Projection results for vehicle routing

Abstract

Abstract.A variety of integer programming formulations have been proposed for Vehicle Routing Problems (VRPs), including the so-called two-and three-index formulations, the set partitioning formulation, and various formulations based on extra variables representing the flow of one or more commodities. Until now, there has not been a systematic study of how these formulations relate to each other. An exception is a paper of Luis Gouveia, which shows that a one-commodity flow formulation of Gavish and Graves yields, by projection, certain 'multistar' inequalities in the two-index space.We give a survey of formulations for the capacitated VRP, and then present various results of a similar flavour to those of Gouveia. In particular, we show that:-the three-index formulation, augmented by certain families of valid inequalities, gives the same lower bound as the two-index formulation, augmented by certain simpler families of valid inequalities, -the two-commodity flow formulation of Baldacci et al. gives the same lower bound and the same multistar inequalities as the one-commodity Gavish and Graves formulation, -a certain non-standard multi-commodity flow formulation, with one commodity per customer, implies by projection certain 'hypotour-like' inequalities in the two-index space, -the set partitioning formulation implies by projection both multistar and hypotour-like inequalities in the two-index space.We also briefly look at some other variants of the VRP, such as the VRP with time windows, and derive multistar-like inequalities for them. We also present polynomial-time separation algorithms for some of the new inequalities.

New techniques for cost sharing in combinatorial optimization games

“…It is easy to show that all of the inequalities which are assignable for the traveling salesman game are also assignable for the vehicle routing game. Moreover, further assignable inequalities were presented in [25]. (They were called decomposable inequalities in that paper.)…”

“…Other assignable inequalities discussed in [25] include the Knapsack Large Multistar (KLM) inequalities [24], which are a generalization of the GLM inequalities, and the so-called hypotour-like inequalities. While the complexity of the separation problem for the KLM inequalities is unknown, a generalization of the hypotour-like inequalities can be separated efficiently [25].…”

New Lower Bounds for the Vehicle Routing Problem with Simultaneous Pickup and Delivery

“…Dell'Amico et al [3] basically extended the one-commodity flow formulation proposed by Gavish and Graves [7] for the CVRP by adding constraints (6) and (9), and the term P ij in (7). Gouveia [8] showed that it is possible to obtain stronger inequalities for D ij by using the tighter bounds (11) instead of (8) in the Gavish and Graves formulation.…”

“…Gouveia [8] showed that it is possible to obtain stronger inequalities for D ij by using the tighter bounds (11) instead of (8) in the Gavish and Graves formulation. Accordingly, we can apply the same idea to develop stronger inequalities for P ij by replacing (9) with (12) and for D ij + P ij by replacing (7) with (13).…”

“…Letchford and Salazar-Gonzalez [9] have shown that the one-commodity formulation and the directed two-commodity flow formulation with their respective stronger inequalities are equivalent for the CVRP. However, this fact is not verified for the VRPSPD as stated by Proposition 1.…”