A Bundle Method for Integrated Multi-Depot Vehicle and Duty Scheduling in Public Transit

  • Ralf Borndörfer
  • Andreas Löbel
  • Steffen Weider
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 600)

Abstract

This article proposes a Lagrangean relaxation approach to solve integrated duty and vehicle scheduling problems arising in public transport. The approach is based on a version of the proximal bundle method for the solution of concave decomposable functions that is adapted for the approximate evaluation of the vehicle and duty scheduling components. The primal and dual information generated by this bundle method is used to guide a branch-and-bound type algorithm.

Computational results for large-scale real-world integrated vehicle and duty scheduling problems with up to 1,500 timetabled trips are reported. Compared with the results of a classical sequential approach and with reference solutions, integrated scheduling offers remarkable potentials in savings and drivers’ satisfaction.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ball, M. O., Bodin, L., and Dial, R. (1983). A matching based heuristic for scheduling mass transit crews and vehicles. Transportation Science, 17, 4–31.CrossRefGoogle Scholar
  2. Borndörfer, R., Grötschel, M., and Löbel, A. (2003). Duty scheduling in public transit. In W. Jäger and H.-J. Krebs, editors, MATHEMATICS — Key Technology for the Future, pages 653–674. Springer Verlag, Berlin. http://www.zib.de/PaperWeb/abstracts/ZR-01-02.Google Scholar
  3. Daduna, J. R. and Völker, M. (1997). Fahrzeugumlaufbildung im ÖPNV mit unscharfen Abfahrtszeiten (in German). Der Nahverkehr, 11/1997, pages 39–43.Google Scholar
  4. Daduna, J. R. and Wren, A., editors (1988). Computer-Aided Transit Scheduling, volume 308 of Lecture Notes in Economics and Mathematical Systems. Springer.Google Scholar
  5. Daduna, J. R., Branco, I., and Paixão, J. M. P., editors (1995). Computer-Aided Transit Scheduling, volume 430 of Lecture Notes in Economics and Mathematical Systems. Springer.Google Scholar
  6. Darby-Dowman, K., J. K. Jachnik, R. L. L., and Mitra, G. (1988). Integrated decision support systems for urban transport scheduling: Discussion of implementation and experience. In J. R. Daduna and A. Wren, editors, Computer-Aided Transit Scheduling, volume 308 of Lecture Notes in Economics and Mathematical Systems, pages 226–239, Berlin. Springer.Google Scholar
  7. Desrochers, M. and Rousseau, J.-M., editors (1992). Computer-Aided Transit Scheduling, volume 386 of Lecture Notes in Economics and Mathematical Systems. Springer.Google Scholar
  8. Desrochers, M. and Soumis, F. (1989). A column generation approach to the urban transit crew scheduling problem. Transportation Science, 23(1), 1–13.Google Scholar
  9. Falkner, J. C. and Ryan, D. M. (1992). Express: Set partitioning for bus crew scheduling in Christchurch. In M. Desrochers and J.-M. Rousseau, editors, Computer-Aided Transit Scheduling, volume 386 of Lecture Notes in Economics and Mathematical Systems, pages 359–378, Berlin. Springer.Google Scholar
  10. Freling, R. (1997). Models and Techniques for Integrating Vehicle and Crew Scheduling. Ph.D. thesis, Erasmus University Rotterdam, Amsterdam.Google Scholar
  11. Freling, R., Huisman, D., and Wagelmans, A. P. M. (2001a). Applying an integrated approach to vehicle and crew scheduling in practice. In S. Voß and J. R. Daduna, editors, Computer-Aided Scheduling of Public Transport, volume 505 of Lecture Notes in Economics and Mathematical Systems, pages 73–90, Berlin. Springer.Google Scholar
  12. Freling, R., Wagelmans, A. P. M., and Paixao, J. M. P. (2001b). Models and algorithms for single-depot vehicle scheduling. Transportation Science, 35, 165–180.CrossRefGoogle Scholar
