Solving an Uncapacitated Exam Timetabling Problem Instance

using a Hybrid NSGA-II

Nuno Leite

1∗

, Rui Neves

, Nuno Horta

, Fernando Mel

ıcio

and Agostinho Rosa

3†

ISEL - Lisbon Polytechnic Institute, Rua Conselheiro Em

ıdio Navarro 1, 1959-007 Lisboa, Portugal

Telecommunications Institute/IST, TU-Lisbon, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

LaSEEB-System and Robotics Institute, Department of Bioengineering/IST, TU-Lisbon,

Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Keywords:

Exam Timetabling Problem, Evolutionary Algorithms, Multi-objective Optimization, Combinatorial Prob-

lems.

Abstract:

This paper describes the construction of an university examination timetable using a hybrid multi-objective

evolutionary algorithm. The problem instance that is considered is the timetable of the Electrical, Telecom-

munications and Computer Department at the Lisbon Polytechnic Institute, which comprises three bachelor

degree programs and two master degree programs, having about 80 courses offered and 1200 students enrolled.

The task of manually construct the exam timetable for this instance is a complex one due essentially to the

high number of combined degree courses. This manual process takes, considering a two-person team, about

one week long. A hybrid multi-objective evolutionary algorithm, based on the Non-dominated Sorting Genetic

Algorithm-II (NSGA-II), is proposed for solving this problem instance, incorporating two distinct objectives:

one concerning the minimization of the number of occurrences of students having to take exams in consecutive

days, and a second one concerning the minimization of the timetable length. The computational results show

that the automatic algorithm achieves better results compared to the manual solution, and in negligible time.

1 INTRODUCTION

The construction of school and university examina-

tion timetables is one of the most important tasks

taking place in educational institutions. Many in-

stitutions still elaborate their timetables in a manual

form, involving a great deal of time and human re-

sources and leading to suboptimal solutions. The task

of automatically constructing examination timetables

is known as the Exam Timetabling Problem (ETTP),

and is an extensive studied optimization problem. The

basic problem consists in distributing a set of exams

by temporal periods, satisfying a set of hard and sec-

ond order (or soft) constraints. Constraints of the

ﬁrst type cannot be violated as this results in an in-

feasible timetable. Constraints of the second type

represent institution’s view of what makes a good

timetable and should be satisﬁed as many as possible.

∗

Work partially supported by the FCT

SFRH/PROTEC/67953/2010 grant.

†

Work partially supported by the FCT

SFRH/BSAB/1101/2010 and PEst-OE/EEI/LA0009/2011

grants.

Examples of constraints include: not scheduling ex-

ams with common students in the same period (hard

constraint); sufﬁcient seating capacity for all exams

(hard constraint); leave at least a two day interval be-

tween exams for all students (soft constraint). More-

over, depending if seating capacity hard constraint is

considered or not, the ETTP is further classiﬁed in

Capacitated ETTP and Uncapacitated ETTP, respec-

tively. The actual program curricula seen at universi-

ties are designed to offer a great degree of diversity

and ﬂexibility to the students, letting them choose a

considerable number of free or optional courses. In

order to make this possible with available teacher,

faculty staff, and university resources (rooms, equip-

ment, etc.), courses are being offered in multiple re-

lated programs. This growing number of combined

courses imposes extra difﬁculties in solving the ETTP.

The development of systems to automate the con-

struction of university examination timetables has

begun in the 1960 decade. The paper (Qu et al.,

2009) constitute a survey of the recent (from 1995

to 2008) techniques and algorithmic approaches used

to solve this problem. These techniques are clas-

106

Leite N., Neves R., Horta N., Melício F. and Rosa A..

Solving an Uncapacitated Exam Timetabling Problem Instance using a Hybrid NSGA-II.

DOI: 10.5220/0004166001060115

In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 106-115

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

siﬁed in the following groups: Graph based se-

quential techniques, Constraint based, Local search

based (e.g. Tabu search, Simulated Annealing),

Population based (e.g. Evolutionary algorithms,

Ant algorithms), Multi-criteria techniques, Hyper-

heuristics and Decomposition/clustering techniques.

More recently, the ETTP has been approached like

a Multi-objective/Multi-criteria Optimization prob-

lem, acknowledging the true dimensions of real

world problems, that typically have many facets to

consider (proximity costs between student exams,

timetable lengths, room assignment, invigilator avail-

ability, etc.). Multi-criteria techniques were proposed

in (Burke et al., 2001) and (Petrovic and Bykov,

2003). Other recent works (C

e et al., 2004), (Wong

et al., 2004), (Cheong et al., 2009) and (Mumford,

2010), applied Multi-Objective Evolutionary Algo-

rithms (MOEAs) to solve the ETTP. Evolutionary ap-

proaches are well suited to solve Multi-objective Op-

timization (MOO) problems because a population of

solutions is already being manipulated in each iter-

ation of the evolutionary algorithm. Therefore, the

population-approach of evolutionary algorithms can

be effectively used to ﬁnd the multiple trade-off so-

lutions of MOO problems. In MOO the solutions

are characterized by optimum sets of alternative non-

dominated solutions, known as Pareto sets. Several

MOEA have been proposed in the literature (Deb,

2001). It is known that metaheuristics, like evo-

lutionary algorithms, work better if hybridized with

other techniques (Raidl, 2006). In fact, the most suc-

cessful applications of MOEA to the ETTP are hy-

brid approaches, being usually hybridized with some

form of Local Search procedures. Moscato and Nor-

man (Moscato and Norman, 1992) introduced the

term memetic algorithm to describe evolutionary al-

gorithms in which local search is used. Following

this stream several authors developed hybridizations

of MOEA with other metaheuristics (Ehrgott and

Gandibleux, 2008). In (Burke and Landa Silva, 2005)

the authors present design guidelines of memetic al-

gorithms for scheduling and timetabling problems.

In this work we propose a novel hybrid MOEA

and show its application on a real world ETTP in-

stance. The considered problem instance is the ex-

amination timetable of the Electrical, Telecommuni-

cations and Computer Department (DEETC) at the

Lisbon Polytechnic Institute. The proposed MOEA

is based on the Elitist Non-dominated Sorting Ge-

netic Algorithm-II (NSGA-II) (Deb et al., 2002).

The NSGA-II procedure is one of the popularly used

MOEA which attempt to ﬁnd multiple Pareto-optimal

solutions in a multi-objective optimization problem.

Like the works (C

e et al., 2004), (Wong et al.,

Table 1: Characteristics of the DEETC dataset.

Exams Students Enrolment Periods

80 1238 4637 18

Table 2: Number of exams per program in the winter

semester of DEETC department.

LEETC LEIC LERCM MEIC MEET

32 30 29 19 25

2004) (Cheong et al., 2009) and (Mumford, 2010), we

also consider two objectives: one that maximizes each

student free time between exams, and a second objec-

tive that considers the minimization of the timetable

length.

The paper is organized as follows: the next sec-

tion describes the DEETC department ETTP instance

and its formulation as a multi-objective optimization

problem. Section 3 presents the algorithmic ﬂow of

the proposed MOEA. Section 4 presents simulation

results and analysis of the proposed algorithm. Fi-

nally, conclusions and future work are presented in

Section 5.

2 PROBLEM DESCRIPTION

The problem instance considered in this work is the

DEETC timetable of the winter semester of the last

academic year. The DEETC timetable comprises ﬁve

programs: three B.Sc. programs (named LEETC,

LEIC and LERCM) and two M.Sc. programs (named

MEIC and MEET). B.Sc. and M.Sc. programs have

six and four semesters duration, respectively. The

DEETC dataset characteristics are listed in Table 1.

The number of exams per program is listed in Ta-

ble 2. About 34 of the 80 courses lectured in DEETC

are shared by different programs, as depicted in Ta-

ble 3. The high complexity of the timetable is due

mainly to two reasons: (1) high degree of course shar-

ing in different programs and different semesters (e.g.

LSD course is offered in the 1st and 2nd semesters

of LEIC and LEETC programs, respectively); (2) the

courses of the even semesters (summer semesters)

are also being lectured in the winter semester, thus

increasing the timetable complexity, because there

are students attending courses in the even and odd

semesters. To get an idea of the number of students

involved in each semester, we present in Table 4 the

number of classes proposed for the winter semester

for each program. Each class of the 1st to the 3rd

semester has on average 30 students and the remain-

der semesters have 20 students per class on average.

