Computers, Materials & Continua
DOI:10.32604/cmc.2023.034051
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Article

Optimization Techniques in University Timetabling Problem: Constraints, Methodologies, Benchmarks, and Open Issues

Abeer Bashab1, Ashraf Osman Ibrahim2,*, Ibrahim Abakar Tarigo Hashem3, Karan Aggarwal4, Fadhil Mukhlif5, Fuad A. Ghaleb5 and Abdelzahir Abdelmaboud6

1Faculty of Computer Science & Information Technology, Alzaiem Alazhari University, 13311, Khartoum, Sudan
2Faculty Computing and Informatics, Universiti Malaysia Sabah, Kota Kinabalu, 88400, Sabah, Malaysia
3Department of Computer Science, University of Sharjah, Sharjah, 27272, United Arab Emirates
4Electronic and Communication Engineering Department, Maharishi Markandeshwar Deemed to be University, Mullana, Ambala, 133207, Haryana, India
5Information Assurance and Security Research Group (IASRG), School of Computing, Faculty of Engineering, Universiti Teknologi Malaysia, 81310, Johor, Malaysia
6Department of Information Systems, King Khaled University, Muhayel, 61913, Aseer, Saudi Arabia
*Corresponding Author: Ashraf Osman Ibrahim. Email: ashrafosman@ums.edu.my
Received: 05 July 2022; Accepted: 10 October 2022

Abstract: University timetabling problems are a yearly challenging task and are faced repeatedly each semester. The problems are considered non-polynomial time (NP) and combinatorial optimization problems (COP), which means that they can be solved through optimization algorithms to produce the aspired optimal timetable. Several techniques have been used to solve university timetabling problems, and most of them use optimization techniques. This paper provides a comprehensive review of the most recent studies dealing with concepts, methodologies, optimization, benchmarks, and open issues of university timetabling problems. The comprehensive review starts by presenting the essence of university timetabling as NP-COP, defining and clarifying the two formed classes of university timetabling: University Course Timetabling and University Examination Timetabling, illustrating the adopted algorithms for solving such a problem, elaborating the university timetabling constraints to be considered achieving the optimal timetable, and explaining how to analyze and measure the performance of the optimization algorithms by demonstrating the commonly used benchmark datasets for the evaluation. It is noted that meta-heuristic methodologies are widely used in the literature. Additionally, recently, multi-objective optimization has been increasingly used in solving such a problem that can identify robust university timetabling solutions. Finally, trends and future directions in university timetabling problems are provided. This paper provides good information for students, researchers, and specialists interested in this area of research. The challenges and possibilities for future research prospects are also explored.

Keywords: University timetabling; timetabling approaches; meta-heuristics; combinatorial optimization

1  Introduction

University timetabling belongs to a class of problems called NP (non-deterministic polynomial)-COP (combinatorial optimization problem). This class has distinctively recognized features such as:

•   A method to solve this kind of problem in a specific reasonable time is yet to be found.

•   The computational time required to achieve a viable solution grows exponentially with the problem size.

•   It is commonly done by hand; a human can repeat the work routine, and it is time-consuming.

•   The main objective concentrates on fulfilling all the stated hard and soft constraints, which increases the complexity.

•   The exact solution is only achievable for modest cases of optimization problems. The majority of cases adopt approximation algorithms, which do not guarantee the optimal solution but are hopefully able to obtain “good enough” solutions.

The above features help clarify why university timetabling is a very complicated task, since it aims to obtain the ideal feasible timetable among various institutions, and each institute has specific rules, conditions, and structure. Moreover, these institutions continue to grow rapidly every year as the number of registered students increases, as more students mean more teachers, subjects, buildings, and hard manual work. Furthermore, the reached solution may be unsatisfactory in some respects. For these reasons, university timetabling problems require automating the gained solutions to help accelerate the search process for the optimal timetable. Over the years, several algorithms have been presented to solve university timetabling problems. Many publications argue about completely automating university timetabling. However, the human touch is still required to guide and reinforce the search process. Moreover, preferences in the candidate solutions need to be considered that are unable to be made and expressed through automated systems. So, in general, it is a tailor-made process [1].

