Hybridizing Integer Programming Models with an Adaptive Decomposition Approach for Exam Timetabling Problems

Rong QuFang HeEdmund K. Burke Abstract The idea of decomposition has been successfully applied to address large combinatorial optimization problems across a range of applications. However, in timetabling, it has not been widely applied. One major difficulty of course, is that early assignment in one sub-problem may lead to later conflict in solving interrelated sub-problems. In our previous work, timetabling problems were adaptively decomposed into a difficult set and an easy set of exams. They were generated by using the information gathered from previous iterations during the problem solving. The approach obtained promising results and showed somewhat unsurprisingly, that a small set of difficult exams contributed to a much larger portion of the total cost of the solution constructed. An interesting issue, which is explored in this paper, is to investigate the effect of constructing complete solutions based on an optimal solution of this difficult sub-problem. In this paper, we first present an IP formulation for solving the difficult sub-problem. To have a tighter initial formulation, a well-known inequality called the Clique Inequality is utilized. We then examine the combinatorial properties of the problem to introduce a new family of cutting planes. These are shown to be helpful in obtaining a solution which is within a gap of less than 10% of optimal for the sub-problem, based on which the final solution is constructed. Promising results have been obtained on several benchmark exam timetabling problems in the literature.