Access to this full-text is provided by Wiley.
Content available from Journal of Optimization
This content is subject to copyright. Terms and conditions apply.
Research Article
A NNIA Scheme for Timetabling Problems
Yu Lei and Jiao Shi
School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China
Correspondence should be addressed to Yu Lei; leiy@nwpu.edu.cn
Received 29 December 2016; Revised 6 March 2017; Accepted 16 March 2017; Published 30 May 2017
Academic Editor: Linqiang Pan
Copyright © Yu Lei and Jiao Shi. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
is paper presents a memetic multiobjective optimization algorithm based on NNIA for examination timetabling problems. In
this paper, the examination timetabling problem is considered as a two-objective optimization problem while it is modeled as a
single-objective optimization problem generally. Within the NNIA framework, the special crossover operator is utilized to search
in the solution space; two local search techniques are employed to optimize these two objectives and a diversity-keeping strategy
which consists of an elitism group operator and an extension optimization operator to ensure a sucient number of solutions in the
pareto front. e proposed algorithm was tested on the most widely used uncapacitated Carter benchmarks. Experimental results
prove that the proposed algorithm is a competitive algorithm.
1. Introduction
e examination timetabling problem has long been a
challenging area for researchers in the elds of operational
research and articial intelligence, especially at the time that
the Toronto benchmark dataset was stated by Carter and
Laporte()[].eproblemhasbeenmoredicult
because universities are recruiting more students into a larger
variety of courses with a growing number of combined
degree courses [] (Merlot et al. ). In the past
years there are many methods that have been applied to
this problem. e represented techniques include constraint-
based techniques [], population-based techniques including
genetic algorithms [], graph coloring techniques [, ],
ant colony optimization [], scatter search [], local search
methods including tabu search [] and simulated annealing
[, ], variable neighborhood search [], and hybrid and
hyperheuristic approaches []. Generally, this problem is
modeled as a single-objective optimization problem; only the
number of clashes is considered by researchers.
To minimize the number of clashes in an exam timetable,
Burke and Newall () [] stated that the clashes can
be eliminated if a large number of periods were allocated.
Burke et al. () [] also stated that longer timetables
are needed to decrease the number of clashes. It is obvious
that the ETTP is denitely a two-objective optimization
problem:thenumberofclashesandthenumberofperiods.
Within the reasonable scope of the number of periods, the
number of clashes must be minimized as much as possible.
Hence, it is needed to minimize multiple conicting cost
functions, which can be best solved through the method
of multiobjective optimization [] that imported several
features from the research on the graph coloring problem and
used a variable-length chromosome representation that this
paper also adopts.
Evolutionary multiobjective optimization (EMO), whose
main goal is to handle multiobjective optimization problems
(MOPs), has become a hot topic in the eld of evolutionary
computation. By simultaneously optimizing more than one
objective, Multiobjective Optimization Evolutionary Algo-
rithms (MOEAs) can acquire a set of solutions considering
the inuence of all the objective functions. Each of those
solutions cannot be said better than the other and corre-
sponds to the tradeos between those dierent objectives.
Multiobjective examination timetabling problem as a MOP
has two contradictory objectives. e optimization of one
objective tends to minimize the number of clashes; the other
objective tends to decrease the number of time periods. Many
MOEAs have been proposed in recent years. Malim et al.
() [] studied three dierent Articial Immune systems
and indicated that the algorithms can be appropriate for
both course and exam timetabling problems. However, aer
Hindawi
Journal of Optimization
Volume 2017, Article ID 5723239, 11 pages
https://doi.org/10.1155/2017/5723239
Journal of Optimization
published they were found to represent a mistake in the code,
and it is invalid [].
Many of the existing methods for exam timetabling
problems are applicable to single-objective exam timetabling
problems. By calculation, these single-objective optimization
algorithmsonly can obtain one result, and the computational
eciency is poor. In this paper, we proposed a novel MOEA-
based approach for multiobjective examination timetabling
problem. By calculation, multiple results can be obtained
by our proposed algorithms. In order to simultaneously
optimize the two objectives, we adopt the framework of
multiobjective immune algorithm NNIA [] with some
modications. e NNIA simulates the phenomenon of the
multifarious antibodies symbiosis and a small number of
antibody’s activation in the immune response according
to a method of selecting the nondominated neighborhood
individuals. It chooses the small number of relatively isolated
individuals as active antibodies and clones according to the
crowding-distance value and then applies on the operators of
recombination and mutation to strengthen the searching of
the sparse area of the pareto front. e reasons we adopt the
frame of the NNIA are that NNIA is proposed by ourselves
andtheclonestrategycanmaketheparetofrontuniform
andgetthesatisedsolutions.emaincontributionsare
that we adopt elitism group strategy to keep the diversity of
the group and the two vertical local search operators to get
the optimized solutions. Experiments show that the proposed
algorithmisabletondasetoftradeosolutionsbetweenthe
two objectives.
e paper is organized as follows. Section describes
the background information with problem formulation and
related works. Section gives a description of the proposed
algorithm in detail. Section presents the experimental study.
