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The Vehicle Routing Problem (VRP) is a class of well-known NP-hard combinatorial optimisation problems. The VRP is concerned with the design of the optimal routes, used by a fleet of identical vehicles stationed at a central depot to serve a set of customers with known demands. In the basic version of the problem, known as a Capacitated VRP (CVRP), only capacity restrictions for vehicles are considered and the objective is to minimize the total cost (or length) of routes. The Capacitated Vehicle Routing Problem with Time Windows (CVRPTW), which is a generalization of the CVRP, is one of the most studied variants of the VRP. In the CVRPTW, the vehicles must comply with constraint of time windows associated with each customer in addition to capacity constraints. The study of the VRP is very important. The VRP contributes directly to a real opportunity to reduce costs in the important area of logistics. Logistics can be roughly described as the delivery of goods from one place (supplier) to others (consumers). Transportation management, and more specifically vehicle routing, has a considerable economical impact on all logistic systems. Due to the nature of the problem, it is not viable to use exact methods for large instances of the VRP. Therefore, most approaches rely on heuristics that provide approximate solutions. Some specific methods have been developed to this problem. Another option is to apply standard optimization techniques, such as tabu search, simulated annealing, constraint programming, genetic algorithms and ant systems.Our main interest is about the metaheuristics used to solve the VRP and more particularly about the ant colony system. The first algorithm based on the ant colony system, applied to the CVRP, was proposed by [Bullnheimer & al. 1999] known as « Ant System » (AS), applied first for the TSP in [Dorigo & al. 1996]. The pheromone and the nearest neighbor heuristic are used to build the routes of the vehicles. To improve the routes, a heuristic of 2-OPT is combined with the AS. This algorithm was tested on 40 benchmark but the performances are inferior to those relative to the tabu search algorithm. Yet, in terms of time accomplishment, the AS is a serious rival for the existing metaheuristics.

A Hybrid Ant Colony System Approach for the Capacitated Vehicle Routing Problem and the Capacitated Vehicle Routing Problem with Time Windows