School Timetabling Problem Research Articles

An Effective Local Particle Swarm Optimization-Based Algorithm for Solving the School Timetabling Problem

This paper deals with the school timetabling problem. The problem was formulated as encountered in a typical Greek high school. A local version of the particle swarm optimization algorithm was developed and applied to the problem at hand. Results on well-established benchmark instances showed that the proposed algorithm achieved the proven optima provided from an integer programming method presented in earlier research. In almost all cases, the current algorithm beat the integer programming method, either concerning the lower bound yielded or the execution time needed.

Two-phase tabu search algorithm for solving Chinese high school timetabling problems under the new college entrance examination reform

Upon the latest reform to the college entrance examination in China (i.e., Gaokao), high schools began implementing an optional class system. Under this scheme, students’ time slots become complex, thereby increasing the difficulty in formulating a suitable timetable from the available ones. To address this problem, the course-scheduling model was improved. On the basis of the original hard constraints, the “concurrent group” was considered, and the softer constraints were regarded as optimization goals, such as “teaching plans synchronously”, “no idle periods in the timetables of teachers”, and “evenly distributed lessons”. Given these soft constraints, the model becomes more practical. In this study, a two-phase tabu search algorithm was proposed to solve the problem. The proposed algorithm uses the characteristics of the graph coloring model to eliminate redundant calculations in the neighborhood search process, thereby effectively improving computational efficiency. Fifteen practical instances of different scales were selected for testing to verify the effectiveness of the algorithm. The proposed algorithm can formulate high-quality available timetables (The average satisfaction rate of soft constraints is more than 71%) in a short period.

Case—School Timetabling Problem: A Scheduling Problem for High-School Institutions

Funding: This work was supported by the Fondo Nacional de Desarrollo Científico y Tecnológico [Grant 1200221] and the Agencia Nacional de Investigación y Desarrollo [Grant AFB180003].

Case Article—School Timetabling Problem: A Scheduling Problem for High-School Institutions

We present a comprehensive case study to integrate students into several concepts related to integer linear programming. The case article starts with a relatively simple scheduling/assignment problem. Then, the problem incorporates new elements to present various modeling principles incrementally. Each variation of the case enables the instructor to engage in insightful discussion about the nature of the optimal solutions and how the changes made to the objective function or key constraints affect these solutions. The case article also describes different ways to use the case, which range from a comprehensive introduction to a concluding capstone project in an undergraduate or graduate course in linear and integer programming. Funding: This work was supported by Fondo Nacional de Desarrollo Científico y Tecnológico [Grant 1200221] and Agencia Nacional de Investigación y Desarrollo [Grant AFB180003]. Supplemental Material: The Teaching Note and data file are available at https://www.informs.org/Publications/Subscribe/Access-Restricted-Materials .

High quality timetables for Italian schools

This work introduces a complex variant of the timetabling problem, which is motivated by the case of Italian schools. The new requirements enforce to (i) provide teachers with the same idle times, (ii) avoid consecutive days with heavy workload, (iii) limit multiple daily lessons for each class, (iv) introduce shorter time units to differentiate entry and exit times. We present an integer programming model for this problem, which is denoted by Italian High School Timetabling Problem (IHSTP). However, requirements (i), (ii), (iii) and (iv) cannot be expressed according to the current XHSTT standard. Since the IHSTP model is very hard to solve by an off-the-shelf solver, we present a two-step optimization method: the first step optimally assigns teachers to lesson times and the second step assigns classes to teachers. An extensive experimentation is performed on the model by realistic and real instances from Italian schools, as well as benchmark instances from the literature. Finally, the experiments show that the method is effective in solving both this new problem and the simplified problem without the new requirements.

