A study of genetic algorithms for solving the school timetabling problem.

The experimental work described in this dissertation was carried out at the School of Computer Science, University of Natal, Pietermaritzburg, from January 2007 to April 2013, under the supervision of Professor Nelishia Pillay. Where the work of others has been used, it is duly acknowledged in the text.

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The school timetable problem is a common optimization problem that many elementary schools and high schools face. The second goal is to evaluate a genetic algorithm that uses an indirect representation (IGA) when solving the school timetable problem. When comparing the performance of the two approaches, IGA outperformed DGA for all the school schedule problems tested.

Introduction

Purpose of the study
Objectives
Contributions to the study
Thesis layout

To develop and evaluate a genetic algorithm approach that uses indirect representation in solving the school timetable problem. There has been no previous investigation of indirect representations in genetic algorithms in solving the school timetable problem, and it was found that this genetic algorithm (IGA) performed better than a genetic algorithm using direct representation (DGA). This chapter describes the school timetable problems that various authors have tried to solve, the methods that have been used to solve the problem, and the results obtained.

An overview of the school timetabling problem

The School timetabling problem
Common hard constraints

Lesson requirements
Clashes
Consecutive period requirements
Co-teaching and subclasses
Specialized venues
Teacher availability

Common soft constraints

Daily teacher/subject limits
Compact timetables
Replacement teachers
Resource preferences

The school timetabling problem as a multi-objective problem
Chapter summary

So assigning the principal to the last few periods of the day would be a violation. A general requirement is the limitation of the number of lessons a teacher is allocated per day. In the three studies, the soft constraints were considered as sub-goals and the number of violations for each of the sub-goals had to be minimized.

Previous work on the school timetabling problem

The tabu search

Tabu search description
Applications of tabu search to the school timetabling problem

Integer or linear programming

Integer or linear programming description
Application of integer programming to the school timetabling problem

Tiling algorithms
Simulated Annealing

Introduction to simulated annealing
Applications of simulated annealing to the school timetabling problem

Constraint programming

Constraint programming description
Applications of constraint programming to the school timetabling problem

Particle Swarm Optimization

Introduction to particle swam optimization
Application of particle swarm optimization to the school timetabling problem

Hybrid algorithms
Comparative studies
Chapter Summary

Additionally, a set of CLUs were combined to form the week's lessons (i.e. the class schedule) for a given class. All free periods for classes must be allocated to the last period of the day. Tabu search was then applied to reduce the cost of hard constraints and the cost of soft schedule constraints.

Introduction to Genetic Algorithms

Introduction
Biological background
Genetic Algorithm Overview

Initial population generation
Fitness function
Selection method
Genetic operators
Control parameters

Advances in Genetic Algorithms

Control models
Individual representation
Initial population creation
Tournament Selection
Genetic operators

Chapter Summary

In the genetic algorithm presented by Goldberg [GOLD89], all individuals in the initial population are generated randomly. In the genetic algorithm used by [GOLD89], the offspring are created by combining the string fragments (parts of the string) that are obtained from each of the parents. If there are no duplicate individuals in the initial population, the diversity of the population increases and allows for greater coverage of the search space.

Genetic algorithms and the School Timetabling Problem

Evolutionary or genetic algorithms
Genetic algorithms with other techniques
Comparative studies
Chapter Summary

Representation
Control model
Initial population generation
Selection method
Genetic operators
Single phase versus multiphase
School Timetabling Problems

Each free period must be allocated at the end of the day or at the beginning of the day. The authors found feasible schedules in the initial population, so the main objective of the genetic algorithm was to optimize the quality of these feasible schedules. Free periods for class schedules should be moved to either the beginning or end of the day.

An individual in the population was represented as a two-dimensional matrix with the rows representing the days of the week and the columns representing the periods. The cost of the grid was determined by the weighted sum of the constraint violations. Hill climbing ensured that moves did not result in an increase in the constraint cost of the grid.