Even though there are several algorithms for solving university timetabling, this study focuses on optimization algorithms, particularly because they overcome some common classes of problems. Meta-heuristic algorithms are categorized as population-based approaches and are widely applied in solving university timetabling problems due to their approved strength factors. Recently, some articles on timetabling problems were published, which studied systematic mapping for meta-heuristic for university timetabling problems [2] and a survey for optimization methods in school timetabling [3]. Another study introduced a survey focusing on trends and opportunities in university course timetabling [4]. Most recent studies have focused on one category of timetabling problems, such as school, course, and examination timetabling. In this study, university timetabling problems are discussed in detail, considering both course and examination timetabling problems.

This paper aims to comprehensively review optimization approaches for solving university timetabling problems. The classification of current and recent studies is investigated and discussed for both university course timetabling and university examination timetabling problems. In addition, the adopted meta-heuristic optimization algorithms are demonstrated in solving this kind of problem. The constraints related to university timetabling that need to be considered to obtain a good solution are explained, while different methodologies used in previous studies are explored. In addition, methods to analyze and measure the performance of the solution by demonstrating the commonly used benchmark datasets are also elaborated. Moreover, the study provides some robust university timetabling solutions, trends, and future directions.

This study is structured as follows: Section 2 provides context for the problem’s nature, classifications, and constraints. In Section 3, the university timetabling methodologies and approaches are presented. Section 4 addresses the common techniques for analyzing optimization algorithms. Section 5 describes the benchmarking standard. After that, Sections 6 and 7 show the discussion, future direction, and open issues, respectively. Finally, Section 8 concludes the paper.

2  Background

The term “scheduling” is considered a comprehensive term covering various types of problems such as timetabling, sequencing, and rostering. A brief description of these problems is explained below:

Scheduling is the allocation, subject to constraints, of resources to objects being placed in space-time in such a way as to minimize the total cost of some set of the resources used.

Timetabling is the allocation, subject to constraints, of given resources to objects being placed in space-time in such a way as to satisfy as many as possible of a set of desirable objectives.

Sequencing is the construction, subject to constraints, of an order in which activities are to be carried out or objects are to be placed in some representation of a solution.

Rostering is the placing, subject to constraints, of resources into slots in a pattern. One may seek to minimize some objectives or simply to obtain a feasible allocation. Often, the resources will rotate through a roster [5]. Based on that, the issue of timetabling can be described as shown in Fig. 1.

Figure 1: University timetabling is part of the scheduling problem

The timetabling approach covers several types of real problems, such as companies (employees and staff), schools, universities, the military, hospitals (nurses and doctors), transportation (flight, train), and sports. The focus of this paper is particularly on university timetabling and its obstructions and challenges. University timetabling is a difficult optimization problem that is faced every year. Finding the optimal timetable becomes hard and time-consuming due to:

•   There is an increasing number of educational institutions, each with their own ideas about how and when their courses or exams should be completed or performed.

•   The massive number of subjects presented by these institutions.

•   The growing numbers of students, however, mean that these days they can combine subjects from various tracks, possibly even in diverse faculties.

•   There are several hard constraints to be fulfilled.

2.1 University Timetabling

University timetabling problems can be classified as combinatorial optimization problems (COP). The overall goal of this type of problem is to select the best solution from a set of possible solutions. For example, timetabling is intended to assign a timeslot for each event [6]. In certain cases, the problem of timetabling consists of locating any timetable that meets all the stated constraints, in which case the problem is formulated as a search problem. In other cases, what is needed is a timetable that meets all the hard constraints and minimizes or maximizes a declared objective function that embeds the soft constraints; in these instances, the problem is formulated as an optimization problem [1]. The challenge of optimization is to find the best values according to a stated objective function while respecting a set of constraints. In addition, the complexity of the problem comes from the nature of the variables, which are either discrete or continuous [7]. University timetabling is a broad concept with many subfields. However, the following clarification focuses on the two most common essential educational timetabling fields: examination and course timetabling.

2.1.1 Examination Timetabling

Examination timetabling can be formally defined as “the assigning of examinations to a limited number of available time periods in such a way that there are no conflicts or clashes” [8]. Examination timetabling is a substantial educational timetabling concept. In most examination timetabling, the common objective is to avoid clashing and conflicting, where a student must not take two exams at the same time and make sure that there will be only one exam assigned to a room at any time slot. Also, the room capacity is taken into consideration, as the room must be large enough to contain all the students as well as avoid time overlap in the same room [9]. University examination timetabling can be classified into capacitated and un-capacitated problems. The capacitated problem is committed to ensuring the number of examined students does not exceed the capacity of the scheduled room. On the other hand, the un-capacitated problem disregards room capacity [9].