Finally, conclusions are given in Section .
2. Background
2.1. Mathematical Model. As previously mentioned, the for-
mat of examination timetabling problems described in this
paper was rst formulated by Carter and Laporte [] in .
In this problem, a set of exams ={
1,2,...,|𝐸|}need to be
scheduledintoasetofperiods = {1,2,...,||} with each
period having a seating capacity . ere are three periods
per weekday and a Saturday morning period. No exam is
held on Sundays. It is assumed that the exam period starts
on a Monday. e problem can be formally specied by rst
dening the following:
min
|𝐸|−1
𝑖=1
|𝐸|
𝑗=𝑖+1
|𝑃|−1
𝑝=1 𝑖𝑝𝑗(𝑝+1)𝑖𝑗 ()
min ||()
sub.|𝐸|−1
𝑖=1
|𝐸|
𝑗=𝑖+1
|𝑃|−1
𝑝=1 𝑖𝑝𝑗𝑝𝑖𝑗 =0, ()
|𝑃|
𝑝=1𝑖𝑝 ≤1, ∀∈{1,...,||},()
where 𝑖𝑝 is one if exam 𝑖is allocated to period ;otherwise,
𝑖𝑝 equals zero. 𝑖𝑗 is the number of students registered for
exams 𝑖and 𝑗.
Equations () and () are the two objectives of minimizing
the number of clashes and timetable length, respectively.
Equation () is the constraint that no student is to be
scheduled to take two exams at any one time, while () states
that every exam can only be scheduled once in any timetable.
To evaluate the quality of one feasible timetable, a func-
tion evaluating the average cost for per student based on so
constraints has been proposed. It can be presented as follows:
tness =∑4
𝑠=0 𝑠×
𝑠
,()
where 𝑠=2
𝑠( = 0,1,2,3,4) is the weight that represents
the importance of scheduling exams with common students
either , , , , or timeslots away in one timetable and 𝑠
is the number of students involved in the violation of the so
constraint. is the total number of students in the problem.
For this reason that () emphasizes the most important
indicators, that is, whether the exams in the timetable are allo-
cated throughout the timetable equally, we use this function
as one of the objectives in our algorithm. e two objectives
of our algorithm optimized are described as follows:
min 1=||
min 2=∑4
𝑠=0 𝑠×
𝑠
.()
2.2. Related Works. e ETTP is a semiannual or annual
problem for colleges and is studied by many operational
researches widely due to its complexity and utility. ere have
been proposed a large range of approaches to solve the prob-
lem, discussed in the existing literature. ese approaches can
be classied into the following broad categories []: graph-
based sequential techniques, local search-based techniques,
population-based techniques, and hyperheuristics.
e graph coloring heuristics are one of the earliest
algorithms. Welsh and Powell [] in proposed a bridge
that is built between graph coloring and timetabling and
made a great contribution to the eld of the timetabling. e
ve ordering strategies which extended from graph coloring
heuristics on examination timetabling problems and a series
of examination timetabling problems were introduced by
Carter,Laporte,andLeein,calledUniversityofToronto
Benchmark Data. By developing two variants of selection
strategies, Burke et al. [] studied the inuence of bringing
a random element into the employment of graph heuristics
in . ese simple strategies showed improved pure graph
heuristics on the sides of both the quality and diversity of
the solutions when tested on three of the Toronto datasets.
Asmuni et al. [] in employed fuzzy logic to order
the exams to be scheduled on account of graph coloring
heuristics on the Toronto datasets and indicated that it is an
appropriate evaluation for arranging the exams. Corr et al.
[] investigated a neural network, the objective of which is
to arrange the most dicult exams at an early stageof solution
Journal of Optimization
construction. e work has showed the feasibility of using
neural network as a generally adaptive applicable technique
on timetabling problems.
e local search-based techniques represent a large por-
tion of the work which has appeared in the last decade [].
Mainly because various constraints can be handled relatively
easily, they have been applied on a variety of timetabling
problems. Di Gaspero and Schaerf () [] carried out
a valuable investigation on a family of tabu search based
techniques whose neighborhoods concerned those which
contributed to the violations of hard and so constraints.
Burke et al. [] investigated variants of Variable Neighbour-
hood Search and obtained the best results in the literature
across some of the problems in the Toronto datasets. Caramia
et al. [, ] developed a ne-tuned local search method
where a greedy scheduler assigned examinations into the
least timeslots and a penalty decrease improved the timetable
without increasing the number of timeslots.
e genetic algorithm is one of the most typical repre-
sentatives of the population-based techniques. It is noticed
that the algorithm has a good performance in the literatures.
