Skip to main content
SpringerLink
Log in

A Computational Study of a Cutting Plane Algorithm for University Course Timetabling

  • Papers
  • Published:
Journal of Scheduling Aims and scope Submit manuscript

Abstract

In this paper, we describe a case-study where a Branch-and-Cut algorithm yields the “optimal” solution of a real-world timetabling problem of University courses (University Course Timetabling problem).

The problem is formulated as a Set Packing problem with side constraints. To tighten the initial formulation, we utilize well-known valid inequalities of the Set Packing polytope, namely Clique and Lifted Odd-Hole inequalities. We also analyze the combinatorial properties of the problem to introduce new families of cutting planes that are not valid for the Set Packing polytope, and their separation algorithms. These cutting planes turned out to be very effective to yield the optimal solution of a set of real-world instances with up to 69 courses, 59 teachers, and 15 rooms.

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

  • International timetabling competition, http://www.idsia.ch/Files/ttcomp2002/.

  • Abdennadher, S. and M. Marte, “University course timetabling using constraint handling rules,” Applied Artificial Intelligence, 14(4), 311–325 (2000).

    Article  Google Scholar 

  • Abramson, D., “A very high speed architecture for simulated annealing,” IEEE Computer, 25, 27–36 (1992).

    Google Scholar 

  • Akkoyunlu, E. A., “A linear algorithm for computing the optimum university timetable,” The Computer Journal, 16(4), 347–350 (1973).

    Article  Google Scholar 

  • Bardadym, V. A., “Computer aided school and university timetabling, The new wave,” in Burke and Ross (eds.), Practice and Theory of Automated Timetabling: First International Conference, Edinburgh, UK, 1995. Selected Papers, volume 1153 of Lecture Notes in Computer Science Series. Springer-Verlag, 1996, pp. 22–45.

  • Birbas, T., S. Daskalaki, and E. Housos, “Timetabling for greek high school,” Journal of the Operational Research Society, 48, 1191–1200 (1997).

    Article  Google Scholar 

  • Borndorfer, R. and R. Weismantel, “Set packing relaxations of some integer programs,” Mathematical Programming, 88, 425–450 (2000).

    Google Scholar 

  • Burke, E. K. and M. Carter (eds.), Practice and Theory of Automated Timetabling II: Second International Conference, PATAT'97, Toronto, Canada, August 1997. Selected Papers, volume 1408 of Lecture Notes in Computer Science Series. Springer-Verlag, 1998.

  • Burke, E. K. and P. Causmaecker (eds.), Practice and Theory of Automated Timetabling IV: 4th International Conference, PATAT 2002, Gent, Belgium, August 21–23, 2002, Selected Revised Papers, volume 2740 of Lecture Notes in Computer Science Series. Springer-Verlag, 2003.

  • Burke, E. K., D. de Werra, and J. Kingston, “Applications to timetabling,” in Gross and Yellen (eds.), Handbook of Graph Theory. CRC Press, 2003, pp. 445–474.

  • Burke, E. K. and W. Erben (eds.), Practice and Theory of Automated Timetabling III: Third International Conference, PATAT 2000 Konstanz, Germany, August 16-18, 2000, Selected Papers, volume 2079 of Lecture Notes in Computer Science Series. Springer-Verlag, 2001.

  • Burke, E. K. and P. Ross (eds.), Practice and Theory of Automated Timetabling: First International Conference, Edinburgh, UK, 1995. Selected Papers, volume 1153 of Lecture Notes in Computer Science Series. Springer-Verlag, 1996.

  • Burke, E. K. and M. Trick (eds.), The Practice and Theory of Automated Timetabling V: 5th International Conference, PATAT 2004, Pittsburgh, PA USA, August 18–20, 2004.

  • Burke, E. K., K. S. Jackson, J. H. Kingston, and R. F. Weare, “Automated timetabling: The state of the art,” The Computer Journal, 40(9), 565–571 (1997).

    Article  Google Scholar 

  • Burke, E. K., G. Kendall, and E. Soubeiga, “A tabu-search hyperheuristic for timetabling and rostering,” Journal of Heuristics, 9(6), 451–470 (2003).

    Article  Google Scholar 

  • Burke, E. K., J. P. Newall, and R. F. Weare, “A memetic algorithm for university exam timetabling,” in Burke and Ross (eds.), Practice and Theory of Automated Timetabling: First International Conference, Edinburgh, UK, 1995. Selected Papers, volume 1153 of Lecture Notes in Computer Science Series. Springer-Verlag, 1996, pp. 241–250.

