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Projection results for vehicle routing

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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.

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Authors and Affiliations

  1. Department of Management Science, Lancaster University, Lancaster, LA1 4YW, UK

    Adam N. Letchford

  2. DEIOC, Universidad de La Laguna, 38271, Tenerife, Spain

    Juan-José Salazar-González

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  1. Adam N. Letchford
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  2. Juan-José Salazar-González
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Correspondence to Adam N. Letchford.

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Received: May, 2004

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Letchford, A., Salazar-González, JJ. Projection results for vehicle routing. Math. Program. 105, 251–274 (2006). https://doi.org/10.1007/s10107-005-0652-x

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