Particularly, the hybridizations of genetic algorithms with
local search methods, memetic algorithms, have an excel-
lent performance in this area. In , Corne et al. []
introduced genetic algorithms to solve general educational
timetabling problems. e function of this work is that
certain problem structures in some particularly generated
graph coloring problems cannot be handled by obtaining
direct representation in the genetic algorithms. Ross et al. []
in indicated that by testing on specially generated graph
coloring problems of dierent homogeneity and connectivity
the transition regions were existent in solvable timetabling
problems.estudycanmakeresearchersunderstandhow
dierent algorithms perform on complex timetabling prob-
lems. Terashima-Marin et al. [] in indicated a clique-
basedcrossoveronthetimetablingproblemswhichwas
turned into graph problems. Erben [] () indicated
a grouping genetic algorithm with appropriate encoding,
crossover and mutation operators, and tness functions
studied. is method requires less computational time than
someofthemethodsintheliterature.BurkeandLandaSilva
[] discussed some issues concerning the design of memetic
algorithms for scheduling and timetabling problems. Burke
et al. [] developed a memetic algorithm to reassign single
exams and sets of exams and employed light and heavy
mutation operators. However, neither of these mutations on
their own improved the solution quality. Malim et al. []
developed three variants of Articial Immune systems and
indicated that the algorithms can be suitable for course and
exam timetabling problems. However, there was a problem
in the results; aer publication they were showed to represent
an error in the code and invalidness.
More and more researchers pay attention to the hyper-
heuristics approach. In Ahmadi et al. [] investigated
a variable neighborhood search, aiming to nd good com-
binations of heuristics for dierent examination timetabling
problem. Kendall and Hussin [] in developed a tabu
search hyperheuristic; they adopted moving strategies and
constructive graph heuristics to be the low level heuristics.
Start
Popul ation
initialization
Crossover and
mutation
Nondominated sorting
Local search operators
Itermax?
End
Ye s
Clone
No
Diversity-keeping
operator
F : e ow of algorithm.
InBurkeetal.[]researchedobtainingtabusearchto
nd sequences of graph heuristics to construct solutions for
timetabling problems and considered the eects of various
numbers of low level graph heuristics on the examination
timetabling problems. By conducting an empirical study on
both benchmark functions and exam timetabling problems,
Bilgin et al. [] () studied heuristic selection meth-
ods and acceptance criteria within a hyperheuristic. e
memetic algorithm hyperheuristic with a single hill climber at
a time showed that it performed better on approaches tested.
For the interested readers, more details can be referred from
[].
In summary, during the recent years, there are an
increasing number of excellent algorithms; almost all of these
algorithms were tested on either benchmark datasets or in
real applications, which had made quite good achievements.
In this paper, we also proposed a multiobjective optimization
algorithm, called Nondominated Neighbor Immune Algo-
rithm (NNIA) in []. NNIA adopts an immune inspired
operator, a nondominated neighbor-based selection tech-
nique, two heuristic search operators, and elitism. It indicates
that NNIA is an eective method for solving MOPs by a
number of experiments. Due to its good performance, we
will adopt the framework of NNIA with some modications,
which will be described in the following section. e con-
tribution of this paper is that we solve this task by using
multiobjective optimization technique.
3. The Proposed Algorithm
3.1. Algorithmic Flow of MOEA Based on NNIA. e algo-
rithmic ow of our algorithm is presented in Figure . At
the start of the algorithm, a conict matrix was created
according to Burke and Newall [], which has dimensions ||
by || with the denition 𝑖𝑗 from Section . being its (,)th
Journal of Optimization
element. is matrix can check and eliminate the conicts in
the timetables eciently.
Becauseofthelarge-scaleindividualofthepopulation,
itisinadvisabletosearchinthenormalpopulation.Elitism
strategy and crowded selection optimization mechanism are
put forward. In our algorithm we adopt elitism strategy two
times to reduce the computation burden and extend the
range of nondominated solutions in elitism group. In the
normal population, the children population aer crossover
and mutation is mixed with parent population. en the
nondomined solutions of this new population are put into
the elitism group. e purpose of the strategy is to oer more
nondominated solutions to the elitism group.
