Skip to main content
SpringerLink
Log in

A set partitioning reformulation of a school bus scheduling problem

  • Published:
Journal of Scheduling Aims and scope Submit manuscript

Abstract

We present an integer programming model for the integrated optimization of bus schedules and school starting times, which is a single-depot vehicle scheduling problem with additional coupling constraints among the time windows. For instances with wide time windows the linear relaxation is weak and feasible solutions found by an ILP solver are of poor quality. We apply a set partitioning relaxation to compute better lower bounds and, in combination with a primal construction heuristic, also better primal feasible solutions. Integer programs with at most two non-zero coefficient per constraint play a prominent role in our approach. Computational results for several random and a real-world instance are given and compared with results from a standard branch-and-cut approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Applegate, D., Bixby, R., Chvatal, V., & Cook, W. (1998). On the solution of traveling salesman problems. Documenta Mathematica ICM III, 645–656.

    Google Scholar 

  • Aspvall, B., & Shiloach, Y. (1980). A polynomial time algorithm for solving systems of linear inequalities with two variables per inequality. SIAM Journal on Computing, 9, 827–845.

    Article  Google Scholar 

  • Balinski, M., & Quandt, R. (1964). On an integer program for a delivery problem. Operations Research, 12, 300–304.

    Article  Google Scholar 

  • Bar-Yehuda, R., & Rawitz, D. (2001). Efficient algorithms for integer programs with two variables per constraint. Algorithmica, 29(4), 595–609.

    Article  Google Scholar 

  • Bierlaire, M., Liebling, T., & Spada, M. (2003). Decision-aid methodology for the school bus routing and scheduling problem. In Conference paper STRC 2003, 3rd Swiss transport research conference, Ascona.

    Google Scholar 

  • Bodin, L., & Berman, L. (1979). Routing and scheduling of school buses by computer. Transportation Science, 13(2), 113–129.

    Article  Google Scholar 

  • Bowerman, R. L., Hall, G. B., & Calamai, P. H. (1995). A multi-objective optimisation approach to school bus routing problems. Transportation Research. Part A, Policy and Practice, 28(5), 107–123.

    Article  Google Scholar 

  • Braca, J., Bramel, J., Posner, B., & Simchi-Levi, D. (1997). A computerized approach to the New York city school bus routing problem. IIE Transactions, 29, 693–702.

    Google Scholar 

  • Bramel, J., & Simchi-Levi, D. (2002). Set-covering-based algorithms for the capacitated VRP. In P. Toth & D. Vigo (Eds.), The vehicle routing problem. SIAM monographs on discrete mathematics and applications (pp. 85–108). Philadelphia: SIAM.

    Chapter  Google Scholar 

  • Cohen, E., & Megiddo, N. (1994). Improved algorithms for linear inequalities with two variables per inequality. SIAM Journal on Computing, 23, 1313–1347.

    Article  Google Scholar 

  • Corberan, A., Fernandez, E., Laguna, M., & Marti, R. (2000). Heuristic solutions to the problem of routing school buses with multiple objectives (Technical report TR08-2000). Dep. of Statistics and OR, University of Valencia, Spain.

  • Cordeau, J.-F., Desaulniers, G., Desrosiers, J., Solomon, M. M., & Soumis, F. (2002). VRP with time windows. In P. Toth & D. Vigo (Eds.), The vehicle routing problem. SIAM monographs on discrete mathematics and applications (pp. 157–193). Philadelphia: SIAM.

    Chapter  Google Scholar 

  • Desrosiers, J., & Lübbecke, M. (2005). Selected topics in column generation. Operations Research, 53(6), 1007–1023.

    Article  Google Scholar 

  • Desrosiers, J., Dumas, Y., Solomon, M. M., & Soumis, F. (1995). Time constrained routing and scheduling. In M. O. Ball, T. L. Magnanti, C. L. Monma, & G. L. Nemhauser (Eds.), Network routing. Handbooks in operations research and management science (Vol. 8). Amsterdam: Elsevier.

    Google Scholar 

  • Fügenschuh, A. (2005). The integrated optimization of school starting times and public transport (PhD Thesis). Logos Verlag, Berlin, 175 pp.

  • Fügenschuh, A. (2006). The vehicle routing problem with coupled time windows. Central European Journal of Operations Research, 14(2), 157–176.

    Article  Google Scholar 

  • Fügenschuh, A. (2009). Solving a school bus scheduling problem with integer programming. European Journal of Operational Research, 193(3), 867–884.

    Article  Google Scholar 

  • Fügenschuh, A., & Martin, A. (2006). A multicriterial approach for optimizing bus schedules and school starting times. Annals of Operation Research, 147(1), 199–216.

    Article  Google Scholar 

  • Fulkerson, D. R. (1971). Blocking and anti-blocking pairs of polyhedra. Mathematical Programming, 1, 168–194.

    Article  Google Scholar 

  • Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: a guide to the theory of NP-completeness. New York: Freeman.

    Google Scholar 

  • Grötschel, M., Lovász, L., & Schrijver, A. (1988). Geometric algorithms and combinatorial optimization. New York: Springer.

    Google Scholar 

  • Hochbaum, D. S., & Naor, J. (1994). Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM Journal on Computing, 23(6), 1179–1192.

    Article  Google Scholar 

  • Hochbaum, D. S., Megiddo, N., Naor, J., & Tamir, A. (1993). Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Mathematical Programming, 62, 69–83.

    Article  Google Scholar 

  • ILOG CPLEX Division, 889 Alder Avenue, Suite 200, Incline Village, NV 89451, USA. Information available at http://www.cplex.com.

  • Keller, H., & Müller, W. (1979). Optimierung des Schülerverkehrs durch gemischt ganzzahlige Programmierung. Zeitschrift für Operations Research B 23, 105–122 (In German).

    Article  Google Scholar 

  • Lagarias, J.C. (1985). The computational complexity of simultaneous diophantine approximation problems. SIAM Journal on Computing 14(1), 196–209.

    Article  Google Scholar 

  • Megiddo, N. (1983). Towards a genuinely polynomial algorithm for linear programming. SIAM Journal on Computing, 12, 347–353.

    Article  Google Scholar 

  • Nelson, C. G. (1978). An n log n algorithm for the two-variable-per-constraint linear programming satisfiability problem (Technical report AIM-319). Stanfort University.

  • Pratt, V. R. (1977). Two easy theories whose combination is hard (Technical report). Massachusetts Institute of Technology. Cambridge, MA.

  • Padberg, M. W. (1973). On the facial structure of set packing polyhedra. Mathematical Programming, 5, 199–215.

    Article  Google Scholar 

  • Shostak, R. (1981). Deciding linear inequalities by computing loop residues. Journal of the ACM, 28, 769–779.

    Article  Google Scholar 

  • Toth, P., & Vigo, D. (2002). The vehicle routing problem. In SIAM monographs on discrete mathematics and applications. Philadelphia: SIAM.

    Google Scholar 

  • Wikipedia. Online available at URL http://www.wikipedia.org.

Download references

Author information

Authors and Affiliations

  1. Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, 14195, Berlin, Germany

    Armin Fügenschuh

Authors
  1. Armin Fügenschuh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Armin Fügenschuh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fügenschuh, A. A set partitioning reformulation of a school bus scheduling problem. J Sched 14, 307–318 (2011). https://doi.org/10.1007/s10951-011-0234-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10951-011-0234-0

Keywords

Search

Navigation