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Fast feasibility check of the multi-material vertical alignment problem in road design

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Abstract

When building a road, it is critical to select a vertical alignment which ensures design and safety constraints. Finding such a vertical alignment is not necessarily a feasible problem, and the models describing it generally involve a large number of variables and constraints. This paper is dedicated to rapidly proving the feasibility or the infeasibility of a Mixed Integer Linear Program (MILP) modeling the vertical alignment problem. To do so, we take advantage of the particular structure of the MILP, and we prove that only a few of the MILP’s constraints determine the feasibility of the problem. In addition, we propose a method to build a feasible solution to the MILP that does not involve integer variables. This enables time saving to proving the feasibility of the vertical alignment problem and to find a feasible vertical alignment, as emphasized by numerical results. It is on average 75 times faster to prove the feasibility and 10 times faster to build a feasible solution.

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Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through Collaborative Research and Development Grant #CRDPJ 479316-15 sponsored by Softree Technical Systems Inc. Part of the computation in this research was carried out using a software library provided by Softree Technical System Inc. Part of the research was performed in the Computer-Aided Convex Analysis (CA2) laboratory funded by a Leaders Opportunity Fund (LOF, John R. Evans Leaders Fund, Funding for research infrastructure) from the Canada Foundation for Innovation (CFI) and by a British Columbia Knowledge Development Fund (BCKDF). The authors address special thanks to the reviewers for their careful reading and valuable insights.

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Correspondence to Dominique Monnet.

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Appendix

Appendix

See Tables 456 and 7

Table 4 Mean gap between cost of solution computed by Algorithm 1 and cost of the first found solution to the MILP
Table 5 Mean time ratio between time to find a feasible alignment with Algorithm 1 and time to find a feasible alignment to the MILP
Table 6 Mean time ratio between time to prove infeasibility of LP (30) and infeasibility of the MILP subject to null slope constraint
Table 7 Mean time ratio between time to prove infeasibility of LP (30) and infeasibility of the MILP with elevation modification

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Monnet, D., Hare, W. & Lucet, Y. Fast feasibility check of the multi-material vertical alignment problem in road design. Comput Optim Appl 75, 515–536 (2020). https://doi.org/10.1007/s10589-019-00160-3

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Keywords

  • Road design
  • Vertical alignment
  • MILP
  • Feasibility testing