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” [1]. Despite the hardness of graph colouring, it is often easy to generate feasible colourings. However, real-world timetabling systems [2] 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. Throughout the paper, the Udine Course Timetabling Problem is used as an illustrative example of timetabling with soft constraints. The problem has been formulated by Schaerf and Di Gaspero [3, 4] at the University of Udine. Its input can be outlined as follows: