Solving the Examination Timetabling Problem
with the Shuffled Frog-leaping Algorithm
Nuno Leite
1,2,∗
, Fernando Mel
´
ıcio
1,2
and Agostinho Rosa
2,3,†
1
ISEL - Lisbon Polytechnic Institute, Rua Conselheiro Em
´
ıdio Navarro, 1, 1959-007 Lisboa, Portugal
2
LaSEEB - System and Robotics Institute, Av. Rovisco Pais 1, TN 6.21, 1049-001 Lisboa, Portugal
3
Department of Bioengineering/IST, TU-Lisbon, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal
Keywords:
Examination Timetabling Problem, Toronto Benchmarks, Memetic Algorithm, Shuffled Frog-Leaping
Algorithm.
Abstract:
Shuffled Frog-Leaping Algorithm (SFLA) is a recently proposed memetic metaheuristic algorithm for solving
combinatorial optimisation problems. SFLA has both global and local search capabilities, and great conver-
gence speed towards the global optimum. Compared to a genetic algorithm, the experimental results show
an effective reduction of the number of evaluations required to find the global optimal solution. The Exami-
nation Timetabling Problem (ETTP) is a complex combinatorial optimisation problem faced by schools and
universities every epoch. In this work, we apply the Shuffled Frog-Leaping Algorithm to solve the ETTP. The
evolution step of the algorithm, specifically the local exploration in the submemeplex is adapted based on the
standard SFLA. The algorithm was evaluated on the Toronto benchmark instances, and the preliminary exper-
imental results obtained are comparable to those produced by state of art algorithms while requiring much less
time.
1 INTRODUCTION
Examination timetabling is an important practical
problem faced by schools and universities every
epoch. This problem, termed ETTP – Examination
Timetabling Problem, consists in scheduling exams
into a limited number of time slots and rooms sub-
ject to a set of constraints, providing that no students
should attend two or more exams at the same time. If
the room capacity is infinite, the ETTP is classified
as Uncapacitated ETTP, otherwise it is named Capac-
itated ETTP.
The ETTP belongs to the general class of
timetabling problems that includes other educa-
tional timetabling problems (e.g. school and course
timetabling), employee rostering, sports timetabling
and others. These problems, in general, belong to the
class of NP-complete problems, limiting the applica-
tion of optimal deterministic algorithms (e.g. Math-
ematical programming techniques or Dynamic Pro-
gramming) to simplified and small size problem in-
∗
Nuno’s work was partially supported by the FCT
SFRH/PROTEC/67953/2010 grant.
†
This work was also supported by the FCT Project PEst-
OE/EEI/LA0009/2011.
stances. In approaching real timetabling problems,
researchers often have investigated characteristics of
the problem that can be exploited in order to apply
a heuristic method capable of producing satisfactory
results. Proposed methods that combine algorithmic
strategies originating from the Operations Research
and Artificial Intelligence present the state-of-the-art
results.
Several approaches to solve the ETTP have been
proposed since the first works in the 1960’s decade.
In its survey, Carter (Carter, 1986) classified these ap-
proaches into four types: sequential methods, cluster
methods, constraint-based methods and generalised
search. Petrovic and Burke (Petrovic and Burke,
2004) later added more six categories: hybrid evolu-
tionary algorithms, metaheuristics, multi-criteria ap-
proaches, case based reasoning techniques, hyper-
heuristics and adaptive approaches. A recent and de-
tailed overview of the proposed methods to solve the
ETTP can be found in (Qu et al., 2009).
The Shuffled Frog-Leaping Algorithm (SFLA)
is a memetic metaheuristic proposed in 2003 (Eu-
suff and Lansey, 2003) (Eusuff et al., 2006). The
SFLA was applied to many areas and problems,
namely: solving TSP (Xue-hui et al., 2008), Cluster-
175
Leite N., Melício F. and Rosa A..
Solving the Examination Timetabling Problem with the Shuffled Frog-leaping Algorithm.
