Title: Solving the market split problem via branch-and-cut

Authors: Haibo Wang, Gary Kochenberger, Bahram Alidaee, Mark Lewis

Addresses: AR Sanchez Jr. School of Business, Texas A&M International University, Laredo, TX 78045, USA. ' School of Business Administration, University of Colorado at Denver, Denver, CO 80217, USA. ' School of Business Administration, University of Mississippi, University, MS 38677, USA. ' School of Business Administration, University of Mississippi, University, MS 38677, USA

Abstract: In 1999 some researchers put forth some small but extremely difficult 0/1 problems derived from the market split (market sharing) problem originally proposed by a researcher in 1978. These problems, offered as a challenge to the optimisation community, motivated a series of research efforts trying several competing methodologies. As part of their testing, they employed both branch-and-bound and branch-and-cut algorithms and reported that branch-and-bound performed poorly and branch-and-cut did even worse. Based on this result they concluded that |standard approaches to integer programming are not well suited for this class of problems|. Other researchers offered similar assessments. In this note we demonstrate that current state of the art branch-and-cut methods, unavailable earlier, is able to solve these problems to optimality. Then we also solve 40 numerical problems to optimality and also find that it works quite nicely on even larger problem instances.

Keywords: integer programming; small hard instances; branch-and-bound; branch-and-cut; market split; market sharing; optimisation.

DOI: 10.1504/IJMMNO.2009.030091

International Journal of Mathematical Modelling and Numerical Optimisation, 2009 Vol.1 No.1/2, pp.121 - 127

Published online: 09 Dec 2009 *

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