5th International Vilnius Conference
EURO Mini Conference
“Knowledge-Based Technologies and OR Methodologies
for Strategic Decisions of Sustainable Development”
(KORSD-2009)
September 30–October 3, 2009, Vilnius, LITHUANIA
ISBN 978-9955-28-482-6
M. Grasserbauer, L. Sakalauskas,
E. K. Zavadskas (Eds.): KORSD-2009
Selected papers. Vilnius, 2009, pp. 439–444
© Institute of Mathematics and Informatics, 2009
© Vilnius Gediminas Technical University, 2009
Application of Optimization Methods in
Profiled School Scheduling
Lina Pupeikiene1, Eugenijus Kurilovas2
Vilnius Gediminas Technical University,
Sauletekio al. 11, 10223 Vilnius, Lithuania
2
Institute of Mathematics and Informatics,
Akademijos st. 4, 08663 Vilnius, Lithuania
E-mail: [email protected];
2
[email protected]
1
Abstract: Sustainable development does not focus solely on environmental issues. Education
and training build the critical foundation for sustainable development. One of the priorities of
the Lithuanian national sustainable development strategy is the enhancement of accessibility
and effectiveness of education and science systems. The main problem investigated in the paper
is the enhancement of school management by the means of creation and optimization of a profiled school schedule. This kind of task does not have any algorithms of polynomial complexity
that is why the principal attention is paid to heuristic methods. The paper reports on the results
of experimentation with the local search techniques (such as Local, Local Random, MonteCarlo, Simulated Annealing and Bayes) for optimization of the profiled school schedule. A new
element of this work is optimization of Simulated Annealing (SA) parameters using special
Bayes (BA) methods. Another new element is application of vectorial optimization theory by
fixing Pareto optimal schedules such that would satisfy the subjective criteria of the parameters
according to that particular condition of the spot. Both these new elements make the given
work distinct from all the other earlier works on the profiled school schedule optimization. The
results of each technique are analyzed.
Keywords: profiled school scheduling, optimization, local search techniques, parameters,
Simulated Annealing, Bayes method
1
Introduction
The paper investigates the problem of a profiled school schedule optimization (for the previous investigations in the field see, e.g., Appleby et al., 1961; Cooper and Kingston, 1993;
Gaidukevičienė and Kurilovas, 2005; Willemen, 2002).
A profiled school is referred here as the last grades of a secondary school or gymnasium
where the pupils can mostly choose their preferred learning profile subjects. Therefore this task
is more complex in comparison with a secondary school scheduling without profiled classes.
This kind of task does not have any methods of polynomial complexity that is why the
principal attention is paid to heuristic methods.
The efficiency of optimization programs depends on the choice of a method for the given
class of tasks. The paper reports on the results of the experimentation with the several methods –
local search techniques (such as Local, Local Random, Monte-Carlo, Simulated Annealing and
Bayes) – for optimization of the profiled school schedule.
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L. Pupeikiene, E. Kurilovas
2
Goal and tasks of the work
3
Heuristics for efficient decision
The main goal of the research presented in the paper is to investigate heuristic optimization
methods for polynomially unsolvable tasks using profiled school schedules as examples.
To reach this goal the following tasks were realised:
1. To investigate the school schedule creation and evaluation criteria used in the current
general education institutions. Using these criteria, to evaluate the results of the application of different optimization methods.
2. To investigate the impact of the chosen heuristic parameters on the speed and accuracy
of solution selecting a proper optimization method for optimizing these parameters.
3. To evaluate the influence of subjective and objective indices on different optimization
algorithms and the effect of these indices while evaluating school schedules with different requirements.
4. After analysing the principles of schedule creation, to prepare recommendations for the
estimation and application in choosing and optimising heuristic parameters.
5. To implement and test the functioning of school schedule optimization, i.e., the program aimed at school schedule optimization in Lithuania.
6. To implement the results in the internet environment with as good as possible conditions of practical use and to prepare proposals for future development and improvement
of the optimization program.
This section deals with heuristics that allow more efficient decision to be found. It can be
calculated by generalizing as much as possible heuristics used in Lithuanian profiled schools.
Then main problem for school schedule optimization is described as follows:
min F (τ ),
(1)
τ ∈Θ
where
F(τ) is the total penalty of some schedule τ.
A is the set of schedules satisfying the physical constraints.
The penalties F(τ) depend on the subjective expert evaluations, therefore we regard them as
heuristics.
Penalty points are calculated:
F = Ff + Fn
(2)
where
Ff is the sum of the penalties for the physical restrictions.
Fn is the sum of the penalties for the general inconveniences.
All physical restrictions and inconveniences are shown in Fig. 1.
The Formula (3) has been used here to make a schedule from all used heuristics parameters
(see Fig. 1 and Formula 2):
(3)
τ = schedule [ D [ M ]][V ][ G [ S ]][ K ]
where
D[M] is the matrix of all subjects;
M is the total number of all teachers;
V denotes the total number of weekly working hours;
G[S] is the matrix of all groups;
S is the total number of all pupils;
K is the number of school rooms.
