Penalising Patterns in Timetables: Novel Integer Programming Formulations
Many complex timetabling problems have an underpinning bounded graph colouring component, a pattern penalisation component and a number of side constraints. The bounded graph colouring component corresponds to hard constraints such as “students are in at most one place at one time” and “there is a limited number of rooms” . Despite the hardness of graph colouring, it is often easy to generate feasible colourings. However, real-world timetabling systems  have to cope with much more challenging requirements, such as “students should not have gaps in their individual daily timetables”, which often make the problem over-constrained. The key to tackling this challenge is a suitable formulation of “soft” constraints, which count and minimise penalties incurred by matches of various patterns. Several integer programming formulations are presented and discussed in this paper.
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