An investigation of fuzzy multiple heuristic orderings in the construction of university examination timetables

In this paper, we present an investigation into using fuzzy methodologies to guide the construction of high quality feasible examination timetabling solutions. The provision of automated solutions to the examination timetabling problem is achieved through a combination of construction and improvement. The enhancement of solutions through the use of techniques such as metaheuristics is, in some cases, dependent on the quality of the solution obtained during the construction process. With a few notable exceptions, recent research has concentrated on the improvement of solutions as opposed to focusing on investigating the ‘best’ approaches to the construction phase. Addressing this issue, our approach is based on combining multiple criteria in deciding on how the construction phase should proceed. Fuzzy methods were used to combine three single construction heuristics into three different pair wise combinations of heuristics in order to guide the order in which exams were selected to be inserted into the timetable solution. In order to investigate the approach, we compared the performance of the various heuristic approaches with respect to a number of important criteria (overall cost penalty, number of skipped exams, number of iterations of a rescheduling procedure required and computational time) on 12 well-known benchmark problems. We demonstrate that the fuzzy combination of heuristics allows high quality solutions to be constructed. On one of the 12 problems, we obtained lower penalty than any previously published constructive method and for all 12 we obtained lower penalty than when any of the single heuristics were used alone. Furthermore, we demonstrate that the fuzzy approach used less backtracking when constructing solutions than any of the single heuristics. We conclude that this novel fuzzy approach is a highly effective method for heuristically constructing solutions and, as such, has particular relevance to real-world situations in which the construction of feasible solutions is often a difficult task in its own right.     

课程表问题的一种模拟退火超启发式算法

课程表问题是经典的组合优化问题,属于NP-hard问题.长期以来人们一直都在寻求快速高效的近似算法,以便在合理的计算时间内准确解决大规模课程安排问题,并提出许多有效且实用的启发式和元启发式算法.在此基础上提出了一种基于多个图染色启发式规则的模拟退火超启发式算法.在超启发式算法的框架中,用模拟退火算法作为高层搜索算法,多个图染色启发式规则为底层的构造算法.与现有的方法相比,该算法具有很好的通用性,可以很容易推广到考试时间表、会议安排.旅行商问题、背包问题等应用领域.实验表明,该算法是可行有效的,且无一例时间、空间冲突.     

Timetabling through Constrained Heuristic Search and Genetic Algorithms

Hybrid genetic algorithms are presented that use constrained heuristic search and genetic techniques for the timetabling problem (TP). The TP is an NP-hard problem for which a general polynomial time deterministic algorithm is not known. The paper describes the classification of constraints and the constraint ordering to obtain the minimization of backtracking and the maximization of parallelism. The school timetabling problem is discussed in detail as a case study. The genetic algorithm approach is particularly well suited to this kind of problem, since there exists an easy way to assess a good timetable, but not a well structured automatic technique for constructing it. So, a population of timetables is created that evolves toward the best solution. The evaluation function and the genetic operators are well separated from the domain-specific parts, such as the knowledge of the problem and the heuristics, i.e. from the timetable builder. The present paper illustrates an approach based on the hybridization of constrained heuristic search with novel genetic algorithm techniques. It compares favourably with known programs to solve decision problems under logic constraints. The cost of the new algorithm and the quality of the solutions obtained in significant experiments are reported.     

Iterated local search using an add and delete hyper-heuristic for university course timetabling

Hyper-heuristics are (meta-)heuristics that operate at a higher level to choose or generate a set of low-level (meta-)heuristics in an attempt of solve difficult optimization problems. Iterated local search (ILS) is a well-known approach for discrete optimization, combining perturbation and hill-climbing within an iterative framework. In this study, we introduce an ILS approach, strengthened by a hyper-heuristic which generates heuristics based on a fixed number of add and delete operations. The performance of the proposed hyper-heuristic is tested across two different problem domains using real world benchmark of course timetabling instances from the second International Timetabling Competition Tracks 2 and 3. The results show that mixing add and delete operations within an ILS framework yields an effective hyper-heuristic approach.     