2.1.2 Course Timetabling

Course timetabling can be formally defined as “a multi-dimensional assignment problem in which students and teachers (or faculty members) are assigned to courses, course sections, or classes; events (individual meetings between students and teachers) are assigned to classrooms and times [10].” Course timetabling is a significant educational timetabling field. Course, lecture, or class timetabling is a multipart scheduling problem in which students are assigned to courses, and consequently, those courses must be allocated to specific rooms and timeslots [11]. Furthermore, course scheduling can be divided into subproblems like student scheduling, course scheduling, teacher assignment, and classroom assignment [12]. The differences between courses and exam timetabling are, firstly, that, commonly, each course has one assigned exam; secondly, the exam could be assigned to one room or divided across several rooms. On the other hand, courses must be scheduled in one room; nothing else [13]. University course timetabling is divided into curriculum-based and enrollment-based course timetabling. First, the issue is to develop a weekly lecture schedule and then use the enrollment-based approach to develop the timetable later on [14].

2.2 Timetabling Constraints

The constraints can be considered as borders when a feasible timetable is found. Constraints are classified into two types: hard and soft constraints. Once forming the timetable, the variation of the hard and soft constraints should be considered, as well as considering that these constraints differ from one institution to another.

•   Hard constraints: these cannot be violated, and hard constraints must be met for the solution to be valid for the problem.

•   Soft constraints: can be violated, they are not required, but their fulfillment is critical to producing a good quality timetable and is used to measure the quality of the solution. Every violation denotes a penalty for the solution that is added to the cost.

Fig. 2 lists common examples of both examination and course timetabling for more specifics; check [7,15], for examination constraints as well as [12,15] for university course constraints.

Figure 2: University timetabling scheduling constraints

3  University Timetabling Problem Methodologies

This section demonstrates different methodologies utilized in university timetabling using meta-heuristic optimization algorithms.

3.1 Timetabling Methodologies

The diverse proposed methods for solving university timetabling problems can be divided into four agreed categories, in addition to six extended categories:

•   Sequential methods

•   Cluster methods

•   Constraint-based methods

•   Meta-heuristic methods

•   Generalized search

•   Hybrid evolutionary algorithms

•   Multi-criteria approaches

•   Case-based reasoning techniques

•   Hyper heuristics

•   Adaptive approaches

The majority of approaches have been previously suggested to solve timetabling problems. In the following, only three substantial and significant timetabling approaches will be described in detail: heuristic algorithms, meta-heuristic algorithms, and hybridization methods. Fig. 3 illustrates the hierarchy and clarifies the common framework of university timetabling approaches.

Figure 3: University timetabling scheduling methods

3.2 Heuristic Algorithms

Sequential heuristics algorithms are simple and easy methods to solve timetabling problems. The primary idea is to schedule the events by arranging them in a series, starting with the most difficult event [12]. It has proven capable of producing quick solutions. However, the performance of sequential heuristic approaches is relatively weak as compared to meta-heuristic efficiency [11]. Local search is one example of heuristic algorithms and is described as follows:

LS is a functional algorithm for solving combinatorial optimization problems (COPs). Most approaches that tackle NP/COPs problems include local search. Technically, local search takes the current best solution from the search space and repeatedly replaces it with a newer, better solution from the neighborhood [7]. Local search is presented to maximize the conditions between several candidate solutions in the search space. Because of the success of LS, recently, it has been applied to meta-heuristic algorithms such as PSO [16].

3.3 Meta-heuristic Algorithms

Meta-heuristic algorithms are elaborated approaches that are improved from the sequential heuristic method. Meta-heuristics have the capability to produce superior and better solutions as compared to sequential heuristic algorithms [12]. Broadly, in university timetabling, the initial solution is constructed using an appropriate heuristic mechanism, and the rest of the optimization process is carried out using an elected meta-heuristics algorithm. The performance of meta-heuristic algorithms may vary from one instance to another, depending on various factors. Therefore, the development of an overall comprehensive framework that remains stable and balanced under different circumstances with no need for expensive adaptation has been a research subject in recent years [17,18]. Diverse meta-heuristic approaches have been represented and developed lately, inspired by different fields; for example, analogy to scientific domains such as physics, biology, neurology, and sociology [7]. A meta-heuristic is categorized into single-solution and population-based approaches as follows:

3.3.1 Single-Solution Based Approaches

It addresses and handles one solution during the search process to obtain the best solution [11]. Instead of using a set of candidate solutions, it processes a single solution, picked up according to a set of criteria, and replaces it repeatedly with an improved solution until the termination step is accessed when the final conditional criterion is satisfied [17].

