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Rapid branching

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Abstract

We propose rapid branching (RB) as a general branch-and-bound heuristic for solving large scale optimization problems in traffic and transport. The key idea is to combine a special branching rule and a greedy node selection strategy in order to produce solutions of controlled quality rapidly and efficiently. We report on three successful applications of the method for integrated vehicle and crew scheduling, railway track allocation, and railway vehicle rotation planning.

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Algorithm 1
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Algorithm 2
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Notes

  1. The relative gap is defined between the best integer objective UB and the objective of the best lower bound LB as \(100\cdot\frac{\mathit{UB}-\mathit{LB}}{\mathit{UB}+10^{-10}}\).

References

  • Applegate D, Bixby R, Chvátal V, Cook W (1995) Finding cuts in the TSP (a preliminary report). Technical report, Center for Discrete Mathematics and Theoretical Computer Science (DIMACS). DIMACS Technical Report 95-05

  • Beale EML, Forrest JJH (1976) Global optimization using special ordered sets. Math Program 10:52–69

    Article  Google Scholar 

  • Beale E, Tomlin J (1970) Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In: Lawrence J (ed) Proceedings of the 5th international operations research conference. Tavistock Publications, London, pp 447–454

    Google Scholar 

  • Borndörfer R, Schlechte T (2007) Models for railway track allocation. In: Liebchen C, Ahuja RK, Mesa JA (eds) ATMOS 2007—7th workshop on algorithmic approaches for transportation modeling, optimization, and systems, Dagstuhl seminar proceedings. internationales begegnungs- und forschungszentrum für informatik (IBFI), Schloss Dagstuhl, Germany. http://drops.dagstuhl.de/opus/volltexte/2007/1170

    Google Scholar 

  • Borndörfer R, Grötschel M, Lukac S, Mitusch K, Schlechte T, Schultz S, Tanner A (2006) An auctioning approach to railway slot allocation. Compet Regul Netw Ind 1(2):163–196. ZIB Report 05-45 at http://opus.kobv.de/zib/volltexte/2005/878/

    Google Scholar 

  • Borndörfer R, Löbel A, Weider S (2008) A bundle method for integrated multi-depot vehicle and duty scheduling in public transit. In: Hickman M, Mirchandani P, VoßS (eds) Computer-aided systems in public transport. Lecture notes in economics and mathematical systems, vol 600. Springer, Berlin, pp 3–24. ZIB Report 04-14 at http://opus.kobv.de/zib/volltexte/2004/790/

    Chapter  Google Scholar 

  • Borndörfer R, Grötschel M, Jäger U (2010a) Planning problems in public transit. In: Grötschel M, Lucas K, Mehrmann V (eds) Production factor mathematics. Springer, Berlin Heidelberg, pp 95–122, acatech. ZIB Report 09-13

    Chapter  Google Scholar 

  • Borndörfer R, Schlechte T, Weider S (2010b) Railway track allocation by rapid branching. In: Erlebach T, Lübbecke M (eds) Proceedings of the 10th workshop on algorithmic approaches for transportation modelling, optimization, and systems. OpenAccess series in informatics (OASIcs), Dagstuhl, Germany, vol 14, pp 13–23. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik

    Google Scholar 

  • Borndörfer R, Reuther M, Schlechte T, Weider S (2011) A hypergraph model for railway vehicle rotation planning. In: Caprara A, Kontogiannis S (eds) 11th workshop on algorithmic approaches for transportation modelling, optimization, and systems. OpenAccess series in informatics (OASIcs), Dagstuhl, Germany, vol 20, pp 146–155. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik

    Google Scholar 

  • Cambini R, Gallo G, Scutellà M (1997) Flows on hypergraphs. Math Program, Ser B 78(2):195–217

    Article  Google Scholar 

  • Caprara A, Toth P, Fischetti M (2000) Algorithms for the set covering problem. Ann Oper Res 98(1–4):353–371

    Article  Google Scholar 

  • Caprara A, Monaci M, Toth P, Guida PL (2006) A Lagrangian heuristic algorithm for a real-world train timetabling problem. Discrete Appl Math 154(5):738–753

