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Algorithms 2013 , 6 , 227-244; doi:10.3390/a
algorithms
ISSN 1999-
www.mdpi.com/journal/algorithms Article
Solving University Course Timetabling Problems Using
Constriction Particle Swarm Optimization with Local Search
Ruey-Maw Chen * and Hsiao-Fang Shih
Department of Computer Science and Information Engineering, National Chinyi University of Technology, Taichung, Taiwan; E-Mail: [email protected]
***** Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +886-4-2392-4505; Fax: +886-4-2391-7426.
Received: 19 February 2013; in revised form: 25 March 2013 / Accepted: 8 April 2013 / Published: 19 April 2013
Abstract: Course timetabling is a combinatorial optimization problem and has been confirmed to be an NP-complete problem. Course timetabling problems are different for different universities. The studied university course timetabling problem involves hard constraints such as classroom, class curriculum, and other variables. Concurrently, some soft constraints need also to be considered, including teacher’s preferred time, favorite class time etc. These preferences correspond to satisfaction values obtained via questionnaires. Particle swarm optimization (PSO) is a promising scheme for solving NP-complete problems due to its fast convergence, fewer parameter settings and ability to fit dynamic environmental characteristics. Therefore, PSO was applied towards solving course timetabling problems in this work. To reduce the computational complexity, a timeslot was designated in a particle’s encoding as the scheduling unit. Two types of PSO, the inertia weight version and constriction version, were evaluated. Moreover, an interchange heuristic was utilized to explore the neighboring solution space to improve solution quality. Additionally, schedule conflicts are handled after a solution has been generated. Experimental results demonstrate that the proposed scheme of constriction PSO with interchange heuristic is able to generate satisfactory course timetables that meet the requirements of teachers and classes according to the various applied constraints.
OPEN ACCESS
Keywords: course timetabling; soft constraint; particle swarm optimization; constriction factor; interchange
1. Introduction
Course timetabling is a scheduling problem. However, it is far more complicated than general scheduling as it involves teachers, students, classrooms, and courses. From a broader perspective, course arrangement includes many interrelated issues, such as exams, meetings, administrative allocation, etc. [1]. The study by Even and Itai [2] proved that the course timetabling problem is an NP-complete problem. Conventionally, course timetabling has been conducted manually. Due to the large variety of constraints, resource limitations and complicated human factors involved, course timetabling often takes a lot of time and manpower. Using computers to perform course timetabling, however, can not only consolidate the preferences of the people concerned but can also enable achievement of high satisfaction in spite of the many constraints. Obviously, this results in saving a lot of time and thus manpower. Ever since the 1960s, scholars have been studying course timetabling problems. Hertz [3] proposed using tabu search as the study method to deal with course timetabling problems in two stages (TATI/TAG) and emphasized that the approach is suitable for handling problems in large-scale course timetabling and test scheduling. Mooney et al. [1] came up with a nonlinear integer programming model, setting preference values according to the status of each teacher, to establish the maximum classroom utilization rate and teachers’ priority in course timetabling. Masood [4] employed a two-stage multi-objective scheduling model for allocation of teachers, courses, and time. Werra [5] suggested a restricted coloring theory and used mathematical concepts to apply the theory in course timetabling. Ross et al. discussed the state of the art in applying evolutionary algorithms to tackle timetabling problems of various kinds [6]. Cambazard et al. [7] used local search and constraint programming techniques for solving post enrolment-based course timetabling problems. An initial solution generated by Largrangian relaxation and improved by simulated annealing for solving master course timetable was proposed by Gunawan et al. [8]. Genetic algorithm (GA) was applied to deal with university timetabling derived from various limitations and constraints [9]. Ant colony optimization (ACO) algorithm was used to solve the post enrollment course timetabling problem by Nothegger et al. [10]. Meanwhile, Shiau [11] solved course scheduling problems by applying hybrid particle optimization. Tassopoulos and Beligiannis [12] applied hybrid particle swarm optimization to create feasible different Greek high school timetables. A memetic computing scheme named hybrid harmony search (HHS) algorithm for solving a university course timetabling problem was proposed by Al-Betar et al. [13]. The HHS integrates particle swarm optimization and hill climbing to balance exploitation and exploration search. Moreover, a hybrid genetic algorithm and tabu search approach was suggested for solving post enrolment course timetabling problems by Jat and Yang [14]. Considering the great impact of course timetabling results on teaching quality and learning interest, this study allows teachers and classes to express their expected course timeslots and preferences to help produce course timetabling results that can better meet the expectations of all involved as well as
(2) Course timetabling should be conducted according to the timeslot preferences of teachers and students as well as the levels of preference. (3) Each teacher may not teach more hours than the limit stipulated by the undergraduate department or graduate institute. (4) Each full-time teacher must teach classes for at least three days a week.
