Graph-based Heuristics

The examination timetabling problem can be represented as a graph colouring problem where the vertices represent the examinations, the edges represent the conflict between two examinations and the colour of the vertices represent different time-slots in the timetable. The discussion of the timetabling problem as a graph theoretical model is described in studies by de Werra (1985), de Werra (1997) and Burke et al. (2004c).

A definition of the concepts and terms that relate to a graph can be found in Burke et al. (2004c). An undirected graph G = (V, E) is a representation that consists of a set of vertices, V = v₁, ..., vn, and a set of edges, E. If (vi, vj) is an edge in a graph G = (V, E), then vertex v_i is adjacent to vertex v_j (Burke et al., 2004c). The graph colouring method creates a timetable by using a sequential strategy. Most of the early timetabling studies used the sequential technique to solve the problem because it was the simplest and easiest to implement. It is based on an ordering strategy which allows the examination with the most difficulty to be chosen first, by trying to place the examinations sequentially into time-slots in order to produce a timetable. Burke et al.

(2004c) also listed the most commonly used graph colouring heuristics for examination timetabling, i.e. largest degree, largest weighted degree, colour degree and saturation degree. From this point onwards, the graph-based ordering strategies in this thesis are referred to as graph colouring heuristics.

A study by Broder (1964) was one of the earliest that utilised the ordering strategy of graph colouring theory for solving examination timetabling problems. The use of the

‘largest degree’ heuristic as an ordering strategy was based on the difficulty of the exam-ination to be scheduled. This difficulty was measured by the number of edges connecting to vertices. The largest number of edges of a vertex shows which examination is the most difficult to schedule due to the large number of conflicts with other examinations.

A study by Cole (1964) also employed the largest degree heuristic to examination timetabling problems. A table of conflict matrix N × N was presented which listed the conflict by courses and in order to allocate the courses into suitable time-slots, the

largest degree heuristic was employed. Further, Peck and Williams (1966) presented the graph colouring procedure when using the largest degree heuristic for examination timetabling. The ordering was modified by partitioning and rearranging the assignment of examinations to time-slots. Welsh and Powell (1967) also made use of the graph colouring heuristic in order to find the least number of colours (chromatic number ) ap-plied to the vertices of a graph. The main idea was to create a non-conflict graph where there was no matching colour for two adjacent vertices. The largest degree heuristic was used in identifying the colour of the vertices.

The relationship between the examination timetabling and the graph colouring problem was studied by Wood (1969) who identified the chromatic number of a graph so that the number of time-slots needed in a certain examination session could be obtained. In another study, Wood (1968) constructed a system for university timetabling for the Uni-versity of Manchester. In order to determine examinations to be fitted into timetables, an ordering strategy called ‘largest enrolment’ was introduced. This strategy attempted to fit the examinations with the largest student enrolment in the first time-slot in the timetable. Three different methods of identifying the examination session of the shortest duration were compared, i.e. largest degree heuristic, similarity matrix and the upper bound approach proposed by Welsh and Powell (1967). In this study, the similarity matrix produced the best result among them. However, the approach was tested on a small problem involving only twenty vertices in the graph colouring problem.

A dynamic ordering strategy, known as ‘saturation degree’, was first introduced by Br´elaz (1979). This method is very effective compared with other heuristic methods because it dynamically colours the vertex. This heuristic has been applied successfully to examination timetabling (Abdul Rahman et al., 2009, Burke and Newall, 2004, Carter and Laporte, 1996). It works by giving priority to the vertex with the least colour available to be coloured first. Practically, the vertex with the least available colour is the most difficult to be scheduled since it has the smallest saturation degree. A study by Mehta (1981) was one of the earliest investigations implementing this dynamic ordering strategy to solve the examination timetabling problem. The examinations were ordered based on the number of time-slots in conflict, the examinations with the highest conflicting time-slot being the first to be fitted into the schedule.

Laporte and Desroches (1984) developed a system named HORHEC for solving exami-nation timetabling problems at the University of Montreal Business School. During the construction phase, the solution used several graph colouring heuristics: largest degree, largest weighted degree, largest enrolment and random ordering. This was repeated several times in order to obtain a feasible examination timetable. Whenever there were examinations that could not be assigned into the timetable, a backtracking procedure

was incorporated in order to make sure that the problematic examinations could be fitted in.

In another study, Burke et al. (1994) produced a university timetabling spreadsheet type system for examination and course timetabling problems based on graph colouring and constraint manipulation. The events (examination/course) were scheduled in the appropriate time-slot and at the same time fulfilled the room capacity requirement.

The user could interact with the system to obtain a desired solution and if no solution was found, the backtracking strategy of the system was employed in order to develop a complete timetable and to improve the solution quality. The examination timetabling system applied to the University of Nottingham dataset with real world features was presented in Burke et al. (1993).