DOI: 10.5220/0004636001750180
In Proceedings of the 5th International Joint Conference on Computational Intelligence (ECTA-2013), pages 175-180
ISBN: 978-989-8565-77-8
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
ing (Amiri et al., 2009), Flow-shop Scheduling (Xu
et al., 2013), multiobjective optimization (Rahimi-
Vahed and Mirzaei, 2007), and others.
To the best of our knowledge, the application of
SFLA to the ETTP include only the work of Wang et
al. (Wang et al., 2009) (in chinese). The authors pro-
pose a Discrete SFLA (DSFLA) and apply it to the
ETTP. Solutions are encoded using a time permuta-
tion scheme suited to be manipulated by the DSFLA.
The algorithm is evaluated on four datasets of the Ca-
pacited Toronto benchmark (Toronto variant c in (Qu
et al., 2009)). In our work we present a novel appli-
cation of SFLA to the ETTP. Specifically, we adapt
the SFLA model by proposing a new evolution oper-
ator which is capable of exploring the search space
conveniently. The proposed algorithm is applied to
the complete set of Uncapacitated Toronto benchmark
data (Toronto variant b in (Qu et al., 2009)), and com-
pared with other techniques in the literature. The
original version (version I (Carter et al., 1996)) of
the datasets was evaluated. The ETTP mathematical
specification can be found in (Abdullah et al., 2010).
The rest of the paper is organized as follows. The
next section presents the mathematical model of the
SFLA followed by the adaptation to the ETTP in Sec-
tion 3. Experimental results of comparing the adapted
SFLA with other algorithms from the literature are re-
ported and discussed in Section 4. Finally, Section 5
present conclusions of the research undertaken and
discussion on the future work.
2 SFLA MODEL
In the SFLA, a population of F frogs, denoted
U(i), i = 1,. .. ,F, with identical structure, but dif-
ferent adaptation to the environment, is maintained.
The F frogs are divided into m substructures called
memeplexes, where they “search for food” (they are
optimized, in the algorithm sense) and meanwhile,
exchange information (exchange memes) with other
frogs, trying to reach the food localisation (global op-
timum). Each memeplex comprise n frogs, so that
F = mn. After searching locally in their memeplex,
the frogs are ranked and shuffled in order to go, even-
tually, to a different memeplex and exchange their
memes with the frogs located there (Figure 1). The
ranking comprises sorting the frogs in descending or-
der of performance. The partition of frogs is as fol-
lows. The first frog (the frog with the best fitness) in
the sorted list is allocated to the memeplex 1; the sec-
ond frog is allocated to the memeplex 2, and so on, so
that the frog m will go to memeplex m; then, the m +1
frog will go to memeplex 1 again, and the process is
repeated for the remainder frogs. The local search
employed corresponds to the so called Frog-Leaping
local search (Figure 2).
The ith frog fitness is denoted by f (i). To prevent
being trapped in a local optimum, a submemeplex of
size q < n is constructed in each memeplex, which
consists of frogs selected according to their perfor-
mance. For each submemeplex, P
b
and P
w
denote, re-
spectively, the best and worst frog. In the end of each
iteration of the Frog-Leaping local search, the worst
frog in the submemeplex is updated according to the
following rule:
S =
min
{
int[rand ∗(P
b
− P
w
)], S
max
}
,
for a positive step
max
{
int[rand ∗(P
b
− P
w
)], −S
max
}
,
for a negative step
(1)
U(q) = P
w
+ S (2)
where S denotes the updated step size of the shuffled
frog-leaping, rand represents a random number be-
tween (0,1) and S
max
is defined as the maximum step
size that any frog can take. The idea of this step is to
update the worst frog position towards the direction of
the best frog in the memeplex (or towards the direc-
tion of the global best frog, as indicated in Figure 2).
3 APPLYING SFLA TO ETTP
In this section we describe how the SFLA was adapted
in order to solve the ETTP.
3.1 Solution Representation
Each individual frog (solution) is represented by a
vector of dimension equal to the number of periods,
where each position contains the list of exams sched-
uled at that period. Figure 3 shows the graphical rep-
resentation of three solutions (the t
i
’s are the time slots
and the e
j
’s are the exams).
3.2 Initialisation of the Frog Population
The initial frog population is created using a construc-
tion algorithm which is based on the saturation de-
gree graph colouring heuristic (Carter and Laporte,
1996). To construct each solution, the approach be-
gins with an empty timetable and the most difficult
exams to insert (exams with the least number of avail-
able periods) should be selected next for insertion.
The remainder exams to be inserted have more feasi-
ble available periods, and so they are more easy to in-
sert. In this heuristic only the hard constraints are met
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
176
Start
Population size (F)
Counter of memeplexes (im)
Iteration within each memeplex (iN)
Number of memeplexes (m)
Generate population Pf randomly
Evaluate the fitness of Pf
Sort Pf in descending order
Partition Pf into m memeplexes
Local Search (A)
Shuffle the m memeplexes
Convergence
criteria satisfied?
Shuffle the m memeplexes
Shuffle the m memeplexes
Determine the best solution
End
No
Yes
Figure 1: SFLA steps (Illustration adapted from (Amiri
et al., 2009)).
when searching for feasible periods where to schedule
the exams.
3.3 Construction of Submemeplexes
As mentioned previously, the individual frogs in the
memeplex are selected to form a submemeplex ac-
cording to their fitness (Eusuff et al., 2006). The
selection strategy is to give higher weights to frogs
that have higher performance values and less weight
to those with lower performance values. The weights
are assigned following a triangular probability distri-
bution:
P
j
=
2(n + 1 − j)
n(n + 1)
, j = 1,2,...,n (3)
where n is the number of frogs in the memeplex.
As frogs within the memeplex have been previously
sorted in descending order of the fitness value, the
frog with the best performance has the highest prob-
ability p
1
= 2/(n + 1) of being selected for the sub-
memeplex, and the frog with the worst performance
has the lowest probability p
n
= 2/n(n + 1). As men-
tioned before, q distinct frogs are selected randomly
A
im=im+1 im=0
iN=0 iN=iN+1
Determine P
b
, P
w
and P
g
Apply Equations (1) and (2)
Is new frog better
than worst?
Apply Equations (1) and (2)
replacing P
b
by P
g
Is new frog better
than worst?
Generate a frog randomly
Replace worst frog
im=number
of memeplexes
iN=number
of iterations
No
Yes
Yes
No
Yes
Yes
No
No
Figure 2: Frog-Leaping local search. (Illustration adapted
from (Amiri et al., 2009).)
from n frogs in each memeplex to form the submeme-
plex.
3.4 Update the Worst Performance Frog
Inside each submemeplex, the worst performance
frog is updated according to the procedure depicted
on Figure 3. Part of this procedure is an adaptation of
the crossover operator of (Abdullah et al., 2010). As
can be seen from figure, this operator produces feasi-
ble solutions, so no special constraint-handling tech-
niques such as repairing schemes or penalty function
constraint handling are needed.
Executing the steps of the SFLA (Figure 2), the
new frog is going to replace the worst frog if it is bet-
ter than this last one. Otherwise, the procedure of Fig-
ure 3 is repeated but substituting P
b
by the global best
frog, P
g
. If the new frog doesn’t still improve over the
worst frog U(q), then a random solution is generated
as the new U(q), replacing the worst frog.
