Enhanced Genetic Algorithm for Solving the School Timetabling Problem

Transcription

1 Enhanced Genetic Algorithm for Solving the School Timetabling Problem Tan Lay Leng and I.A. Karimi Department of Chemical and Environment Engineering National University of Singapore 10 Kent Ridge Crescent Singapore ABSTRACT The time tabling problem is well known and much research has been done on it. This paper shows how a difficult instance of the time tabling problem with many parameters and loose constraints can be handled using constraint satisfaction. Genetic Algorithm (GA) is one of the most-commonly applied optimization methods on natural selection problems. The process of natural selection results in the appearance of various potentially acceptable solutions to some optimization problems. However, GA can be very slow in the generation of outcomes. The application of GA to the school timetabling problem using Microsoft Excel Visual Basic for Applications is explored in this paper. Various parameter constraints require the evolution of special methods for the dealing with the problem. These methods are focused on the reduction of the generation time and satisfaction of the constraints at the same time. The favorable outcomes introduce a new area of research, as GA running with the new methods appears to emulate manually designed timetables. INTRODUCTION The process of natural selection is shown in Genetic Algorithm (GA) and is often used as a method for solving complex optimization problems. One of the main differences between GA and many other methods, many individual solutions in the form of a population are maintained using GA. Parents are chosen from the population and are then mated to form a new child when necessary. The child is further mutated to introduce diversity into the population. In GA, a child is formed which carries some of the properties of their parents using mating. The properties of a child are further modified by mutation to introduce diversity in population. Mutation can bring about unique properties that cannot be found in either of the parents. If the combined properties of a child are more suited to the constrained environment given, then the chances that the child will survive is increased. The properties of the survived child will then be used for future generation of the population where the desirable properties will be maintained. However, if a child is form with undesirable properties, it is most likely that the child will be eliminated and not be used for future generation. Thus the less desirable properties are removed from future generations. The ultimate aim of GA will be to have the average 1

2 fitness of the population to increase with each generation under the constrained environment. In cost terms, it would mean to have the cost of the parameters that is being considered to decrease with each generation. In order to find the most desirable solution to the constrained environment, the principle of survival of the fittest is used. Only the fittest out of the population will remain and be presented as the solution to the constraint problem. TIMETABLING SOVLING PROBLEM A timetabling program is done to find an optimal solution to the timetabling problem. Microsoft Excel Visual Basic for Applications is used to create the program. In order to facilitate the debugging process, the entire program can be split and written in different modules and macros. Every module served a different purpose in executing the program. The optimal solution of a timetabling problem is whereby there will be no clashes in terms of teachers, classes and room during the scheduling. That means no teacher, class or room is used more than once per period 1,2,3. A tuple is a combination of a teacher, a subject, a room and a class. It is sometimes necessary to schedule the tuple more than once per week. In the program, a tuple is represented as a row in Microsoft Excel Worksheet. An optimal solution of timetabling problem would thus mean a proper scheduling of a number of tuples in a period such that there are no clashes. Tuples are formed based on who are the teachers teaching a particular subject and the requirements of the subjects in terms of number of lessons per week and size of the students taking the subject. In order for us to determine whether a particular combination of tuples in a period is suitable, we need to define an objective or cost parameter. This cost parameter serves as a qualitative measure for us to determine whether a particular combination is suitable. An appropriate cost parameter is used to calculate the number of clashes in any given timetable in terms of teachers, rooms and classes. A desirable timetable is reflected by a cost of zero for all periods. The period cost is calculated as the sum of the parameters cost. The parameters cost consists of the class cost, teacher cost and room cost. A parameter cost is counted as nonzero when a same value of parameter is appeared more than once in a period. The total cost incurred by the timetable is the sum of all the period cost. The ultimate aim of the optimization problem is to find the best combination whereby all the total cost of the timetable is zero or if not the lowest. 2

