Solving examination timetabling problem within a hyper- heuristic framework

Received Jul 25, 2020 Revised Sep 1, 2020 Accepted Apr 7, 2021 Scheduling exams in colleges are a complicated job that is difficult to solve conventionally. Exam timetabling is one of the combinatorial optimization problems where there is no exact algorithm that can answer the problem with the optimum solution and minimum time possible. This study investigated the University of Toronto benchmark dataset, which provides 13 real instances regarding the scheduling of course exams from various institutions. The hard constraints for not violate the number of time slots must be fulfilled while paying attention to fitness and running time. Algorithm of largest degree, hill climbing, and tabu search within a hyper-heuristic framework is investigated with regards to each performance. This study shows that the Tabu search algorithm produces much lower penalty value for all datasets by reducing 18-58% from the initial solution.


INTRODUCTION
In academic fields, such as universities, they have to manage with the examination timetabling. This timetabling is allocating process space and exam time to the college student. Recurring activities at the examination timetabling create an issue for each college student to determine a scheduling time that takes that course. Therefore, timetabling is one of the interesting issues in the field of Combinatorics optimization. In general, the problem of combinatorial optimization is a mathematical study to find an optimal solution to the preparation, sorting, grouping or selection of discrete objects that usually have finite number [1].
The benchmark dataset is from Toronto or commonly referred to as the Toronto data set. Toronto's data sets contain agencies that correspond to the real-world problem variables because they come from different educational institutions and are categorized as stable data sets and are often used in research [2]. This study investigates on solving the problem of Carter data sets (Toronto) using a hyper-heuristic approach where the search is more focused on the heuristic space. The algorithm used within the hyper-heuristic framework is the largest degree, hill-climbing, and tabu search.
This experimental was conducted by presenting several previous studies by the corresponding approach to solve Toronto's data sets problem. These studies including K. Graham and M. Naimah [3] who used tabu search hyper-heuristic, their research shown that their generic method is able to generate good solutions compared to other studies. Di Gaspero and Schaerf [4] using the Tabu search algorithm. Carter et al. [19] use constructive heuristics with backtracking. Caramia et al. [5] use greedy constructive heuristics, their research shows that combined of the backtracking strategy and saturation degree sorting rule can create shorter results solutions in shorter computing time. Burke and Newall [6] use local search method, their research prove that simulated annealing and the great deluxe method is able to increase a good-quality solutions. I. Gabriella & P. Etria [7] using the tabu search algorithm can produce decent solutions with 100,000 iterations. The purpose of this experimental results is to show that the tabu search algorithm is able to produce good penalty results even though the value produced is not the best results.

RESEARCH METHOD 2.1. Combinatorial optimization
Combinatoric optimization research is carried out by entering each possible value or developing a search algorithm to choose the best value. Combinatorial optimization is used to determine the minimum or maximum value based on the problem being discusses. Algorithms in combinatorial optimization can solve quite complex problems with a broad scope [8].

Toronto benchmark dataset
The Toronto benchmark dataset dataset consists of 13 real-time exam schedule problems comprised of three high schools in Canada, five universities in Canada, one university in America, one university in the UK, and one university in the Middle East. It is listed in Table 1 that each dataset shows the final score, number of college students, and timeslot that had been executed in the previous research. The dataset has two boundaries: two examinations with the same participants but different schedules [9].

Hyper-heuristic
An approach that uses machine learning is called hyper-heuristic. Hyper-heuristic is used to automate the selection process and the combination of existing heuristic components. Heuristic usage results are common frameworks that can be used to troubleshoot cross-domain problems [10]. Hyper-heuristics frameworks have a feature with selection of mechanisms and moving acceptance criteria. The first thing to do is to choose a heuristic to apply to a single candidate solution, then determine whether the result of that solution yields an acceptable soultions or not [11].
Hyper-heuristic (HH) is introduced as a method of general optimization, able to explore the heuristic space rather than directly to the solution space. HH support traditional algorithms of heuristic [12]. Such ideas can be developed from the excess of each algorithm or combine two or more algorithms, known as the metaheuristic hybrid. At metaheuristic standard and hybrid metaheuristic, the process focuses on a search space solution for a problem. The difference lies in the number of heuristic strategies used. At the standard metaheuristic, the approach used only one, but not with a metaheuristic hybrid. On the other hand, hyperheuristic focuses on the heuristic search space (strategy being general) [1]. The main component in the problem section of the hyper-heuristic domain is the objective function, formation of the solution's initials, and has a low-level heuristic set. Low-level heuristic is used by hyperheuristics to specific optimization problems. In this case, a hyperheuristic is able to choose low-level heuristic to be used at some point of decision until the termination conditions are met [13]. The methodology part has two phases: the selection of the low-level heuristic (heuristic selection) and the move acceptance. The HH framework can be seen in Figure 1.