SolvinganUncapacitatedExamTimetablingProblemInstanceusingaHybridNSGA-II

107

Table 3: Courses shared among the ﬁve programs offered in the DEETC. The number of shared courses sums to 34 (out of 80

courses with exam). The ﬁrst ﬁve columns contain the semesters where the course is offered. Semesters in M.Sc. courses are

numbered 7 to 10 (four semester master program).

MEIC MEET LERCM LEIC LEETC Course Acronym

1 1 1 Linear Algebra ALGA

1 1 Mathematical Analysis I AM1

1 1 1 Programming Pg

1 2 Logic and Digital Systems LSD

2 2 Mathematical Analysis II AM2

2 2 2 Object Oriented Programming POO

2 2 3 Probability and Statistics PE

2 3 Computer Architecture ACp

3 and 5 Computer Graphics CG

3 and 5 Computation and Logic LC

3 and 5 Functional Programming PF

3 3 4 Imperative Programming in C/C++ PICC/CPg

7 3 5 Digital Communication Systems SCDig

4 4 4 Computer Networks RCp

7 4 5 Virtual Execution Systems AVE

8 4 Multimedia Signal Codiﬁcation CSM

4 5 Operating Systems SOt

7 5 Unsupervised Learning AA

8 5 Database Systems BD

8 5 6 Internet Programming PI

8 5 6 Distributed Computational Systems SCDist

7 7 5 5 5 Internet Networks RI

7 5 Compilers Cpl

7 5 Control Ctrl

7 5 Radio Communications RCom

7 5 Security Informatics SI

7 5 Telecommunication Systems ST

7 7 5 5 Embedded Systems I SE1

7 7 6 Multim. Comm. Networks & Services RSCM

7 6 Distributed Systems SD

7 6 Software Engineering ES

7 to 9 8 6 3 to 6 6 Project Management and Economics EGP

7 to 9 8 6 3 to 6 6 Enterprise Management OGE

7 to 9 8 6 3 to 6 6 Management Systems SG

Table 4: Number of classes proposed for the winter semester for each program.

Sem. LEIC LEETC LERCM MEIC MEET

1st 5 5 3 2 2

2nd 3 3 1 - -

3rd 3 3 2 2 2

4th 2 2 1 - -

5th 3 3 1

6th - - -

Total 16 16 8 4 4

2.1 Problem Formulation

This paper considers an instance of the ETTP that was

ﬁrst formulated in (Burke et al., 1996). In their for-

mulation, if a student is scheduled to take two exams

in any one day there should be a free period between

the two exams. Violation of this constraint is referred

as a clash. In their work it is considered the Capaci-

tated problem, whereas in our work we ignore the ca-

pacity constraint. The corresponding Uncapacitated

problem is formulated as:

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

108

Minimize f

|E|−1

∑

i=1

|E|

∑

j=i+1

|P|−1

∑

p=1

j(p+1)

i j

(1)

=|P| (2)

subject to

|E|−1

∑

i=1

|E|

∑

j=i+1

|P|

∑

p=1

i j

= 0, (3)

|P|

∑

p=1

= 1,∀i ∈ {1,...|E|}, (4)

where:

• E = {e

,...,e

|E|

} is the set of exams to be

scheduled,

• P = {1,2,...,|P|} is the set of periods,

• a

is one if exam e

is allocated to period p, zero

otherwise.

• c

i j

(Conﬂict matrix) is the number of students reg-

istered for exams e

and e

Eqs. (1) and (2) are the two objectives of minimizing

the number of clashes and timetable length, respec-

tively. Constraint (3) is the (hard) constraint that no

student is to be scheduled to take two exams in the

same period. Constraint (4) indicates that every exam

can only be scheduled once in any timetable.

3 HYBRID MULTI-OBJECTIVE

GENETIC ALGORITHM

As mentioned in the introduction, we solve the

DEETC ETTP instance using a hybrid MOEA based

on the NSGA-II algorithm. NSGA-II has the fol-

lowing features: (1) it uses an elitist principle, (2)

it uses an explicit diversity preserving mechanism,

and (3) it emphasizes non-dominated solutions. The

basic NSGA-II was further transformed to include a

step where a Local Search procedure is performed.

The general steps of the hybrid algorithm (named

HMOEA) are depicted in Figure 1. In the following

subsections we describe each block of the HMOEA in

detail.