The efficiency of this approach depends on the definition of the neighborhood alternative candidate solutions to replace the current best one. Even though this notion represents the strength of this approach, the main shortcoming is that single-solutions are easily stuck in local optima as compared with the second category [11]. Some applicable single-solution based methods for solving university timetabling problems include:

•   Simulated Annealing (SA)

SA is considered an LS approach that stimulates the heating of solids in physics. It goes through a different strategy of origin local search, so instead of replacing the current solution frequently with another candidate, it creates a random initial solution and for each iteration, it is replaced by an alternative random solution, which increases the possibility of avoiding trapping in local optima [17].

•   Tabu Search (TS)

TS is based on stating a tabu list as a criterion. First, it starts from the initial result and moves toward a set of neighboring results to select the superior one. If the neighboring result is preferable to the available one, the algorithm will take that route. The transition from the current best result to the neighboring candidate result will be repeated until the termination condition is met [17].

3.3.2 Population Based Approaches

The population-based approaches first initiate a combination of population solutions. This initial set goes through numerous alterations and repetitions to obtain the optimal solution. For each restoration, a selection technique will be applied to choose the fittest solution from the presented population. After that, based on the implemented meta-heuristic algorithm, several modifications are performed to the selected solutions, so refinements are obtained to the solution. This procedure will continue until the optimal solution is obtained [17]. Usually, in university timetabling, the premier solution is formatted by utilizing a suitable heuristic approach. Nevertheless, the improvement was accomplished by using a meta-heuristic algorithm [12]. The following population-based approaches are examples of applicable algorithms for solving university timetabling problems:

Evolutionary Algorithms (EAs)

EAs are stochastic search approaches that imitate natural evaluation in addition to the social behavior of some species. EAs are population-based methods of assessing potential solutions. They consist of three phases: selection, regeneration, and replacement [19]. There are so many evolutionary algorithms available that have been successful and effective in solving optimization problems, like the well-known and commonly used genetic algorithm:

•   Genetic Algorithm (GA)

The genetic algorithm is a stochastic search technique inspired by molecular biology based on the natural evolutionary process and works on the process of natural selection [20]. GA has a great ability to search in large spaces, is a flexible algorithm, and is commonly characterized by robustly solving complex combinatorial problems [21]. The algorithm treats the solution as a chromosome carrying good and bad phenotypes, then uses the re-production technique.

Swarm Intelligence (SI)

Swarm intelligence mimics the social attitudes of species in nature, like bird flocks, beehives, and ant colonies. This kind of algorithm has proven to be effective and efficient in solving complicated optimization problems. A swarm is usually formed from a group or population of individuals that communicate and interact between individuals without any central control, yet leads to a final harmony in the whole colony [20]. The swarm’s formative behavior is based on some simple rules like:

•   central control of the individual’s behavior; decentralization reinforces the robustness of collective behavior, interactions, and performance.

•   Each individual in the swarm has a specified declared role based on the whole objective.

•   Individuals interact and communicate directly or indirectly through influencing a local, resulting in the emergence of intelligent global manners.

•   The swarm adopts a self-organization strategy.

The following are examples of popular and widely used swarm intelligence algorithms:

•   Particle Swarm Optimization (PSO)

PSO is inspirited by the coexistence of herds of birds or shoals of fish. In these swarms, there is a leader who has the best value of fitness to guide the whole swarm, so any individual moves are related to and based on the leader’s moves [20]. As a real-life example, in a herd of birds communicating during flight, each bird looks in a specific direction, and then the group will communicate together to identify and determine the bird at the best location. According to that, every bird speeds toward the superior bird’s location, utilizing velocity based on its immediate position. After that, it explores the search area from its fresh position. The birds can utilize their own expertise, which represents the local search process, as well as the expertise of the entire flock, which represents the global search [22].

