Solving Routing Problems with Branch-Cut-and-Price

Marcus Poggi de Aragão Dep.de Informática Pontif́ıcia Universidade Católica do Rio de Janeiro [email protected] Eduardo Uchoa Dep. Eng. de Produção Universidade Federal Fluminense [email protected] January 15th, 2005 The Capacitated Vehicle Routing Problem (CVRP) is the most basic variant of a vehicle routing problem: homogeneous fleet, single depot, only deliveries, no time-windows, and a one-dimensional capacity constraint on the load of each vehicle. It was recently shown that a careful combination of cut generation and column generation, in the so-called branch-cut-and-price algorithms, can be significantly more effective than previous approaches for the CVRP. This talk is a discussion over the potential of branch-cut-and-price algorithms on more generic and complex variants of vehicle routing. We include pointers to recent literature and present our current lines of research on the subject. Computational results on some classical variants, including the Vehicle Routing Problem with Time-Windows (VRPTW) and the Capacitated Arc Routing Problem (CARP) are given. The talk starts by presenting an approach we have initially developed for the Capacitated Minimum Spanning Tree problem ([3], [10]). Next, it is shown how it could be extended to the CVRP. The corresponding MIP formulation written below, in the form of what we called Explicit Master, combines the edge based and the column generation formulations for that problem. Suppose a complete undirected graph G = (V,E) with vertex set V = {0, 1, . . . , n}. The depot is assigned to vertex 0. The client vertex set V+ = {1, . . . , n} is associated to demands d(·). Each edge e ∈ E has a nonnegative length l(e). There are K vehicles of equal capacity C. For each one, a route, starting and ending at the depot, shall be determined assuring that total demand of the clients in the route do not exceed C. Finally, the set of all K routes must visit each client exactly once and total length is to be minimized.