  13. Freling, R., Huisman, D., and Wagelmans, A. P. M. (2003). Models and algorithms for integration of vehicle and crew scheduling. Journal of Scheduling, 6, 63–85.CrossRefGoogle Scholar
  14. Friberg, C. and Haase, K. (1999). An exact algorithm for the vehicle and crew scheduling problem. In N. H. M. Wilson, editor, Computer-Aided Transit Scheduling, volume 471 of Lecture Notes in Economics and Mathematical Systems, pages 63–80, Berlin. Springer.Google Scholar
  15. Gaffi, A. and Nonato, M. (1999). An integrated approach to extra-urban crew and vehicle scheduling. In N. H. M. Wilson, editor, Computer-Aided Transit Scheduling, volume 471 of Lecture Notes in Economics and Mathematical Systems, pages 103–128, Berlin. Springer.Google Scholar
  16. Haase, K., Desaulniers, G., and Desrosiers, J. (2001). Simultaneous vehicle and crew scheduling in urban mass transit systems. Transportation Science, 35(3), 286–303.CrossRefGoogle Scholar
  17. Hanisch, J. (1990). Die Regionalverkehr Köln GmbH und HASTUS (in German). http://www.giro.ca/Deutsch/Publications/publications.htm.Google Scholar
  18. Helmberg, C. (2000). Semidefinite programming for combinatorial optimization. Technical report ZR00-34. Zuse Institute Berlin.Google Scholar
  19. Hintermüller, M. (2001). A proximal bundle method based on approximate subgradients. Computational Optimization and Applications, (20), 245–266.CrossRefGoogle Scholar
  20. Huisman, D., Freling, R., and Wagelmans, A. P.M. (2005). Multiple-depot integrated vehicle and crew scheduling. Transportation Science, 39, 491–502.CrossRefGoogle Scholar
  21. Kiwiel, K. C. (1990). Proximal bundle methods. Mathematical Programming, 46(123), 105–122.CrossRefGoogle Scholar
  22. Kiwiel, K. C. (1995). Approximation in proximal bundle methods and decomposition of convex programs. Journal of Optimization Theory and Applications, 84(3), 529–548.CrossRefGoogle Scholar
  23. Lemarechal, C. (2001). Lagrangian relaxation. In M. Jünger and D. Naddef, editors, Computational Combinatorial Optimization, volume 2241 of Lecture Notes in Computer Science, pages 112–156, Berlin. Springer.CrossRefGoogle Scholar
  24. Löbel, A. (1997). Optimal Vehicle Scheduling in Public Transit. Ph.D. thesis, TU Berlin. http://www.zib.de/bib/diss/index.en.html.Google Scholar
  25. Löbel, A. (1999). Solving large-scale multi-depot vehicle scheduling problems. In N. H. M. Wilson, editor, Computer-Aided Transit Scheduling, volume 471 of Lecture Notes in Economics and Mathematical Systems, pages 195–222, Berlin. Springer.Google Scholar
  26. Patrikalakis, I. and Xerocostas, D. (1992). A new decomposition scheme of the urban public transport scheduling problem. In M. Desrochers and J.-M. Rousseau, editors, Computer-Aided Transit Scheduling, volume 386 of Lecture Notes in Economics and Mathematical Systems, pages 407–425, Berlin. Springer.Google Scholar
  27. Scott, D. (1985). A large scale linear programming approach to the public transport scheduling and costing problem. In J.-M. Rousseau, editor, Computer Scheduling of Public Transport 2. Amsterdam, Elsevier.Google Scholar
  28. Tosini, E. and Vercellis, C. (1988). An interactive system for extra-urban vehicle and crew scheduling problems. In J. R. Daduna and A. Wren, editors, Computer-Aided Transit Scheduling, pages 41–53, Berlin. Springer.Google Scholar
  29. Voß, S. and Daduna, J. R., editors (2001). Computer-Aided Scheduling of Public Transport, volume 505 of Lecture Notes in Economics and Mathematical Systems. Berlin, Springer.Google Scholar
  30. Wilson, N. H. M., editor (1999). Computer-Aided Transit Scheduling, volume 471 of Lecture Notes in Economics and Mathematical Systems. Berlin, Springer.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ralf Borndörfer
    • 1
  • Andreas Löbel
    • 1
  • Steffen Weider
    • 1
  1. 1.Zuse Institute BerlinBerlinGermany

Personalised recommendations