New integer linear programming approaches for course timetabling

The most complete form of academic timetabling problem is the population and course timetabling problem. In this problem, there may be multiple classes of each subject, and the decision on which students are to constitute each class is made in concert with the decision on the timetable for each class. In order to solve this problem, it is normally simplified or decomposed in some fashion. One simplification commonly used in practice is known as blocking: it is assumed that the classes can be partitioned into sets of classes (or blocks) that will be timetabled in parallel. This restricts clashing to occur only between classes in the same block, and essentially removes the timetabling aspect of the problem, which can be carried out once the blocks are constituted and the classes populated. The problem of constituting the blocks and populating the classes, known as the course blocking and population problem, is nevertheless a challenging problem, and provides the focus of this paper. We demonstrate, using data provided by a local high school, that integer linear programming approaches can solve the problem in a matter of seconds. Key features include remodelling to remove symmetry caused by students with identical subject selection, and the observation that in practice, only integrality of the block composition variables needs to be enforced; the class population aspects of the model have strong integrality properties.     

School timetabling for quality student and teacher schedules

The school timetabling problem, although less complicated than its counterpart for the university, still provides a ground for interesting and innovative approaches that promise solutions of high quality. In this work, a Shift Assignment Problem is solved first and work shifts are assigned to teachers. In the sequel, the actual Timetabling Problem is solved while the optimal shift assignments that resulted from the previous problem help in defining the values for the cost coefficients in the objective function. Both problems are modelled using Integer Programming and by this combined approach we succeed in modelling all operational and practical rules that the Hellenic secondary educational system imposes. The resulting timetables are conflict free, complete, fully compact and well balanced for the students. They also handle simultaneous, collaborative and parallel teaching as well as blocks of consecutive lectures for certain courses. In addition, they are highly compact for the teachers, satisfy the teachers’ preferences at a high degree, and assign core courses towards the beginning of each day. Dr. Birbas is currently the Director for Primary and Secondary Education in the Region of Western Greece.     

改进的D_时刻表算法在大学课程表问题中的应用

课程表的自动生成是大学课程表编排中最复杂的一个环节,是一个典型的时间规划问题。针对这一问题。本文设计了一种全新的使用改进的D_时刻表的设计方法,提出一套大学课程表白动生成方法。该方法是以文献[5]中介绍的有效解决时间规划问题的D_时刻表算法为基础,并通过试验说明了该方法具有一定的可行性和有效性。     

新蚁群算法模型在大学课程时间表问题中的应用

分析大学课程时间表问题的特征,结合已有蚁群算法的求解策略,构建了新的问题求解模型,提出了一种基于蚁群算法和改进过程的求解算法,并在不同规模的问题实例上进行实验。结果表明,算法在目标函数解的质量上有明显改进。     

An integer program and a hybrid genetic algorithm for the university timetabling problem

The university timetabling problem (UTP) has been studied by numerous research groups for decades. In addition to addressing hard and soft constraints, we extend the UTP by considering consecutiveness and periodicity constraints of multi-session lectures, which are common in many eastern Asian universities. Because schedulers can decide the consecutiveness and periodicity constraints for the multi-session lectures within a limited ratio, we consider these novel decision variables in our model. We develop a mixed integer linear program for the UTP. For the analysis, we convert the UTP into the three-dimensional container packing problem (3DCPP) and create a hybrid genetic algorithm (HGA), which has been shown to be efficient in solving the 3DCPP. We also develop a tabu search algorithm based on the existing UTP literature and compare the findings with that of our HGA. The results show that our HGA obtains a better solution than the tabu search algorithm in a reasonable amount of time.     

udpSkeduler: A Web architecture based decision support system for course and classroom scheduling

A key process for post-secondary educational institutions is the definition of course timetables and classroom assignments. Manual scheduling methods require enormous amounts of time and resources to deliver results of questionable quality, and multiple course and classroom conflicts usually occur. This article presents a scheduling system implemented in a Web environment. This system generates optimal schedules via an integer-programming model. Among its functionalities, this system enables direct interaction with instructors in order to gather data on their time availability for teaching courses. The results demonstrate that significant improvements over the typical fully manual process were obtained.     

Scheduling the professional soccer leagues of Austria and Germany

Generating a regular season schedule is a demanding task for any sports league. In Europe, the creation of a suitable schedule for every national top soccer league not only has to address numerous conflicting inner-league requirements and preferences. Additionally, the games of the European Cup matches (Champions League, UEFA Cup, National Cup Winners) have to be taken into account. In this paper we consider the case of Austria and Germany, that is the planning problem the “Deutsche Fußball-Bund” (DFB) and the “Österreichische Fußball-Bund” (ÖFB) are confronted with. For both leagues we develop models and algorithms which yield reasonable schedules quickly. The models borrow their expressive power from so-called partially renewable resources. Our approach generates schedules which have been accepted for play once by the DFB and six times by the ÖFB.     

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 Hall's 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.     

Exact formulations and algorithm for the train timetabling problem with dynamic demand

In this paper we study the design and optimization of train timetabling adapted to a dynamic demand environment. This problem arises in rapid train services which are common in most important cities. We present three formulations for the problem, with the aim of minimizing passenger average waiting time. The most intuitive model would consider binary variables representing train departure times but it yields to non-linear objective function. Instead, we introduce flow variables, which allow a linear representation of the objective function. We provide incremental improvements on these formulations, which allows us to evaluate and compare the benefits and disadvantages of each modification. We present a branch-and-cut algorithm applicable to all formulations. Through extensive computational experiments on several instances derived from real data provided by the Madrid Metropolitan Railway, we show the advantages of designing a timetable adapted to the demand pattern, as opposed to a regular timetable. We also perform an extensive computational comparison of all linear formulations in terms of size, solution quality and running time.     

基于校院两级任务的计算机排课算法研究

结合多年的排课经验，基于校、院两级教务管理模式和减小教室流动性，提出了一种简单有效的排课算法．实践证明该算法切实可行，效果较好．