3. Particle Swarm Optimization Algorithm
Particle Swarm Optimization (PSO) was developed by Kennedy and Eberhart [15] in 1995. In PSO, a bird of a flock is represented as a particle, and the swarm is composed of a group of particles. The position of each particle can be regarded as the Candidate Solution to an optimization problem. Every particle is given a Fitness Function designed in correspondence with the corresponding problem. When each particle moves to a new position in the search space, it will remember its personal best ( P best). In addition to remembering its own information, each particle will also exchange information with the other particles and remember the global best ( G best). Then, each particle will revise its velocity and direction in accordance with its P best and the G best to move toward the optimal value and find the optimal solution. With the advantages of simple and easy application, less parameter setting required, and decent performance, PSO has been adopted in many fields, such as TSP [16], flowshop [17], VRP [18], task-resource assignment in grid [19], special scheduling [20,21], etc. Hence, it has also been applied in this study to establish the optimal timetabling for university courses. To begin a PSO algorithm, the initial velocity and position of each particle in a group of particles are randomly determined. Then, the evolving process is as follows:
(1) The initial position and velocity of each particle in the N th dimension are determined randomly. (2) The fitness value of each particle is assessed according to the defined objective function. (3) If the fitness value of each particle’s current location is better than its P best, the P best is set to the current position. (4) The fitness value of the particle is compared with that of the G best. If it is better, the G best is updated. (5) Equation (1) as shown below is applied to update the velocity and position of each particle. (6) The process is repeated from Step 2 until the termination criterion is met or the optimal solution in the universe is obtained. (^11 ) 1 1
idt^ idt^1 (^ id id )^2 (^ gd id ) idt^ tid^ idt
V V c Rand P X c Rand P X X X V
In the equation, Vid is the velocity component of the i th particle in the d th dimension. Xid is the position component of the i th particle in the d th dimension. c 1 is the cognitive learning factor. c 2 is the social learning factor. Pid is the position component of the P best of the i th particle in the d th dimension. Pgd is the position component of the G best in the d th dimension. Rand () is a random number between [0, 1].
3.1. The Inertia Weights
Since PSO was founded, many derivative algorithms have been developed. In Equation (1), the particle advancement guided by cognitive and social learning factors belongs to the local search ability, whereas Term 1 on the right hand side of Equation (1) is the advancement along the velocity direction of the particle itself and belongs to the global search ability. Therefore, Shi and Eberhart (1998) proposed the inertial weight value concept [22] and added an inertial weight value ( w ) to the original PSO algorithm, Equation (1), to balance the global search ability and the local search ability, as shown in Equation (2) and thereby boost the capability to locate the optimal solution and convergence rate. (^11 ) 1 1
idt^ idt^1 (^ id id )^2 (^ gd id ) idt^ idt^ idt
V V c Rand P X c Rand P X X X V
This version of PSO is the most commonly used and is referred to as the conventional PSO in this study.
