Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem

Abstract

During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50-80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.

An exact solution framework for a broad class of vehicle routing problems

“…In this section, we report a computational comparison of the results obtained by Lysgaard et al (2004), Fukasawa et al (2006) and Baldacci et al (2008bBaldacci et al ( , 2009d on six classes of CVRP instances from the literature, called A, B, E, M and P. These instances are available at http://branchandcut.org/VRP/data. Classes A, B and P were proposed by Augerat (1995).…”

“…Lysgaard et al (2004) proposed a BC algorithm improving the method proposed by Augerat et al (1995); they used a variety of valid inequalities including capacity, framed capacity, comb, partial multistar, hypotour and Gomory cuts. Fukasawa et al (2006) described a BCP for solving the SP model of the CVRP strengthened by the valid inequalities introduced by Lysgaard et al (2004). The lower bound is computed by a column-and-cut generation method that uses q-routes (see Christofides et al 1981a) instead of feasible CVRP routes.…”

“…Because this method may not be competitive with the BC of Lysgaard et al (2004), they combined these two methods. Thus, the resulting algorithm of Fukasawa et al (2006) decides at the root node to use either the BC of Lysgaard et al (2004) or the new BCP. The computational results show that 26 out of 74 CVRP instances are solved using the BC algorithm, and the remaining 48 instances are solved by the new BCP algorithm.…”

“…The most effective exact algorithms for the CVRP are due to Baldacci et al (2004aBaldacci et al ( , 2008b, Lysgaard et al (2004) and Fukasawa et al (2006). Baldacci et al (2004a) described a BC algorithm based on a Two-Commodity network flow formulation of the CVRP.…”

A novel heuristic algorithm for capacitated vehicle routing problem

“…The proposed algorithm was compared with other heuristics on the literature in a different set of problems and the effectiveness of the algorithm was illustrated. Fukasawa et al (2006) proposed an algorithm based on branch and bound cut and traditional Lagrangian relaxation over q routes. The algorithm was tested on more than a hundred of instances.…”