3.1 Chromosome Encoding

In order to optimize for the second objective (see

Eq. (2)), each timetable is represented by a variable-

length chromosome as proposed by (Cheong et al.,

2009), and illustrated in Figure 2. A chromosome en-

codes a complete and feasible timetable, and contain

the periods and exams scheduled in each period. Valid

timetables should have a number of periods belong-

ing to a valid interval, initially given by the timetable

Procedure HMOEA

(t)

: parent population at iteration t

(t)

: offspring population at iteration t

(t)

: combined population at iteration t

L: local search operator

OUTPUT

(t)

: archive of non dominated timetables

Initialize P

and Q

of size N with random

timetables

For each iteration t ← 0,1,...,I

max

− 1 do

(Step 1) Form the combined population,

(t)

= P

(t)

, of size 2N.

(Step 2) Classify R

(t)

into different

non-domination classes.

(Step 3) If t ≥ 1, use local search

procedure L to improve elements of R

(t)

(Step 4) Form the new population P

(t+1)

with solutions of different non-dominated

fronts, sequentially, and use the crowding

sort procedure to choose the solutions

of the last front that can be accommodated.

(Step 5) N

(t+1)

← NonDominated(P

(t+1)

If t = I

max

− 1 then Stop.

(Step 6) Create offspring population Q

(t+1)

from P

(t+1)

by using the crowded tournament

selection, crossover and mutation operators.

(Step 7) Repair infeasible timetables.

Figure 1: Hybrid NSGA-II procedure.

planner. However, the operation of crossover and mu-

tation could produce invalid timetables, because of

extra periods added to the timetable as a result of these

operations. Thus, a repairing scheme must be applied

in order to repair infeasible timetables. The adopted

scheme is explained in detail in Section 3.4.

Figure 2: Variable length chromosome representation. A

chromosome encodes a complete and feasible timetable.

SolvinganUncapacitatedExamTimetablingProblemInstanceusingaHybridNSGA-II

109

Figure 3: Illustration of Day-exchange crossover based

on (Cheong et al., 2009). The shaded periods represent

the chromosomes best days. These are exchanged between

chromosomes, being inserted into randomly chosen periods.

Duplicated exams are then removed.

3.2 Population Initialization

It is known that the basic examination timetabling,

of minimizing the number of slots considering the

hard constraint of not having students with overlap-

ping exams, is equivalent to the graph colouring prob-

lem (C

e et al., 2004). As such, several heuristics

of graph colouring have been applied to the ETTP.

These heuristics inﬂuence the order in which exams

are inserted in the timetable. In this work, we use the

following two heuristics, in the initialization and mu-

tation processes:

• Saturation Degree (SD): Exams with the fewest

valid periods, in terms of satisfying the hard con-

straints, remaining in the timetable are reinserted

ﬁrst.

• Extended Saturation Degree (ESD): Exams with

the fewest valid periods, in terms of satisfying

both hard and soft constraints, remaining in the

timetable are reinserted ﬁrst.

The ESD heuristic is used in the population initializa-

tion procedure, while the SD heuristic is used in the

reinsertion process of the mutation operator (detailed

in Section 3.3). These two procedures are similar to

the procedures applied in (Cheong et al., 2009). The

use of the SD heuristic in the initialization process

has been experimented but with worse results than the

ESD heuristic.

Table 5: HMOEA parameters and environmental settings.

Parameter Value

Population size 40

Number of iterations I

max

=125

Crossover probability 1.0

Mutation probability 0.2

Reinsertion rate 0.02

SHC no. iterations t

max

= 5

SHC temperature T = 0.0001

Computer Intel Core i7-2630QM,

2.0 GHz, 8 GB RAM

OS Win 7.0 Pro, 64 bit

Matlab version Version 7.9 (R2009b)

Table 6: Clashes per course for the manual and automatic

solutions with 18 periods.

Number of clashes

Timetable Manual sol. Automatic sol.

LEETC 287 215

LEIC 197 163

LERCM 114 71

MEIC 33 23

MEET 50 40

Combined 549 393

In the initialization process, a timetable with a ran-

dom (valid) length is generated for each chromo-

some. Then, the unscheduled exams are ordered ac-

cording to the ESD heuristic and a candidate exam

is selected randomly being then scheduled into a ran-

domly chosen period (chosen from the set of periods

with available capacity while respecting the feasibil-

ity constraint). If no such period exists, a new pe-

riod is added to the end of the timetable to accommo-

date the exam. In the ESD heuristic used, a candidate

exam can be scheduled in a period if it does not vio-

late feasibility and if the number of clashes is bellow

or equal to 70. This process is repeated until all exams

have been scheduled.