3.2. Hyper-Heuristic Initialization. As a common step, initial-
ization is to produce an initial population. In our algorithm,
there are a large number of feasible solutions that need
to be optimized, but the process of generating the feasible
solutions is hard for most of the examination timetabling
problems. e diculty level that feasible solutions generated
for dierent examination timetabling problems is dierent. It
is hard to make some unfeasible solutions be the feasible ones
with some conventional ways in the subsequent operation.
e result of the algorithm is inuenced by the number of
feasible solutions in the initial population for some issues.
e initialization in our algorithm produces a set of solutions
randomly and updates the random solutions to be the feasible
ones with the simple genetic algorithm. e details are shown
in the following.
e hyperheuristic initialization is that the exams are
selected for insertion with the help of some heuristic infor-
mation when used in the graph coloring problem [, ]. e
heuristics we use in this paper are as follows:
() Largest degree (LD): exams with the largest number
of conicts with other exams are inserted rst.
() Largest weighted degree (LWD): it is the same as LD
but weighted by the number of students involved.
() Saturation degree (SD): exams with the fewest valid
timeslots, in terms of satisfying the hard constraints,
remaining in the timetable are inserted rst.
ere are two terminated conditions, maximum iteration
number and the maximum number of feasible solutions,
which can make the whole algorithm achieve the stable
result for most examination timetabling problems. But our
algorithm has obvious superiority for the problems that the
feasible solutions are hardly generated, because of this strat-
egy of the initialization. e advantage of the hyperheuristic
initialization is that we can get the timetables with the lengths
being close to the demands of the users. e result can be seen
in the next section.
3.3. Local Search Operators. Some researchers indicated that
adopting local search within evolutionary algorithms is
an much eective approach for nding high quality exam
timetables which can also contribute to the intensication of
the optimization results [, ]. A description of the two-
direction local search operators this paper adopted is given
below.
e rst kind is to minimize the timetable length as far as
possible without concerning the conict number, aiming at
the operator in the local search to minimize the time periods
among the nondominated elitism group.
e selected individual is , the search depth is SD, the
maximum mutation probability is max, the original mutation
probability is ori,theprobabilityincreasingsteplengthis,
the exam timetable is , the maximum iteration number is
Itermax, and the number of examinations is num.
Step 1. Set the original mutation probability ori.
Step 2. Set search depth variable sign =0.
Step 3. Randomly select ori ×
numexams in ,whichis
denoted as 𝑚. Delete all the selected exams in .
Step 4. Rearrange the exams in 𝑚according to the maxi-
mum number of conicts and the timeslots in according to
the heuristics. en insert the exams in 𝑚into the timetable
; if you can not insert them into the inherent timeslots,
extend the timeslots until all exams are arranged; the newly
produced timeslots are new.
Step 5. If the number of new is less than the number of ,
replace with new,andstop;otherwise,=+1,andgo
to Step .
Step 6. If =depth, then ori =
ori +andgotoStep.
Step 7. If ori =
max, compare the timeslots of new and
;thenputthesmalleroneintotheelitistgroupandstop;
otherwise, go to Step and go on.
e second kind aims at minimizing the number of
conicts without concerning the number of timeslots. e
details are described below.
e selected probability is , the search depth is SD, the
maximum selected probability is max, the original mutation
probability is ori,theprobabilityincreasingsteplengthis,
the exam timetable is , the maximum iteration number is
Itermax, and the number of examinations is num.
Step 1. Set the original examination selected probability
ori =.
Step 2. Set search depth variable sign =0.
Step 3. Randomly select ori ×
numexams in and the
newly set temp.
Step 4. Rearrange exams in temp according to the number
of conicts, delete all the exams in temp,andsortallthe
timeslotsinthechangedtimetable.
Step 5. Insert the exams in temp into the timetable at the
premise of not aecting the number of timeslots; the newly
produced individual is new.
Step 6. Compare the number of conicts in new with
which, in according to the formulation, if the former is
Journal of Optimization
Clashes
Timeslots
B
A
i
F : Elitism group local search.
smaller, replace original individuals with new and then stop.
Otherwise, =+1;gotoStep.
Step 7. If =SD, judge the number of conicts in new and
; if the former is smaller, replace with new ;putthesmaller
one into the elitist group; otherwise, =+;gotoStep.
Step 8. If =
max, judge the number of conicts in new
and ; if the former is smaller, replace with new ;thenput
the smaller one into the elitist group; otherwise, remain
unchanged.
3.4. Diversity-Keeping Operators
3.4.1. Elitism Group Strategy. Although the local search can
intensify the optimization results, the discrete optimization
is dierent from the continuous optimization that a small
disturbanceindecisiondomainmayprobablyletindividuals
transform irregularly and even result in deterioration. As
such, in order to avoid this phenomenon we put forward a
novel local search exploitation with an extra elitism group to
save nondominated solutions in every generation. However,
normal population is just oering a space of updating the
nondominated solutions. In our algorithm we also introduce
a corresponding elitism strategy and a crowded selection
optimization mechanism; the details will be introduced in the
next part.
elocalsearchoperatorsareappliedaerthestrategy
of extension and optimization of the elitism group. e
operators avoid an objective in an individual deterioration
and then minimize the other objective and get the new elitist
solutions mixed with original elitist solutions to conduct
a nondominated sorting. e frontier may extend to the
two dierent directions as far as possible according to the
operators.elocalsearchissearchingverticallybetweentwo
objectives in order which is shown as Figure . It indicates that
one objective is optimized prior and then the other is shown
as rout A or rout B of the individual in the gure.
