SOLVING THE UNIVERSITY COURSE TIMETABLING

PROBLEM BY HYPERCUBE FRAMEWORK FOR ACO

Jose Miguel Rubio L, Broderick Crawford L. and Franklin Johnson P.

Escuela de Ingenieria Informática, Pontificia Universidad Católica de Valparaíso, Valparaiso, Chile

Keywords: University Course Timetabling Problem, Ant Colony Optimization, Ant Systems.

Abstract: We present a resolution technique of the University Course Timetabling Problem (UCTP), this technique is

based in the implementation of Hypercube framework using the Max-Min Ant System. We show the

structure of the problem and the design of resolution using this framework. A simplification of the UCTP

problem is used, involving three types of hard restrictions and three types of soft restrictions. We solve

experimental instances and competition instances, the results are presented of comparative form to other

techniques. We present an appropriate construction graph and pheromone matrix representation. Finally the

conclusions are given.

1 INTRODUCTION

The Timetabling problems are faced periodically by

each school, college and university in the world. In a

basic problem, a set of events (particular classes,

conferences, classes, etc) must be assigned to a set

of hours of a way that all the students can attend all

of their respective events. With the reservation of

which restrictions of hard type which necessarily

they must be satisfied and soft restrictions exist that

deteriorate the quality of the generated schedule. Of

course, the difficulty of any particular case of the

UCTP (Cooper and Kingston, 1996) (ten Eikelder

and Willemen, 2001) depends on many factors and

in addition the assignment of rooms perceivably

makes the problem more difficult in general.

Many techniques have been used in the

resolution of this problem, among these we can find

evolutionary algorithms, simulated annealing, and

tabu-search. Other technique has presented good

results is the genetic algorithms (Burke, Newall and

Weare, 1996). But here specifically we represent the

resolution through the ant colony optimization

(ACO) and through the implementation of

Hypercube framework for Max-Min Ant System

(abbreviation in Spanish MTH-SHMM). We give a

representation for the problem, generating an

appropriate construction graph and the respective

pheromone matrix associated.

In the following sections we present the UCTP

problem, the problem design for Hypercube

framework, the instances of the problem used and

the results of the experimentation. Finally the

conclusions of the work.

2 UCTP DESCRIPTION

The problem timetabling considered to make this

study is similar to one presented initially by Paechter

in (Paechter, 2001). Timetabling of university

courses is a simplification of a typical problem

(Paechter, Rankin, Cumming and Fogarty, 1998). It

consists of a set of events E and must to be

scheduled in a set of timeslots T ={t

,…,t

} (k = 45,

they correspond to 5 days of 9 hours each), a set of

rooms R in which the events will have effect, a set

of students S who attend the events, and a set of

features F required by the events and satisfied by the

rooms. Each student attends a number of events and

each room has a maximum capacity. A feasible

timetable is one in which all the events have been

assigned a timeslot and a room so that the following

hard constraints are satisfied:

 No student attends more than one event at the

same time;

 The rooms must be big enough for all students

who attend a class and to satisfy all the

features required by the event;

 Only one event is in each room at any

timeslot.

531

Miguel Rubio L J., Crawford L. B. and Johnson P. F. (2008).

SOLVING THE UNIVERSITY COURSE TIMETABLING PROBLEM BY HYPERCUBE FRAMEWORK FOR ACO.

In Proceedings of the Tenth International Conference on Enterprise Information Systems - AIDSS, pages 531-534

DOI: 10.5220/0001705505310534

 SciTePress

In short, All possible timetable generated is

penalized for each occurrence according to the

number of violations that exists of the soft constraint

of problem. Some of these restrictions appear next:

 A student has a class in the last slot of the day;

 A student has more than two classes in a row;

 A student has exactly one class on a day.

Feasible solutions are always considered to be

superior to infeasible solutions, independently of the

numbers of soft constraint violations. In fact, in any

comparison, all infeasible solutions are to be

considered equally worthless. The objective is to

minimize the number of soft constraint violations in

a feasible solution.