    Article  Google Scholar 

  • Eckstein J, Nediak M (2007) Pivot, cut, and dive: a heuristic for 0-1 mixed integer programming. J Heuristics 13(5):471–503

    Article  Google Scholar 

  • Erol B, Klemenz M, Schlechte T, Schultz S, Tanner A (2008) TTPlib 2008—a library for train timetabling problems. In: Tomii A, Allan J, Arias E, Brebbia C, Goodman C, Rumsey A, Sciutto G (eds) Computers in railways XI. WIT Press, Southampton

    Google Scholar 

  • Fischer F, Helmberg C (2010) Dynamic graph generation and dynamic rolling horizon techniques in large scale train timetabling. In: Erlebach T, Lübbecke M (eds) Proceedings of the 10th workshop on algorithmic approaches for transportation modelling, optimization, and systems. OpenAccess series in informatics (OASIcs), Dagstuhl, Germany, vol 14, pp 45–60. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik

    Google Scholar 

  • Fischer F, Helmberg C, Janßen J, Krostitz B (2008) Towards solving very large scale train timetabling problems by Lagrangian relaxation. In: Fischetti M, Widmayer P (eds) ATMOS 2008—8th workshop on algorithmic approaches for transportation modeling, optimization, and systems, Dagstuhl, Germany. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Germany

    Google Scholar 

  • Giacco GL, D’Ariano A, Pacciarelli D (2011) Rolling stock rostering optimization under maintenance constraints. In: Proceedings of the 2nd international conference on models and technology for intelligent transportation systems, Leuven, Belgium, pp 1–5. MT-ITS 2011

    Google Scholar 

  • IVU Plattform (2008). Sequentielle oder Integrierte Optimierung? IVU Plattform. http://www.ivu.de/fileadmin/ivu/pdf/kundenzeitung/ger/2008/PlattformZtg-0802D-web1865.pdf

  • Kiwiel KC (1995) Approximation in proximal bundle methods and decomposition of convex programs. J Optim Theory Appl 84(3):529–548

    Article  Google Scholar 

  • König FG (2009) Sorting with objectives—graph theoretic concepts in industrial optimization. Doctoral thesis, Technische Universität Berlin

  • Löbel A (1997) Optimal vehicle scheduling in public transit. Shaker Verlag, Aachen. Ph.D. thesis, Technische Universität Berlin

  • Lübbecke M, Desrosiers J (2005) Selected topics in column generation. Oper Res 53(6):1007–1023

    Article  Google Scholar 

  • Lusby R, Larsen J, Ehrgott M, Ryan D (2011) Railway track allocation: models and methods. OR Spektrum 33(4):843–883

    Article  Google Scholar 

  • Maróti G (2006) Operations research models for railway rolling stock planning. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands

  • Schlechte T (2012) Railway track allocation: models and algorithms. Doctoral thesis, Technische Universität Berlin

  • Subramanian R, Sheff R, Quillinan J, Wiper D, Marsten R (1994) Coldstart: fleet assignment at delta air lines. Interfaces 24(1):104–120

    Article  Google Scholar 

  • Wedelin D (1995) An algorithm for a large scale 0-1 integer programming with application to airline crew scheduling. Ann Oper Res 57:283–301

    Article  Google Scholar 

  • Weider S (2007) Integration of vehicle and duty scheduling in public transport. PhD thesis, TU, Berlin. http://opus.kobv.de/tuberlin/volltexte/2007/1624/

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Authors and Affiliations

  1. Konrad-Zuse-Zentrum Berlin, Takustr. 7, 14195, Berlin, Germany

    Ralf Borndörfer, Andreas Löbel, Markus Reuther, Thomas Schlechte & Steffen Weider

Authors
  1. Ralf Borndörfer
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  2. Andreas Löbel
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  3. Markus Reuther
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  4. Thomas Schlechte
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  5. Steffen Weider
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Correspondence to Thomas Schlechte.

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Borndörfer, R., Löbel, A., Reuther, M. et al. Rapid branching. Public Transp 5, 3–23 (2013). https://doi.org/10.1007/s12469-013-0066-8

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