Mathematical Programming

, Volume 105, Issue 2–3, pp 251–274 | Cite as

Projection results for vehicle routing

  • Adam N. Letchford
  • Juan-José Salazar-González


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.


Vehicle routing Projection Integer programming 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Adam N. Letchford
    • 1
  • Juan-José Salazar-González
    • 2
  1. 1.Department of Management ScienceLancaster UniversityLancasterUK
  2. 2.DEIOCUniversidad de La LagunaTenerifeSpain

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