A new software for the university class schedules based on evolutionary algorithms and cognitive rhythms

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Castrillon (2014)

http://edure.org/EdureJournal.htm EduRe Journal Vol. 1 Nº 1 (2014): 64-76 | 64

A new software for the university class schedules based

on evolutionary algorithms and cognitive rhythms

Castrillon, Omar D.1*

1_{Universidad Nacional de Colombia, Ingeniería Industrial}

*Corresponding author: Colombia-Manizales-Campus la Nubia, [email protected], +57-3147255921

Received: 2013-12-15; Accepted: 2014-03-18 Abstract

In this article, it is described a software for the university timetables scheduling based on artificial intelligence techniques and student´s cognitive rhythms. This software consisted in four steps. First of all, there were described the artificial intelligence techniques used to solve this problem. Then, the methodology used to develop this software was explained. Later on, a hypothetical problem was solved using this software. Third, the main procedures employed were described. Finally, there was concluded that timetables scheduling techniques are more efficient than others methods (20%) in solving these problems. In this case, the software described, allowed the comparison between three kinds of methods (Random, genetic and sequential). The random algorithm and genetic algorithm generated equal results. However, the results obtained with genetic algorithm were more consistent that the results obtained with random algorithm. Future research lines will include other variables in the fitness function: capacity, costs, computer time and will also include an adaptable combination of different methods to develop a hyper-heuristics methodology.

Keywords

Cognitive Rhythms, Artificial Intelligence, Genetics Algorithms, Fitness function.

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1. Introduction

Many programs have been developed for University Timetables Scheduling. The following lines describe a quick tour for this area and show their main aspects:

Initially, mathematical solutions were proposed by Gotlieb (1962) to solve this problem, later authors like Tripathy (1992), Schönberger et al (2004), Petrovic et al (2007), Pongcharoen et al (2008); Qu et al (2009), Van den Broek et al (2009), Zhang et al (2010), Turabieh & Abdullah (2011), Suarez et al (2013), Sørensen & Dahms (2014) proposed different algorithms to solve this problem.

In general, the literature has shown different solutions for the university timetables scheduling. In Cacchiani etal (2013) is described a new solution through the computation of improved lower boundaries. In the same way, Qauroonia and Akbarzadeh (2013) proposed and studied a specific genetic algorithm. Furthermore, Rahman et al (2014) showed an adaptative linear combination of graph coloring to solve this problem, and Sørensen and Dahms (2014) presented a model based on Integer Programming.

There are many algorithms based on artificial intelligence techniques: Heuristics, taboo search, integer programming, mimic algorithms, intelligent particles, neural networks, knowledge bases, logic programming, randomized algorithms, graph coloring, DNA models, etc. Nevertheless, the students’ cognitive rhythms have not been considered by these methodologies. The best time to teach some subjects was established by these rhythms (Martinez et al 2004).

Previous articles (with the own co-authoring) considered the students’ cognitive rhythms and solved this problem (Suareza et al, 2013; Suarezb et al, 2013; Castrillon 2014). However, the problem was solved without the specific restrictions of high school.

The student’s cognitive rhythms indicate the appropriated time to teach specific subjects. Subjects with a high level of difficulty should be taught in the middle of the school day. Those with moderate difficulty should be taught at the beginning of the school day. In the same way, the easier subjects should be taught at the end of the school day (Martinez et al, 2004). The consideration of these rhythms should improve the quality of education, an important variable in the higher education (Hernández, 2011).

The software described in this job expose that the timetables are scheduled with the subjects in the best time to teach (when it is possible). Two methodologies were employed: the first one was based on genetic algorithms and the second one used random algorithms.

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http://edure.org/EdureJournal.htm EduRe Journal Vol. 1 Nº 1 (2014): 64-76 | 66 This software considered new constraints that must be included in this problem to consider students’ cognitive rhythms. There was concluded that this software is more efficient (20%) than others techniques in the university’s timetables scheduling. Future lines of investigation will determinate the impact of these techniques in the students’ learning process. Finally, other applications can be founded in (Castiñeiras, and Sáenz-Pérez, 2013; Guyon et al 2010; Dahl 2009; Zhou & Zhong 2007; Liebchen, and Peeters 2009; Barrera 2012; Heydar 2013; Castillo et al 2011; Cacchiani 2012 ).

2. Methods

In this methodology, three different algorithms were employed: genetic, random and sequence algorithm. They were based on artificial intelligence techniques. This problem considered some hard constraints: Professors, rooms, the quantity of hours by day, the subjects of the same semester could not be crossed, there were some unavailable time period for teachers and rooms. Similarly, there were other soft constraints: The subjects should be programmed on an ideal time; the timetable should be the most compact possible. The methodologies were structured with a parallel vector that establishes the order to schedule the subjects. This sequence was going to define the value of the fitness function. In the first algorithm, the vector was always generated randomly. In the second algorithm, the vector was generated with three genetic operators: mutation (3%), combination (97%) and generation. If the algorithm could not find a solution, a new father was generated with the generation operator, named "new father". In the sequence algorithm the vector was defined with the initial order. The timetables were scheduled with three main variables: professors, classrooms and timetables. These variable were a three dimensional structure whose axes are: days, hours and professors or classrooms or semesters.