Mathematical model
The problem of exam scheduling, especially in the Toronto dataset, aims to minimize the proximity cost P with the following calculation: In which, N=number of students. Cij=number of students taking courses i and j together.
| − | =weight of the penalty when i and j are scheduled at timeslot ti and tj.

Algorithm implementation
This process implements an algorithm that will be used at the University of Toronto benchmark datasets. In order to fulfill the objective function, three methods are used, namely the largest degree algorithm, hill climbing, and tabu search. The largest degree algorithm is used to make initial solutions; the results of these algorithms will be evaluated using move acceptance in finding solutions that are better than the initial solution. When the solution received is not feasible, then a penalty will be introduced from the next two algorithms, namely the hill-climbing algorithm and tabu search.
Hyper-heuristic usually has one LLH selection method, with no additional policies and combined with a single-step acceptance approach [14]. This study using two types of LLH, namely "swap" and "move." Swap-move is applied swap operator then followed by the operator of the move. Move-swap implements a displacement operator followed by a swap operator [15]. This case, swap randomly select the exam, then the timeslot from the exam is moved randomly to another timeslot. Meanwhile, the move is made by selecting an exam then moving the exam randomly from the previous timeslot to another timeslot.

Largest degree algorithm
As a general step, initialization refers to the initial step to generate the initial population. A large number of solutions need to be optimised, but the process that produces viable solutions is difficult for Domain-independent information acquisition and processing to selection mechanism of heuristic for apply the heuristic considerable scheduling problems, so some unworthy solutions become viable in some conventional ways such as the largest degree method of finding initial feasible solutions. The largest degree algorithm is able to use for exam with the largest courses of conflicts than other courses are entered first [16]. This method has the concept of sorting courses based on the number of relationships with other subjects. If the number of courses increases, the opportunity is not scheduled at the same time, and the subject with the highest number of degrees will be scheduled in advance. Pre-scheduled courses minimize or avoid the existence of a timeslot that contains several courses taken by the same student [17].

Hill climbing algorithm
The hill-climbing algorithm is a well-know algorithm for its simplicity. This algorithm relatively produces worse results than using the metaheuristic. It starts the search for the exploration process from the 'room' section to find highest point [18]. For each iteration from the solution is used to find a new candidate solution which is compared to the current solution and will be accepted if the cost function is not worse [19].
The hill-climbing method is a heuristic search that aims to solve the problem of finding the closest distance. This method works by determining the next step (node), which will show up as close as possible to the target [20]. For optimization, hill climbing requires shuffling, shifting and block swapping [21]. This algorithm uses a move acceptance-based approach finding solutions that are better than the initial solution. In hill-climbing, each iteration will be chosen randomly to be placed randomly on the ticket lot. When the solution received is not feasible, a penalty will be imposed [20]. The pseudocode of hill-climbing is shown as Figure 2.

Tabu search algorithm
Optimization method that explains local search is search tabu [22]. Tabu search is also an adaptive memory programming method to solve problems in the field of optimization [23]. The selection of the best solution is decided by looking for one solution to the next. The latest quality selection has no obligation to be better than the previous quality. if the latest solution has more benefits and uses than the previous solution, then it can be concluded that the solution is the best [24].
Tabu lists have fundamental algorithms that can prevent searches on previously searched solutions. tabu lists are used to record attributes of some previously applied gestures. it is also used to prevent the search process on the same side [25]. The pseudocode of tabu search is shown as Figure 3.

Testing algorithm implementation
This stage is testing each algorithm in each instance. The testing of these three algorithms was carried out using the Java programming language in NetBeans IDE 8.0, Windows 10 operating system, Intel® Core i7 processor, and 16.0 GB RAM.

Implementation using the largest degree
At this stage, the initial penalty value is implemented when using the largest degree algorithm. The results of the penalty value for using the largest degree algorithm can be seen in Table 2. Based on Table 2, each instance has different fitness and times. If observed based on time, the STA 83 dataset fulfills the constraint with the fastest time of 0.0025727 seconds. STA 83 can run quickly because the number of students and examinations is less compared to other datasets, so it does not require a relatively long running time. It can be seen that the CAR 92 instance is an instance with better performance than other instances. Although based on times, this instance is not the fastest, but this instance can meet hard constraints by not exceeding the timeslot.