SolvingtheExaminationTimetablingProblemwiththeShuffledFrog-leapingAlgorithm
177
t
1
e
2
e
14
e
10
e
3
e
16
t
2
e
1
e
11
e
4
t
3
e
9
e
20
e
5
e
18
t
4
e
6
e
13
e
7
t
5
e
8
e
12
e
15
e
17
e
19
(a) Solution P
0
b
t
1
e
15
e
20
t
2
e
9
e
2
e
12
e
10
e
7
t
3
e
6
e
1
e
17
e
13
t
4
e
5
e
18
e
4
e
16
t
5
e
8
e
14
e
11
e
3
e
19
(b) Solution P
w
t
1
e
2
e
14
e
10
e
3
e
16
t
2
e
1
e
11
e
4
e
8
e
14
e
11
e
3
e
19
t
3
e
9
e
20
e
5
e
18
t
4
e
6
e
13
e
7
t
5
e
8
e
12
e
15
e
17
e
19
(c) New solution P
0
w
Figure 3: Worst frog improvement procedure. (a) First, we copy the best frog, P
b
, and eventually mutate it, producing P
0
b
.
The new frog is generated by inserting in P
0
b
, at a random period (shown shaded), exams chosen from a random period from
solution P
w
(as in (b)), producing the new frog P
0
w
(c). When inserting exams, some could be infeasible or already existing in
that time slot (respectively, the case of e
8
and e
11
in (c)). These exams are not inserted. The duplicated exams in the other
time slots are removed. Then this new solution P
0
w
is further mutated (according to some probability).
Some final details about the procedure illustrated
in Figure 3 are now explained. The frog P
0
b
is a copy
of the frog P
b
with the following mutation. Two peri-
ods of P
0
b
are selected randomly and the exams are
swapped between them. The mutation probability
used is mp
1
= 0.1. The number of periods from P
w
where exams are selected for insertion into P
0
b
is not
one (as depicted in the procedure, for sake of simplic-
ity), but three. Finally, the new solution P
0
w
is also
mutated with the same mutation operator described,
but the mutation probability is set to mp
2
= 0.05.
3.5 Generation of the Random Frog
To generate the random frog, needed when the new
frog is not better than the worst frog, we use the
construction method (based on the graph colouring
heuristic), described in Subsection 3.2.
4 EXPERIMENTAL RESULTS
In this section we present simulation results of the
SFLA on the Toronto benchmark instances. The al-
gorithm was implemented in the C++ language. The
parameters of SFLA are: Population size F = 6000,
Memeplex count m = 20, Memeplex size n = 300,
Submemeplex size q = n/10 = 30, Number of time
loops L = 15. Hardware and software specifications
are: Intel Core i7-2630QM, CPU @ 2.00 GHz × 8,
with 8 GB RAM; OS: Ubuntu 12.04, 32 bit; Compiler
used: GCC v. 4.6.3.
In the experiment carried out we’ve set the meme-
plex size, n, to be much greater than the submemeplex
size, q, in an effort for the algorithm be able to escape
from local optima. The parameter values were chosen
empirically, in a way to achieve a reasonable balance
between global and local exploration (Eusuff et al.,
2006). The number of time loops L is the number
of generations or the number of iterations used as the
stopping criteria in the algorithm (Figure 1).
Table 1 present a comparison of the SFLA and
other published algorithms. The best results are pre-
sented in bold. The SFLA is capable to find feasi-
ble timetables with penalty costs comparable to the
costs of solutions produced by state-of-the-art algo-
rithms. Figure 4 show the SFLA evolution for three
Toronto instances, namely, rye92, sta83, and yor83.
The penalty cost and time required to optimize in
the tested computer are also shown. With respect
to time limits, the fastest instance to be solved was
hec92, which took 56 seconds, and the slowest one
was pur93, which took 2784 seconds (≈ 46 min-
utes). Demeester et al. (Demeester et al., 2012),
which achieved the best results on several benchmark,
have set execution of the algorithm between 1 hour
and 12 hours, for the easier and difficult instances,
respectively. It is not possible to compare the algo-
rithms because the hardware used is different, but as
SFLA obtained the presented results in a very short
time, there’s still considerable processing time that
can be used for application of a more complex and
efficient approach.
IJCCI2013-InternationalJointConferenceonComputationalIntelligence
178
Table 1: Selection of the best results from the literature compared with the best obtained values from the SFLA approach.
