Aggregation Methods for Railway Network Design Based on Lifted Benders Cuts

  • Andreas BärmannEmail author
  • Frauke Liers
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 268)


Rail freight traffic in Germany has experienced significant growth rates over the last decade, and recent forecasts expect this tendency to continue over the next 20 years due to the increases in national and international trade. Internal predictions of Deutsche Bahn AG, the most important German railway enterprise, assume a mean increase of 2% per year for rail freight traffic until 2030. At this pace, the German railway network in its current state would reach its capacity limit way before this date. As investments into the network infrastructure bear a very high price tag, it is crucial to use the available budget in the most efficient manner. Furthermore, the large size of the networks under consideration warrants the development of efficient algorithms to handle the complex network design problems arising for real-world data. This led us to the development of network aggregation procedures which allow for much shorter solution times by compressing the network information. More exactly, our framework works by clustering the nodes of the underlying graph to components and solving the network design problem over this aggregated graph. This kind of aggregation may either be used as a quick heuristic, or it can be extended to an exact method, e.g. by iterative refinement of the clustering, The latter results in a cutting plane algorithm, which also introduces new variables with each refinement. This idea developed in Bärmann et al. (Math Program Comput 7(2):189–217, 2015) is extended in this chapter such that it is able to incorporate the costs for routing flow through the network via lifted Benders optimality cuts. Altogether, our algorithm can be seen as a novel kind of Benders decomposition which allows to shift variables from the subproblem to the master problem in the process. Computations on several benchmark sets demonstrate the effectiveness of the approach.



We thank our co-authors from [1], Alexander Martin, Maximilian Merkert, Christoph Thurner and Dieter Weninger, for the interesting discussions we had on the topic. Furthermore, we gratefully acknowledge the computing resources provided by the group of Michael Jünger in Cologne. In particular, we thank Thomas Lange for his technical support. Last but not least, we thank the BMBF for its financial support under research grant 05M10WEC.


  1. 1.
    Bärmann A, Liers F, Martin A, Merkert M, Thurner C, Weninger D (2015) Solving network design problems via iterative aggregation. Math Program Comput 7(2):189–217. (cited on pages 50, 51, 52, 55, 56, 58, 62, 71)
  2. 2.
    Bärmann A, Martin A, Schülldorf H (2017) A decomposition method for multi-period railway network expansion – with a case study for Germany. Transp Sci. (cited on pages 50, 61)
  3. 3.
    Beck MJ (2013) Kapazitätsmanagement und Netzentwicklung: Erfahrungen mit Kompromissen zum Fahrplan 2018. Presentation by DB Netz AG. Available at: (cited on page 49)Google Scholar
  4. 4.
    Boland N, Ernst A, Kalinowski T, Rocha de Paula M, Savelsbergh M, Singh G (2013) Time aggregation for network design to meet time-constrained demand. In: Piantadosi J, Anderssen RS, Boland J (eds) Proceedings of the 20th international congress on modelling and simulation (MODSIM2013), pp 3281–3287 (cited on page 50)Google Scholar
  5. 5.
    Borndörfer R, Erol B, Graffagnino T, Schlechte T, Swarat E (2014) Optimizing the Simplon railway corridor. Ann Oper Res 218(1):93–106. (cited on page 50)CrossRefGoogle Scholar
  6. 6.
    Borndörfer R, Reuther M, Schlechte T (2014) A coarse-to-fine approach to the railway rolling stock rotation problem. In: Proceedings of the 14th workshop on algorithmic approaches for transportation modelling, optimization, and systems (ATMOS’14), vol 42, pp 79–91. (cited on page 50)
  7. 7.
    Caimi GC (2009) Algorithmic decision support for train scheduling in a large and highly utilised railway network. PhD thesis, Eidgenössische Technische Hochschule Zürich. (cited on page 50)
  8. 8.
    Crainic TG, Li Y, Toulouse M (2006) A first multilevel cooperative algorithm for capacitated multicommodity network design. Comput Oper Res 33(9):2602–2622. (cited on page 50)CrossRefGoogle Scholar
  9. 9.
    DB Netz AG (2013) Netzkonzeption 2030: Zielnetz der DB Netz AG für die Schieneninfrastruktur im Jahr 2030. Information Booklet (cited on page 49)Google Scholar
  10. 10.
    Dempster MAH, Thompson RT (1998) Parallelization and aggregation of nested benders decomposition. Ann Oper Res 81:163–187. (cited on page 51)CrossRefGoogle Scholar
  11. 11.
    Gamst M, Spoorendonk S (2013) An exact approach for aggregated formulations. Technical report, DTU Management Engineering Report 3.2013. Technical University of Denmark (cited on page 51)Google Scholar
  12. 12.
    Gurobi Optimization, Inc. (2017) Website of Gurobi. June 2017. (cited on pages 63, 65)
  13. 13.
    Lumbreras S, Ramos A (2013) Optimal design of the electrical layout of an offshore wind farm applying decomposition strategies. IEEE Trans Power Syst 28:1134–1441. (cited on page 51)Google Scholar
  14. 14.
    Pruckner M, Thurner C, Martin A, German R (2014) A coupled optimization and simulation model for the energy transition in Bavaria. In: Fischbach K, Großmann M, Krieger UR, Staake T (eds) Proceedings of the international workshops SOCNET 2014 and FGENET 2014, pp 97–104 (cited on page 51)Google Scholar
  15. 15.
    Schlechte T, Borndörfer R, Erol B, Graffagnino T, Swarat E (2011) Micro-macro transformation of railway networks. J Rail Transp Plann Manage 1(1):38–48. (cited on page 50)CrossRefGoogle Scholar
  16. 16.
    Shapiro JF (1984) A note on node aggregation and Benders’ decomposition. Math Program 29:113–119. (cited on page 51)CrossRefGoogle Scholar
  17. 17.
    Statistisches Bundesamt S (2017) Website of the Federal Statistical Office, June (2017). (cited on page 48)
  18. 18.
    Trukhanov S, Ntaimo L, Schaefer A (2010) Adaptive multicut aggregation for two-stage stochastic linear programs with recourse. Eur J Oper Res 206:395–406. (cited on page 51)CrossRefGoogle Scholar
  19. 19.
    Zipkin PH (1977) Aggregation in linear programming. PhD thesis, Yale University (cited on page 51)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department MathematikFAU Erlangen-Nürnberg, Lehrstuhl für WirtschaftsmathematikErlangenGermany

Personalised recommendations