The order of subjects, defined in the initial vector, were scheduled by the function named programming. Each subject was organized in the different variables named professors, classrooms and timetables according its ideal time. When it was not possible to schedule it during the ideal time, a penalty was defined as the difference between the ideal hour (IT) and the programed hour (PT). When it was impossible scheduling the subject (in anytime) the penalty was defined as infinite. The fitness function was calculated with these penalties. Programming. The figure 1 shows the interface employed for the algorithms programming. Three different algorithms were linked into the following interface: Random algorithm, genetic algorithm and sequential algorithm.

Experimentation. In the table 1 was defined the feeding structure of the above software. This structure was made in Excel. One specific problem was solved in this article using the three algorithms linked in the software. In this problem, there were defined 13 subjects.

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http://edure.org/EdureJournal.htm EduRe Journal Vol. 1 Nº 1 (2014): 64-76 | 67 They were scheduled then, according to the best time to teach. This time was defined in the table 1, column 10. The problem defined in the table 1 considered two room types. The subjects in the table 1 were scheduled in the variables named: professors, classrooms and timetables with a specific order, according to the methodology described. This order defined the value of the fitness function. The results of this problem are shown in the following section.

Figure 1. Interface. Own source. Programmed in Matlab, Mathworks 2014. Table 1. Problem Definition. Possible Solutions 13! = 6.227.020.800.

Sem Subject Name subject Hrs Gr

Cod Prof Prof Max/ Hours Room Type Ideal Hour Prof 2 Name 1 1 C. DIFERENCIAL 4 1 1 Profeso 1 2 1 9 0 1 2 L. PROGRAMACIÓ 4 1 2 Profeso 2 2 2 9 0 1 2 L. PROGRAMACIÓ 4 2 2 Profeso 2 2 2 9 0 1 3 INTRO A LA II 4 1 3 Profeso 3 2 1 7 2 Profesor 2

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Sem Subject Name subject Hrs Gr

Cod Prof Prof Max/ Hours Room Type Ideal Hour Prof 2 Name

1 3 INTRO A LA II 4 2 3 Profeso 3 2 1 7 2 Profesor 2 1 4 E. COMUNICACIÓN 6 1 4 Profeso 4 3 1 11 0 1 4 E. COMUNICACIÓN 6 2 4 Profeso 4 3 1 11 0 1 5 SOCIOLOGIA 4 1 5 Profeso 5 2 1 16 0 2 6 CALCULO INTEGRAL 4 1 6 profeso 6 2 1 9 0 2 7 ALGEBRA LINEAL 4 1 7 Profeso 7 2 1 9 0 2 8 FISICA MECANICA 4 1 8 Profeso 8 2 1 9 0 2 9 OP. S. HUMA 4 1 9 Profeso 8 2 1 16 0 1 1 C. DIFERENCIAL 4 1 1 Profeso 1 2 1 9 0

3. Result

The tables 2 and 3 show the problem solution described in the table 1. The best fitness value was 8 and the worst 19.

Table 2. Best solutions. I semester

Hour Monday Tuesday Wednesday Thursday Friday

7 I. A LA II-1-AUL:1 I. A LA II-2-AUL:1 I. A LA II-2-AUL:1 I. A LA II-1-AUL:1

8 I. A LA II-1-AUL:1 I. A LA II-2-AUL:1 I. A LA II-2-AUL:1 I. A LA II-1-AUL:1

9 L. PROG-1-AUL:2 L. PROGR-1-AUL:2 C. DIFER-1-AUL:1 L. PROGR-2-AUL:2 L. PROGR-2-AUL:2

10 L. PROGR-1-AUL:2 L. PROGRA-1-AUL:2 C. DIFER-1-AUL:1 L. PROGR -2-AUL:2 L. PROGR-2-AUL:2

11 E. COMU-2-AUL:1 E. COMU-1-AUL:1 C. DIFER-1-AUL:1 E. COMU-2-AUL:1

12 E. COMU-2-AUL:1 E. COMU-1-AUL:1 C. DIFERE-1-AUL:1 E. COMU-2-AUL:1

13 E. COMU-2-AUL:1 E. COMUN-1-AUL:1 E. COMU-1-AUL:1 E. COMU-2-AUL:1

14 E. COMUN-1-AUL:1

15 E. COMUN-1-AUL:1

20 SOCIOL-1-AUL:1 SOCIOL-1-AUL:1

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Table 3. Best solutions. II semester

7 ALGE LIN-1-AUL:1

8 ALG LIN-1-AUL:1

9 CAL INT-1-AUL:1 FIS MECA-1-AUL:1 ALGE LIN-1-AUL:1 FIS MECA-1-AUL:1

10 CAL INT-1-AUL:1 FIS MEC-1-AUL:1 ALGE LIN-1-AUL:1 FIS MEC-1-AUL:1

11 CAL INT-1-AUL:1

12 CALC INT-1-AUL:1

20 OP. HUM-1-AUL:1 OP. S. HUM-1-AUL:1

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Software:

Figure 2. Software Model.

