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A Lagrangian relaxation approach for expansion of a highway network

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Abstract

This paper deals with the problem of improving an existing road network in the context of strategic planning through the creation of new highway corridors. To address this problem we analyse three mixed integer programming models. The so-called [P1] is the classical capacitated multicommodity network design model. The model named [P2] imposes on [P1] the location of a single (main path) highway corridor in the road network and [P3] adds to [P2] a set of sub-tour breaking constraints. The stated goal is to minimize the total travel time for a known origin-destination demand matrix with a given budget. In this paper we propose an efficient method for [P3], based on a Lagrangian relaxation, to obtain easily-solved sub-problems. A cutting-plane method for solving the Lagrangian sub-problems is proposed. This method generates valid cuts until an optimal solution is found. The Lagrangian dual problem is solved using the sub-gradient optimization method. A case study has been carried out for the region of Castilla-La Mancha (Spain). Computational comparisons between the proposed method and a state-of-the-art mixed-integer code are presented. The Lagrangian relaxation approach is found to be capable of generating good feasible solutions to the case study within a reasonable computational time.

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Notes

  1. Note that if a path \(P^\ell \) is not optimal, then all sub-paths of \(P^\ell \) are not optimal because \(\lambda \ge 0\).

References

  • Angulo, E., Castillo, E., García-Ródenas, R., & Sánchez-Vizcaíno, J. (2012). Determining highway corridors. Journal of Transportation Engineering, 138(5), 557–570.

    Article  Google Scholar 

  • Angulo, E., Castillo, E., García-Ródenas, R., & Sánchez-Vizcaíno, J. (2014). A continuous bi-level model for the expansion of highway networks. Computers & Operations Research, 41(2014), 262–276.

    Article  Google Scholar 

  • Boffey, B., & Narula, S. C. (1998). Models for multi-path covering-routing problems. Annals of Operations Research, 82, 331–342.

    Article  Google Scholar 

  • Bruno, G., Gendreau, M., & Laporte, G. (2002). A heuristic for the location of a rapid transit line. Computers & Operations Research, 29(1), 1–12.

    Article  Google Scholar 

  • Bruno, G., Ghiani, G., & Improta, G. (1998). A multi-modal approach to the location of a rapid transit line. European Journal of Operational Research, 104(2), 321–332.

    Article  Google Scholar 

  • Contreras, I., Díaz, J. A., & Fernández, E. (2009). Lagrangean relaxation for the capacitated hub location problem with single assignment. OR Spectrum, 31(3), 483–505.

    Article  Google Scholar 

  • Contreras, I., & Fernández, E. (2012). General network design: A unified view of combined location and network design problems. European Journal of Operational Research, 219(3), 680–697.

    Article  Google Scholar 

  • Crainic, T. G., Frangioni, A., & Gendron, B. (2001). Bundle-based relaxation methods for multicommodity capacitated fixed charge network design. Discrete Applied Mathematics, 112(1–3), 73–99.

    Article  Google Scholar 

  • Crainic, T. G., Gendreu, M., & Farvolden, J. M. (2000). A simplex-based tabu search method for capacitated network design. INFORMS Journal on Computing, 12(3), 223–236.

    Article  Google Scholar 

  • Current, J., & Marsh, M. (1993). Multiobjective transportation network design and routing problems: Taxonomy and annotation. European Journal of Operational Research, 65(1), 4–19.

    Article  Google Scholar 

  • Current, J., Pirkul, H., & Rolland, E. (1994). Efficient algorithms for solving the shortest covering path problem. Transportation Science, 28(4), 317–327.

    Article  Google Scholar 

  • Current, J. R., ReVelle, C. S., & Cohon, J. L. (1984). The shortest covering path problem: An application of locational constraints to network design. Journal of Regional Science, 24(2), 161–183.

    Article  Google Scholar 

  • Current, J. R., ReVelle, C. S., & Cohon, J. L. (1985). The maximum covering/shortest path problem: A multiobjective network design and routing formulation. European Journal of Operational Research, 21(2), 189–199.

    Article  Google Scholar 

  • Current, J. R., ReVelle, C. S., & Cohon, J. L. (1986). The hierarchical network design problem. European Journal of Operational Research, 27(1), 57–66.

    Article  Google Scholar 

  • Dufourd, H., Gendreau, M., & Laporte, G. (1996). Locating a transit line using tabu search. Location Science, 4, 1–19.

    Article  Google Scholar 

  • Duin, C., & Volgenant, A. (1989). Reducing the hierarchical network design problem. European Journal of Operational Research, 39(3), 332–344.

    Article  Google Scholar 

  • Erdemir, E. T., Batta, R., Spielman, S., Rogerson, P. A., Blatt, A., & Flanigan, M. (2008). Location coverage models with demand originating from nodes and paths: Application to cellular network design. European Journal of Operational Research, 190(3), 610–632.

    Article  Google Scholar 

  • Escudero, L. F., & Muñoz, S. (2009). An approach for solving a modification of the extended rapid transit network design problem. Top, 17(2), 320–334.

    Article  Google Scholar 

  • Fernández, P., & Marín, A. (2003). A heuristic procedure for path location with multisource demand. INFOR: Information Systems and Operational Research, 41(2), 165–178.

    Google Scholar 

  • Gutíerrez-Jarpa, G., Donoso, M., Obreque, C., & Marianov, V. (2010). Minimum cost path location for maximum traffic capture. Computers & Industrial Engineering, 58, 332–341.

    Article  Google Scholar 

  • Holmberg, K., & Yuan, D. (2000). A lagrangian heuristic based branch-and-bound approach for the capacitated network design problem. Operations Research, 48(3), 461–481.