Current state of the higher school timetabling problem

Current state of the higher school timetabling problem

School Timetabling Optimisation Using Artificial Bee Colony Algorithm Based on a Virtual Searching Space Method

Although educational timetabling problems have been studied for decades, one instance of this, the school timetabling problem (STP), has not developed as quickly as examination timetabling and course timetabling problems due to its diversity and complexity. In addition, most STP research has only focused on the educators’ availabilities when studying the educator aspect, and the educators’ preferences and expertise have not been taken into consideration. To fill in this gap, this paper proposes a conceptual model for the school timetabling problem considering educators’ availabilities, preferences and expertise as a whole. Based on a common real-world school timetabling scenario, the artificial bee colony (ABC) algorithm is adapted to this study, as research shows its applicability in solving examination and course timetabling problems. A virtual search space for dealing with the large search space is introduced to the proposed model. The proposed approach is simulated with a large, randomly generated dataset. The experimental results demonstrate that the proposed approach is able to solve the STP and handle a large dataset in an ordinary computing hardware environment, which significantly reduces computational costs. Compared to the traditional constraint programming method, the proposed approach is more effective and can provide more satisfactory solutions by considering educators’ availabilities, preferences, and expertise levels.

Mathematical model and simulated annealing algorithm for Chinese high school timetabling problems under the new curriculum innovation

As the first attempt, this paper proposes a model for the Chinese high school timetabling problems (CHSTPs) under the new curriculum innovation which was launched in 2014 by the Chinese government. According to the new curriculum innovation, students in high school can choose subjects that they are interested in instead of being forced to select one of the two study directions, namely, Science and Liberal Arts. Meanwhile, they also need to attend compulsory subjects as traditions. CHSTPs are student-oriented and involve more student constraints that make them more complex than the typical “Class-Teacher model”, in which the element “Teacher” is the primary constraint. In this paper, we first describe in detail the mathematical model of CHSTPs and then design a new two-part representation for the candidate solution. Based on the new representation, we adopt a two-phase simulated annealing (SA) algorithm to solve CHSTPs. A total number of 45 synthetic instances with different amounts of classes, teachers, and levels of student constraints are generated and used to illustrate the characteristics of the CHSTP model and the effectiveness of the designed representation and algorithm. Finally, we apply the proposed model, the designed two-part representation and the two-phase SA on10 real high schools.

A survey of the state-of-the-art of optimisation methodologies in school timetabling problems

Educational timetabling is an ongoing challenging administrative task that is required in most academic institutions. This is mainly due to a large number of constraints and requirements that have to be satisfied. Educational timetabling problems have been classified as NP-hard problems and can be divided into three types: exam timetabling, course timetabling and high school timetabling. The domain of high school timetabling is not well developed when compared to other fields of educational timetabling such as university exam timetabling and course timetabling. As the evolution of the educational systems are continuous, new challenges often arise, requiring new models and solution methodologies. Over the years, a number of methodologies have been developed to address high school timetabling problems. However, there are no comparative studies or rigorous analysis of these methodologies. This survey paper aims to provide a scientific review of high school timetabling. The paper presents a categorisation of the methodologies conducted in recent years based on chronology, category and application (dataset). We first present comparative studies on the success of proposed methodologies. The components and mechanisms of different methodologies are analysed and compared. We also discuss their performance, advantages, disadvantages and potential for improvement. Methodology wise, a shift of popularity from meta-heuristic to mathematical optimisation is observed in recent years. Another observation is that more researchers are opting for XHSTT formatted datasets as a testbed for their algorithms. Finally, we outline the industrial perspective, trends and future direction in high school timetabling optimisation problems.

Hybrid particle swarm optimization with particle elimination for the high school timetabling problem

In this paper, a PSO-based algorithm that hybridized Particle Swarm Optimization (PSO) and Hill Climbing (HC) is applied to high school timetabling problem. This hybrid has two features, a novel solution transformation and particle elimination. The proposed methodologies are tested on the XHSTT-2014 dataset (which is relatively new for the school timetabling problem) plus other additional instances. The experimental results show that the proposed algorithm is effective in solving small and medium instances compared to standalone HC and better than the conventional PSO for most instances. In a comparison to the state of the art methods, it achieved the lowest mean of soft constraint violations for 7 instances and the lowest mean of hard constraint violations for 1 instance.