3 DUAL-MUTATION SYSTEM A Dual-Mutation System is introduced in order to minimize the problem of exhaustive reruns. This system is effective in that two types of mutations are being introduced concurrently. Diversity in the population is maintained, thus allowing better attributes to be preserved. The first mutation is used when the number of tuples in the database used for GA becomes less than it is needed for mating to occur. When this occurs, the cost parameters of the tuples are checked. If the cost is not zero, the first mutation is used to make sure that the cost is optimal ultimately. The tuple that results in the cost parameters of the remaining tuples to be nonzero is identified and removed from the database. This tuple is then placed to one of the successful processed periods. The particular period is to have a cost of zero. After the tuple is being placed, cost checking is carried out again. If the cost is zero, the mutation is considered successful. Otherwise, the tuple is removed from the period and placed to the next non-cost period. The process is acceptable when the all the periods in the database have optimal cost. The first mutation method is further illustrated in Figure 1. Figure 1: First Mutation Method The second mutation is used when any of the processed periods that were formed from GA have non-optimal cost. Any periods whose cost parameters are non-optimal are identified. The second mutation is used on these particular periods to ensure that all the periods will eventually have zero cost. 3

4 Similar to the first mutation, the tuple that is resulted in the cost parameters of period to be nonzero is identified and removed. This tuple is then mutated to one of the periods that has zero cost. Cost checking procedures are being carried out. If the cost is zero, the mutation is considered successful. Otherwise, the row is removed from the period and placed to the next successfully processed period. The process is continued until all the periods have zero cost. The second mutation method is displayed in Figure 2. Figure 2: Second Mutation Method The checking of optimal parameter cost is made easy using Microsoft Excel Visual Basic for Application. Each tuple is represented by a row in Microsoft Excel and the checking of cost in a period means checking the parameter values among the rows that are available in each period. Since a tuple can be easily represented as a row in Microsoft Excel, this makes Microsoft Excel Visual Basic for Application more suitable as the software to implement the program. ASSIGNMENTS OF LABORATORY PERIODS The concept being used here is seen as a staggered-timing method such that, any student not having a tutorial class at any one time is allowed to attend a laboratory lesson. His counterparts in other clusters may be attending a tutorial class when he himself is in a laboratory class. Tutorial classes of each module are grouped into one or more tutorial clusters depending on the number of classes there are for each module. The maximum number of classes per tutorial clusters is dependent on the situation but should be limited to 5. Laboratories lessons for each year are also grouped into one or more laboratories clusters depending on the number of lab classes per year. The maximum number of laboratories classes per laboratory cluster is dependent and should be limited to 3. The assignment of laboratory 4

5 clusters to periods is related to the periods which the tutorial clusters are being assigned to. If a tutorial cluster of a particular year is being assigned to a particular period, the assignment of laboratory cluster of that same year to that period is to be such that students in that particular laboratory cluster do not belong to the tutorial cluster that is being assigned. If the lab cluster were to belong to a different year as the tutorial cluster that is being assigned to the period, the lab cluster assigned to that period can be from any of the lab clusters. CONCLUSION Genetic Algorithm (GA) is applied to a number of optimization problems with much success. The major disadvantage being that the execution time is slow. The software used to implement the timetabling program in this paper is Microsoft Excel Visual Basic for Application. Although it may give optimal results to the timetabling problem but it may not be sufficient for users who prefer very good user interface. The methods discussed in this paper focused on the reduction of the generation time and satisfaction of the constraints at the same time using Microsoft Excel Visual Basic for Application. Using these methods, significant amount of execution time of running GA can be reduced and this makes GA a more attractive method in solving timetabling problems. REFERENCES 1. D. Abramson & J. Abela, A Parallel Genetic Algorithm for Solving the School Timetabling Problem, Technical report, Division of Information Technology, C.S.I.R.O., (April 1991). 2. Gary Lewandowski, Simultaneous Construction of Student Schedules and Timetable, (1996). 3. Colorni A., M. Dorigo & V. Maniezzo, A Genetic Algorithm to Solve the Timetable Problem, Technical Report No , Politecnico di Milano, Italy, (1990). 5