  • Burke, E. K. and S. Petrovic, “Recent research trends in automated timetabling,” European Journal of Operational Research, 140(2), 266–280 (2002).

    Article  Google Scholar 

  • Carter, M. W. “A survey of practical applications of examination timetabling algorithm,” Operations Research, 34(2), 193–202 (1986).

    MathSciNet  Google Scholar 

  • Carter, M. W. “A lagrangean relaxation approach to the classroom assignment problem,” INFOR, 27(2), 230–246 (1989).

    Google Scholar 

  • Carter, M. W., “Timetabling,” in Gass and Harris (eds.), Encyclopedia of Operations research and Management Science. Kluwer, 2001, pp. 833–836.

  • Carter, M. W. and G. Laporte, “Recent developments in practical examination timetabling,” in Burke and Ross (eds.), Practice and Theory of Automated Timetabling: First International Conference, Edinburgh, UK, 1995. Selected Papers, volume 1153 of Lecture Notes in Computer Science Series. Springer-Verlag, 1996, pp. 3–21.

  • Carter, M. W. and G. Laporte, “Recent developments in practical course timetabling,” in Burke and Carter (eds.), Practice and Theory of Automated Timetabling II: Second International Conference, PATAT'97, Toronto, Canada, August 1997. Selected Papers, volume 1408 of Lecture Notes in Computer Science Series. Springer-Verlag, 1998, pp. 3–19.

  • Dascalki, S., T. Birbas, and E. Housos, “An integer programming fromulation for a case study in university timetabling,” European Journal of Operational Research, 153, 117–135 (2004).

    Article  Google Scholar 

  • de Werra, D. “An introduction to timetabling,” European Journal of Operational Research, 19, 151–162 (1985).

    Article  Google Scholar 

  • de Werra, D. “The combinatorics of timetabling,” European Journal of Operational Research, 96, 504–513 (1997).

    Article  Google Scholar 

  • de Werra, D., A. S. Asratian, and S. Durand, “Complexity of some types of timetabling problems,” Journal of Scheduling, 5, 171–183 (2002).

    Article  Google Scholar 

  • Dimopoulou, M. and P. Miliotis, “Implementation of a university course and examination timetabling system,” European Journal of Operational Research, 130, 202–213 (2001).

    Article  Google Scholar 

  • Gass, S. and C. M. Harris (eds.), Encyclopedia of Operations research and Management Science. Kluwer, 2001.

  • Gross, J. and J. Yellen (eds.), Handbook of Graph Theory. CRC Press, 2003.

  • Grötschel, M., L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1993.

  • Gueret, C., N. Jussien, P. Boizumault, and C. Prins, “Minimal perturbation problem in course timetabling,” in Burke and Ross (eds.), Practice and Theory of Automated Timetabling: First International Conference, Edinburgh, UK, 1995. Selected Papers, volume 1153 of Lecture Notes in Computer Science Series. Springer-Verlag, 1996, pp. 130–145.

  • Hoffman, K. L. and M. Padberg, “Solving airline crew scheduling problems by branch-and-cut,” Management Science, 39(6), 657–682 (1993).

    Google Scholar 

  • Hultberga, T. H. and D.M. Cardosob, “The teacher assignment problem: A special case of the fixed charge transportation problem,” European Journal of Operational Research, 101(3), 463–473 (1997).

    Article  Google Scholar 

  • ILOG, ILOG CPLEX 8.0 Reference manual, 2002.

  • Kostuch, P., “The university course timetabling problem with a 3-phase approach,” in Burke and MTrick (eds.), The Practice and Theory of Automated Timetabling V: 5th International Conference, PATAT 2004, Pittsburgh, PA USA, August 18–20, 2004, pp. 251–266.

  • Lawrie, N. L. “An integer linear programming model of a school timetabling problem,” The Computer Journal, 12(4), 307–316 (1969).

    Article  Google Scholar 

  • Leung, J. (ed.), Handbook of Scheduling: Algorithms, Models, and Performance Analysis. CRC Press, 2004.

  • Lopez-Garcia, L. and A. Posada-Bolivar, “A course timetabling system in a mexican university,” in Burke and MTrick (eds.), The Practice and Theory of Automated Timetabling V: 5th International Conference, PATAT 2004, Pittsburgh, PA USA, August 18–20, 2004, pp. 527–530.

