Lin, Z and Kwan, RSK (2016) Local convex hulls for a special class of integer multicommodity flow problems. Computational Optimization and Applications, 64 (3). pp. 881-919. ISSN 0926-6003
Abstract
Based on previous work in rolling stock scheduling problems (Alfieri et al. in Transp Sci 40:378–391, 2006; Cacchiani et al. in Math Progr B 124:207–231, 2010; Lin and Kwan in Electron Notes Discret Math 41:165–172, 2013; Schrijver in CWI Q 6:205–217, 1993; Ziarati et al. in Manag Sci 45:1156–1168, 1999), we generalize a local convex hull method for a class of integer multicommodity flow problems, and discuss its feasibility range in high dimensional cases. Suppose a local convex hull can be divided into an up hull, a main hull and a down hull if certain conditions are met, it is shown theoretically that the main hull can only have at most two nonzero facets. The numbers of points in the up and down hull are explored mainly on an empirical basis. The above properties of local convex hulls have led to a slightly modified QuickHull algorithm (the “2-facet QuickHull”) based on the original version proposed by Barber et al. (ACM Trans Math Softw 22:469–483, 1996). As for the feasibility in applying this method to rolling stock scheduling, our empirical experiments show that for the problem instances of ScotRail and Southern Railway, two major train operating companies in the UK, even in the most difficult real-world or artificial conditions (e.g. supposing a train can be served by any of 11 compatible types of self-powered unit), the standard QuickHull (Barber et al. in ACM Trans Math Softw 22:469–483, 1996) can easily compute the relevant convex hulls. For some even more difficult artificial instances that may fall outside the scope of rolling stock scheduling (e.g. a node in a graph can be covered by more than 11 kinds of compatible commodities), there is evidence showing that the “2-facet QuickHull” can be more advantageous over the standard QuickHull for our tested instances. When the number of commodity types is even higher (e.g. >19), or the number of points in a high dimensional space (e.g. 15 dimensions) is not small (e.g. >2000), the local convex hulls cannot be computed either by the standard or the 2-facet QuickHull methods within practical time.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016, The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
Keywords: | Integer multicommodity network flow; Convex hull computation; Rolling stock scheduling |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) > Institute for Computational and Systems Science (Leeds) |
Funding Information: | Funder Grant number EPSRC EP/M007243/1 First Rail Holdings Ltd 4700245342 |
Depositing User: | Symplectic Publications |
Date Deposited: | 10 Mar 2016 14:21 |
Last Modified: | 11 Apr 2018 15:37 |
Published Version: | http://dx.doi.org/10.1007/s10589-016-9831-3 |
Status: | Published |
Publisher: | Springer Verlag |
Identification Number: | 10.1007/s10589-016-9831-3 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:95521 |