The last two approaches combined the hill-climbing (RNA) algorithm with each of the genetic algorithms. The fitness of a schedule was determined by computing the weighted sum of the constraint violations. Another important contributing factor was the use of the tabu search incorporated with the mutation operator.

Studies by [DIST01], [ABRA91a], [CERD08], and [RAHO06] include hill climbing as part of the genetic operators to solve their respective problems.

Methodology

Introduction
Fulfilling the objectives of the study

Objective One
Objective Two

Hypothesis testing
The School Timetabling Problems

The Hard Defined TimeTable (HDTT) school timetabling problem
The Valouxis Greek school timetabling problem
The Beligiannis Greek school timetabling problem
The Woodlands Secondary school timetabling problem
The W.A. Lewitt primary school timetabling problem

System implementation details
Chapter Summary

The mathematical formulation of the fitness function for this problem is simply the sum of all the collisions of a given lattice. This increases the difficulty of the problem as these constraints must be met rather than minimized. The formulation of the fitness function to calculate the soft constraint cost is a summation of the number of violations for each of the listed soft constraints.

There are also four soft constraints that need to be minimized, which is more than most problems discussed in the literature. Thus, this problem is one of the most challenging problems compared to the other problems in the literature. Based on the above constraints, the fitness function formulation to calculate the hard constraint cost is shown below.

Lower costs for soft constraints indicate better schedule quality. So the goal is to minimize the value of the above function. Unlike the Beligiannis problem, the number of classes and teachers involved in the subclassing and co-teaching requirements is more than two, which increases the difficulty of the problem. The goal is to minimize the cost of the schedule and a feasible schedule is one with a hard constraint cost of 0.

Thus, the fitness value is determined by the sum of the number of hard constraint violations for each hard constraint HCX.

Table 6.1: Levels of significance, critical values and decision rules

A Genetic Algorithm Approach using a Direct Representation

Overall algorithm
Initial population creation

Representation
Initial population creation process
Converting the class-teacher lessons list into a list of tuples
The sequential construction method (SCM)
Initial population creation during Phase 2

Evaluating a timetable for feasibility and quality

Evaluating the feasibility of a timetable (Phase 1)
Evaluating the quality of a timetable (Phase 2)

Selecting a parent

Standard tournament selection
Variant tournament selection (VTS)

Genetic operators for Phase 1

Two violation mutation (2V)
One violation mutation (1V)
Hill climbing versus non-hill climbing operators

Genetic operators for Phase 2

Random Swap
Row swap
One violation mutation (1V)
Two violation mutation (2V)

Control parameters
Chapter Summary

As discussed in Section 5.4 (Chapter 5), using an intersection operator would require repair operators to remove duplicate tuples and add missing tuples. The schedule with the best fitness (i.e. the least number of strong constraint violations) is added to the initial population. The random heuristic is repeatedly applied until all tuples have been placed on the schedule.

Any one of the seven tuples from assembly number 3 will then be assigned to a period in the grid. The tuples are assigned to periods in the grid in this order, potentially reducing the number of violations when filling the grid. Initially, all tuples have the same degree of saturation (i.e. the number of periods) since the grid is empty.

At the next iteration of the allocation process, all tuples with a saturation degree of 29 are prioritized, and one of these tuples (selected using secondary heuristics) will be allocated to the schema. The two teachers (or venues) are then swapped within the schedule, possibly resulting in the removal of hard constraints. Three of the four operators discussed in this section are based on the mutation operators used in the literature.

Another cell is randomly selected and the contents of the two cells are swapped, resulting in the possibility that the soft constraint violation is removed.