•   Ant Colony Optimization (ACO)

The basic notion of the ACO algorithm is to find the shortest path from a food source to an ant hill by smelling pheromones. The colony is managed by hundreds of individuals. When collecting food, if there are two possible paths to access the food source, the ants choose randomly. Basically, half of them choose the first direction and the other half choose the other one. The shortest path obtains a huge amount of pheromone. So next time, the ants will recognize the shorter path by smelling the pheromones [20]. Looking at ACO, a random route or path creation is basically a mutation process, while pheromone concentration selection affords a technique for electing the shortest path, and crossover processes are not declared in this algorithm [23].

•   Artificial Bee Colony (ABC)

ABC mimics the natural behavior of bees during pollen collection. It is categorized into three bee sets: employed/forager bees; onlooker bees or observer bees; and scouts. The process starts by sending the scouts to randomly search around for promising areas. After searching, they return to the hive and express themselves by dancing. The information about the found site is vital, and this information guides the colony to evaluate the amount of energy needed to harvest. Based on that, the colony could send bees directly to the most promising place [20]. Both the scout and employed bees are mainly considered as mutation processes, where selection is based on the objective, which is the honey. There is no explicit crossover process [23].

Memetic Algorithm (MA)

The memetic algorithm is basically formed by hybridizing meta-heuristic algorithms that represent a global search with a local search [7]. A memetic algorithm is like a genetic algorithm, except the components that form the chromosome are called memes instead of genes. By the same token, there is a contrastive way to classify meta-heuristic algorithms. This standpoint divides meta-heuristics into three categories based on the hard and soft constraints considered according to declared requirements [18]:

•   One-stage optimization algorithms: where the satisfaction of both hard and soft constraints is conducted simultaneously.

•   Two-stage optimization algorithms: where the satisfaction of the soft constraints is only conducted when achieving the feasible timetable.

•   Algorithms that allow relaxations: where violations of the hard constraints are disallowed from the beginning by relaxing some other features of the problems, and then trying to satisfy the soft constraints.

3.3.3 Multi-Objective Based Approaches

A multi-objective optimization problem (MOP) is a process to simultaneously optimize two or more conflicting objectives. In university timetabling, sometimes it is needed to solve more than one constraint simultaneously. In this case, MOP is the best choice. For example, in exam timetabling problems, sometimes students can take exams in as many consecutive intervals as possible but at the same time reduce the schedule length and meet the difficult constraints like seat capacity and no overlapping exams. However, recent studies have been conducted successfully using multi-objective optimization algorithms in university timetabling problems [24–27], as well as in other applications [28].

3.4 Hybridization Methods

A hybrid algorithm is a combination of at least two algorithms complementing each other that are performed and executed together to generate a gainful synergy from their integration. In general, operating an effective and functional hybrid approach is a difficult mission [29]. The hybridization technique plays an outstanding role in creating a robust hybrid algorithm by integrating the features of each combined algorithm while simultaneously minimizing any substantial shortcomings [30]. The hybridization between single and population-based methods aims to utilize the preference features of population-based methods, which have enough capability to declare the potential promising zones in the exploration area, and single-based methods, which have enough capability to exploit the prospective areas [31]. For more information about this rising area, refer to the following references [32–35]. The allocation of the covered publications according to the adopted algorithm sort is presented in Table 1.

4  Optimization Algorithms Evaluation

Since university timetabling problems are categorized as NP-COP, the most probable way to solve and handle this kind of problem is through optimization algorithms. The following section clarifies some major concepts about optimization algorithms. To evaluate optimization algorithms, two phases should be explored: first, analyzing the algorithm; and second, measuring the total performance of an algorithm.

4.1 Optimization Algorithms Analysis

Optimization algorithms can be analyzed by different procedures. It follows two common and fundamental approaches: the exploration and exploitation method and the evolutionary operators’ method.

4.1.1 Exploration and Exploitation

Since most optimization algorithms are naturally inspired, they are analyzed according to their exploitation and exploration functionality in the search space. An exploitation process tries to create new solutions that are better than the existing ones, and the search process for better candidates is done on a local scale. From the same aspect, the exploration process tries to create solutions with enough variety from the existing solutions, and the search is done on a global level, making it less probable to get trapped in local mode. The exploration process indicates more efficiency than the exploitation process in the space search operation. But this comes with some limitations; since the search goes on a global scale, the time of convergence is affected, making it slower [23,151].

However, fulfilling this balance is still an open problem. Furthermore, such a balance depends on various factors; for instance, the setting of tuning and controlling parameters. Additionally, such a balance may not occur globally and may differ from one problem to another. It is consistent with “no-free-lunch (NFL)” theorems, which indicate that for a specific problem, one algorithm may record high performance; at the same time, it may perform badly for other problems. Essentially, there is no algorithm that has absolute priority over other algorithms [23]. To learn more about the interesting NFL theorem, refer to [152].

