Our computations were done with a number of goals in mind. The main objective was to provide a schedule which satisfied all hard constraints and minimized the cost of medium and soft constraints, using real-life data sets for a large university. We also aimed to find an acceptable set of annealing parameters and move strategies for general timetabling problems of this kind, and to study the effect of using a preprocessor to provide the annealing program with a good starting point. Finally, we wanted to make a comparison of the performance of the three different cooling schedules, geometric cooling, adaptive cooling, and reheating based on cost.
We spent quite some time finding optimal values for the various parameters
for the annealing schedule, such as the initial temperature,
the parameters controlling the rate of cooling
( for geometric cooling, a for adaptive cooling)
and reheating (K),
and the number of iterations at each temperature
(for more details, see Ref. [11]).
Johnson et al. [19] noted in their SA implementation for the
traveling salesman problem (TSP)
that the number of steps at each temperature (or the size of the Markov
chain) needed to be at least proportional to the ``neighborhood'' size
in order to maintain a high-quality result. From our experiments
we found the same to be true for the scheduling problem, even though it is
very different from the TSP.
Furthermore, in a few tests for one semester we fixed the number of
classes and professors but varied the number of rooms and time slots, and
found that the final result improves as the number of iterations in
the Markov chain becomes
proportional to a combination of the number of classes, rooms and time slots.
We also observed the same behavior when we fixed the number of rooms and
time slots but varied number of classes.
Our study case involved real scheduling data covering three semesters at Syracuse University. The size and type of the three-semester data is shown in Table 1. Nine types of rooms were used: auditoriums, classrooms, computer clusters, conference rooms, seminar rooms, studios, laboratories, theaters, and unspecified types. Staff and teaching assistants are considered part of the set of professors. Third semester (summer) data was much smaller than other semesters, however, there were additional space and time constraints and fewer available rooms. Our data was quite large in comparison to data used by other researchers. For example, high school data used by Peterson and colleagues [12, 13] consists of approximately 1000 students, 20 different possible majors, and an overall periodic school schedule (over weeks). In the case of Abramson et al. [2], their data set was created randomly and was relatively small, and they stated that problems involving more than 300 tuples were very difficult to solve.
Table 1 lists all major components of the data we have used.
Timetabling problems can be characterized by their sparseness.
After the required number of lessons have been scheduled, there will be
spare space-time slots, hence, the
sparseness ratio of the problem is defined as the ratio
.
The denser the problem, the lower the sparseness ratio, and the
harder the problem is to solve.
Table 2 shows the sparseness of the three-semester data.
For university scheduling, the sparseness ratio generally decreases as the
data size (particularly the number of classes) increases,
so the problem becomes harder to solve.
Including student preferences makes the problem much harder, but these are
viewed as medium constraints and thus are not necessarily satisfied in a
valid solution.
Our overall results are shown in Tables 3 and 4. These tables show the percentage of classes that could be scheduled in accordance with the hard constraints. In each case (apart from the expert system, which is purely deterministic), we have done 10 runs (with the same parameters, just different random numbers), and the tables show the average of the 10 runs, as well as the best and worst results. The MFA results are different only due to having different initial conditions. Each simulated annealing run takes about 10-20 hours on a Unix workstation.
As expected, each of the methods did much better for the third (summer) semester data, which has a higher sparseness ratio. Our results also confirm what we expected for the different cooling schedules for simulated annealing, in that adaptive cooling performs better than geometric cooling, and reheating improves the result even further.
When a random initial configuration is used, simulated annealing performs very poorly, even worse than the expert system (ES). However, there is a dramatic improvement in performance when a preprocessor is used to provide a good starting point for the annealing. In that case, using the best cooling schedule of adaptive cooling with reheating as a function of cost, we are able to find a valid class schedule every time.
In the case of mean-field annealing, the overall results are generally below those of SA and ES. In addition, we have found in the implementation of this method that the results were quite sensitive to the size of the data as well the type of constraints involved. If we confine ourselves to the set of hard constraints, the results are as good as or even better than the other methods. However if we take into account the medium and soft constraints, that is, the overall cost function, this method does not perform as well.
Student preferences are included only as medium constraints in our
implementation, meaning that these do not have to be satisfied for a valid
solution, but they have a high priority.
For the valid schedules we have produced, approximately of the student
preferences were satisfied. This is reasonably good (particularly since
other approaches do not deal with student preferences at all), but we are
working to improve upon this result.