Clashes
Timeslot
123457
AB
i+1
i
i
F : e computation of congestion degree.
3.4.2. Extension Optimization Strategy. Due to the normal
selection and mutation operators making a little contribution
to the nondominated elitism group, we present a strategy
to extend and optimize the elitism group based on the
congestion degree which is shown in Figure . Compute the
dierence of timetable length between every individual and
its right side one. If the -value is , then extend a time period
of this individual and randomly select some exams into the
time period. We can get a uniform frontier by this means. As
is shown in Figure , the crowding degree of individual is
the length of the line segment AB. Assume that the crowding
degree of individual is ; according to our theory, expend
the individual and get andthenputitintotheoriginal
elitism group. Finally, we take local search operators to the
generations aer extending the time periods.
4. Experimental Analysis
Our algorithm is programmed in Matlab and simulations
are performed on the . GHz Core Personal Computer. We
use uncapacitated benchmark examinations timetabling
datasets proposed by Carter and Laporte [] to evaluate the
eectiveness of our algorithm. e details of the proposed
benchmarks are shown in Table . As no dataset is designed
to evaluate the multiobjective timetabling algorithms, we just
use the datasets used in the single objective to evaluate the
feasibility and reasonableness of our algorithm. e param-
eter settings are presented in Table . e population size is
and the maximum iteration is . e other parameters’
choosing reason will be described in the following parts.
In the following sections, we will study our algorithm
in two sides. One is to make analysis on the contribution
of diversity-keeping local search operators searching in two
dierent directions orderly which is shown with four compar-
ative experiments. e other one is to discuss the contribution
of elitism group strategy applied on our algorithm two times
with four experiments presented.
Journal of Optimization
3
4
5
6
7
8
9
10
11
Yor83
withE withoutE
3
4
5
6
7
8
9
10
Tre91
withE withoutE
4
5
6
7
8
9
10
11
withE withoutE
Sta83
1
2
3
4
5
6
withE withoutE
Ute92
withE withoutE
Lse91
8
7
6
5
4
31
2
3
4
5
6
7
8
withoutE
Kfu93
withE
2
3
4
5
6
7
8
withE withoutE
Hec92
3
4
5
6
7
8
9
10
withE withoutE
Ear83
F : Performance comparison for MOEA with and without diversity-keeping strategy.
Journal of Optimization
118
114
110
106
102
98
none doublewithSwithF none doublewithSwithF
Yor83
90
85
80
75
70
Ute92
28
27
26
25
24
23
22
Tre91
510
490
470
450
430
Sta83
56
54
52
50
48
46
44
42
Lse91 Kfu93
1550
1500
1450
1400
1350
1300
1250
80
78
76
74
72
70
68
Hec92
230
225
220
215
210
205
200
Ear83
none doublewithSwithF none doublewithSwithF
none doublewithSwithF none doublewithSwithF
none doublewithSwithF none doublewithSwithF
F : Performance comparison for MOEA with dierent local search settings.
Journal of Optimization
T : Characteristics of datasets.
Dataset Timeslots Exams Students Conict density
Car .
Car .
Ear .
Hec .
Kfu .
Lse .
Rye .
Sta .
Tre .
Ute .
York .
T : Parameter setting for simulation study.
Parameter Value
Population size
: the number of exams scheduled by each heuristic
𝑐: the crossover probability .
𝑚: the mutation probability .
SD: the search depth of the local search
Itermax: the maximum iteration number
4.1. Contribution of Diversity-Keeping Strategy. is section
presents the performance of the diversity-keeping operators.
To assess the eectiveness of the strategies, a comparison was
conducted as in Figure . is saying that the algorithm
runs with the strategies while is indicating that
the algorithm runs without the strategies. e boxplots
were drawn according to the statistic number of solutions.
e eight data in this experiment were running ten times
independently. From Figure , it can be observed obviously
that the operator of does better than does
in every dataset. e results well proved the eciency of
diversity-keeping operator.
4.2. Contribution of Local Search Operators. In this section,
we use the hypervolume as an indicator to estimate the eec-
tiveness of the algorithm; in our comparable experiments
the indicator 𝑝isthemaximumvalueofthedatasettobe
compared in all dimensions.
