New Lower Bounds on the Number of Vehicles for the Vehicle Routing Problem with Time Windows

The Vehicle Routing Problem with TimeWindows (VRPTW) consists in determining the routing plan of vehicles with identical capacity in order to supply the demands of a set of customers with predefined time windows. This complex multi-constrained problem has been widely studied due to its industrial, economic and environmental implications. In this work, we are interested in defining the number of vehicles needed to visit all the customers. This objective is very important to evaluate the fixed costs for operating the fleet. In this paper, we provide an analysis of several lower bounds based on incompatibility between customers and vehicle capacity constraints. Then we develop an adaptation of Energetic Reasoning algorithm for VRPTW with a limited fleet. The proposed approach focuses on some time-intervals and exploits time constraints, incompatibility graph and bin packing models in order to obtain new valid lower bounds for the fleet size. Experiments conducted on the standard benchmarks show that our algorithms outperform the classical lower bound techniques and prove the optimality for 339 out of 468 instances.