Feature-based tuning of simulated annealing applied to the curriculum-based course timetabling problem

We consider the university course timetabling problem, which is one of the most studied problems in educational timetabling. In particular, we focus our attention on the formulation known as the curriculum-based course timetabling problem (CB-CTT), which has been tackled by many researchers and for which there are many available benchmarks.The contribution of this paper is twofold. First, we propose an effective and robust single-stage simulated annealing method for solving the problem. Second, we design and apply an extensive and statistically-principled methodology for the parameter tuning procedure. The outcome of this analysis is a methodology for modeling the relationship between search method parameters and instance features that allows us to set the parameters for unseen instances on the basis of a simple inspection of the instance itself. Using this methodology, our algorithm, despite its apparent simplicity, has been able to achieve high quality results on a set of popular benchmarks.A final contribution of the paper is a novel set of real-world instances, which could be used as a benchmark for future comparison.     

Design, engineering, and experimental analysis of a simulated annealing approach to the post-enrolment course timetabling problem

The post-enrolment course timetabling (PE-CTT) is one of the most studied timetabling problems, for which many instances and results are available. In this work we design a metaheuristic approach based on simulated annealing to solve the PE-CTT. We consider all the different variants of the problem that have been proposed in the literature and we perform a comprehensive experimental analysis on all the available public instances. The outcome is that our solver, properly engineered and tuned, performs very well on all cases, providing the new best known results on many instances and state-of-the-art values for the others.     

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.     

Adaptive selection of heuristics for assigning time slots and rooms in exam timetables

This paper presents an iterative adaptive approach which hybridises bin packing heuristics to assign exams to time slots and rooms. The approach combines a graph-colouring heuristic, to select an exam in every iteration, with bin-packing heuristics to automate the process of time slot and room allocation for exam timetabling problems. We start by analysing the quality of the solutions obtained by using one heuristic at a time. Depending on the individual performance of each heuristic, a random iterative hyper-heuristic is used to randomly hybridise the heuristics and produce a collection of heuristic sequences to construct solutions with different quality. Based on these sequences, we analyse the way in which the bin packing heuristics are automatically hybridised. It is observed that the performance of the heuristics used varies depending on the problem. Based on these observations, an iterative hybrid approach is developed to adaptively choose and hybridise the heuristics during solution construction. The overall aim here is to automate the heuristic design process, which draws upon an emerging research theme which is concerned with developing methods to design and adapt heuristics automatically. The approach is tested on the exam timetabling track of the second International Timetabling Competition, to evaluate its ability to generalise on instances with different features. The hyper-heuristic with low-level graph-colouring and bin-packing heuristics approach was found to generalise well over all the problem instances and performed comparably to the state of the art approaches.     

An effective hybrid algorithm for university course timetabling

The university course timetabling problem is an optimisation problem in which a set of events has to be scheduled in timeslots and located in suitable rooms. Recently, a set of benchmark instances was introduced and used for an ‘International Timetabling Competition’ to which 24 algorithms were submitted by various research groups active in the field of timetabling. We describe and analyse a hybrid metaheuristic algorithm which was developed under the very same rules and deadlines imposed by the competition and outperformed the official winner. It combines various construction heuristics, tabu search, variable neighbourhood descent and simulated annealing. Due to the complexity of developing hybrid metaheuristics, we strongly relied on an experimental methodology for configuring the algorithms as well as for choosing proper parameter settings. In particular, we used racing procedures that allow an automatic or semi-automatic configuration of algorithms with a good save in time. Our successful example shows that the systematic design of hybrid algorithms through an experimental methodology leads to high performing algorithms for hard combinatorial optimisation problems.     

Solving a timetabling problem using hybrid genetic algorithms

All over the world, human resources are used on all kinds of different scheduling problems, many of which are time-consuming and tedious. Scheduling tools are thus very welcome. This paper presents a research project, where Genetic Algorithms (GAs) are used as the basis for solving a timetabling problem concerning medical doctors attached to an emergency service. All the doctors express personal preferences, thereby making the scheduling rather difficult. In its natural form, the timetabling problem for the emergency service is stated as a number of constraints to be fulfilled. For this reason, it was decided to compare the strength of a Co-evolutionary Constraint Satisfaction (CCS) technique with that of two other GA approaches. Distributed GAs and a simple special-purpose hill climber were introduced, to improve the performance of the three algorithms. Finally, the performance of the GAs was compared with that of some standard, nonGA approaches. The distributed hybrid GAs were by far the most successful, and one of these hybrid algorithms is currently used for solving the timetabling problem at the emergency service. © 1997 John Wiley & Sons, Ltd.     