3.3 Selection, Crossover and Mutation

The offspring population is created from the parent

population by using the crowded tournament selec-

tion operator (Deb et al., 2002). This operator com-

pares two solutions and returns as the winner of the

tournament the one which as a better rank, or if the

solutions have the same rank, the one who has a bet-

ter crowding distance (the one which is more far apart

from their direct neighbours).

The crossover and mutation operations were

adapted from the ones introduced in (Cheong et al.,

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

110

Figure 4: Evolution of the Pareto front.

2009). In the crossover operator, termed Day-

exchange crossover, the best days, selected based on

the crossover rate, are exchanged between chromo-

somes. The best day of a chromosome consist of the

day (a period, in our case) which has the lowest num-

ber of clashes per student. This operation is illustrated

in Figure 3. To ensure feasibility after the crossover

operation, the duplicated exams are deleted. Notice

that, as mentioned before, the result of inserting a new

period in a chromosome could produce a timetable

with a number of periods larger than the valid upper

limit. If this is the case, a repair scheme is applied in

order to compact the timetable.

The mutation operator removes a number of ex-

ams, selected based on the reinsertion rate, and rein-

serts them into other randomly selected periods while

maintaining feasibility. We use the SD graph colour-

ing heuristic to reorder the exams, prior to reinserting

them. As in the case of the crossover operator, the

mutation operator could also add extra periods to the

timetable, for the exams that could not be rescheduled

without violating the hard constraints.

3.4 Repairing Scheme

The repair scheme adopted is similar to the period

control operator of (Cheong et al., 2009), consisting

of the following two operations: (1) Period expan-

SolvinganUncapacitatedExamTimetablingProblemInstanceusingaHybridNSGA-II

111

Table 7: Manual solution for the LEETC examination timetable. The courses marked in bold face are shared with other

programs, as shown in Table 3. The number of clashes of this timetable is 287.

Sem. Course 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

ALGA x

Pg x

1st AM1 x

FAE x

ACir x

POO x

AM2 x

2nd LSD x

E1 x

MAT x

PE x

ACp x

3rd EA x

E2 x

SS x

RCp x

PICC/CPg x

4th PR x

FT x

SEAD1 x

ST x

RCom x

RI x

5th SE1 x

AVE x

SCDig x

SOt x

PI x

6th SCDist x

EGP x

OGE x

SG x

sion, used when the timetable has a number of periods

below the lower limit, and (2) Period packing, used

when the timetable has a number of periods above the

upper limit. In the period expansion operation, empty

periods are ﬁrst added to the end of the timetable such

that the timetable length is equal to a random number

within the period range. A clash list, comprising all

exams involved in at least one clash, is maintained.

Then, all the exams in the clash list are sweep in a

random order and rescheduled into a random period

without causing any clashes while maintaining fea-

sibility. Exams which could not be moved are left

intact. The period packing operation proceeds as fol-

lows: ﬁrst, the period with the smallest number of stu-

dents is selected; then the operation searches in order

of available period capacity, starting from the small-

est, for a period which can accommodate exams from

the former while maintaining feasibility and without

causing any clashes. The operation stops when the

timetable length is reduced to a random number in

the desired range or when it goes one cycle through

all periods without rescheduling any exam.

3.5 Ranking Computation

The non-dominated sorting procedure used in NSGA-

II use the evaluation of the two objective functions to

rank the solutions. We adopt a simple penalization

scheme in order to penalize solutions with an invalid

number of periods. The penalization is enforced ac-

cording to the following pseudo-code:

If timetable length > max length Then

Pen

= f

+ α

(timetable length − max length)

Pen

= f

+ α

(timetable length − max length)

Else If timetable length < min length Then

Pen

= f

+ α

(min length − timetable length)

Pen

= f

+ α

(min length − timetable length).

We set α

= 1000 and α

= 10 to introduce a high

penalization on the number of clashes and number of

periods, respectively.

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

112

Table 8: Automatic solution for the LEETC examination timetable. The courses marked in bold face are shared with other

programs, as shown in Table 3. The number of clashes of this timetable is 215.

