Abstract
The International Timetabling Competition 2019 (ITC 2019) posed a university timetabling problem involving assigning classes to times and rooms for an entire semester while assigning students to required classes. We propose a new mixed integer programming (MIP) formulation of the problem. The MIP formulation takes advantage of different graph structures in conflict graphs to construct a strong formulation of the constraints. In addition, we introduce a reduction algorithm that removes redundancies from the input data. We show that the reduction algorithm, combined with the graph-based MIP formulation, outperforms the MIP formulated by Holm et al. (A MIP formulation of the International Timetabling Competition 2019 problem, 2020) and thus becomes the new state-of-the-art MIP formulation for the ITC 2019. This paper reports the graph-based MIP formulation, which we used during the ITC 2019, and discusses additional approaches that one can use to strengthen the MIP.
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The authors would like to thank Niels-Christian Fink Bagger for providing Python code for finding clique covers. Dennis S. Holm’s Ph.D. project is part of the Data Science for University Management project (dsumsoftware.com) funded by MaCom A/S and Innovation Fund Denmark (IFD). Rasmus Ø. Mikkelsen’s industrial Ph.D. project is funded by IFD. IFD has supported the work solely financially and has not participated in any research-related activities.
Appendix
Appendix
1.1 A Notations
1.2 B Reduced distribution constraints
Table 12 shows the number of reduced distribution constraints by each method.
1.3 C Reduced variables
Table 13 shows the number of vertices reduced by fixed vertices and cliques for each type of graph.
1.4 D Specified MIP size
This appendix presents the numbers of the specific constraints in the model. Table 14 gives the number of constraints generated by the conflict graphs.Table 15 shows the number of distribution constraints not modeled by the class–time conflict graph. Table 16 presents the number of variables and constraints used to model student sectioning and student conflicts.
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Holm, D.S., Mikkelsen, R.Ø., Sørensen, M. et al. A graph-based MIP formulation of the International Timetabling Competition 2019. J Sched 25, 405–428 (2022). https://doi.org/10.1007/s10951-022-00724-y
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DOI: https://doi.org/10.1007/s10951-022-00724-y