Matias Sørensen

Integer programming for the generalized high school timetabling problem

Recently, the XHSTT format for high school timetabling was introduced. It provides a uniform way of modeling problem instances and corresponding solutions. The format supports a wide variety of constraints, and currently 38 real-life instances from 11 different countries are available. Thereby, the XHSTT format serves as a common ground for researchers within this area. This paper describes the first exact method capable of handling an arbitrary instance of the XHSTT format. The method is based on a mixed-integer linear programming (MIP) model, which is solved in two steps with a commercial general-purpose MIP solver. Computational results show that our approach is able to find previously unknown optimal solutions for 2 instances of XHSTT and proves optimality of 4 known solutions. For the instances not solved to optimality, new non-trivial lower bounds were found in 11 cases, and new best known solutions were found in 9 cases. Furthermore, the approach is compared with the finalists of Round 2 of the International Timetabling Competition 2011 and is shown to be competitive with one of the finalists.

A Two-Stage Decomposition of High School Timetabling applied to cases in Denmark

Integer Programming (IP) has been used to model educational timetabling problems since the very early days of Operations Research. It is well recognized that these IP models in general are hard to solve, and this area of research is dominated by heuristic solution approaches. In this paper a Two-Stage Decomposition of an IP model for a practical case of high school timetabling is shown. This particular timetabling problem consists of assigning lectures to both a timeslot and a classroom, which is modeled using a very large amount of binary variables. The decomposition splits this model into two separate problems (Stage I and Stage II) with far less variables. These two separate problems are solved in sequence, such that the solution for the Stage I model is given as input to the Stage II model, implying that irreversible decisions are made in Stage I. However, the objective of the Stage II model is partly incorporated in the Stage I model by exploiting that Stage II can be seen as a minimum weight maximum matching problem in a bipartite graph. This theoretically strengthens the decomposition in terms of global optimality. The approach relies on Halls theorem for the existence of matchings in bipartite graphs, which in its basic form yields an exponential amount of constraints in the Stage I model. However, it is shown that only a small subset of these constraints is needed, making the decomposition tractable in practice for IP solvers. To evaluate the decomposition, 100 real-life problem instances from the database of the high school ERP system Lectio are used. Computational results show that the decomposition performs significantly better than solving the original IP, in terms of both found solutions and bounds.

A fix-and-optimize matheuristic for university timetabling

University course timetabling covers the task of assigning rooms and time periods to courses while ensuring a minimum violation of soft constraints that define the quality of the timetable. These soft constraints can have attributes that make it difficult for mixed-integer programming solvers to find good solutions fast enough to be used in a practical setting. Therefore, metaheuristics have dominated this area despite the fact that mixed-integer programming solvers have improved tremendously over the last decade. This paper presents a matheuristic where the MIP-solver is guided to find good feasible solutions faster. This makes the matheuristic applicable in practical settings, where mixed-integer programming solvers do not perform well. To the best of our knowledge this is the first matheuristic presented for the University Course Timetabling problem. The matheuristic works as a large neighborhood search where the MIP solver is used to explore a part of the solution space in each iteration. The matheuristic uses problem specific knowledge to fix a number of variables and create smaller problems for the solver to work on, and thereby iteratively improves the solution. Thus we are able to solve very large instances and retrieve good solutions within reasonable time limits. The presented framework is easily extendable due to the flexibility of modeling with MIPs; new constraints and objectives can be added without the need to alter the algorithm itself. At the same time, the matheuristic will benefit from future improvements of MIP solvers. The matheuristic is benchmarked on instances from the literature and the 2nd International Timetabling Competition (ITC2007). Our algorithm gives better solutions than running a state-of-the-art MIP solver directly on the model, especially on larger and more constrained instances. Compared to the winner of ITC2007, the matheuristic performs better. However, the most recent state-of-the-art metaheuristics outperform the matheuristic.

A Strategic View of University Timetabling

Abstract University timetabling has traditionally been studied as an operational problem where the goal is to assign lectures to rooms and timeslots and create timetables of high quality for students and teachers. Two other important decision problems arise before this can be solved: what rooms are necessary, and in which teaching periods? These decisions may have a large impact on the resulting timetables and are rarely changed or even discussed. This paper focuses on solving these two strategic problems and investigates the impact of these decisions on the quality of the resulting timetables. The relationship and differences between operational, tactical and strategic timetabling problems are reviewed. Based on the formulation of curriculum-based course timetabling and data from the Second International Timetabling Competition (ITC 2007), three new bi-objective mixed-integer models are formulated. We propose an algorithm based on the ϵ-constraint method to solve them. The algorithm can be used to analyze the impact of having different resources available on most timetabling problems. Finally, we report results on how the three objectives – rooms, teaching periods, and quality – influence one another.

Benders’ Decomposition for Curriculum-Based Course Timetabling

Abstract In this paper we applied Benders’ decomposition to the Curriculum-Based Course Timetabling (CBCT) problem. The objective of the CBCT problem is to assign a set of lectures to time slots and rooms. Our approach was based on segmenting the problem into time scheduling and room allocation problems. The Benders’ algorithm was then employed to generate cuts that connected the time schedule and room allocation. We generated only feasibility cuts, meaning that most of the solutions we obtained from a mixed integer programming solver were infeasible, therefore, we also provided a heuristic in order to regain feasibility. We compared our algorithm with other approaches from the literature for a total of 32 data instances. We obtained a lower bound on 23 of the instances, which were at least as good as the lower bounds obtained by the state-of-the-art, and on eight of these, our lower bounds were higher. On two of the instances, our lower bound was an improvement of the currently best-known. Lastly, we compared our decomposition to the model without the decomposition on an additional six instances, which are much larger than the other 32. To our knowledge, this was the first time that lower bounds were calculated for these six instances.