Chvátal-Gomory Rank-1 Cuts Used in a Dantzig-Wolfe Decomposition of the Vehicle Routing Problem with Time Windows

This chapter shows how Chvatal-Gomory (CG) rank-1 cuts can be used in a Branch-and-Cut-and-Price algorithm for the Vehicle Routing Problem with Time Windows (VRPTW). Using Dantzig-Wolfe decomposition we split the problem into a Set Partitioning Problem as master problem and an Elementary Shortest Path Problem with Resource Constraints as pricing problem. To strengthen the formulation we derive general CG rank-1 cuts based on the master problem formulation. Adding these cuts to the master problem means that an additional resource is added to the pricing problem for each cut. This increases the complexity of the label algorithm used to solve the pricing problem since normal dominance tests become weak when many resources are present and hence most labels are incomparable. To overcome this problem we present a number of improved dominance tests exploiting the step-like structure of the objective function of the pricing problem. Computational experiments are reported on the Solomon test instances showing that the addition of CG rank-1 cuts improves the lower bounds significantly and makes it possible to solve a majority of the instances in the root node of the branch-and-bound tree. This indicates that CG rank-1 cuts may be essential for solving future large-scale VRPTW problems where we cannot expect that the branching process will close the gap between lower and upper bounds in reasonable time.