Hybrid genetic algorithms for timetabling

Hybrid genetic algorithms are presented that use optimization heuristics and genetic techniques to outperform all existing programs for the timetabling problem. The timetabling problem is very hard (NP-complete) and a general polynomial time deterministic algorithm is not known. An artificial intelligence approach, in a logic programming environment, may be useful for such a problem. The decomposition and classification of constraints and the constraint ordering to obtain the minimization of the backtracking and the maximization of the parallelism are illustrated. The school timetabling problem is discussed in detail as a case study. The genetic algorithm approach is particularly well suited for this kind of problem, since there exists an easy way to assess a good timetable but not a well-structured automatic technique for constructing it. So, a population of timetables is created that evolves toward the best solutions. The evaluation function and the genetic operators are well separated from the domain-specific parts, such as the problem knowledge and the heuristics, i.e., from the timetable builder. A fundamental issue and a general problem in the decision process and automated reasoning is how to efficiently obtain logic decisions under disjunctive constraints. Logic constraint satisfaction problems are in general NP-hard and a general deterministic polynomial time algorithm is not known. The present article illustrates an approach based on the hybridization of constrained heuristic search with novel genetic algorithm techniques. It compares favorably with the best-known programs to solve decisions problems under logic constraints. Complexity of the new algorithms and results of significant experiments are reported. © 1996 John Wiley & Sons, Inc.     

Solving the high school timetabling problem using a hybrid cat swarm optimization based algorithm

In this contribution we present the application of a hybrid cat swarm optimization (CSO) based algorithm for solving the school timetabling problem. This easy to use, efficient and fast algorithm is a hybrid variation of the classic CSO algorithm. Its efficiency and performance is demonstrated by conducting experiments with real-world input data. This data, collected from various high schools in Greece, has also been used as test instances by many other researchers in their publications. Results reveal that this hybrid CSO based algorithm, applied to the same school timetabling test instances using the same evaluation criteria, exhibits better performance in less computational time compared to the majority of other existing approaches, such as Genetic Algorithms (GAs), Evolutionary Algorithms (EAs), Simulated Annealing (SA), Particle Swarm Optimization (PSO) and Artificial Fish Swarm (AFS). The algorithm's main process constitutes a variation of the classic CSO algorithm, properly altered so as to be applied for solving the school timetabling problem. This process contains the main algorithmic differences of the proposed approach compared to other algorithms presented in the respective literature.     

A genetic algorithm for a university weekly courses timetabling problem

The timetabling problem is concerned with the allocation, subject to constraints, of given resources to objects in space and time in such way as to satisfy as nearly as possible a set of desirable objectives. This problem is known to be NP–complete and as such only combinatorial optimization methods can guarantee an optimal timetable. In this paper we propose a sector–based genetic algorithm for solving a university weekly courses timetabling problem. Preliminary experimental results indicate that the algorithm is promising.     

A survey of approaches for university course timetabling problem

Scheduling is one of the problems which so many researches have been conducted on it over the years. The university course timetabling problem which is an NP-hard problem is a type of scheduling problem. Timetabling process must be done for each semester frequently, which is an exhausting and time consuming task. The allocation of whole of events in timeslots and rooms performs by the university course timetabling process considering the list of hard and soft constraints presented in one semester, so that no conflict is created in such allocations. In the university course timetabling problem (UCTTP), the hard constraints should not be violated under any conditions; soft constraints also should not be violated as much as possible. The aim of the present paper is to analyze available approaches in the study of university course timetabling problems, including operational researches, metaheuristic methods and intelligent novel methods; also the distributed multi agent systems based approach (Cooperative Search method) is investigated due to its scalability which enables the timetabling of common events between departments. In addition, in this work a complete introduction of reliable datasets has been given to test and evaluation of the structure of considered algorithms.     