3.2. Constriction Factors
Clerc [23] (1999) proposed the constriction factor method; a constriction factor is added into the PSO prototype to ensure stable convergence. Eberhart and Shi [24] conducted a comparative analysis on inertia weights and constriction factors and discovered that the best convergence effect can be achieved by defining V max = X max (the maximum velocity is confined to the maximum position range) when a constriction factor is applied. Bratton and Kennedy later defined this as the Standard PSO (SPSO) [25]. The particle velocity update equation modified according to the constriction factor based PSO is as shown below: (^11 )
2 1 2
[ 1 ( ) 2 ( )] (^2) , , 4 2 4
Vid t^ K Vid t c Rand Pid X (^) id c Rand Pgd Xid
K c c
(^)
In the equation, K is the constriction factor and constant φ is the sum of learning factors c 1 and c 2. Kennedy and Clerc [26] discovered in their study that when constant φ is smaller than four, convergence of the entire swarm cannot be guaranteed. On the other hand, when it is larger than four, not only do the particles move quickly and achieve convergence, but the convergence can also take place at an appropriate velocity. Normally, the settings are K = 0.72984, and c 1 = c 2 = 2.05. In this study, experiments were conducted with the conventional PSO and the SPSO and the results comparatively analyzed.
3.3. Particle Encoding
The four main factors in course timetabling are teachers, courses, students (classes), and classrooms together with other teaching facilities. The combination of these four factors is defined as the particle position and each particle represents a solution group. The objective is to obtain the optimal particle position, (the optimal course timetabling solution). Due to teachers’ preference, the 3 h of 3-credit
Figure 2. Particle encoding scheme.
If there are 10 teachers, the particle position vector then has the value of 20 × 10 = 200 elements. The elements in the particle are between 1 and 200. There are 5 days a week and 4 timeslots per day. Elements 1–20 stand for the class schedule of the first teacher, 21–40 for the second teacher, 41 to 60 for the third teacher, and so on as shown in Figure 2. First, each particle is initialized and the range of the initial position is set between 0 and 9 as shown in Figure 3. The range of the initial velocity is set between −0.5 and 0.5. The position vector Xh = [ Xhi ], i = 1–200 is established to represent the particle position; h is the particle and i is the element. Next, the element value of Xhi is rounded to the nearest integer [0, 9] and temporarily stored in Shi. Then, sequencing is conducted according to the initial value of Shi as shown in Figure 4 to extract numbers of maximum values of the Shi , the quantity to be extracted depending on the number of hours each teacher teaches. Suppose the first and the second teachers teach three courses each and three maximum values are then extracted as shown in Figure 5. The timeslot arrangement for these two teachers is made according to the position numbers of these three maximum Shi values to find out whether there is any conflict between the teacher, class, and classroom schedules. If there is no conflict, the course code is put in the teaching schedule of the first teacher, the class schedule, and the schedule of the classroom. If there is any conflict, the time slot arrangement is then exchanged with the timeslot in the fourth arrangement as shown in Figure 6. If any conflict appears again, the exchange is made with the timeslots in the fifth arrangement, and so on. The teacher’s preference values corresponding to the teacher schedule and the class’s preference values corresponding to the class schedule are then added up to calculate the fitness value. Next, the fitness value of each particle is assessed, the personal best ( P best) and global best ( G best) are updated, and finally the velocity and position of each particle are updated.
Figure 3. Particle initialization.
Figure 4. Shi after sorting.
