPTW by branch-price-and-cut (BPC). First, we develop a sharper form of the limited-memory subset-row inequalities by representing the memory as an arc subset rather than a node subset. Second, from the elementary inequalities of Balas (1977) [Balas, E. 1977. Some valid inequalities for the set partitioning problem. In B.H. Korte, P.L. Hammer, E.L. Johnson, G.L. Nemhauser, eds., Studies in Integer Programming, Annals of Discrete Mathematics, vol. 1, Elsevier, 13-47], we derive a family of inequalities that dominate them. These enhancements are embedded into an exact BPC algorithm that includes state-of-the-art features such as bi-directional labeling, decremental state-space relaxation, completion bounds, variable fixing, and route enumeration. Computational results show that these enhancements are particularly effective for the most difficult instances and that our BPC algorithm can solve all 56 Solomon instances with 100 customers and 51 of the 60 Gehring and Homberger instances with 200 customers.
Published February 2016 , 23 pages