Johnson (1990) constructed examination timetables based on the ‘difficulty factor’ ob-tained from graph colouring heuristics. During the first phase of the implemented ap-proach, the combination of largest enrolment and largest degree was used as an ordering strategy to assign examinations to time-slots. Several variations of the relative weights of each criterion were considered in order to produce a number of different timetables.

In the next phase, the simulated annealing approach was used to improve the solution quality of the obtained timetable.

Essentially, the sequential heuristics have proved to be very efficient when incorporating a backtracking procedure (Carter et al., 1994). This procedure is implemented whenever some examinations (violated examinations) cannot not be assigned to a timetable due to conflict with other examinations and they should therefore be rescheduled into the available time-slots. Examinations that conflict with the current infeasible examination are unscheduled in order to allow the current violated examination to be scheduled first.

Then, the unscheduled examinations are rescheduled back into new time-slots. There are a number of studies relating to the backtracking procedure implemented with graph colouring heuristics in order to obtain a feasible timetable (Asmuni et al., 2009, Burke and Newall, 1999, Carter et al., 1996, Gogos et al., 2008, Laporte and Desroches, 1984).

In 1996, Carter et al. developed a commercial software for examination timetabling, named EXAMINE. Their study incorporated a backtracking procedure in order to re-solve the infeasibility problem during the timetable construction. The approach was found to reduce the length of the examination session by half compared with sequential techniques without backtracking. Five graph colouring heuristics were implemented to order the examinations based on its scheduling difficulties (see Table 2.1). The study found that the saturation degree heuristic provided a good sequence ordering of exami-nations, while experimental results on the Toronto benchmark datasets showed that the

quality of the final solution was enhanced when combining a backtracking strategy and tabu list.

Table 2.1: The graph colouring heuristics in examination timetabling

Heuristic Descriptions

Largest degree (Broder, 1964)

The largest number of edges/conflicting examinations is scheduled first.

Saturation degree (Br´elaz, 1979)

Ordering is based on the number of time-slots in conflict where the examination with the fewest time-slots is sched-uled first.

Largest weighted degree (Carter and Laporte, 1996)

The examination with the largest number of students who are involved in the conflict is scheduled first.

Largest enrollment (Wood, 1969)

The largest number of students registered for the examina-tions is scheduled first.

Random ordering The ordering is random for the purpose of benchmarking and comparison with other sequencing strategies.

Current research trends in examination timetabling include the use of an adaptive or-dering technique combined with graph colouring heuristics. The adaptive approach in Burke and Newall (2004) was based on the concept of ‘squeaky wheel optimisation’, proposed by Joslin and Clements (1999). The approach could adapt to any given prob-lem by adding a heuristic modifier to the basic graph colouring heuristic technique and by promoting difficult examinations to be scheduled first at each iteration based on its order. The study took into account different considerations of hard and soft constraints in order to test the effectiveness of heuristic modifiers. The details of this approach are discussed in Chapter 3. The technique introduced a good initialisation strategy for ex-amination timetabling problems, the results demonstrating that the adaptive heuristic ordering approach could improve the quality of the obtained solution compared with using only a basic graph colouring heuristic approach.

The adaptive ordering strategy proposed by Burke and Newall (2004) was further studied by Abdul Rahman et al. (2009) who incorporated shuffling strategies, i.e. block and top-window, to different graph coloring heuristics. These strategies acted as a stochastic component and ordered the examinations within a group of examinations. Different sizes of examinations in a group and diverse modifier types have been tested with the Toronto benchmark datasets. Their study found that the saturation degree heuristic produced better results compared with the largest degree heuristic. It was concluded that the dynamic nature of saturation degree caused this heuristic to work very effectively on the tested problem. For more details on the study see Chapter 3.

A recent study by Carrington et al. (2007) used the weighted graph model proposed by Kiaer and Yellen (1992) for solving examination/course timetabling. The vertices and

edges of a graph usually hold much information related to the objective function that help in the ordering of examinations. The weighted graph model was enhanced by introducing several new heuristics for vertex-selection and time-slot-selection and the implementation was varied with various combinations and partitions. The vertex introduced by the graph was selected and coloured until it was finished. The approach was tested on the Toronto benchmark datasets and showed promising results compared with the pure graph colouring heuristics. The study was extended by Burke et al. (2010e) where the introduced heuristics were combined with a linear approach. Moreover, weights were adapted to each heuristic that contributed to the ordering of the vertices.

Kahar and Kendall (2010) solved the examination timetabling problem at the Universiti Malaysia Pahang, using four graph colouring heuristics, the time-slots being chosen based on the best time-slot from the candidate list of five. The approach was able to produce a feasible solution to the problem and was found to be better than the university’s current software.