Solving Vehicle Routing Problems Using Constraint Programming and Lagrangean Relaxation in a Metaheuristics Framework D. Guimarans*, R. Herrero, J.J. Ramos, S. Padrón Dpt. Telecommunication and Systems Engineering, Universitat Autònoma de Barcelona, Spain ABSTRACT This paper presents a methodology based on the Variable Neighbourhood Search metaheuristic, applied to the Capacitated Vehicle Routing Problem. The presented approach uses Constraint Programming and Lagrangean Relaxation methods in order to improve algorithm’s efficiency. The complete problem is decomposed into two separated subproblems, to which the mentioned techniques are applied to obtain a complete solution. With this decomposition, the methodology is able to provide a quick initial feasible solution which is rapidly improved by metaheuristics’ iterative process. Constraint Programming and Lagrangean Relaxation are also embedded within this structure to ensure constraints satisfaction and to reduce the calculation burden. By means of the proposed methodology, promising results have been obtained. Remarkable results presented in this paper include a new best-known solution for a rarely solved 200-customers test instance, as well as a better alternative solution for another benchmark problem. Keywords: Vehicle Routing, Constraint Programming, Lagrangean Relaxation, Metaheuristics, Variable Neighbourhood Search, Hybrid Algorithms. INTRODUCTION Routing vehicles to collect or delivery goods is a problem which many companies face each day, laying at the heart of many distribution systems. In practice, objectives and constraints are highly variable and, most of times, complex. In fact, real problems often require a specific modelling and solving methodology. On the other hand, most research is focused on well-known sets of academic problems including certain characteristics. However, since flexible and efficient algorithms are likely to be adapted to various practical contexts, these prototype problems become a nice reference where to test developed methodologies. This class of logistics problems, usually known as the Vehicle Routing Problem (VRP), is among the most popular research areas in combinatorial optimization. Since it was first defined by Dantzig and Ramser (1959), several variants of the basic problem have been proposed and studied. The most basic VRP is the Capacitated Vehicle Routing Problem (CVRP) that assumes a fleet of vehicles with homogeneous capacity housed in a single depot. It is so a generalization of the Travelling Salesman Problem (TSP) and is therefore NP-hard (Savelsbergh, 1985). For such problems, finding an optimal solution requires a high computational effort.