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APPLICATION OF OPTIMIZATION METHODS IN PROFILED SCHOOL SCHEDULING
Limit weekly working days
No simultaneous lessons for teachers
Limited number of teachers lesson hours
Required
No simultaneous lessons for students
Limited number of students lesson hours
Limited number of daily lessons
Limited number of students in a lesson
Timetable
Different lessons in different classrooms
Special lesson in special classroom
Desired
Compact timetables for students
Compact timetables for teachers
Convenient lesson hours for students
Convenient lesson hours for teachers
Convenient sequence of lessons
Stability of subject-groups
Figure 1. Restrictions for a creation of the profiled school schedule
There are no schedules that satisfy all restrictions and personal preferences because they
contradict each other as usual. A compromise solution is reached by defining penalties for violation of constraints and disregarding inconveniences.
To make the optimal profiled school schedule several optimization methods are probably
needed. To evaluate the efficiency of each method the detailed analysis of these methods was
performed.
4
Evaluation and comparison of the applied optimization methods
This section deals with several optimization methods suitable for school scheduling: Local
Deterministic (LD), Local Randomized (LR) (for local search techniques see, e.g., Schaerf A.
and Schaerf M., 2005; for Mathematical Programming see Žilinskas, 2005), Simulated Annealing (SA) (see, e.g., Abramson et. al., 1996; Yang, 2000), and SA with the parameters optimized
using the Bayesian approach (BA) (see, e.g., Mockus, 2002; Mockus, 2004; Mockus et. al.,
2000). The efficiency of each method has been explored.
The experiment has shown that the most efficient method is BA which has optimized the
parameters of SA. The scientific novelty of this approach is the increasing efficiency of the
Simulated Annealing by the application of the Bayesian method for automatic optimization of
parameters.
Recommendations have been prepared for choosing the default parameters. It has intervals
of SA parameters (for initial temperature and cooling rate), iterations of SA and the own itera441
L. Pupeikiene, E. Kurilovas
tions. The stabile convergence of BA method begins when the iterations of SA are more then
500. The efficiency of this method is represented in Fig. 2 and Fig. 3.
Figure 2. Average improvements obtained while applying Bayes method
Figure 3. Top improvements obtained while applying Bayes method
Choosing and optimizing of the heuristic parameters in the profiled school is very important. The optimal parameters must be chosen. The quality of the optimal schedule and the speed
of optimization depend on these parameters. Two conditions for choosing and optimizing heuristics are stressed: (1) Heuristics must comply with the different internal rules of a profiled school;
(2) Heuristics must be oriented to the fast and convenient search for an optimal schedule.
Evaluation and formalization of local conditions can be achieved using the scalarization
method of vectorial optimization theory. This is convenient for practical applications and provides Pareto-optimal solutions. This approach was not used in other publications on school
scheduling.
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APPLICATION OF OPTIMIZATION METHODS IN PROFILED SCHOOL SCHEDULING
5 Experimental scheduling program implementation
The possibilities of practical use of aforementioned methods depend on the realisation of
the program. We have examined various popular programming languages that could be used for
school schedule optimization, with a view to choose the most user-friendly, and their pros and
cons has been evaluated. The most user friendly programming language has been established
after evaluating: (1) Dependence on the operating system; (2) Speed of the work; (3) Maintainability; (4) Multilingual support; (5) Security and flexibility. The analysis of the programming
languages that could provide the most user-friendly environment has shown that Java Servlet
technology seems to be the most suitable.
As a result, a unique web application was created at the Institute of Mathematics and Informatics. We have chosen a platform independent architecture and used the Java Servlet technology to develop the application.
Thus Bayes optimization method that optimizes SA parameters has been applied and opensource school schedule optimization program “Optima” has been developed and implemented.
To this end we have used Java technologies (min v1.6.0_07-b06), XML technologies, the
Apache web server of the Institute of Mathematics and Informatics (min v6.0.16) as well as
Linux and Windows operating systems.
We have created a user-friendly internet interface for the school schedule optimization program. With reference to data, used to create schedules in Lithuanian schools, an easily comprehensible template for the initial data file has been created in which all the data used for the
creation an optimal schedule have been described.
Judging by the publications to date, this is a first case of Java Servlet technology usage in
school schedule optimization. In this case, the school schedule optimization is performed very
simply: just by using a browser, without any additional tools.
The results of the work could be used in Lithuanian comprehensive institutions (basic
schools, high schools and gymnasia). Heuristic algorithms with optimization of heuristic parameters, described in this work, have been used at the Marijampolė Rygiškių Jono gymnasium
and Trakų Vytauto Magnus gymnasium.
6
Conclusion
The exact methods of optimal scheduling of profiled schools are of exponential time. Thus
heuristic methods need to be applied. The efficiency of optimization depends on the parameters
of heuristics therefore automated procedures should be applied to optimize these parameters.
The research has shown that optimization of SA parameters using Bayesian methods is a
new and efficient way for solving school scheduling problems by the heuristic methods. The
proposed optimal scheduling approach is sufficiently efficient and can be recommended for optimization of the profiled school schedules.
Creation and usage of the effective optimization solution for the profiled school scheduling
should enhance the school management, the effectiveness of the whole education system, and
thus the effectiveness of sustainable development.
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