Tabu Search, Partial Elementarity, and Generalized k-Path Inequalities for the Vehicle Routing Problem with Time Windows

Abstract

The vehicle routing problem with time windows consists of delivering goods at minimum cost to a set of customers using an unlimited number of capacitated vehicles assigned to a single depot. Each customer must be visited within a prescribed time window. The most recent successful solution methods for this problem are branch-and-price-and-cut algorithms where the column generation subproblem is an elementary shortest-path problem with resource constraints (ESPPRC). In this paper, we propose new ideas having the potential to improve such a methodology. First, we develop a tabu search heuristic for the ESPPRC that allows, in most iterations, the generation of negative reduced cost columns in a short computation time. Second, to further accelerate the subproblem solution process, we propose to relax the elementarity requirements for a subset of the nodes. This relaxation, however, yields weaker lower bounds. Third, we introduce a generalization of the k-path inequalities and highlight that these generalized inequalities can, in theory, be stronger than the traditional ones. Finally, combining these ideas with the most recent advances published in the literature, we present a wide variety of computational results on the Solomon's 100-customer benchmark instances. In particular, we report solving five previously unsolved instances.

Exact algorithms for different classes of vehicle routing problems

“…The decremental state-space relaxation introduced by Righini and Salani [2008] and the partial elementarity relaxation introduced by Desaulniers et al [2008] are special cases of the ng-path relaxation. These relaxations can be obtained by setting N i = V , for each vertex i ∈ V , where V ⊆ V is a selected set of vertices.…”

“…Jepsen et al [2008] described a BCP algorithm based on the SP model and Subset-Row (SR) inequalities. This method was improved by Desaulniers et al [2008] by adding both SR and generalized k-path inequalities and by using a tabu search heuristic, instead of dynamic programming (DP) algorithms, to rapidly generate negative reduced cost routes. Their method outperforms all other algorithms, remarkably decreasing the computational time, and solving 5 of the 10 open Solomon instances.…”

“…GenR is an extension of the algorithm described by Baldacci et al [2008] for the symmetric CVRP that, in turn, is based on the method proposed by Mingozzi et al [1994] and adapted by Baldacci et al [2006] for the asymmetric CVRP on a multigraph. Similar methods have been used by Righini and Salani [2008], Jepsen et al [2008], and Desaulniers et al [2008].…”

“…H 4 differs from the methods of Jepsen et al [2008], Desaulniers et al [2008], and Baldacci et al [2008] for the pricing algorithm.…”

A Scalable Heuristic for Pickup-and-Delivery of Splittable Loads and Its Application to Military Cargo-Plane Routing

“…Various heuristics (Indra-Payoong et al, 2009;Savelsbergh and Sol, 1998;Xu et al, 2003) or meta-heuristics (Desaulniers et al, 2008;Moreno et al, 2006) have been proposed for an NP-hard pricing subproblem arising in a vehicle routing problem. We will use a tabu search based heuristic as tabu search seems most convenient to deal with the closed walks.…”