Introduction

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.