Sem. Course 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

ALGA x

Pg x

1st AM1 x

FAE x

ACir x

POO x

AM2 x

2nd LSD x

E1 x

MAT x

PE x

ACp x

3rd EA x

E2 x

SS x

RCp x

PICC/CPg x

4th PR x

FT x

SEAD1 x

ST x

RCom x

RI x

5th SE1 x

AVE x

SCDig x

SOt x

PI x

6th SCDist x

EGP x

OGE x

SG x

3.6 Local Exploitation

The Local Exploitation step employs a Local Search

procedure to improve locally some elements of the

population. First, 2N/4 groups of fours are formed

by randomly selecting into each group elements of

the population R

(t)

. Then, tournaments between el-

ements of each group are taken. The chromosome

which has the lower rank (the one who belongs to

the front with lower number) wins the tournament

and is then scheduled for the improve step. With

this procedure, about N/2 of the chromosomes are se-

lected for improvement. Also, it is guaranteed that at

least one element of the non-dominated front is se-

lected for improvement. The selected chromosomes

are improved locally using a short iteration Stochastic

Hill Climber (SHC) procedure, with objective func-

tion f

= minimization of the number of clashes. We

set a low temperature T in the SHC. In this way, our

SHC works like a standard Hill Climber but with only

one neighbour, instead of evaluating a whole neigh-

bourhood of solutions. The random neighbour is se-

lected according to the following operation. Firstly, a

clash list for the selected chromosome is built. Then,

the neighbour chromosome is the one which results

from applying the best move of an exam in the clash

list into a feasible period. The best move is the one

that leads to the highest decrease in the number of

clashes.

4 COMPUTATIONAL RESULTS

In our experiments we applied the proposed HMOEA

to the DEETC dataset speciﬁed in Section 2. Table 5

gives the algorithmic parameters and environmental

settings used in the experiments. The algorithm was

programmed in the Matlab language.

Firstly, we present the results on the performance

of HMOEA and then compare it with the available

manual solution. In the experiment made, the initial

period range was set to the interval [14,22], that is,

four periods below and upper the number of periods

SolvinganUncapacitatedExamTimetablingProblemInstanceusingaHybridNSGA-II

113

set in the manual solution. The performance of the

HMOEA in terms of the evolution of the Pareto is

illustrated in Figure 4. We can observe that the al-

gorithm converges rapidly as in iteration 25 it has al-

ready a complete ﬁrst front that is a good approxi-

mation of the ﬁnal Pareto front. After that iteration,

the individual solutions are further optimized but to

a lesser extent. The running time for this experience

(the best result out of ﬁve runs) was 296 seconds or ≈

5 minutes.

In Table 6 we compare the number of clashes per

course obtained by the manual and automatic (con-

sidering the obtained solution with 18 periods) pro-

cedures. As we can conclude from this table, the au-

tomatic solution improved the number of clashes in

all the timetables, which corresponds to a lower num-

ber of clashes in the optimized merged timetable. Ta-

bles 7 and 8 present the timetables for the most difﬁ-

cult program: the LEETC program. We can see that,

qualitatively, the timetable produced by the automatic

procedure has a reasonable layout as the exams within

the same semester are well distributed.

5 CONCLUSIONS

In this paper we solved a real instance of the exam

timetabling problem using a hybrid multi-objective

evolutionary algorithm. The instance considered

comprises ﬁve programs with high degree of course

sharing between programs, which difﬁcult the manual

construction of the timetable. In the manual elabora-

tion of the timetable actually ﬁve timetables are opti-

mized concurrently, one for each program. The auto-

matic algorithm solves this instance by optimizing the

combined timetable. With the application of the pro-

posed hybrid MOEA, the present instance was solved

effectively, with lower number clash conﬂicts com-

pared to the manual solution and in negligible time.

The current results were obtained without special ﬁne

tuning. Moreover, in experiences made, we obtained

lower number of clashes than the actual results, but

the optimization in each timetable was not balanced,

as some timetables were more optimized than others.

This is explained by the intrinsic difﬁculty in optimiz-

ing each timetable, e.g. the LEETC is more difﬁcult

to optimize than the the LERCM timetable, because

it has a greater number of shared courses and more

students registered on those courses.

5.1 Future Work

Several improvements could be made to the algo-

rithm. Some are listed next:

• In order to prevent the algorithm to optimize in

an unbalanced way, we could considering adding

has an objective a measure of program balance,

in order to guide the algorithm to prefer solutions

were the number of clashes is minimized and the

balance in programs is achieved.

