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 practical three‐phase ILP approach for solving the examination timetabling problem

A practical mathematical programming based approach is introduced for solving the examination timetabling problem at the German Jordanian University (GJU), whereby the complex process of acquiring a feasible examination timetable is simplified by subdividing it into three smaller sub‐problems (phases). Accordingly, the exams are initially allocated to time slots in phase one, the time slots are then allotted to days in phase two, and finally in phase three the exams are assigned to rooms based on the number of students taking each exam and capacities of the rooms. The solution for each phase is acquired based on an integer linear programming (ILP) formulation, while satisfying a set of hard constraints that ensure comfortable exam timetables for all students and meet the desired requirements set by GJU administrative staff. Furthermore, the solver can be controlled and launched from a student information system named MyGJU Admin, which enabled registrars at the university to easily, quickly, and accurately generate final exam timetables in several standard formats. Moreover, the approach was validated based on recent GJU registration information as well as real‐world benchmark data.     

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 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.     

天课时分配下的多阶段新高考走班制排课算法

伴随新高考改革,高中排课过程需要考虑学生的科目选择。潜在的学生上课时间冲突提高了排出可行课表的难度,排课过程中对课表的复杂要求也更难得到满足。针对这些挑战提出一种多阶段优化算法解决高中“走班制”教学课程时间表优化问题。优化侧重点从课表时段分配转为天课时分配,即对每个课程班每天的课时数目进行决策。除需要满足课时不冲突的约束条件外,主要优化目标为“课时分布均匀”“教案平齐”“同时上课”。根据问题特点设计了三种课表变换算子用于在教学班天课时分配阶段提升新设计的爬山算法的寻优能力。在三组不同难度和规模的实验数据上,多阶段优化算法以高于85%的概率排出可行课表。相较大规模真实案例,人工生成案例和中规模真实案例在目标函数上得到较为理想的优化。整体课表的教案平齐违反主要源于行政班课表。发现同时上课的设置具有指导其他目标函数优化的能力。     

A two-phase heuristic evolutionary algorithm for personalizing course timetables: a case study in a Spanish university

This paper presents, as a case study, the application of a two-phase heuristic evolutionary algorithm to obtain personalized timetables in a Spanish university. The algorithm consists of a two-phase heuristic, which, starting from an initial ordering of the students, allocates students into groups, taking into account the student's preferences as a primal factor for the assignment. An evolutionary algorithm is then used in order to select the ordering of students which provides the best assignment.The algorithm has been tested in a real problem, the timetable of the Telecommunication Engineering School at Universidade de Vigo (Spain), and has shown good performance in terms of the number of constraints fulfilled and groups assigned to students.     

Applying evolutionary computation to the school timetabling problem: The Greek case

In this contribution, an adaptive algorithm based on evolutionary computation techniques is designed, developed and applied to the timetabling problem of educational organizations. Specifically, the proposed algorithm has been used in order to create feasible and efficient timetables for high schools in Greece. The algorithm has been tested exhaustively with real-world input data coming from many different high schools and has been compared with several other effective techniques in order to demonstrate its efficiency and superior performance. Simulation results showed that the algorithm is able to construct a feasible and very efficient timetable more quickly and easily compared to other techniques, thus preventing disagreements and arguments among teachers and assisting each school to operate with its full resources from the beginning of the academic year. Except from that, due to its inherent adaptive behavior it can be used each time satisfying different specific constraints, in order to lead to timetables, thus meeting the different needs that each school may have.     

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.     

Fairness in examination timetabling: Student preferences and extended formulations

Variations of the examination timetabling problem have been investigated by the research community for more than two decades. The common characteristic between all problems is the fact that the definitions and datasets used all originate from actual educational institutions, particularly universities, including specific examination criteria and the students involved. Although much has been achieved and published on the state-of-the-art problem modelling and optimisation, a lack of attention has been focussed on the students involved in the process. This work presents and utilises the results of an extensive survey seeking student preferences with regard to their individual examination timetables, with the aim of producing solutions which satisfy these preferences while still also satisfying all existing benchmark considerations. The study reveals one of the main concerns relates to fairness within the student's cohort; i.e. a student considers fairness with respect to the examination timetables of their immediate peers, as highly important. Considerations such as providing an equitable distribution of preparation time between all student cohort examinations, not just a majority, are used to form a measure of fairness. In order to satisfy this requirement, we propose an extension to the state-of-the-art examination timetabling problem models widely used in the scientific literature. Fairness is introduced as a new objective in addition to the standard objectives, creating a multi-objective problem. Several real-world examination data models are extended and the benchmarks for each are used in experimentation to determine the effectiveness of a multi-stage multi-objective approach based on weighted Tchebyceff scalarisation in improving fairness along with the other objectives. The results show that the proposed model and methods allow for the production of high quality timetable solutions while also providing a trade-off between the standard soft constraints and a desired fairness for each student.     

Generation of classes of robust periodic railway timetables

In this paper we discuss the problem of randomly sampling classes of fixed-interval railway timetables from a so-called timetable structure. Using a standard model for the timetable structure, we introduce a natural partitioning of the set of feasible timetables into classes. We then define a new probability distribution where the probability of each class depends on the robustness of the timetables in that class. Due to the difficulty of sampling directly from this distribution, we propose a heuristic sampling method and illustrate using practical data that our method indeed favors classes containing robust timetables over others.     