4.1.2 Evolutionary Operators

Crossover, mutation, and selection present the essential and primary evolutionary operators for optimization algorithms. Crossover and mutation processes drive variation to offer new solutions. In addition, the selection operator has a double role, choosing the best solutions from a subspace and reinforcing the self-organization and convergence processes. However, mostly all algorithms use the mutation and selection operators over the crossover operator [23].

4.2 Performance Measurement

Despite the massive number of studies and publications on diverse meta-heuristic algorithms, there is still a lack of qualified and dependable performance measures to compare between the distinct algorithms. Commonly, there are four prime approaches to comparing two algorithms.

•   First approach compares the accuracy of two algorithms when solving the same problem for a stable number of function evaluations.

•   Second approach compares the required function evaluations of two various algorithms for a stated accuracy.

•   Third approach compares the execution times for two algorithms.

•   Fourth approach equalizes the results of algorithms to generate a ratio for comparison [23].

5  Benchmarking

Many optimization algorithms claim superiority over other comparable algorithms. Consequently, benchmark functions can be used as indicators to determine the most powerful and reliable algorithms to prove their effectiveness [153]. No exact algorithm can beat all comparable algorithms in all test cases and conditions [154].

5.1 Experimental Settings

The evolutionary approaches provide experimental settings with very substantial issues since they can influence the expected outcomes of the experiments. The setting must be flawless to obtain an optimal output; otherwise, imperfect settings may lead to un-optimal results. In order to have an adjusted assessment between various algorithms, it is crucial to set the value of each algorithm to optimum in order to achieve the best possible result.

5.2 Benchmark Functions

Each benchmark function identifies its features, such as unimodal, multimodal, separable, or non-separable. It is noteworthy that the combination of characteristics defines the function of complexity [153]. The intent of the optimization tactic is to find the global optimum; thus, the zone around the local optimum must be avoided to prevent getting stuck in the local optimum as far as possible [153].

5.2.1 University Course Timetabling Benchmark Data

University course timetabling faces an absence of benchmark data. The International Timetabling Competition (ITC) 2007 encloses data for enrollment-based course timetabling and curriculum-based course timetabling. The following is an explanation of the recognized benchmark of course timetabling:

•   ITC2007 Track2-Enrollment-based Course Timetabling

This dataset compares twenty-four existing datasets; all feature at least one optimal solution with no hard or soft constraint infringements. The main shortcoming of this benchmark is un-reality of datasets; they were generated by the competition organizers [14].

•   ITC2007 Track3-Curriculum-based Course Timetabling

To date, there are twenty-one examples available. All data examples are from real life, not the same as in the previous track. They were collected by the University of Udine, and there is at least one functional solution for each example [14]. The datasets are available on the competition website: http://www.cs.qub.ac.uk/itc2007/index.htm.

•   The Purdue benchmark data

The University of Purdue offers access to real data on its official website. The datasets are available for each department of the university [14]. Purdue datasets are available at https://www.unitime.org/uct_datasets.php.

5.2.2 University Examination Timetabling Benchmark Data

The university examination timetabling benchmark datasets are interesting data. These have been gathered by researchers, which has led to some established diversity of benchmark problems. The following is a brief explanation of the examination timetabling benchmark:

•   The Toronto benchmark data

The incapacitated dataset for the examination timetabling problem is presented in [155]. This data contains a clash-free constraint as a hard constraint, in which the student cannot sit for two exams at the same time. Likewise, the soft constraint is to spread exams as consistently as possible throughout the examination period [143]. By the same token, 13 real-world problems are available. Two objectives are provided by the Toronto benchmark data; 1) to minimize the number of assigned time slots and 2) to minimize the average cost per student [14]. The studies in examination timetabling tasks directed to establishing diverse options for the problem are called Toronto a, Toronto b, Toronto c, Toronto d and Toronto e [145]. They are illustrated based on a set of objectives and formed as thus:

Toronto a Objective: To minimize the total of desired timeslots.

Toronto b Objective: To space out contradictory exams’ limited timeslots.

Toronto c Objective: To minimize the impact of assigning two exams to students on one day.

Toronto d Objective: To minimize assigning two exams respectively for students on one day and overnight.