Table 1. Preference timeslot example. Timeslot Mon Tue Wed Thu Fir 1 1 5 3 5 4 2 1 4 2 5 3 Lunch Break 3 2 4 2 − 10 2 4 2 4 − 10 − 10 2
The principal objective of this study was to find the optimal satisfaction of teachers and classes with the results of course timetabling. Since soft constraints can increase such satisfaction, the fitness function (satisfaction value) is defined as the result of subtracting the soft constraint penalty function from the total value of teacher and class satisfactions of the course timetable. Hence, the definition of the fitness function in this work is as shown in Equation (4).
fitness (^) t (^) preference _ levelt (^) (^) c preference _ levelc iPenaltyi (4)
In the equation, preference_levelt represents t teacher’s preference level, and preference_levelc the c class’s preference level. Penaltyi is the penalty function for violation of the i th soft constraint. The fitness function is applied in the iteration to measure the fitness value. The larger it is, the better the solution the particle represents and the fewer are the soft constraints violated.
3.5. PSO with Local Search
As the search by PSO in the solution space evolves, movement is made according to the personal best of each particle and the global best. Therefore, after generation evolution, the search area decreases and there is a high probability that each particle will be trapped on the local optimal solution and become unable to escape from the local optima. Although the particles have good searching capability in local areas, they are unable to find better solutions outside these areas and thus lose the opportunity to locate the optimal solution in the universe. Hence, in this study, local search mechanisms are consolidated and a disturb mechanism [27] is added in the movement of the particles to increase the probability of particles escaping from the local optimal solution and to find the optimal solution in the universe quickly. Local search is performed after the movement and position change of particles in PSO. The areas around the position of each particle obtained using PSO are examined in the hope of finding a better solution to update the p best and the g best. The interchange heuristic, which swaps two randomly selected elements, is applied to the local search to explore the adjacent solution space, thereby improving the quality of the solution by helping to avoid convergence to the locally optimal solution. Such complementation boosts the performance of PSO through optimization of the solution search.
4. Experimental Results and Discussion
This study was conducted by analyzing a situation involving 16 teachers, 10 classes, and 10 classrooms. The goal was to produce the most satisfactory class schedule to meet the various constraints as well as the expectations of the teachers and the classes. The influence of the different
parameters in the algorithm on the level of satisfaction with the course timetabling results was also examined and the experimental results proved the adopted framework was effective. The course timetabling process was further analyzed as follows: (1) Prioritization of teachers The prioritization of teachers in course timetabling will have a significant effect on the results of course timetabling, therefore, no specific teacher’s timetable should be assigned first and they are thus assigned randomly. (2) Course data The data regarding the courses a teacher will teach needs to be established before the semester begins. Such data includes the teacher code, course code, class code of the class to take the course in question, teacher’s status, number of class hours, classroom, etc. (3) Classroom data Classrooms involve certain particular factors. A teacher is assigned to maintain and manage each one of the classrooms of the Computer Science and Information Engineering Department where this study was performed and all the classes to be taught by a teacher are normally arranged to be conducted in the classroom that the teacher is assigned to maintain and manage. Restated, if classroom A is placed under the management of teacher A, the courses that teacher A teaches will be preferably conducted in classroom A. Since the number of teachers in the department is larger than the number of classrooms, some classrooms are placed under the management of two or more teachers. (4) Handling of teacher, class and classroom schedules conflicts A class time conflict may happen when two teacher timeslots, classroom timeslots and course timeslots overlap and it becomes an infeasible solution. To avoid an endless search for solutions to class time conflicts in all timeslots, such conflicts are singled out from timeslots as being infeasible solutions and assessed whether they should be discarded according to the satisfaction level.
4.1. Particle Quantity Analysis
Figure 7 shows the results of analysis of the evolution of 5, 10, 15, 20, 25, 30 and 35 particles to observe the influence of different numbers of particles on the overall fitness value of course timetabling. The response of standard PSO (denoted by SPSO) and conventional PSO (denoted by PSO) to the quantity of particles is tested. Apparently, larger numbers of particles do not result in better fitness values. When the evolution of each quantity of particles is conducted for 6000 generations, the best fitness value comes from when the number of particles is 30.
Figure 9. PSO comparison between c 1 = c 2 = 2 and c 1 = c 2 = 2.05 ( w = 0.8).