Computing delay resistant railway timetables

In the past, much research has been dedicated to compute optimum railway timetables. A typical objective has been the minimization of passenger waiting times. But only the planned nominal waiting times have been addressed, whereas delays as they occur in daily operations have been neglected. Delays have been rather treated mainly in an online context and solved as a separate optimization problem, called delay management.We provide the first computational study which aims at computing delay resistant periodic timetables. In particular we assess the delay resistance of a timetable by evaluating it subject to several delay scenarios to which optimum delay management will be applied.We arrive at computing delay resistant timetables by selecting a new objective function which we design to be somehow in the middle of the traditional simple timetabling objective and the sophisticated delay management objective. This is a slight extension of the concept of “light robustness” (LR) as it has been proposed by Fischetti and Monaci [2006. Robust optimization through branch-and-price. In: Proceedings of AIRO]. Moreover, in our application we are able to provide accurate interpretations for the ingredients of LR. We apply this new technique to real-world data of a part of the German railway network of Deutsche Bahn AG. Our computational results suggest that a significant decrease of passenger delays can be obtained at a relatively small price of robustness, i.e. by increasing the nominal travel times of the passengers.     

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 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.     

Multiobjective multiproduct parcel distribution timetabling: a real‐world application

Multiobjective multiproduct parcel distribution timetabling problem is concerned with generating effective timetables for parcel distribution companies that provide interdependent services (products) and have more than one objective. A parcel distribution timetabling problem is inherently multiobjective because of the multitude of criteria that can measure the performance of a timetable. This paper provides the mathematical formulation of the problem and applies the model to a real‐world case study. The application shows that without a common ground with the practitioners, it would be impossible to define the actual requirements and objectives of the company; problem definition is as important as model construction and solution method.     

Compromise ratio with weighting functions in a Tabu Search multi-criteria approach to examination timetabling

University examination scheduling is a difficult and heavily administrative task, particularly when the number of students and courses is high. Changes in educational paradigms, an increase in the number of students, the aggregation of schools, more flexible curricula, among others, are responsible for an increase in the difficulty of the problem. As a consequence, there is a continuous demand for new and more efficient approaches. Optimisation and Constraint Programming communities have devoted considerable attention to this difficult problem. Just the definition of a satisfactory, not to mention optimal, timetabling may be complex. In fact, to characterise a timetabling solution, a single criteria may not be enough, since what may be considered good for one group of students may be regarded inappropriate for other students, or teachers. In this paper, four criteria were used to characterise the spreading of the exams over the examination period. A set of constraints regarding the non-overlapping of exams with students in common was considered. A multi-objective optimisation program was used to handle the four criteria and a Tabu Search was implemented to find a good feasible solution for this problem. Two new features to increase the automation of the algorithm were proposed. First, it uses a Fuzzy Inference Ruled Based System to choose the tabu tenure of the elements in the tabu list. Secondly, a modified version of the Compromise Ratio (CR) is proposed, where the usual fixed weights are replaced by weighting functions to rank the neighbourhood solutions in each iteration. Sufficient conditions which guarantee the monotonicity of the weighting functions are presented.     

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 informed genetic algorithm for the examination timetabling problem

This paper presents the results of a study conducted to investigate the use of genetic algorithms (GAs) as a means of inducing solutions to the examination timetabling problem (ETP). This study differs from previous efforts applying genetic algorithms to this domain in that firstly it takes a two-phased approach to the problem which focuses on producing timetables that meet the hard constraints during the first phase, while improvements are made to these timetables in the second phase so as to reduce the soft constraint costs. Secondly, domain specific knowledge in the form of heuristics is used to guide the evolutionary process. The system was tested on a set of 13 real-world problems, namely, the Carter benchmarks. The performance of the system on the benchmarks is comparable to that of other evolutionary techniques and in some cases the system was found to outperform these techniques. Furthermore, the quality of the examination timetables evolved is within range of the best results produced in the field.     

Model-based problem solving for university timetable validation and improvement

Constraint satisfaction problems can be expressed very elegantly in state-based formal methods such as B. But can such specifications be directly used for solving real-life problems? In other words, can a formal model be more than a design artefact but also be used at runtime for inference and problem solving? We will try and answer this important question in the present paper with regard to the university timetabling problem. We report on an ongoing project to build a curriculum timetable validation tool where we use a formal model as the basis to validate timetables from a student’s perspective and to support incremental modification of timetables. In this article we describe the problem domain, the formalization in B and our approach to execute the formal model in a production system using ProB.     

Classroom assignment for exam timetabling

We consider the problem of assigning a set of independent (non-conflicting) exams having a given size to a set of classrooms having certain capacities. We formulate the problem as a zero–one linear integer program for both cases where each classroom contains no more than one exam and for the case where this constraint is relaxed. For the second case, the problem is also formulated as a transportation problem. We already presented two working papers that consider the problem of assigning exams to timeslots at the Faculty of Economics and Management Sciences of Sfax using several heuristics based on graph colouring. This paper is a supplement of the earlier ones, where the solution obtained in these papers (i.e., timetables composed of non-conflicting exams) serves as an input to the problem treated here. Some elementary results are proved and a simple heuristic procedure is developed to solve the problem.     

A hybrid particle swarm optimization based algorithm for high school timetabling problems

In this contribution a hybrid particle swarm optimization (PSO) based algorithm is applied to high school timetabling problems. The proposed PSO based algorithm is used for creating feasible and efficient high school timetables. In order to demonstrate the efficiency of the proposed PSO based algorithm, experiments with real-world input data coming from many different Greek high schools have been conducted. Computational results show that the proposed hybrid PSO based algorithm performs better than existing approaches applied to the same school timetabling input instances using the same evaluation criteria.