Toronto e Objective: To minimize assigning more than one exam during the day.

•   The Nottingham benchmark data

The study [77] modified six data sets from Carter et al. [155] and presented them as benchmarks alongside the examination timetabling data from the University of Nottingham. The objective was to minimize the impact of assigning two consecutive exams to students [14].

•   The Melbourne benchmark data

Study [115] provided two new datasets with two time slots for five workdays from the University of Melbourne with the objective of minimizing assigning two exams to students on one day as well as overnight [14]. All the mentioned benchmark data can be assessed at http://www.cs.nott.ac.uk/~pszrq/data.htm.

•   The Purdue benchmark data

Müller [15] provided the newest benchmark in examination timetabling. Nine datasets from Purdue University were presented; each data contained 29 examination periods for all two hours long [14]. The Purdue benchmark datasets are available online at https://www.unitime.org/exam_datasets.php.

6  Discussion

This survey presents an extensive integral review of university timetabling cases by submitting four bases as platforms to form and organize the review, and consequently trying to conduct a compatible and functional outcome covering all timetabling cases. Firstly, the survey started by clarifying the nature of timetabling problems to accurately achieve the optimal timetable through the extracted functional and efficient guided solution. Secondly, the problems were classified to provide a good perspective of the intended objectives considering the stated constraints. Thirdly, indicating and observing the applied approaches for solving timetabling problems were indicated and observed to show the main junction of these methods, the concept, the primary mechanism, and illustrate the primary bifurcation of these approaches as well as the concept, the core mechanism, and features of each individual method. Finally, the standards and specifications to adopt the method of candidates were described to ensure the optimality of the selected method, which indicates accomplishing the possible timetable. Therefore, two important questions have been answered at this stage: what analytic methods have been used to assess, measure, and compare the approach’s performance? In addition, what kinds of benchmark datasets are utilized for the experimental tests? To elucidate the preferences over methods. Fig. 4 displays the number of publication types distributed in journals, conferences, reports, books, and thesis dissertations.

Figure 4: Number of publications by source type

The prime aim of this class of problems is to satisfy all problem constraints rather than optimizing a certain objective. The complication of this type of problem is progressively increasing with the growing number of increasing stated constraints. Every time more constraints are added, it gets harder to solve the problem, until it gets to a point where it is impossible to reach the optimal solution.

The most adjustment with the convergence dilemma is by conceding some of the stated constraints through the solving process to make it possible to reach a feasible solution. That is exactly why university timetabling is such a complex problem, because constraints differ from one institution to another. It should be considered that no timetable tackles all the hard and soft constraints as well as the indicated objectives. Only a few of these constraints are under control, while the rest are ignored. Therefore, a more general timetable is important. For this reason, the suggested timetable should further generalize and meet all the hard constraints and minimize (or maximize) a particular objective function that surrounds the soft constraints. The timetable cannot tackle all the hard and soft constraints as well as the indicated objectives. Most timetables can only handle part of the constraints and ignore the remaining ones.

A meta-heuristic is a technique for solving optimization problems that involves exploring the optimal solutions to a set of constraints and objectives, as well as suggesting candidate or alternate solutions to the stated problem. In general, meta-heuristics can be categorized into single-solution-based methods and population-based methods. Concerns about meta-heuristic algorithms have grown dramatically over the last decade. Despite the huge number of publications and studies, there is still a lack of performance measures for comparing different algorithms. The former clarified and illustrated meta-heuristic algorithms were selected based on their commonness and broad implementations. Fig. 5 illustrates the publication ratio distribution of the meta-heuristic algorithms used to optimize university timetabling problems. Performance and experimental outcomes have demonstrated operative success. In particular, the genetic algorithm as compared to other methods has shown great publication (16%). In recent years, more studies tend to implement hybrid methods due to their being fruitful in the final outcomes, with a 35% higher value as compared with the mentioned methods.