• Consider room assignment, by solving the Capac-

itated ETTP.

Finally, in order to evaluate the performance of the

HMOEA, we intend to run the algorithm in the set of

ETTP benchmarks available - the Toronto and Not-

tingham benchmarks (Qu et al., 2009) - and compare

with other approaches.

REFERENCES

Burke, E., Bykov, Y., and Petrovic, S. (2001). A multicrite-

ria approach to examination timetabling. In Burke, E.

and Erben, W., editors, Practice and Theory of Auto-

mated Timetabling III, volume 2079 of Lecture Notes

in Computer Science, pages 118–131. Springer Berlin

/ Heidelberg.

Burke, E. and Landa Silva, J. (2005). The design of

memetic algorithms for scheduling and timetabling

problems. In Hart, W., Smith, J., and Krasnogor, N.,

editors, Recent Advances in Memetic Algorithms, vol-

ume 166 of Studies in Fuzziness and Soft Computing,

pages 289–311. Springer Berlin / Heidelberg.

Burke, E. K., Newall, J. P., and Weare, R. F. (1996). A

memetic algorithm for university exam timetabling. In

(Eds.), E. K. B. . P. R., editor, Lecture notes in com-

puter science: Proceedings of the 1st international

conference on the practice and theory of automated

timetabling, PATAT 1995, volume 1153, pages 241–

250, Edinburgh, Scotland. Berlin: Springer.

Cheong, C., Tan, K., and Veeravalli, B. (2009). A

multi-objective evolutionary algorithm for examina-

tion timetabling. Journal of Scheduling, 12:121–146.

e, P., Wong, T., and Sabourin, R. (2004). Application of

a hybrid multi-objective evolutionary algorithm to the

uncapacitated exam proximity problem. In eds): Pro-

ceedings of the 5th International Conference on Prac-

tice and Theory of Automated Timetabling (PATAT

2004), pages 151–167.

Deb, K. D. (2001). Multi-objective optimization using evo-

lutionary algorithms. Chichester: Wiley.

Deb, K. D., Pratap, A., Agarwal, S., and Meyarivan, T.

(2002). A fast and elitist multiobjective genetic algo-

rithm : NSGA-II. IEEE Transactions on Evolutionary

Computation, 6(2):182–197.

Ehrgott, M. and Gandibleux, X. (2008). Hybrid meta-

heuristics for multi-objective combinatorial optimiza-

tion. In Blum, C., Aguilera, M., Roli, A., and Sam-

pels, M., editors, Hybrid Metaheuristics, volume 114

of Studies in Computational Intelligence, pages 221–

259. Springer Berlin / Heidelberg.

IJCCI2012-InternationalJointConferenceonComputationalIntelligence

114

Moscato, P. and Norman, M. G. (1992). A ”memetic” ap-

proach for the traveling salesman problem implemen-

tation of a computational ecology for combinatorial

optimization on message-passing systems. In Pro-

ceedings of the International Conference on Parallel

Computing and Transputer Applications, pages 177–

186. IOS Press.

Mumford, C. (2010). A multiobjective framework for

heavily constrained examination timetabling prob-

lems. Annals of Operations Research, 180:3–31.

Petrovic, S. and Bykov, Y. (2003). A multiobjective op-

timisation technique for exam timetabling based on

trajectories. In Burke, E. and Causmaecker, P., edi-

tors, Practice and Theory of Automated Timetabling

IV, volume 2740 of Lecture Notes in Computer Sci-

ence, pages 181–194. Springer Berlin Heidelberg.

Qu, R., Burke, E. K., McCollum, B., Merlot, L. T. G., and

Lee, S. Y. (2009). A survey of search methodologies

and automated system development for examination

timetabling. Journal of Scheduling, 12:55–89.

Raidl, G. R. (2006). A uniﬁed view on hybrid metaheuris-

tics. In Hybrid Metaheuristics, pages 1–12.

Wong, T., C

e, P., and Sabourin, R. (2004). A hy-

brid MOEA for the capacitated exam proximity prob-

lem. In Congress on Evolutionary Computation, 2004.

CEC2004., volume 2, pages 1495 – 1501 Vol.2.

SolvinganUncapacitatedExamTimetablingProblemInstanceusingaHybridNSGA-II

115