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A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

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Abstract

We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. The methodology proposed allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour and a half, outperforming other state-of-the-art methods, which exhausted the (144 GB of) available memory in the largest instances.

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Acknowledgments

The first author has been supported by MINECO/FEDER Grants MTM2012-31440 and MTM2015-65362-R of the Spanish Ministry of Economy and Competitiveness; the second author has been supported by the European Research Council-ref. ERC-2011-StG 283300-REACTOPS; the third author has been supported by the Portuguese Science Foundation (FCT-Fundação para a Ciência e Tecnologia) under the Project UID/MAT/04561/2013 (CMAF-CIO/FCUL). The authors would like to thank the two anonymous reviewers for their valuable comments, suggestions and insights that helped improving the manuscript.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, Universitat Politècnica de Catalunya, Jordi Girona 1–3, 08034, Barcelona, Catalonia, Spain

    Jordi Castro

  2. Department of Marketing, IESEG School of Management (LEM-CNRS 9221), 3 rue de la Digue, 59000, Lille, France

    Stefano Nasini

  3. Department of Statistics and Operations Research/Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016, Lisbon, Portugal

    Francisco Saldanha-da-Gama

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  1. Jordi Castro
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  3. Francisco Saldanha-da-Gama
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Corresponding author

Correspondence to Jordi Castro.

Appendix: Tables of numerical experiments of Sect. 4.1

Appendix: Tables of numerical experiments of Sect. 4.1

Tables 8, 9 and 10 contain the CPU times (seconds) required by the Benders decomposition and the branch-and-cut to solve instances of (1)–(8), with one, three and six time periods respectively. The parameter specification has been defined in Table 1, with different combinations of \(\alpha \) and \(\beta \) and for two sizes \(m = n = 500\) and \(m = n = 1000\).

Table 8 CPU time of instances of two sizes \(m = n = 500\) and \(m = n = 1000\), with one time period (\(k = 1\))
Full size table

1.1 One period

See Table 8.

1.2 Three periods

See Table 9.

Table 9 CPU time of instances of two sizes \(m = n = 500\) and \(m = n = 1000\), with three time periods (\(k = 3\))
Full size table

1.3 Six periods

See Table 10.

Table 10 CPU time of instances of two sizes \(m = n = 500\) and \(m = n = 1000\), with six time periods (\(k = 6\))
Full size table

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Castro, J., Nasini, S. & Saldanha-da-Gama, F. A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method. Math. Program. 163, 411–444 (2017). https://doi.org/10.1007/s10107-016-1067-6

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