Exam Timetabling Problem Research Articles

The classroom assignment problem: Complexity, size reduction and heuristics

In this paper, we investigate a compound of the exam timetabling problems which consists of assigning a set of independent exams to a certain number of classrooms. We can define the exam timetabling problem as the scheduling of exams to time slots in first stage and at a second stage, the assignment of a set of exams extracted from one time slot to some available classrooms.Even though the formulation of this problem looks simple as it contains only two sets of constraints including only binary variables, we show that it belongs to the class of NP hard problems by reduction from the Numerical Matching with Target Sum problems (NMTS).In order to reduce the size of this problem and make it efficiently solvable either by exact method or heuristic approaches, a theorem is rigorously demonstrated and a reduction procedure inspired from the dominance criterion is developed. The two methods contribute in the search for a feasible solution by reducing the size of the original problem without affecting the feasibility. Since the reduction procedures do not usually assign all exams to classrooms, we propose a Variable Neighbourhood Search (VNS) algorithm in order to obtain a good quality complete solution. The objective of VNS algorithm is to reduce the total classroom capacity assigned to exams. A numerical result concerning the exam of the main session of the first semester of the academic year 2009–2010 of the Faculty of Economics and Management Sciences of Sfax shows the good performance of our approach compared with lower bound defined as the sum of the total capacity of all assigned classrooms and the total size of the remaining exams after reduction.

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.

Graph Colouring and Clustering Heuristic Approach for Minimizing Examination Duration: A Case Study

Exam timetable is difficult to be done manually due to several factors such as dual academic calendar, larger student enrolments, constraints among invigilators and limitations of resources. At tertiary education, preparing exam timetable is very critical in order to ensure that all students are able to sit for the exam of every subject that they have registered without any clashing and only sit for one exam at one time. The lecturers who are also the invigilators as well need to be considered as one of the elements in the development of exam timetable as they are required to be in one venue at one time. Therefore, a good time table needs to ensure that the students and invigilators are able to commit their roles accordingly during the exam period. But the main problem is the duration of the exam which will be extended to fulfil all the requirements. This study presents a solution method intended for reducing exam duration in Centre for Foundation Studies and Extension Education (FOSEE), Multimedia University (MMU), Malaysia. The method of solution is using heuristic approaches that include graph colouring, clustering and sequential heuristic. The discussions were focused on constraints among invigilators and the approach is tested on real-world exam timetabling problems. Keyword : exam timetable, heuristic approach, graph colouring.

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.

Integrating neural networks and logistic regression to underpin hyper-heuristic search

A hyper-heuristic often represents a heuristic search method that operates over a space of heuristic rules. It can be thought of as a high level search methodology to choose lower level heuristics. Nearly 200 papers on hyper-heuristics have recently appeared in the literature. A common theme in this body of literature is an attempt to solve the problems in hand in the following way: at each decision point, first employ the chosen heuristic(s) to generate a solution, then calculate the objective value of the solution by taking into account all the constraints involved. However, empirical studies from our previous research have revealed that, under many circumstances, there is no need to carry out this costly 2-stage determination and evaluation at all times. This is because many problems in the real world are highly constrained with the characteristic that the regions of feasible solutions are rather scattered and small. Motivated by this observation and with the aim of making the hyper-heuristic search more efficient and more effective, this paper investigates two fundamentally different data mining techniques, namely artificial neural networks and binary logistic regression. By learning from examples, these techniques are able to find global patterns hidden in large data sets and achieve the goal of appropriately classifying the data. With the trained classification rules or estimated parameters, the performance (i.e. the worth of acceptance or rejection) of a resulting solution during the hyper-heuristic search can be predicted without the need to undertake the computationally expensive 2-stage of determination and calculation. We evaluate our approaches on the solutions (i.e. the sequences of heuristic rules) generated by a graph-based hyper-heuristic proposed for exam timetabling problems. Time complexity analysis demonstrates that the neural network and the logistic regression method can speed up the search significantly. We believe that our work sheds light on the development of more advanced knowledge-based decision support systems.