The algorithms’s authors are: C96 (Carter et al., 1996), M03 (Merlot et al., 2003), B04 (Burke and Newall, 2004), Y05 (Yang
and Petrovic, 2005), B08 (Burke and Bykov, 2008), B10 (Burke et al., 2010), D12 (Demeester et al., 2012).
Instance SFLA C96 M03 B04 Y05 B08 B10 D12
car91 6.04 7.10 5.10 5.00 4.50 4.58 4.90 4.52
car92 5.08 6.20 4.30 4.30 3.93 3.81 4.10 3.78
ear83 37.31 36.40 35.10 36.20 33.71 32.65 33.20 32.49
hec92 11.38 10.80 10.60 11.60 10.83 10.06 10.30 10.03
kfu93 16.57 14.00 13.50 15.00 13.82 12.81 13.20 12.90
lse91 13.60 10.50 10.50 11.00 10.35 9.86 10.40 10.04
pur93 6.81 3.90 – – – 4.32 – 5.67
rye92 10.96 7.30 8.40 – 8.53 7.93 – 8.05
sta83 157.66 161.50 157.30 161.90 158.35 157.03 156.90 157.03
tre92 9.21 9.60 8.40 8.40 7.92 7.72 8.30 7.69
uta92 4.00 3.50 3.50 3.40 3.14 3.16 3.30 3.13
ute92 27.12 25.80 25.10 27.40 25.39 27.79 24.90 24.77
yor83 38.52 41.70 37.40 40.80 36.35 34.78 36.30 34.64
5 CONCLUSIONS
In the research undertaken we investigated the ap-
plication of the Shuffled Frog-Leaping Algorithm to
the examination timetabling problem. The SFLA is
a memetic metaheuristic with global and local search
capabilities, and providing a lower number of evalu-
ations of the fitness function compared to genetic al-
gorithms. The key issues in the ETTP are the feasible
exploration of the search space and minimisation of a
costly fitness evaluation.
The simple worst frog improvement method im-
plemented was able to search the solution space and
without disrupting heavily the new frogs found.
The preliminary results obtained in the Toronto
benchmark data indicate that the proposed approach
give results that are comparable to the ones obtained
by state-of-the-art algorithms. On some instances, the
algorithm converges prematurely and cannot progress
easily (e.g. yor83). In other instances (e.g. rye92 and
pur93), due to the characteristics of the dataset, for
the tested parameters, the population does not saturate
and the algorithm could improve the results further, if
more time was given. Another aspect to mention is
the time taken by the algorithm to solve the Toronto
instances. This time was between approximately 1
minute and 46 minutes for the easiest and difficult in-
stances, respectively. The times reported in literature
vary between 1 hour and 12 hours, respectively. Al-
though is not possible to compare algorithms because
the hardware is different, we think there’s a consider-
able time gap that could be used in order to apply a
more complex approach capable of generating better
results.
Our future work will be aimed to improve the
algorithm further by incorporating a mechanism for
maintaining the memeplex diversity, avoiding the
population saturation. Also, we intend to study and
implement an efficient neighbourhood or set of neigh-
bourhoods to be applied when creating the new so-
lutions. Finally, we intend to test the algorithm on
the International Timetabling Competition datasets
(ITC2007).
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0 50 100 150 200 250 300
9
10
11
12
13
14
15
16
17
18
Number of Generations
Penalty Cost
(a)
0 50 100 150 200 250 300
156
157
158
159
160
161
162
163
164
165
Number of Generations
Penalty Cost
(b)
0 50 100 150 200 250 300
38
39
40
41
42
43
44
45
46
47
48
49
50
Number of Generations
Penalty Cost
(c)
Figure 4: Convergence of the SFLA on the following
Toronto datasets (exact penalty cost achieved and total time
taken are also shown): (a) rye92 dataset (took 393 sec-
onds, obtaining a penalty cost of 10.96); (b) sta83 dataset
(took 66 seconds, obtaining a penalty cost of 157.66); (c)
yor83 dataset (took 189 seconds, obtaining a penalty cost
of 39.33).
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