The main procedures are showed in the following graph (Figure 2). This program was fed by table 1 (excel table), this table was the main entry for the procedures named: random, genetic and sequential.

Programming: The subject was scheduled in three multidimensional structures: professors, semesters and rooms. The procedure had to find a place to program the subjects.

Random: This procedure received the table 1, which was described earlier in this job, with the sequence established in the initial vector of the algorithm that used the procedure named "programming" to sequence all the subjects into the multidimensional structures (figures 2, 3 and 4). The father and the sons were generated randomly by the procedure named get_random. The best result founded was 8.

Genetic: Similar to the last procedure, it received the table 1 with the sequence established in the initial vector, where the algorithm used the procedure named "programming" to

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http://edure.org/EdureJournal.htm EduRe Journal Vol. 1 Nº 1 (2014): 64-76 | 71 sequence all subjects in the multidimensional structures. The father and the sons were generated by two genetic operators: mutation and combination. The first genetic operator had a probability of 3%; two random positions in the father were changed by this operator. The second operator had a probability of 97%; two different parts of two fathers were combined by this operator to produce a new son. The best result founded was 8.

Sequential: Only one solution was generated by this procedure. The solution was defined according to the initial sequence in the table 1, which constituted the initial vector. Sometimes, it was not possible to find one solution with the initial order and the solution is unfeasible. The result founded was 10.

The results were selected and exported by the procedures named: best results, worst results and export results. The first two procedures chose the best and worst solution respectively, the third procedure exported the best and the worst solution to XLS format. Finally, the file was organized by a macro named format. The procedures showed in the figure 2 were organized on user interface (UI), this interface is showed in the figure 3.

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http://edure.org/EdureJournal.htm EduRe Journal Vol. 1 Nº 1 (2014): 64-76 | 72 Figure 3. Software Interface, Own source. Programmed in Matlab (Matworks 2014) 4. Discussion

The literature review shows many articles regarding the application of artificial intelligence techniques in different kinds of problems (Caicedo, 2010; Gamarra and Quintero, 2013; Martinez and Castiblanco, 2009; Rodriguez y Rivera, 2008; Wilson et al, 2011). However, the application of these techniques in solving this specific issue is low.

The software that was explained in this paper solved this problem with three different methods. Two based on artificial intelligence and one based on traditional techniques. The first two showed similar results in the fitness function. However, the results of the third were bigger (20%) than the first two, in the same function.

Many solutions were generated with the random algorithm, but without specific methodology (random generation). Some of them, were very good, other solutions were bad. For this reason, this algorithm is not always reliable. On the other hand, the solutions generated by the genetic algorithms were most consistent.

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http://edure.org/EdureJournal.htm EduRe Journal Vol. 1 Nº 1 (2014): 64-76 | 73 Today, many techniques have been developed to solve this problem (see bibliography). However, in the literature does not exist one solution for university’s timetables scheduling that consider students' cognitive rhythms, the application of them in solving this specific issue is low. In this job three evolutionary methodologies were made to find the best solution.

The methodologies based on traditional techniques showed that is not always possible to find a feasible solution that satisfies all hard constraints, because these methodologies do not change the initial father. An unfeasible solution is produced by a specific order in the father.

All hard constraints must be satisfied; however when it is not possible the solution generation is interrupted and a new father has to be constructed to elaborate a new solution. Only, the feasible solutions are penalized. This aspect improves the efficient of the algorithm.

5. Conclusions

The algorithms proposed in this article were 20% more efficient than the traditional techniques and the methodology used for them were easy to replicate. An appropriated coding of the algorithms described in the figure 1 and 2 will generate the same results if the genetic operators are used in the form described. The results obtained through this methodology were highly consistent. This is shown in the analysis of variance.

The results produced by the methodologies based on artificial intelligence were similar. However, the genetic algorithm was more reliable and stable than the random algorithm. Future research lines will include others variables in the fitness function: capacity and cost.

Futures lines

An adaptive heuristic will be developed as future investigation line. This meta-heuristics will combine several methodologies to find a good solution. The new solution will be compared with the artificial intelligence techniques employed in this job

.

Acknowledgment

This work was done as a research byproduct carried out during the author's sabbatical year and was funded by the “Universidad Nacional de Colombia”, industrial engineering department - headquarter Manizales (Caldas - Colombia).

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