A hybrid Ant Colony Algorithm for the exam timetabling problem

Due to increased student numbers and regulation changes educational institutions that allow for greater flexibility, operations researchers and computer scientists have renewed their interest in developing effective methods to resolve the examination timetabling problem. Thus, in the intervening decades, important progress was made in the examination timetabling problem with appearance of adaptation of meta-heuristics. This paper presents a hybridization of the Ant Colony Algorithm and a Complete Local search with Memory heuristic, in order to maximize as much as possible; the free time between consecutive exams for each student, while respecting the conflict constraints, a student cannot sit more than one exam in the same timeslot. Vue l’augmentation du nombre d’étudiants dans les établissements scolaires et universitaires, le nombre d’examens à passer par chaque étudiant et les réformes pédagogiques actuelles, les planifications classiques des cours et des examens ne sont plus suffisantes, ce qui a amené les chercheurs opérationnels et les informaticiens à chercher des nouvelles méthodes pour résoudre le problème d’emploi du temps des examens. Notre travail consiste à planifier les examens de telle sorte à maximiser le temps de séparation entre deux examens consécutifs pour chaque étudiant et ceci en utilisant l’algorithme de colonies de fourmis hybridé avec une technique de recherche locale.

Term-end exam scheduling at United States Military Academy/West Point

Scheduling term-end exams (TEE) at the United States Military Academy in West Point is unlike any other exam timetabling problem we know of. Exam timetabling normally produces a conflict-free timetable covering a reasonably long exam period, where every exam is scheduled exactly once for all the students enrolled in the corresponding class. The situation is quite different at West Point. There are hundreds of exams to schedule over such a short time period that there is simply no feasible solution. The challenge is then to allow something that is not even considered elsewhere, that is, creating multiple sessions of some exams, scheduled at different times within the exam period, to allow each student to take all exams he/she must take. The overall objective is to find a feasible exam schedule with a minimum number of such duplicate exams. The paper describes a system that has been developed at GAMS Development Corp. in close cooperation with the scheduling staff at West Point, and that has been used successfully since 2001. It uses mathematical optimization in several modules, and some of the techniques proposed are new. It is fast and flexible, and allows for human interaction, such as adding initially unexpected constraints, coming for instance from instructors' preferences and dislikes, as well as their hierarchical rankings. It is robust and can be used by people familiar with the organization at West Point, without the need for them to be technically-trained. Overall, using the course and student information databases, it is an effective decision support system that calls optimization tools in an unobtrusive way.

Hybridizations within a graph-based hyper-heuristic framework for university timetabling problems

A significant body of recent literature has explored various research directions in hyper-heuristics (which can be thought as heuristics to choose heuristics). In this paper, we extend our previous work to construct a unified graph-based hyper-heuristic (GHH) framework, under which a number of local search-based algorithms (as the high level heuristics) are studied to search upon sequences of low-level graph colouring heuristics. To gain an in-depth understanding on this new framework, we address some fundamental issues concerning neighbourhood structures and characteristics of the two search spaces (namely, the search spaces of the heuristics and the actual solutions). Furthermore, we investigate efficient hybridizations in GHH with local search methods and address issues concerning the exploration of the high-level search and the exploitation ability of the local search. These, to our knowledge, represent entirely novel directions in hyper-heuristics. The efficient hybrid GHH obtained competitive results compared with the best published results for both benchmark course and exam timetabling problems, demonstrating its efficiency and generality across different problem domains. Possible extensions upon this simple, yet general, GHH framework are also discussed.

