A column generation and a post optimization VNS heuristic for the vehicle routing problem with multiple time windows

The Vehicle Routing Problem with Multiple Time Windows (VRPMTW) is a generalization of the Vehicle Routing Problem (VRP), where the customers have one or more time windows in which they can be visited. In this paper, we propose a Column Generation (CG) algorithm and a post optimization heuristic based on a Variable Neighborhood Search (VNS) to provide both lower and upper bounds for the cost of optimal solutions to VRPMTW. As in CG algorithms for VRP, the master problem is based on a Weighted Set Covering formulation. However, due to the multiple time windows, the pricing subproblem is an Elementary Shortest Path Problem with Multiple Time Windows and Capacity Constraints, which is more difficult to solve than the classical Elementary Shortest Path Problem with a Single Time Window and Capacity Constraints. Computational experiments were performed on 594 instances generated from classical Solomon instances with up to 17 customers. They showed that CG was able to produce lower bounds, within one hour of running time, for 66.7% of the instances. Besides, the post optimization heuristic was able to improve the solution provided by the VNS heuristic in 28.9%, finding integer optimal solutions for 39.9% of the instances. Moreover, for the instances where lower bounds are known, the average optimality gap was 6.0% on average.