3 DESIGN OF HYPERCUBE

FRAMEWORK SHMM FOR

TIMETABLING (MTH-SHMM)

3.1 Resolution Structure

Given the restrictions presented in the previous

section and the characteristics of problem, we can

now consider the option to design an effective

MTH-SHMM for the UCTP. We have to decide how

to transform the assignment problem (to assign

events to timeslots) into an optimal path problem

which the ants can solve (Blum, Dorigo, 2004). To

do this we must create an appropriate construction

graph for the ants to follow. We must then decide on

an appropriate pheromone matrix and heuristic

information to influence the paths that the ants will

take through the graph.

We present the principal elements used to

generate the UCTP solutions, presenting in Figure 1

these three elements.

3.2 Construction Graph

One of the main elements of the ACO metaheurístic

is the power to model to the problem on construction

graph (Dorigo and Di Caro, 1999) (Blum, Dorigo

and Roli, 2001), that way a trajectory through the

graph represents a problem solution. In this

formulation of the UCTP it is required to assign each

one of │E│ events to │T│ timeslots. Where the

direct representation of the construction graph is

given by E × T; given this graph we can then

establish that the ants walk throughout a list of

events, choosing timeslot for each event. The ants

follow one list of events, and for each event and, the

ants decide timeslot t. each event at this single time

in timeslot, thus in each step an ant chooses any

possible transition as show the figure 2.

Figure 1: An instance of the problem is received like

input, this it happens through an association process event-

timeslot, assigns events to a timeslot, later a matching

algorithm ( Socha, 2003) is used for makes the assignation

from rooms to each one of events associated to timeslot. In

this point a solution is complete, but is low quality. Then a

local search algorithm (Rossi-Doria, Blue, Knowles and

Sampels, 2006) is applied that improves the quality of the

solution and gives like final result one optimal solution to

the UCTP.

Figure 2: Each ant follows a list of events, and for each

event e Є E, an ant chooses a timeslot t Є T.

The ants travel through the construction graph

selecting ways of probabilistically way. Using the

following function:

∑

∈

)(

),(

(1)

3.3 The Pheromone Matrix

In search of a pheromone matrix we represented that

pheromones indicates the absolute position where

the events must be placed. With this representation

the pheromone matrix is given by τ (A

) = τ, i=1,…,

|E|, the pheromone does not depend on the partial

assignments A

. It can observe that in this case the

pheromone will be associated with nodes in the

construction graph rather than edges between the

nodes.

A disadvantage of this directs pheromone

representation is that the absolute position of events

in the timeslots it does not matter very much in

producing a good timetable. The relative placement

of events is more important. For example, given a

ICEIS 2008 - International Conference on Enterprise Information Systems

532

perfect timetable, it is usually possible to permute

many groups of timeslots without affecting the

quality of the timetable.

By other side, we defined that for the use of the

heuristic information η it must use a function that

calculates a weighted sum of several or all of the

soft and hard constrains in each assignation, which is

to incur very high computational cost stops this class

of problem (Socha, 2003). For this we will not use

this type of information to orient the route of the

ants.

4 EXPERIMENTATION

The framework is based in an algorithm in which

some modifications are made of presented in (Socha,

Knowles and Sampels, 2003) (Stützle and Hoos,

2000), it was implemented in C++ programming

language, under Linux system using GNU G++

compiler GCC 2.96. The behavior of Hypercube

framework Max-Min Ant System (MTH-MMAS)

was observed in the resolution of the UCTP. The

used instances appear below.

Instances 1: Instances of the UCTP are

structured using a generator described in

http://www.dcs.napier.ac.uk/~benp. This generator

allows generating classes of instances small,

medium, which reflect varied problems of

timetabling of several sizes.

Instances 2: In addition it was used a series of 20

instances created for International Timetabling

Competition, these instances are made with the same

generator used in instances 1.

The parameters study are made initially, to

evaluate the best values than must to assume these

parameters. The small (small1) instances was used

for using the MTH-MMAS without local search

making evaluations with different ants numbers m

and with different evaporations factors ρ, the

parameters of α = 1, number on attempts = 10 and a

maximum time by attempt = 90 seconds for all the

tests. The results are in the following table.

In the table it can observed the best results are

obtained using the parameter m=20 obtaining a

evaluation of 16 in 6.06 seconds. And for the case of

evaporation factor the best value is =0,5 in 8.1

seconds.

