Examination Timetabling with Fuzzy Constraints

The aim of this paper is to consider flexible constraint satisfaction in timetabling problems. The research is carried out in the context of university examination timetabling. Examination timetabling is subject to two types of constraints: hard constraints that must not be violated, and soft constraints that often have to be violated to some extent. Usually, an objective function is introduced to measure the satisfaction of soft constraints in the solution by summing up the number of students involved in the violation of the constraint. In existing timetabling models the binary logic strategy is employed to handle the satisfaction of the constraints, i.e. a constraint is either satisfied or not. However, there are some constraints that are difficult to evaluate using the binary logic: for example, the constraint that large exams should be scheduled early in the timetable. Fuzzy IF–THEN rules are defined to derive the satisfaction degree of this constraint, where both the size of the exam and the time period that the exams are scheduled in are expressed using the linguistic descriptors Small, Medium and Large, and Early, Middle and Late, respectively. In a similar way, the constraint that students should have enough break between two exams is modelled. A number of memetic algorithms with different characteristics are developed where corresponding fitness functions aggregate the satisfaction degrees of both fuzzy constraints. The proposed approach is tested on real-world benchmark problems and the results obtained are discussed.