    Article  Google Scholar 

  • Jha, M. K. (2001). Using a GIS for automated decision-making in highway cost analysis. Transportation Research Record, 1768, 260–267.

    Article  Google Scholar 

  • Jha, M. K., McCall, C., & Schonfeld, P. (2001). Using GIS, genetic algorithms, and visualization in highway development. Computer-Aided Civil and Infrastructure Engineering, 16(6), 399–414.

    Article  Google Scholar 

  • Jha, M. K., & Schonfeld, P. (2000). Integrating genetic algorithsms and geographic information system to optimize highway alignments. Transportation Research Record, 1719, 233–240.

    Article  Google Scholar 

  • Jong, J. C. (1998). Optimizing highway alignments with genetic algorithms. Ph.D. thesis, College Park, University of Maryland

  • Kang, M. W. (2008). An alignment optimization model for a simple highway network. Ph.D. thesis, College Park, University of Maryland.

  • Konak, A. (2012). Network design problem with relays: A genetic algorithm with a path-based crossover and a set covering formulation. European Journal of Operational Research, 218(3), 829–837.

    Article  Google Scholar 

  • Laporte, G., Marín, A., Mesa, J. A., & Ortega, F. A. (2007). An integrated mehodology for the rapid transit network design problem. Lectures notes in computer science (Vol. 4359, pp. 187–199).

  • Laporte, G., Marín, A., Mesa, J. A., & Perea, F. (2011). Designing robust rapid transit networks with alternative routes. Journal of Advanced Transportation, 45(1), 54–65.

    Article  Google Scholar 

  • Laporte, G., Mesa, J. A., & Ortega, F. A. (2000). Optimization methods for the planning of rapid transit systems. European Journal of Operations Research, 122(1), 1–10.

    Article  Google Scholar 

  • Laporte, G., Mesa, J. A., & Ortega, F. A. (2002). Locating stations on rapid transit lines. Computers & Operations Research, 29(6), 741–759.

    Article  Google Scholar 

  • Laporte, G., Mesa, J. A., Ortega, F. A., & Perea, F. (2011). Planning rapid transit networks. Socio-Economic Planning Sciences, 45(3), 95–104.

    Article  Google Scholar 

  • Laporte, G., Mesa, J. A., Ortega, F. A., & Sevillano, I. (2005). Maximizing trip coverage in the location of a single rapid transit alignment. Annals of Operations Research, 136(1), 49–63.

    Article  Google Scholar 

  • Marín, A. (2007). An extension to rapid transit network design problem. Top, 15, 231–241.

    Article  Google Scholar 

  • Marín, A., & García-Ródenas, R. (2009). Location of infrastructure in urban railway networks. Computers & Operations Research, 36(5), 1461–1477.

    Article  Google Scholar 

  • Marín, A., Mesa, J. A., Perea, F. (2009). Integrating robust railway network design and line planning under failures. In R. K. Ahuja et al. (Ed.), Robust and online large-scale optimization LNCS (Vol. 5868, pp. 273–292).

  • Marín, A., & Pelegrín, B. (1999). Applying Lagrangian relaxation to the resolution of two-stage location problems. Annals of Operations Research, 86, 179–198.

    Article  Google Scholar 

  • Mesa, J. A., & Boffey, T. B. (1996). A review of extensive facility location in networks. European Journal of Operational Research, 95(3), 592–603.

    Article  Google Scholar 

  • Ortúzar, Jd D., & Willumsen, L. G. (1994). Modelling transport. Chichester: Wiley.

    Google Scholar 

  • Pirkul, H., Current, J., & Nagarajan, V. (1991). The hierarchical network design problem: A new formulation and solution procedures. Transportation Science, 25(3), 175–182.

    Article  Google Scholar 

  • Polyak, B. T. (1969). Minimization of unsmooth functionals. USSR Computational Mathematics and Mathematical Physics, 9(3), 14–29.

    Article  Google Scholar 

  • Sancho, N. G. F. (1997). The hierarchical network design problem with multiple primary paths. European Journal of Operational Research, 96(2), 323–328.

    Article  Google Scholar 

  • Solanki, R. S., Gorti, J. K., & Southworth, F. (1998). Using decomposition in large-scale highway network design with a quasi-optimization heuristic. Transportation Research Part B: Methodological, 32(2), 127–140.

    Article  Google Scholar 

  • Timothy, J. (2013). The maximal covering/shortest path problem revisited: An examination and reformulation of the problem to allow the elimination or attachment of sub-tours. Ph.D. thesis, M. A. University of California, Santa Barbara

  • Yang, H., & Bell, M. G. H. (1998). Models and algorithms for road network design: A review and some new developments. Transport Reviews: A Transnational Transdisciplinary Journal, 18(3), 257–278.

    Article  Google Scholar 

Download references

Acknowledgments

This research has been financed by the Ministerio de Economia y Competitividad of Spain with the TRA-2011-27791-C03-03 research project.

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Authors and Affiliations

  1. Escuela Superior de Informática, Departamento de Matemáticas, Universidad de Castilla La Mancha, 13071 , Ciudad Real, Spain

    Eusebio Angulo, Ricardo García-Ródenas & José Luis Espinosa-Aranda

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  1. Eusebio Angulo
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  2. Ricardo García-Ródenas
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  3. José Luis Espinosa-Aranda
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Correspondence to Eusebio Angulo.

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Angulo, E., García-Ródenas, R. & Espinosa-Aranda, J.L. A Lagrangian relaxation approach for expansion of a highway network. Ann Oper Res 246, 101–126 (2016). https://doi.org/10.1007/s10479-014-1682-7

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