To prove the eciency of the proposed two local search
operators, this section shows the performance of the algo-
rithms with and without local search operators. As is shown
in Figure , is the setting that does not use local search
at all, and is the setting with the local search aiming
at minimizing the conict number, while the setting is
the local search to minimize the timeslots and incor-
porates two local search operators. Ten independent runs of
thefoursettingsareconductedtoobtainstatisticalresults.
From Figure , the contribution of local search operators is
obvious, since the operators which use two local search are
able to generate solutions with signicantly lower number
of clashes. For the hypervolume value of the nondominated
yor83 ute92 tre91 sta83 lse91 kfu93 hec92 ear83
Number of solutions
3
4
5
6
7
8
9
10
11
F : Number of pareto optimal solutions for the datasets.
solutions of datasets Ute, Sta, Lse, Kfu, Hec, and
Ear, the application of the two local search settings shows
better results than other three settings.
From the statistic boxplots we can see that our algorithm
is robust to the indicator of the students conict numbers.
e outliers are few and the dierences between the highest
and the lowest values are small, which also demonstrate the
robustness of our algorithm.
4.3. Performance of Multiobjective Algorithm Based on NNIA.
is section presents the multiobjective optimization perfor-
mance of the algorithm based on NNIA. On top of showing
the advantages of our algorithm, the role of the two objectives
will be validated as follows. e experiment was conducted
running ten times independently.
e boxplots in Figure show the number of pareto
optimal solutions of the eight datasets. e number of pareto
optimal solutions for the dataset 92 is nearly four. e
number of pareto optimal solutions for other datasets is
almost distributed from seven to ten.
Experiments were conducted to further show the results.
From Figure , we can see the details of both the num-
ber of periods and its corresponding clashes. All of the
datasets tested perform good, the pareto optimal solutions
are distributed uniformly, and the clashes are relatively small.
Journal of Optimization
35
10 15 20
20
25
30
35
Number of periods
Number of clashes
Ute92
20 25 30 35 40
20
30
40
50
60 York 8 3
Number of clashes
Number of periods
20 25 30 35 40
7
8
9
10
11 Tre92
Number of clashes
Number of periods
10 15 20 25
80
100
120
140
160
180 Sta83
Number of clashes
Number of periods
15 20 25 30
10
12
14
16
18 Lse91
Number of clashes
Number of periods
20 25 30 35
12
14
16
18
20
22 Kfu93
Number of periods
Number of clashes
15 20 25 30
8
10
12
14
16
18 Hec92
Number of clashes
Number of periods
20 25 30 35 40
30
35
40
45 Ear83
Number of clashes
Number of periods
F : Pareto optimal solutions for the datasets.
Journal of Optimization
e experiments support our algorithm powerfully and the
multiobjective exam timetabling problem is solved well.
5. Conclusions
In this paper, the exam timetabling problem has been
regarded as a multiobjective optimization problem which
involves the minimizing of the number of clashes and number
of periods in a timetable. A multiobjective evolutionary algo-
rithm, based on NNIA, featured with elitism group strategy,
congestion degree based on extension optimization strategy,
and two local search operators, has been presented.
e proposed MOEA is dierent from most existing
single-objective-based methods in the fact that it optimizes
twoobjectivesconcurrentlyandgetasetofsolutionsreason-
able instead of producing single-length timetables. It has been
proved that such an approach is more universal and would
be able to function eectively. e results also show that the
algorithm can generate relatively short clash-free timetables
and various solutions which are convenient for deciders to
choose on their own preference.
e work we do in this paper focuses on the temporal
aspect of the ETTP, which has solved the problem well in
a sense. However, it still has some shortcomings, how to
balance the diversity and approximation, which can be sub-
jected for future study.
Conflicts of Interest
e authors declare that they have no conicts of interest.
References
[] M. W. Carter and G. Laporte, “Recent developments in practical
examina-tion timetabling,” in Proceedings of the 1st International
Conference on Practice and eoryof Automated Timetabling,E.
K. Burke and P. Ross, Eds., vol. of Springer Lecture Notes in
Computer Science,pp.–,.
[] L.T.G.Merlot,N.Boland,B.D.Hughes,andP.J.Stuckey,“A
hybrid algorithm for the examination timetabling problem,” in
Proceedings of the 4th Internationalconference on the Practice
and eory of Automated Timetabling (PATAT ’02),K.Burkeand
P. De Causmaecker, Eds., vol. of Lecture Notes in Computer
Science, pp. –, Springer, Berlin, Germany, .
[] T. M¨
uller, “ITC solver description: a hybrid approach,”
Annals of Operations Research,vol.,no.,pp.–,.