An ILP based heuristic for a generalization of the post-enrollment course timetabling problem

We consider a new timetabling problem arising from a real-world application in a private university in Buenos Aires, Argentina. In this paper we describe the problem in detail, which generalizes the Post-Enrollment Course Timetabling Problem (PECTP), propose an ILP model and a heuristic approach based on this formulation. This algorithm has been implemented and tested on instances obtained from real data, showing that the approach is feasible in practice and produces good quality solutions.     

A hyperheuristic approach to examination timetabling problems: benchmarks and a new problem from practice

Many researchers studying examination timetabling problems focus on either benchmark problems or problems from practice encountered in their institutions. Hyperheuristics are proposed as generic optimisation methods which explore the search space of heuristics rather than direct solutions. In the present study, the performance of tournament-based hyperheuristics for the exam timetabling problem are investigated. The target instances include both the Toronto and ITC 2007 benchmarks and the examination timetabling problem at KAHO Sint-Lieven (Ghent, Belgium). The Toronto and ITC 2007 benchmarks are post-enrolment-based examination timetabling problems, whereas the KAHO Sint-Lieven case is a curriculum-based examination timetabling problem. We drastically improve the previous (manually created) solution for the KAHO Sint-Lieven problem by generating a timetable that satisfies all the hard and soft constraints. We also make improvements on the best known results in the examination timetabling literature for seven out of thirteen instances for the To ronto benchmarks. The results are competitive with those of the finalists of the examination timetabling track of the International Timetabling Competition.     

A greedy gradient-simulated annealing selection hyper-heuristic

Educational timetabling problem is a challenging real world problem which has been of interest to many researchers and practitioners. There are many variants of this problem which mainly require scheduling of events and resources under various constraints. In this study, a curriculum based course timetabling problem at Yeditepe University is described and an iterative selection hyper-heuristic is presented as a solution method. A selection hyper-heuristic as a high level methodology operates on the space formed by a fixed set of low level heuristics which operate directly on the space of solutions. The move acceptance and heuristic selection methods are the main components of a selection hyper-heuristic. The proposed hyper-heuristic in this study combines a simulated annealing move acceptance method with a learning heuristic selection method and manages a set of low level constraint oriented heuristics. A key goal in hyper-heuristic research is to build low cost methods which are general and can be reused on unseen problem instances as well as other problem domains desirably with no additional human expert intervention. Hence, the proposed method is additionally applied to a high school timetabling problem, as well as six other problem domains from a hyper-heuristic benchmark to test its level of generality. The empirical results show that our easy-to-implement hyper-heuristic is effective in solving the Yeditepe course timetabling problem. Moreover, being sufficiently general, it delivers a reasonable performance across different problem domains.     

A survey of search methodologies and automated system development for examination timetabling

Examination timetabling is one of the most important administrative activities that takes place in all academic institutions. In this paper, we present a critical discussion of the research on exam timetabling which has taken place in the last decade or so. This last ten years has seen a significantly increased level of research attention for this important area. There has been a range of insightful contributions to the scientific literature both in terms of theoretical issues and practical aspects. The main aim of this survey is to highlight the new trends and key research achievements that have been carried out in the last decade. We also aim to outline a range of relevant important research issues and challenges that have been generated by this body of work. We first define the problem and discuss previous survey papers. Within our presentation of the state-of-the-art methodologies, we highlight recent research trends including hybridisations of search methodologies and the development of techniques which are motivated by raising the level of generality at which search methodologies can operate. Summarising tables are presented to provide an overall view of these techniques. We also present and discuss some important issues which have come to light concerning the public benchmark exam timetabling data. Different versions of problem datasets with the same name have been circulating in the scientific community for the last ten years and this has generated a significant amount of confusion. We clarify the situation and present a re-naming of the widely studied datasets to avoid future confusion. We also highlight which research papers have dealt with which dataset. Finally, we draw upon our discussion of the literature to present a (non-exhaustive) range of potential future research directions and open issues in exam timetabling research.     

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.