Two-stage Heuristic for Primary School Timetabling Problem with Combined Classes Consideration

This research focuses on a primary school timetabling problem, a small-scale primary school that is located at Pengerang, Johor. In this small-scale primary school, six classes have been allotted, from standard one until standard six. Most of the primary school timetables are manually developed, which is extremely time-consuming. According to the new policy announced on 12th Dec 2017by the Ministry of Education (MoE) Malaysia, due to the shortage of teachers, combined-classes should be implemented in low-enrolment schools with fewer than 30 students. MoE has introduced another policy on 30th June 2018 that recommends schools to reduce the number of subjects that are being taught in a day to solve the overloaded school bag issue. There is a set of hard constraints in this primary school timetabling problem due to the stipulation that a teacher can only teach one subject at a time; each subject must satisfy the total weekly period(s), and the combined classes can only combine one subject at a time. The main objective of this study is to propose a heuristic solution to this solves primary school timetabling problem with the consideration of combined-classes. A two-stage timetabling heuristic approaches been offered due to its simplicity in dealing with numerous constraints. The two-stage heuristic method was clustered into subject groups in the first stage to ease the timeslots allocation in the second stage. A clash-free timetable can be obtained from this proposed algorithm. The result generated by this proposed solution outperforms the current manual practice in solution quality and computing efficiency.

A Mathematical Programming Model and Enhanced Simulated Annealing Algorithm for the School Timetabling Problem


 Despite significant research efforts for School Timetabling Problem (STP) and other timetabling problems, an effective solution approach (model and algorithm) which provides boundless use and high quality solution has not been developed. Hence, this paper presents a novel solution approach for solving school timetabling problem which characterizes the problem-setting in the timetabling problem of the high school system in Nigeria. We developed a mixed integer linear programming model and meta-heuristic method - Enhanced Simulated Annealing (ESA) algorithm. Our method incorporates specific features of Simulated Annealing (SA) and Genetic Algorithms (GA) in order to solve the school timetabling problem. Both our solution approach and SA approach were implemented using Matrix Laboratory 8.6 software. In order to validate and demonstrate the performance of the developed solution approach, it was tested with the highly constrained school timetabling datasets provided by a Nigerian high school using constraints violation, simulation time and solution cost as evaluation metrics. Our developed solution approach is able to find optimal solution as it satisfied all the specified hard and soft constraints with average simulation time of 37.91 and 42.16 seconds and solution cost of 17.03 and 18.99, respectively, for JSS and SSS to the problem instance. A comparison with results obtained with SA approach shows that the developed solution approach produced optimal solution in smaller simulation time and solution cost, and has a great potential to solve school timetabling problems with satisfactory results. The developed ESA algorithm can be used for solving other related optimization problems.

Modularizing school timetabling problems in different types of classes for Taiwanese elementary and junior high schools

School timetabling problems (STPs) stem from difficult decisions related to assigning teachers and class sessions to various class time slots, subject to different scheduling, event and resource constraints. This research focused on how to efficiently construct a mathematical model of the STP subject to more than 40 class combinations for Taiwanese elementary and junior high schools. Based on the investigation, this study adopted the concept of modularity to classify class constraints into five modules: fundamental constraint, regular program, special education program, gifted performing and visual arts program and talented athlete program. This research proposed the methodology to apply the modularity concept to construct a customised class combination affected by STPs. Using the numerical data shows that the corresponding school timetable could be generated based on the customised class combination of the STPs. In addition, the scenarios with an objective function or with an infeasible solution were analysed and discussed. Finally, a framework of the automated school timetabling system based on the modularity concept was presented for future school timetabling system development.

Modularising school timetabling problems in different types of classes for Taiwanese elementary and junior high schools

Modularising school timetabling problems in different types of classes for Taiwanese elementary and junior high schools

Pattern-based models and a cooperative parallel metaheuristic for high school timetabling problems

High school timetabling problems consist in building periodic timetables for class-teacher meetings considering compulsory and non-compulsory requirements. This family of problems has been widely studied since the 1950s, mostly via mixed-integer programming and metaheuristic techniques. However, the efficient search of optimal or near-optimal solutions is still a challenge for many problems of practical size. In this paper, we investigate mixed-integer programming formulations and a parallel metaheuristic based algorithm for solving high school timetabling problems with compactness and balancing requirements. We propose two pattern-based formulations and a solution algorithm that simultaneously exploits column generation and a team of metaheuristics to build and improve solutions. Extensive computational experiments conducted with real-world instances demonstrate that our formulations are competitive with the best existing high school timetabling formulations, while our parallel algorithm presents superior performance to alternative methods available in the literature.