The values shown in the tables previously

presented they belong to a series of executions that

allow of experimental form to determine as are more

advisable parameters to use in the execution of the

algorithm of MTH-MMAS. This way we compared

the algorithm of the Max-Min Ant System with and

without Hypercube framework, in addition the local

search is included to increment the quality of the

solutions in different instances.

Table 1: It presents the best results obtained when proving

the instance small1.tim varying ants number m and

evaporation factor ρ.

Best solutions MTH-SHMM

Evaluation Tº seg.

5 17 6,79 0,2 15 7,11

10 16 7,46 0,5 13 8,1

20 16 6,06 0,8 17 6,79

4.1 Comparison with other Techniques

Here it present a comparative picture between the

solutions obtained for different instances for the

UCTP doing use of different techniques like

Simulated annealing, advanced search and simulated

annealing with local search. (Rossi-Doria, Blue,

Knowles and Sampels, 2006). The results obtained

for the competition instances appear below.

Table 2: It present the best results obtained when proving

the instances of the International Timetabling Competition

compared with other techniques.

Technique 1 2 3 4 5 6 7 8 9 10

SA 45 25 65 115 102 13 44 29 17 61

257 112 266 441 299 209 99 194 175 308

SA-LS

211 128 213 408 312 169 281 214 164 222

MTH-

MMAS

270 193 294 586 406 221 305 244 201 358

Technique 11 12 13 14 15 16 17 18 19 20

44 107 78 52 24 22 86 31 44 7

273 242 364 156 95 171 148 117 414 113

SA-LS

196 282 315 345 185 185 409 153 281 106

MTH-MMAS

268 312 341 403 222 234 371 184 345 201

For these instances and compared with the

other solutions the MTH-MMAS it present two

characteristics to evaluate; first it has the capacity to

generate feasible solutions for these instances. These

instances are of great you make difficult since they

are for Timetabling competitions. Second the quality

of the generated solutions is of very low category

compared with the technique based on Simulated

Annealing, which has the best found historical

results for these instances, but in comparison with

the other instances do not present great difference.

These evaluations are not feasible in order to decide

if a technique is better than other, since the

differences in variable results can be for different

external variables.

SOLVING THE UNIVERSITY COURSE TIMETABLING PROBLEM BY HYPERCUBE FRAMEWORK FOR ACO

533

To continuation it presents the comparison for

the small and medium instances. We will compare

the algorithm of MTH-MMAS and the MMAS

pure with respect to Ant Colony System algorithm

of Krzysztof Socha (ACS) and to algorithm based on

random restart local search (RRLS).

As it is possible to observe, for these instances

in the MTH-MMAS present superiority in the

quality of the generated solutions (smaller VRS).

Always by on the quality the solutions generated

with the MMAS. We can say that the hypercube

framework it improves the quality of the ant

algorithm applied

Table 3: It present best results obtained when proving the

test instances small and medium.

Technique Small1 small2 small3 medium1 medium2

RRLS

11 8 11 199 202

ACS

1 3 1 195 184

MMAS

3 6 3 152 250

MTH-MMAS

0 4 1 138 186

5 CONCLUSIONS

A formal model was given to apply Hypercube

framework to solve the University course

timetabling problem (UCTP) making use of Max-

Min Ant System, was generated an efficient model

that solves instances of this problem creating good

construction graph of and expressing a good

pheromone matrix.

We presented the test result made for the Max-

Min Ant System making use of Hypercube

framework. We observed traverse of the given

results, that this propose framework is good means

of resolution of combinatorial problems and for the

case of the UCTP it presented good results for

instances of small and medium type. Although the

results were low quality for the instances of the

competition it emphasizes the fact that always it

generates solutions feasible and for instances of

normal difficulty of good evaluation, not obtain the

best results for this problem, but if it improves in

contrast with the Max-Min Ant System without

framework.

REFERENCES

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timetable construction problems. In Proceedings of the

1st International Conference on Practice and Theory of

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H. M. M. ten Eikelder and R. J. Willemen. Some

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problems. In Proceedings of the 3rd International

Conference on Practice and Theory of Automated

Timetabling (PATAT 2000), 2001.

E. K. Burke, J. P. Newall, and R. F. Weare. A memetic

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http://www.dcs.napier.ac.uk/~benp

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