A multi-objective evolutionary algorithm for examination timetabling

This paper considers the scheduling of exams for a set of university courses. The solution to this exam timetabling problem involves the optimization of complete timetables such that there are as few occurrences of students having to take exams in consecutive periods as possible but at the same time minimizing the timetable length and satisfying hard constraints such as seating capacity and no overlapping exams. To solve such a multi-objective combinatorial optimization problem, this paper presents a multi-objective evolutionary algorithm that uses a variable-length chromosome representation and incorporates a micro-genetic algorithm and a hill-climber for local exploitation and a goal-based Pareto ranking scheme for assigning the relative strength of solutions. It also imports several features from the research on the graph coloring problem. The proposed algorithm is shown to be a more general exam timetabling problem solver in that it does not require any prior information of the timetable length to be effective. It is also tested against a few influential and recent optimization techniques and is found to be superior on four out of seven publicly available datasets.

A mixed-integer mathematical modeling approach to exam timetabling

This paper explores mathematical programming models for an exam timetabling problem related to Kuwait University (KU). In particular, we consider two subproblems: (a) the ExamTimetabling Problem (ETP), which is concerned with assigning exams to designated exam-periods and classrooms, and (b) the Proctor Assignment Problem (PAP), which deals with the assignment of proctors to exams. While this exam timetabling problem is ubiquitous in many academic institutions worldwide, idiosyncrasies of the problem related to gender-based policies and having multiple exam centers at KU require novel models. A mixed-integer exam timetabling programming model (ETM) is developed for ETP, which takes into account restrictions related to exam-period conflicts, facility and human resources, and commuting and traffic considerations. Assuming that exam-periods are specified for all exams as determined by ETM, another mixed-integer programming model is formulated for PAP, denoted by PAM, which incorporates the proctors’ preferences for specific days and exam-periods. Computational results are reported and analyzed for solving ETM and PAM directly using the CPLEX Optimization Software (version 9.0), and for implementing a specialized sequential LP-based heuristic for solving PAM. The results obtained significantly improve upon those derived via the existing manual approach implemented at KU, in terms of eliminating conflicts as well as from the overall efficiency and equity points of view.

An enterprise system component based on graph colouring for exam timetabling

PurposeThis paper aims to consider the exam timetabling of the re‐sit session in the Faculty of Economics and Management Sciences of Sfax. The objective is to find a timetable which minimizes the number of timeslots for exams required by the enrolled students.Design/methodology/approachTwo heuristic procedures based on graph colouring are developed and tested on real data to solve the exam timetabling problem at the faculty.FindingsThese heuristics were tested on a simple example which shows the out‐performance of the second heuristic compared with the first one. When tested with the real data of the faculty, exam size heuristic provides a timetable with a shorter timeframe; however, the timetable obtained from the second heuristic is of better quality.Originality/valueThe main contribution of this paper is to create an automated exam timetabling that helps the faculty to manage its own enterprise system.

Improving paper spread in examination timetables using integer programming

One of the desirable attributes of real-life examination timetabling solutions is the maximization of paper spread, which is a measure of the amount of study time that each student has between examinations. In this study, we face with a predefined examination schedule that must be modified in order to maximize paper spread in course of examination. We will show how the integer programming can be employed to achieve this aim. The model presents constraints for the quality of feasible examination timetables and all the requirements found in most academic institutions. The approach is tested on real-world exam timetabling problems. The computational experiments and results will be reported.

A time-predefined local search approach to exam timetabling problems

In recent years the processing speed of computers has increased dramatically. This in turn has allowed search algorithms to execute more iterations in a given amount of real-time. Does this necessarily always lead to an improvement in the quality of final solutions? This paper is devoted to the investigation of that question. We present two variants of local search where the search time can be set as an input parameter. These two approaches are: a time-predefined variant of simulated annealing and an adaptation of the “great deluge” method. We present a comprehensive series of experiments which show that these approaches significantly outperform the previous best results (in terms of solution quality) on a range of benchmark exam timetabling problems. Of course, there is a price to pay for such better results: increased execution time. We discuss the impact of this trade-off between quality and execution time. In particular we discuss issues involving the proper estimation of the algorithm's execution time and the assessment of its importance.