The primary objective of this study is to derive an approximate solution
to a high-dimensional and non-geometric combinatorial optimization
problem, namely the university timetabling problem.
The timetabling problem (TTP) has been tackled by many researchers
[#50##1#], mostly in the field of operations research. This is a very
difficult problem due to its nature as a multi-constraint satisfaction
non-Euclidean combinatorial optimization problem for which no
particular heuristic or method has been shown to produce good results
for large-scale problems. This problem continues to draw interests
from researchers in different quarters: from Leighton's [#33##1#]
approach to a large scale scheduling via methods of graph coloring to
de Werra's [#21##1#] review of the area to Hertz [#29##1#, #38##1#],
Dowsland [#3##1#, #4##1#], Abramson [#18##1#] and others applications of
various cases of the problem. A number of different methods and
heuristics have been tried on different instances from high school to
university course scheduling. For small to medium size problems, some
of the methods outlined in [#50##1#] have been shown to produce
superior results than others, but as far as we aware of at the present
there are no comprehensive experimental studies comparing directly the
performance of those methods on the same set of a large scale
real-life data. What we have in mind here is an experimental work in
the same caliber of the extensive work carried out by Johnson and
colleagues [#19##1#, #20##1#, #42##1#] of simulated annealing.
The timetabling problem we have studied can be summarized as follows.
Given data sets of classes and their days, hours, enrollments, and
instructors; rooms and their capacities, types, and locations;
distances between buildings; priorities of each building for different
departments; and students and their class preferences;
the problem is to construct a feasible class schedule satisfying all
the hard constraints and minimizing the medium and soft constraints.
Hard constraints are space and time constraints that must be
satisfied, such as scheduling only one class at a time for any
professor, student, or classroom.
Medium and soft constraints are student and professor preferences that
should be minimized if not satisfied.
We used data for the academic year 1993-94 for classes at Syracuse
University. Currently this problem is handled by the university
scheduling department in a semi-automated fashion. A scheduling program
is used to find a partial solution, and substantial manual effort is
required to iterate towards a final solution. Also, when scheduling a
certain semester (e.g. autumn 1996), a template of a previous semester
(e.g. autumn 1995) is used as part of the input data.
We tackle this problem using a multi-phase approach. In the first
phase an expert system is used as a preprocessor to derive a partial
solution. The second phase is the use of simulated annealing using three
different cooling schedules: the standard geometric cooling, reheating
as a function of cost, and adaptive cooling. Having a preprocessor
in the application of simulated annealing was found to greatly improve
the quality of the solution. Using this method, we were able to find
feasible solutions to the complex class scheduling problem for a
large university. Prior to the approach of this paper, we have used
mean field annealing of Peterson et al. [#26##1#], an expert system,
and simulated annealing each applied separately to the same
problem but we observed that none of these methods gave us as good
results as reported here for the multi-phase approach.