Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations

Exact Solutions to the Symmetric and Asymmetric Vehicle Routing Problem with Simultaneous Delivery and Pick-Up

“…Class 6 consists of random test instances, where two different geographical scenarios are examined; a uniformly distributed and an urban configuration of customers. The instances of class 7 are derived from the Christofides, Mingozzi, and Toth (1979) benchmark for the CVRP. Both test sets consist of instances with 50 customers.…”

Exact algorithms for different classes of vehicle routing problems

“…Foremost, we propose new models for some of the problems considered and develop new exact algorithms that are highly competitive with the other exact algorithms from the literature. The new algorithms proposed share common ingredients: column generation (see Desaulniers et al [2005]), dynamic programming (see Bellman [1957], Bertsekas [1995], Bertsekas [2000], Art and Mauch [2007]), and state-space relaxation (see Christofides et al [1981c], Christofides et al [1981b]). …”

“…The cost f (q, i) of a least-cost (q, i)-path can be computed as described by Christofides et al [1981b]. A (q, i)-route is a (q, 0)-path.…”

“…Bounding procedure H 1 enlarges the route set R with the t-routes (to solve the VRPTW) or the q-routes (in the CVRP, see Christofides et al [1981b]). The initial route set R of the master problem contains all single-customer routes (0, i, 0), i ∈ V .…”

“…Procedure H 1 finds a dual solution (u 1 , v 1 , g 1 ) of cost LB 1 of problem LRF 1 by adding SC and WSR3 inequalities, ignoring WT inequalities, and replacing the route set R with the set of all q-routes (see Christofides et al [1981b]). Procedure H 2 differs from H 1 as it uses elementary routes instead of q-routes.…”