Solving the Greek school timetabling problem by a mixed integer programming model

This study deals with the school timetabling problem for the case of Greek high schools. At first, the problem is modelled as a Mixed Integer Programming problem for ten instances referring to Greek high schools. Then, the problem is coded using the MathProg programming language. Two different linear programming solvers are employed, Gurobi and CPLEX, to solve the problem for the instances at hand. Two methodologies are proposed. The first one deals with the problem utilising a model that includes all hard and soft constraints, called “monolithic” model, while the second one is based on a decomposition of the problem to six sub-problems. It should be stated that Gurobi and CPLEX did not produced satisfactory results when the monolithic model was the case. Computational results demonstrate the effectiveness of the second proposed methodology, as optimal solutions or new lower bounds were found. In addition, the results produced by Mixed Integer Programming are compared with the best so far published results, obtained by two Nature Inspired algorithms namely Particle Swarm Optimization and Cat Swarm Optimization.

Optimization in timetabling in schools using a mathematical model, local search and Iterated Local Search procedures

Abstract This paper addresses the school timetabling problem, which consists of defining the date and time in which classes will be given by teachers in educational institutions. For this purpose, a tool that uses Operational Research (OR) techniques was developed, focused on generating and optimizing Elementary and High School timetables, taking into account teachers’ preferences for certain days or for sequenced (twinned) classes. Conductive to solving the problem, a Non Linear Binary Integer Programming mathematical model (NLBIP) and Local Search (LS) and Iterated Local Search (ILS) procedures were comparatively applied. A real problem with 14 timetables of public schools in the city of Araucária (in Paraná State, Brazil) was analyzed. The results indicate that the computational time required by the mathematical model is feasible for the problems in question. The ILS technique has the potential for testing larger scale problems, as it presents a dispersion of 3.5% to 7.7% relative to the optimal solution (obtained by the NLBIP) and a computational time that is 15 to 338 times faster.

An Assignment Problem and Its Application in Education Domain: A Review and Potential Path

This paper presents a review pertaining to assignment problem within the education domain, besides looking into the applications of the present research trend, developments, and publications. Assignment problem arises in diverse situations, where one needs to determine an optimal way to assign n subjects to m subjects in the best possible way. With that, this paper classified assignment problems into two, which are timetabling problem and allocation problem. The timetabling problem is further classified into examination, course, and school timetabling problems, while the allocation problem is divided into student-project allocation, new student allocation, and space allocation problems. Furthermore, the constraints, which are of hard and soft constraints, involved in the said problems are briefly elaborated. In addition, this paper presents various approaches to address various types of assignment problem. Moreover, direction and potential paths of problem solving based on the latest trend of approaches are also highlighted. As such, this review summarizes and records a comprehensive survey regarding assignment problem within education domain, which enhances one’s understanding concerning the varied types of assignment problems, along with various approaches that serve as solution.

A Hidden Markov Model Approach to the Problem of Heuristic Selection in Hyper-Heuristics with a Case Study in High School Timetabling Problems.

Operations research is a well-established field that uses computational systems to support decisions in business and public life. Good solutions to operations research problems can make a large difference to the efficient running of businesses and organisations and so the field often searches for new methods to improve these solutions. The high school timetabling problem is an example of an operations research problem and is a challenging task which requires assigning events and resources to time slots subject to a set of constraints. In this article, a new sequence-based selection hyper-heuristic is presented that produces excellent results on a suite of high school timetabling problems. In this study, we present an easy-to-implement, easy-to-maintain, and effective sequence-based selection hyper-heuristic to solve high school timetabling problems using a benchmark of unified real-world instances collected from different countries. We show that with sequence-based methods, it is possible to discover new best known solutions for a number of the problems in the timetabling domain. Through this investigation, the usefulness of sequence-based selection hyper-heuristics has been demonstrated and the capability of these methods has been shown to exceed the state of the art.

Parallel local search algorithms for high school timetabling problems

High school timetabling consists in assigning meetings between classes and teachers, with the goal of minimizing the violation of specific soft requirements. This family of problems has been frequently considered in the literature, but few strategies employing parallelism have been proposed. In this exploratory study, we consider two different parallel frameworks and present a thorough computational study in order to understand algorithmic decisions that are closely related to performance. Our best algorithm outperforms state-of-the-art algorithms for variants of the problem considered, indicating both the efficiency and the flexibility of the method.