[] P. Ross, E. Hart, and E. Corne, “Some observations about
GAbasedexam timetabling,” in Proceedings of the 2nd Inter-
national Conference on Practiceand eory of Automated
Timetabling,E.K.BurkeandM.W.Carter,Eds.,vol.of
Lecture Notes in Computer Science, pp. –, Springer, .
[] E.K.Burke,B.McCollum,A.Meisels,S.Petrovic,andR.Qu,“A
graph-based hyper-heuristic for educational timetabling prob-
lems,” European Journal of Operational Research,vol.,no.,
pp.–,.
[] N.R.Sabar,M.Ayob,G.Kendall,andR.Qu,“Roulettewheel
graph colouring for solving examination timetabling problems,”
in Proceedings of the 3rd International Conference on Com-
binatorial Optimization and Applications,vol.ofLecture
Notes in Comput. Sci.,pp.–,Springer,.
[] M. Eley, “Ant algorithms for the exam timetabling problem,” in
Proceedings of the International Conference on the Practice and
eory of Automated Timetabling (PATAT ’07),E.K.Burkeand
H. Rudova, Eds., vol. of Lecture Notes in Computer Science
(2007),pp.–,Springer.
[] N. Mansour, V. Isahakian, and I. Ghalayini, “Scatter search
technique for exam timetabling,” Applied Intelligence,vol.,
no. , pp. –, .
[] G. De Smet, “ITC—Examination Track,” in Proceedings of
the Practice and eory of Automated Timetabling (PATAT ’08),
Montreal, Canada, .
[] E. Burke, Y. Bykov, J. Newall, and S. Petrovic, “A time-predened
local search approach to exam timetabling problems,” IIE
Transa cti ons, vol. , no. , pp. –, .
[] J. M. ompson and K. A. Dowsland, “A robust simulated
annealing based examination timetabling system,” Computers
and Operations Research,vol.,pp.–,.
[] R. Qu and E. K. Burke, “Hybrid variable neighbourhood hype-
heuristics for exam timetabling problems,” in Proceedings of the
MIC2005: e Sixth Meta-Heuristics International Conference,
Vienna , Austria, .
[] E. K. Burke and J. P. Newall, “Solving examination timetabling-
problems through adaptation of heuristic orderings,” Annals of
Op-erational Research,vol.,pp.–,.
[] E. K. Burke, D. G. Elliman, P. H. Ford, and R. F. Weare,
“Specialisedrecombinative operators for the timetabling prob-
lem,” in Evolutionary Computing: AISB Workshop,pp.–,
Springer, Sheeld, UK, .
[] C. Y. Cheong, K. C. Tan, and B. Veeravalli, “A multi-objective
evolutionary algorithm for examination timetabling,” Journal of
Scheduling,vol.,no.,pp.–,.
[] M. R. Malim, A. T. Khader, and A. Mustafa, “Articial immune-
algorithms for university timetabling,” in Proceedings of e 6th
International Conference on Prac-Tice and eory of Automated
Timetabling,E.K.BurkeandH.Rudova,Eds.,pp.–,
Brno, Czech Republic, August .
[] R.Qu,E.K.Burke,B.McCollum,L.T.Merlot,andS.Y.Lee,“A
survey of search metho dologies and automated system develop-
mentfor examination timetabling,” Journal of Scheduling,vol.
,no.,pp.–,.
[] M. Gong, L. Jiao, H. Du, and L. Bo, “Multiobjective immune
algorithm with nondominated neighbor-based selection,” Evo-
lutionary Computation,vol.,no.,pp.–,.
[] E. K. Burke, J. Kingston, and D. de Werra, “Applications to
timetabling,” in Handbook of Graph eor y,J.GrossandJ.
Yellen,Eds.,pp.–,ChapmanHall,London,UK,.
[] D. J. A. Welsh and M. B. Powell, “An upper bound for the
chromatic number of agraph and its application to timetabling
problems,” e Computer Journal,vol.,no.,pp.-,.
[] E. K. Burke, J. P. Newall, and R. F. Weare, “A memeticalgo-
rithm for university exam timetabling,” in Proceedings of the
1st International Conference on the Practice and eory of
Automated Timetabling (PATAT ’95),E.K.BurkeandP.Ross,
Eds., vol. of Lecture Notes in Computer Science,pp.–,
Springer, Edinburgh, Scotland, .
[] H. Asmuni, E. K. Burke, J. Garibaldi, and B. McCollum,
“Fuzzy multipleordering criteria for examination timetabling,”
in Proceedings of the 5th International Conference on Practice
and eory of Automated Timetabling,E.K.BurkeandM.Trick,
Eds.,vol.ofSpringer Lecture Notes in Computer Science,pp.
–, .