  • Mannino, C. and A. Sassano, “An exact algorithm for the maximum stable set problem,” Computational Optimization and Applications, 3(4), 243–258 (1994).

    Article  Google Scholar 

  • McClure, R. H. and C. E. Wells, “A mathematical programming model for faculty course assignment,” Decision Science, 15, 409–420 (1984).

    Google Scholar 

  • Muller, T. and H. Rudova, “Minimal perturbation problem in course timetabling,” in Burke and MTrick (eds.), The Practice and Theory of Automated Timetabling V: 5th International Conference, PATAT 2004, Pittsburgh, PA USA, August 18–20, 2004, pp. 283–304.

  • Nemhauser, G. L. and L. A. Wolsey, Integer and Combinatorial Optimization. Wiley-Interscience, 1999.

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

    Article  Google Scholar 

  • Paechter, B., A. Cumming, M. G. Norman, and H. Luchian, “Extensions to a memetic timetabling system,” in Burke and Ross (eds.), Practice and Theory of Automated Timetabling: First International Conference, Edinburgh, UK, 1995. Selected Papers, volume 1153 of Lecture Notes in Computer Science Series. Springer-Verlag, 1996, pp. 251–266.

  • Petrovic, S. and E. Burke, “University timetabling,” in Leung (ed.), Handbook of Scheduling: Algorithms, Models, and Performance Analysis. CRC Press, 2004, chapter 45.

  • Rudova, H. and K. Murray, “University course timetabling with soft constraints,” in Burke and Causmaecker (eds.), Practice and Theory of Automated Timetabling IV: 4th International Conference, PATAT 2002, Gent, Belgium, August 21–23, 2002, Selected Revised Papers, volume 2740 of Lecture Notes in Computer Science Series. Springer-Verlag, 2003, pp. 310–328.

  • Schaerf, A., “A survey of automated timetabling,” Artificial Intelligence Review, 13, 87–127 (1999).

    Article  Google Scholar 

  • Schaerf, A. and L. Di Gaspero, “Multi-neighbourhood local search with application to course timetabling,” in Burke and Causmaecker (eds.), Practice and Theory of Automated Timetabling IV: 4th International Conference, PATAT 2002, Gent, Belgium, August 21–23, 2002, Selected Revised Papers, volume 2740 of Lecture Notes in Computer Science Series. Springer-Verlag, 2003, pp. 262–275.

  • Socha, K., J. Knowles, and M. Sampels, “A MAX-MIN Ant System for the University Timetabling Problem,” in M. Dorigo, G. Di Caro, and M. Sampels, editors, Proceedings of ANTS 2002—From Ant Colonies to Artificial Ants: Third International Workshop on Ant Algorithms, volume 2463 of Lecture Notes in Computer Science, pp. 1–13. Springer Verlag, Berlin, Germany, 2002.

  • Tripathy, A., “School timetabling—a case in large binary integer linear programming,” Management Science, 30(12), 1473–1489 (1984).

    Google Scholar 

  • Waterer, H., E. L. Johnson, P. Nobili, and M. W. P. Savelsbergh, “The relation of time indexed formulations of single machine scheduling problems to the node packing problem,” Mathematical Programming, 93, 477–494 (2002).

    Article  Google Scholar 

  • Wren, A., “Scheduling, timetabling and rostering—a special relationship?,” in Burke and Ross (eds.), Practice and Theory of Automated Timetabling: First International Conference, Edinburgh, UK, 1995. Selected Papers, volume 1153 of Lecture Notes in Computer Science Series. Springer-Verlag, 1996, pp. 46–75.

Download references

Author information

Authors and Affiliations

  1. RCOST - Research Center on Software Technology, Università del Sannio, C.so Garibaldi 107, 82100, Benevento, Italy

    Pasquale Avella

  2. Centro di Ricerca in Matematica Pura e Applicata, Università di Salerno, Via Ponte Don Melillo, 84084, Fisciano, (SA), Italy

    Igor Vasil'Ev

  3. Institute of System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences, Lermontov Srt., 134, 664033, Irkutsk, Russia

    Igor Vasil'Ev

Authors
  1. Pasquale Avella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Vasil'Ev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Avella, P., Vasil'Ev, I. A Computational Study of a Cutting Plane Algorithm for University Course Timetabling. J Sched 8, 497–514 (2005). https://doi.org/10.1007/s10951-005-4780-1

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10951-005-4780-1

Keywords

Search

Navigation