Figure 10. SPSO comparison between c 1 = c 2 = 2 and c 1 = c 2 = 2.05 ( χ = 0.72984).
4.4. Performance Comparison between PSO and SPSO with/without Local Search
In this Section, PSO, SPSO, PSOLS (PSO with local search) and SPSOLS (SPSO with local search), associated with corresponding learning factors and V max are applied to analyze the differences in search performance. The results are as shown in Tables 2–5. They indicate that after local search is added, regardless of the differences of learning factors and other parameters, the outcomes are significantly better than those obtained by using PSO or SPSO alone. For PSOLS, when c 1 = 2, c 2 = 2, w = 0.8 and V max = 4, the average satisfaction level (406.8) is higher than when c 1 = 2, c 2 = 2, w = 0. and V max = 3. Additionally, for SPSOLS, c 1 = 2, c 2 = 2, χ = 0.72984 and V max = 3 yields higher average satisfaction level (411.6) than applying c 1 = 2, c 2 = 2, χ = 0.72984 and V max = 4. Moreover, the performance by utilizing SPSOLS is better than that of applying PSOLS. This confirms that SPSOLS can improve the quality of the solutions as proposed in this study.
Table 2. Performance comparison among PSO and PSOLS (PSO with local search) ( c 1 = 2.05, c 2 = 2.05, w = 0.8).
Iteration (^) V PSO= −3~3 V^ PSOLS = −3~3 V^ PSO= −4~4 V^ PSOLS = −4~ 2000 384 398 368 402 3000 370 402 385 414 4000 380 402 400 396 5000 396 416 396 410 6000 375 404 380 402 Average 381 404.4 385.8 404.
Table 3. Performance comparison among PSO and PSOLS ( c 1 = 2, c 2 = 2, w = 0.8).
Iteration (^) V PSO= −3~3 V^ PSOLS = −3~3 V^ PSO= −4~4 V^ PSOLS = −4~ 2000 362 386 386 400 3000 370 403 382 412 4000 396 404 390 412 5000 390 404 374 390 6000 368 402 361 420 Average 377.2 399.8 378.6 406.
Table 4. Performance comparison among SPSO and SPSOLS (SPSO with local search) ( c 1 = 2.05, c 2 = 2.05, w = 0.72984).
Iteration (^) V SPSO = −3~3 V^ SPSOLS = −3~3 V^ SPSO = −4~4 V^ SPSOLS = −4~ 2000 381 398 376 378 3000 371 384 392 410 4000 396 418 394 410 5000 380 384 374 419 6000 392 415 392 418 Average 384 399.8 385.6 407
Table 5. Performance comparison among SPSO and SPSOLS ( c 1 = 2, c 2 = 2, χ = 0.72984).
Iteration (^) V SPSO = −3~3^ SPSOLS V = −3~3 V^ SPSO = −4~4^ SPSOLS V = −4~ 2000 400 412 375 401 3000 408 416 394 412 4000 402 416 410 424 5000 398 390 380 412 6000 406 424 398 408 Average 402.8 411.6 391.4 411.
Additionally, premature convergence was prevented by adding an interchange local search mechanism. The following conclusions have been established:
(1) With local search added in PSO and SPSO, the satisfaction (fitness value) of the teachers and classes is better than the fitness value obtained without adding local search. This confirms that the quality of solutions can be improved by applying the proposed scheme in this paper. (2) The results from application of different parameters show that use of SPSOLS obtains the best solution (the best fitness of 424 and maximum average fitness of 411.6). Restated, SPSOLS outperforms PSOLS. The above conclusions indicate that application of the concept of combining SPSO with local search proposed in this study in finding solutions to optimization problems in course timetabling is sound and results in better solutions.
Acknowledgments
This work was partly supported by the National Science Council, Taiwan, under contract NSC 101-2221-E-167-012.
Conflict of Interest
The authors declare no conflict of interest.
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