Journal of Optimization
[] P. H. Corr, B. McCollum, M. A. J. McGreevy, and P. McMullan,
“A newneural network based construction heuristic for the
examination timetablingproblem,” in Proceedings of the Inter-
national Conference on Parallel Problem Solving From Nature
(PPSN), pp. –, Springer, Reykjavik, Iceland, September
.
[] L. Di Gaspero and A. Schaerf, “Tabu search techniques for
examination timetabling,” in Proceedings of the 3rd International
Conference on Practice and eory of Automated Timetabling III,
E. K. Burke and W. Erben, Eds., vol. of Lecture Notes in
Computer Science, pp. –, Springer, Berlin, Germany, .
[] M. Caramia, P. DellOlmo, and G. F. Italiano, “New algorithms
for examination timetabling,” in Proceedings of the 4th Inter-
national Workshop, on Algorithm Engineering (WAE 2000),
S. Naher and D. Wagner, Eds., vol. of Lecture Notes
in Computer Science, pp. –, Springer, Berlin, Germany,
.
[] M. Caramia, P. Dell’Olmo, and G. F. Italiano, “Novel local-
search-based approaches to university examination timetabl-
ing,” INFORMS Journal on Computing,vol.,no.,pp.–,
.
[] D. Corne, P. Ross, and H. Fang, “Evolutionary timetabling:
Practice, prospects and work in progress,” in Proceedings of the
UK Planning and Scheduling SIG Workshop,P.Prosser,Ed.,.
[] P. Ross, D. Corne, and H. Terashima-Marin, “e phase
transition niche for evolutionary algorithms in timetabling,” in
Proceedings of the 1st International Conference on Practice and
eory of Automated Timetabling,E.K.BurkeandP.Ross,Eds.,
vol. of Lecture Notes in Computer Science, pp. –,.
[] H. Terashima-Marin, P. Ross, and M. Valenzuela-Rendon,
“Clique-based crossover for solving the timetabling problem
with GAs,” in Proceedings of the Congress on Evolutionary
Computation, CEC 1999,pp.–,Washington,DC,USA,
July .
[] W. Erben, “A grouping genetic algorithm for graph colouring
and examtimetabling,” in Proceedings of the 3rd International
Conference on Practice and eory of Automated Timetabling,
E. K. Burke and W. Erben, Eds., vol. of Lecture Notes in
Computer Science, pp. –, Springer, Berlin, Germany, .
[] E. K. Burke and J. D. Landa Silva, “He design of memetic algo-
rithms forscheduling and timetabling problems,” in Proceedings
of the Recent Advances in Memetic Algorithms and Related
Search Technologies. Studies in Fuzziness and So Computing
166,W.E.Hart,N.Krasnogor,andJ.E.Smith,Eds.,pp.–
, Springer, Berlin, Germany, .
[] S. Ahmadi, R. Barone, P. Cheng, P. Cowling, and B. McCollum,
“Erturbation based variable neighbourhood search in heuristic
space for examination time tabling problem,” in Proceedings
of the Multidisciplinary International Scheduling: eory and
Applications (MISTA ’03), pp. –, Nottingham, UK, August
.
[] G. Kendall and N. M. Hussin, “A tabu search hyper-heuristic
approach to the examination timetabling problem at the MARA
university of technology,” in Proceedings of the 5th International
Conference on Practice and eory of Automated Timetabling,
E. K. Burke and M. Trick, Eds., vol. of Lecture Notes in
Computer Science, pp. –, .
[] B. Bilgin, E. Ozcan, and E. E. Korkmaz, “An experimental study
on hyper-heuristics and exam timetabling,” in Proceedings of the
6th International Conference on Practice and eory of Auto-
mated Timetabling,E.K.BurkeandH.Rudova,Eds.,vol.of
Lecture Notes in Computer Science, pp. –, Springer, .
[] E. K. Burke, D. G. Elliman, P. H. Ford, and R. F. Weare, “Exami-
nation timetabling in British universitiesa survey,” in Proceed-
ings of e 1st International Conference on the Practice and
eory of Automated Timetabling (PATAT ’96),E.K.Burkeand
P. Ross, Eds., vol. of Lecture Notes in Computer Science,pp.
–, Springer, Edinburgh, Scotland, .
[]Y.Lei,M.Gong,L.Jiao,W.Li,Y.Zuo,andQ.Cai,“A
double evolutionary pool memetic algorithm for examination
timetabling problems,” Mathematical Problems in Engineering,
vol.,ArticleID,pages,.
[] Y.Lei,M.Gong,J.Zhang,W.Li,andL.Jiao,“Resourceallo-
cation model and double-spherecrowdingdistanceforevo-
lutionary multi-objective optimization,” European Journal of
Operational Research,vol.,no.,pp.–,.
Available via license: CC BY 4.0
Content may be subject to copyright.