Table 7.2: List of class-teacher lessons

A Genetic Algorithm Approach using an Indirect Representation

Overall Algorithm
Initial Population Creation

Instruction String Representation
Algorithm for Initial Population Creation
The Sequential Construction Method (SCM)
Phase 2 Initial Population Creation

Evaluating an individual in the population
Selecting a parent
Genetic Operators

Mutation
Crossover

Control Parameters
Summary

The length of the initial string is equal to the number of tuples to be allocated. Then the deallocate instruction will remove the tuple from the roadmap (resulting in an empty roadmap). The allocation instruction is then applied and an unallocated tuple is placed in the roadmap.

A population of strings is created with each string containing a combination of instructions that will be used to improve the quality of the schedule found in Phase 1. The fitness value of the individual is calculated by counting the number of violations of hard constraints (Phase 1) and violations of soft constraints (Phase 2) of the manufactured schedule. If an allocation instruction ("A") is executed, then the tuple with the lowest degree of saturation (i.e. the tuple with the least number of periods without violation) is allocated to a period that does not violate any hard constraint (if it is possible).

This section describes the crossover and mutation genetic operators used in both phases of IGA. For this operator, a randomly selected crossover point is selected for each of the two individuals. As can be seen in Figure 8.3, the crossover operator also caused the length of the strings to change.

This is followed by the description of how an individual is represented, how the initial population is generated for both phases of the IGA and how each individual is evaluated.

Table 8.1: Instructions used to build a timetable

Results and discussion

Introduction

DGA process evaluation

The Abramson benchmark school timetabling problem (HDTT)

Comparison of low-level construction heuristics
Comparison of selection methods
Comparison of mutation operators

Comparison of Phase 1 selection methods
Comparison of Phase 1 mutation operators

The Beligiannis Greek high school timetabling problem

W.A. Lewitt primary school timetabling problem

The Woodlands secondary school timetabling problem

Best performing DGA processes

The DGA approach using the degree of saturation produces solutions in the smallest number of bands. The DGA using the 2VNH mutation operator performs better than the DGA approach using one of the hill-climbing operators. From Table 9.18, it is concluded that DGA using variant tournament selection performs better than DGA using standard tournament selection.

The DGA using variant tournament selection also produces the schedule with the fewest number of soft constraint violations (39). The average qualities in Table 9.39 indicate that the DGA using standard tournament selection produces better quality schedules than the DGA using variant tournament selection. The hypothesis states that the DGA using standard tournament selection will produce better quality schedules than the DGA using variant tournament selection.

The results of using the DGA approach with each of the three construction heuristics are shown in Table 9.45. On average, DGA using the degree of saturation produces the least number of strong constraint violations. DGA using the degree of saturation heuristic also produces the best quality schedule with the least number of soft constraint violations (7).

The hypothesis states that DGA using standard tournament selection performs better than DGA using variant tournament selection method (in terms of feasibility). The success rates listed in Table 9.50 indicate that the DGA using the hybrid operator is far superior to the DGA using any of the other mutation operators. The average quality of DGA using each mutation operator is given in the table below (Table 9.53).

Table 9.1: Processes and parameter values to test best low-level construction heuristic (HDTT problem)

Fine-tuning of DGA control parameter values

The Abramson benchmark school timetabling problem

Fine tuning the SCM size
Fine-tuning the population size
The best tournament size
Fine-tuning the number of swaps
Maximum number of generations

The Valouxis school timetabling problem

Fine-tuning the SCM size
Fine-tuning the tournament size

The W.A. Lewitt primary school timetabling problem

Summary of fine-tuning

Success rates and the number of generations taken to find a solution are shown in Table 9.72. The following table (Table 9.84) shows the processes and parameter values used to test the best tournament size. The processes and parameter values used to test the tournament sizes are listed in the table below (Table 9.97).

The quality of the timetables produced using the DGA approach with different tournament sizes is listed in Table 9.99. The success rates and average quality of the schedules produced using the DGA approach with each SCM value are shown in Table 9.107. The processes and parameter values used in testing the different population sizes are listed in Table 9.108.

The success rates, average HC costs and average quality for the DGA using each population size are shown in Table 9.109.