Figure 5: Publication of different meta-heuristic algorithms

The interest in swarm intelligence algorithms (SI) has been remarkable in the last decade. Swarm intelligence algorithms are efficient and reliable when applied to solving university timetabling problems. Swarm intelligence (SI) is one of the computational intelligence methods used to solve complex problems like university timetabling. Swarm intelligence deals with groups (collective behaviors) composed of a set of individuals (local interactions of the individuals with each other and with their environment) that coordinate using decentralized control and self-organization. Another confusing point about university timetabling is how to evaluate and examine different swarm intelligence algorithms against each other, considering each institution has specific stated objective functions and declared problem formulations, thus increasing the difficulty of performing a justified comparison. Comparing the algorithms in order to find the superior one is a vague concept since different algorithms achieve various objectives, and that revokes the standard criteria for formal evaluation and testing before comparison. In recent times, hybridization algorithms have attracted the interest of researchers because they provide an effective and outstanding performance in solving optimization problems by utilizing the advantage of a combination of population-based with a single-based methodology. The exploration process is the ability to expand the search over a broad area to reach the unvisited spaces, while the exploitation process focuses on the potentially promising spaces of suitable solutions to optimally utilize and converge. So, the search process must be in equilibrium between the exploitation and the exploration to reach the optimal solution.

7  Future Directions and Open Issues

This part shows numerous essential open research issues hindering the application of meta-heuristic algorithms in university timetabling. Several essential and primary approaches to handling university timetabling problems have not been included in this review because this study covered only some fundamental algorithms. It is noted that there is a gap between presumptive theories and factual practices concerning the employment of optimization tactics as a functional solution to university timetabling problems. Some evolutionary and swarm intelligence algorithms, like Biogeography-Based Optimization, Symbiotic Organisms Search, Grey Wolf Optimizer, Intelligent Water Drops Algorithm (IWD), Chicken Swarm Optimization (CSO), great deluge algorithm (GD), Monkey Algorithm, Bat Algorithm, Sheep Flocks Algorithm, Moth-Flame Optimization and Sine-Cosine Algorithm, Whale Optimization Algorithm, and Harris Hawks Optimization Algorithm, have never been utilized for solving university timetabling problems so far. In addition, multi-objective evolutionary algorithms are promising and have the capability to identify robust university course timetabling solutions. Therefore, multi-objective evolutionary algorithms are good choices to provide better approximation functions. This gap offers scope for researchers to examine and try out more modern optimization algorithms, in addition to running numerous analyses and comparisons over the conducted results to state and clarify the strengths and shortcomings of each algorithm and form a solid and rich database for future studies as a reference and guideline.

Additional recommended scopes for future studies are to keep up-to-date on the submitted novel meta-heuristic algorithms and explore more algorithms; summarize the concluded strengths and limitations for future utilization; hybridize further meta-heuristic algorithms and investigate the combination impact to emphasize the validity and qualified ability of the hybridization method; intensify the publication and research on systematic reviews and surveys to initiate a substantial platform and dependable background for future studies; generalize a global format and standard languages for timetabling; and develop university timetabling software fields; systems, and tools in general. However, there is still a lack of qualified performance measurements to compare between various algorithms. There is still a need for more real-world benchmark datasets to improve the analytical and experimental ways to evaluate the used approaches for solving university timetabling problems. Working on the evaluation techniques and finding a standard can exceed the ambiguous outcome when certain algorithms show preferable performance on some specific optimization problems and less achievement on other problems.

8  Conclusion

An increase in the number of publications addressing the university timetabling problem can be observed over the last decade. University timetabling is a non-polynomial-combinatorial optimization problem (NP-COP) which consists of two essential components: examination and course timetabling. The main goal of this kind of problem is to meet all the problem constraints rather than optimizing a particular objective. Viable solutions require high computational run time, which can increase exponentially with the problem size. Therefore, the use of meta-heuristic algorithms was explored here, along with the identification of open issues and challenges that must be addressed in the future to help solve this problem. The hybridization technique is able to achieve superior performance and robust efficiency by identifying the functional fittest solution for a particular timetabling problem. In addition, it can be concluded that multi-objective evolutionary algorithms can produce good and robust solutions. The intent of conducting this study was to investigate and explore the publishing rate in this field and to identify those issues which need further attention, and which seem promising. Moreover, we also aimed to identify which meta-heuristic algorithm had the highest publication rate. This survey outlined neglected topics in this domain as a foundation for future research, as well as emphasizing the common publishing platform as a standpoint indicator.

The limitations of this study were mainly in the collection of data from database sources, in addition to the search strategy used and the lack of pre-screening tools for the collected data. Moreover, the findings may be impacted by several factors that were not considered in this study. Lastly, the current study was not specifically designed to evaluate features related to datasets and validation measures used in optimization algorithms.

Funding Statement: This research work was supported by the University Malaysia Sabah, Malaysia.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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