Implementation using the hill climbing
Iterations in 13 instances. From the Table 3, it can be noted that the delta value which states the change in penalty increases from the initial solution. The largest increase in value in the HEC 92 dataset was 66,567%, while the smallest value in the STA 83 dataset was 16,915%. Unfortunately, although the HEC 92 instances has the greatest value, the timeslot of this dataset does not meet the hard constraint. The largest delta value that meets the hard constraint is the KFU 93. If observed based on time, the YOR 83 instance has the fastest running time than other instances with a time of 143,208 seconds. From the results of experiments using hill-climbing, states that this algorithm produces a penalty value that is more optimal than the previous penalty value using the largest degree. The hill-climbing algorithm has the concept of accepting a better solution and rejecting a worse solution. If the result of hill-climbing shows the best solution than the initial solution, then the initial solution is replaced by the best solution from hill-climbing.  ISSN: 2302-9285

Implementation using the tabu search
Based on Table 4, the dataset obtained different fitness and time with 10,000 iterations. Each determines the level of goodness of optimization. Based on fitness values, the HEC 92 dataset has the best fitness increase of 58,284%. A dataset with better performance that can be used is HEC 92; this is due to the rise in the value of the best fitness with fast computing time. The experiment was carried out using 10,000 iterations on 13 instances using the hill-climbing algorithm and the tabu search algorithm. From Table 5, it can be noted that the tabu search fitness value is better than the hill-climbing fitness value. The delta value (the percentage increase in penalty values from Hill Climbing to Tabu Search) shows that all datasets have a gain or no decrease. The PUR 93 obtains the highest increase in penalty values with a total of 23,939%, while the low rise is in the UTE 92 with 3,127%.

Comparing hill climbing algorithm and tabu search algorithm using box and whisker plot
Comparison of the tabu search and hill-climbing algorithms is displayed using a box and whisker plot. Each box has lines at one quartile in the bottom line of the box (Q1), the middle line as lower limit (median-Q1), and top-most line as the upper limit (Q3-median). There are also top whiskers (max-Q3) and bottom whiskers (median-Q1). The experiment runs 11 times in two iteration variations in each algorithm (10,000 iterations and 1 million iterations). The whisker is the above line (referred to as the "top whisker") and below (referred to as the "bottom whisker"). From Figure 5, it can be seen from the five data sets that both algorithms tend to have the best penalty value in the iteration 1 million compared to 10,000. It is also shown that the tabu search (TS) algorithm always has the best penalty value compared to hill-climbing (HC).

Perfomance comparison on hill climbing and tabu search
In Figure 6, hill-climbing and tabu search algorithms' performance is run from the iteration steps 10,000 to 1,000,000. From the given line chart, it shows that the 10,000 iterations of the tabu search algorithm are far superior and have lower penalty values than hill-climbing.

Comparison of results with other published results
The performance of the tabu search algorithm was carried out ten times, running at 1,000,000 iterations. The results of the experiment are the average of each instance. In Table 6, the best value of the trial results is displayed and compared with some previous studies on the problems of the Toronto dataset.
The results of this trial were compared with several previous studies, namely, K. Graham and M. Naimah [3] who used tabu search hyper-heuristic. Gaspero and Schaerf [4] using the tabu search algorithm, Carter et al. [26] use constructive heuristics with backtracking, Caramia et al. [5] use greedy constructive heuristics, Burke and Newall [6] use a local search method, I. Gabriella & P. Etria use tabu search algorithm with 100,000 iterations [7]. The purpose of this experimental results is to show that the tabu search algorithm is able to produce good penalty results even though the value produced is not the smallest. The results show that there is one instance that provides the best value, STA 83, compare to previous studies.

CONCLUSION
Based on the results of trials and analysis of the results that have been carried out, it can be concluded that the proposed method produces a fitness value that is much smaller than that of the constructive heuristic. The hill-climbing algorithm can reduce the number of timeslots in several datasets that were not initially feasible as a feasible solution. Furthermore, optimization using the tabu search algorithm results in better performance by giving a smaller penalty or fitness value to Toronto datasets than the hillclimbing algorithm. The difference is 18-58%. Compare to the previous studies; this study result is not the worst, which shows the potential of the proposed method if further investigated.

BIOGRAPHIES OF AUTHORS
Shinta Dewi currently pursuing her bachelor's degree at Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia. She is actively participated in organizational activities and held the position of secretary in the Information Media Department at the Information Systems Student Association (HMSI) for period 2019-2020. She is currently working on her thesis on the topic of optimization with the problem of optimizing the route using a hyperheuristic algorithm.
Raras Tyasnurita received her bachelor's degree in computing from Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia, and the MBA degree from National Taiwan University of  Science and Technology