Decomposition, reformulation, and diving in university course timetabling

Abstract

In many real-life optimisation problems, there are multiple interacting components in a solution. For example, different components might specify assignments to different kinds of resource. Often, each component is associated with different sets of soft constraints, and so with different measures of soft constraint violation. The goal is then to minimise a linear combination of such measures. This paper studies an approach to such problems, which can be thought of as multiphase exploitation of multiple objective-/value-restricted submodels. In this approach, only one computationally difficult component of a problem and the associated subset of objectives is considered at first. This produces partial solutions, which define interesting neighbourhoods in the search space of the complete problem. Often, it is possible to pick the initial component so that variable aggregation can be performed at the first stage, and the neighbourhoods to be explored next are guaranteed to contain feasible solutions. Using integer programming, it is then easy to implement heuristics producing solutions with bounds on their quality. Our study is performed on a university course timetabling problem used in the 2007 International Timetabling Competition (ITC), also known as the Udine Course Timetabling problem. The goal is to find an assignment of events to periods and rooms, so that the assignment of events to periods is a feasible bounded colouring of an associated conflict graph and the linear combination of the numbers of violations of four soft constraints is minimised. In the proposed heuristic, an objective-restricted neighbourhood generator produces assignments of periods to events, with decreasing numbers of violations of two period-related soft constraints. Those are relaxed into assignments of events to days, which define neighbourhoods that are easier to search with respect to all four soft constraints. Integer programming formulations for all subproblems are given and evaluated using ILOG CPLEX 11. The wider applicability of this approach is analysed and discussed

Lower bounds for the ITC-2007 curriculum-based course timetabling problem

“…To the best of our knowledge, only two recently published studies are devoted to the calculation of lower bounds for the CBCT problem ( [4], [12]). These studies are based on different integer programming formulations.…”

“…The bounds from Table 1 are retrieved on November 2010 from this website. In addition, the last column integrates 4 recently improved lower bounds from [4] next to which we include a (*) symbol. The optimal solutions (for a total of 7) are given in bold.…”

“…These studies are based on different integer programming formulations. In [4], the authors additionally propose, together with an IP formulation, a new problem specific branch and cut procedure, where an exponential number of cuts has to be considered to ensure optimality, but the separation can be achieved in reasonable time. This approach has obtained very good lower bounds.…”

“…Soft constraints are used to define the objective function to be minimized. Examples of recent studies on university timetabling can be found in [1,2,4,5,6,10,12,14,16,18].…”

“…Solving their model using CPLEX11 on a 3.4GHz Linux PC with 1GB memory within a time limit of about 4 hours, they obtain lower bounds with an average gap to the best existing upper bounds of 32.63% over the first 14 benchmark instances. In [4], Burke et al devised a problem specific branch-and-cut procedure using CPLEX11 that reaches globally better bounds (average gap of 26.65%) within a time limit of 8 hours on an Intel Pentium 4 processor clocked at 3.20GHz with 4GB of RAM.…”