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Artif Intell RevDOI 10.1007/s10462-013-9399-6
Review of state of the art for metaheuristic techniquesin
Academic Scheduling Problems
Chong Keat Teoh Antoni Wibowo Mohd Salihin Ngadiman
Springer Science+Business Media Dordrecht 2013
Abstract The Academic Scheduling Problems have drawn great
interest from manyresearchers of various fields, such as
operational research and artificial intelligence. Despitethe long
history of literature, the problem still remains as an interesting
research topic as newand emerging metaheuristic techniques continue
to exhibit promising results. This paper sur-veys the properties of
the Academic Scheduling Problems, such as the complexity of
theproblem and the constraints involved and addresses the various
metaheuristic techniques andstrategies used in solving them. The
survey in this paper presents the aspects of solutionquality in
terms of computational speed, feasibility and optimality of a
solution.
Keywords Academic scheduling problem Course scheduling Exam
scheduling Hyper-heuristics Metaheuristic Scheduling
Timetabling
1 Introduction
This paper is written as a continuity from the previous works of
Lewis (2007) who performedan in-depth survey on the
metaheuristic-based techniques for Academic Scheduling Prob-lems.
Essentially, scheduling is defined as the allocation of resources
over time to performa collection of tasks (Baker 1974) and the
objective is to assign a set of entities to a limitednumber of
resources over time, in such a way to meet a set of pre-defined
schedule require-ments. In recent years, a noticeable pattern is
observed in the area of academic schedulingwhere many complex
problems are efficiently solved using the principles of
meta-heuristics.
C. K. Teoh (B) A. Wibowo M. S. NgadimanFaculty of Computer
Science and Information Systems, Universiti Teknologi Malaysia
(UTM),81310 Johor Bahru, Johor, Malaysiae-mail:
[email protected]; [email protected]
A. Wibowoe-mail: [email protected]
M. S. Ngadimane-mail: [email protected]
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C. K. Teoh et al.
Academic scheduling problem is regarded as both a
non-deterministic polynomial-timehard (NP-hard) and
non-deterministic polynomial-time complete (NP-complete)
problem,meaning that the computational time increases exponentially
as the problem size grows(Bardadym 1996). In general, Academic
Scheduling Problems can be classified into twodistinct typeswhich
are either examor course timetabling (Chaudhuri andDe 2010). In
coursescheduling, the ultimate objective is to optimally assign
lecturers to a particular period of timeto teach a particular
course, with regards to the specific constraints placed by the
organizationwhile the objective of examscheduling is tomaximize the
timegapbetween exams.Accordingto Zhipeng and Jin-Kao (2010), course
timetabling can be further separated into 2 categorieswhich are
post-enrolment (post-graduate) based, where scheduling of courses
is based onthe students enrolment data while curriculum based
(undergraduate) is based on the coursesoffered by the university.
Identical to other scheduling problems, its objective is to
assignfrom a limited amount of resources (lecturers, rooms, etc.)
over a period of time (time periodsto a day) to perform a set of
tasks (lectures) and this is one of the most common issues facedin
every institute of education (Baker 1974; Omar et al. 2003).
The nature of the problem can be said to be highly constrained
due to its large size, varietyof variables and subjected under
large amount of constraints, which may differ from oneinstitution
to another (Pongcharoen et al. 2007). In fact, the academic
scheduling problem isalso synonymous to a constraint satisfaction
problem (Mariott and Stuckey 1998).
This paper is organized as follows: In Sect. 2, the background
of the Academic SchedulingProblems and their constraints are
presented. In Sect. 3, the variousmetaheuristic approaches,emerging
metaheuristic algorithms (Ant Colony Optimization and
Hyper-Heuristics) andresults are described. Sect. 4 presents the
suitability of themetaheuristicmethods in achievingcertain solution
quality and concludes the review paper by providing some future
works.
2 Problem background
The following section discusses the problem background of the
Academic Scheduling Prob-lems which encompasses the challenges, the
various metaheuristic categories and concludeswith a general
mathematical model.
2.1 The academic scheduling problem
In generating a good timetable in almost every university, the
primary objective is to optimallyassign lecturers to teach a
specific course at a specific room during a specific time. One of
thegreatest challenge and common problem faced in all Academic
Scheduling Problems is togenerate a conflict-free and a high
quality timetable which are often very difficult to
achieve(Nuntasen and Innet 2007; Zhang et al. 2010). This is due to
the stochastic behaviour of themeta-heuristics algorithm and the
highly-constrained nature of the problem.
The constraints pertaining to the academic scheduling problem
can be categorized into2 categories that are hard constraints and
soft constraints. Basically, the hard constraintsare mandatory
constraints which cannot be violated under any circumstances at
all, lestthe timetable becomes infeasible. On the other hand, the
soft constraints such as lecturerpreferences are secondary
constraints which can be violated, but preferably not as
theyconstitute to the effectiveness and quality of the solution.
These 2 types of constraints canbe further classified into 5
categories namely unary constraints, binary constraints,
capacityconstraints, event spread constraints and agent constraints
(Lewis 2007). One of the notedhard constraints here is that the
problem is bound to the limitation of time and space whereby
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Review of state of the art
there are only 5working days and (depending on institution) 89
usable hours daily. Interestedreaders can refer to Pongcharoen et
al. (2007) for a list of widely used common hard andsoft
constraints shared by various universities. It is worth to note
here that the constraints donot encompass all educational
institutions as they are unique. They are documented in thispaper
to provide a framework for research purposes.
All scheduling problems share a similar behaviour, which is to
generate a feasible scheduleby maximizing (or minimizing) the
objective function value such that the schedule wouldstill remain
in the feasible search space region. This value is also synonymous
to the fitnessfunction value. The ideal fitness value is acquired
through minimizing the violations for thevarious assignments which
are subjected to the hard and soft constraints. The lower the
fitnessfunction value, the better the quality of the solution. An
example of the list of constraints andthe translated mathematical
model, adopted from Tahar (2010) is given as follow:
i. A lecturer can only teach a class at a time (Hard
Constraint).ii. A classroom can only handle a class at a time (Hard
Constraint).iii. Two timeslots for the same course cannot fall on
the same day (Soft Constraint).iv. Courses in the same level cannot
be at the same time (Hard Constraint).v. No courses on Monday
between 11.00a.m. and 12.30p.m. (Hard Constraint).
When translated into mathematical notation, it yields equation
(1) and (2):
MinF (x) =
c
i
f(
Ahi
)+
c
i
f(
Asi)
(1)
s.t.(ci , di , ti , pi , ri , li ) and (c j , d j , t j , p j ,
r j , l j )A1 : (di = d j ) and (ti = t j ) and (pi = p j )A2 : (di
= d j ) and (ti = t j ) and (ri = r j )A3 : (ci = c j ) and (di = d
j )A4 : (li = l j ) and (ti = t j )A5 : (di = Mon) and (ti =
11.0012.30)
(2)
where:
F(x) = F(, , Ahi , Asi ) = fitness function value, = weight
attached to hard constraint, = weight attached to soft
constraint,ci = courses corresponding to the ith course,di = day
corresponding to the ith course,pi = professor corresponding to the
i
th course,ri = room corresponding to the ith course,li = level
corresponding to the ith course,Ahi = hard constraint corresponding
to the the i
th course,Asi = soft constraint corresponding to the the i
th course,i = 1, 2,, N .N = number of courses.
Based on the formulation stated above, the objective is to
locate a feasible solution in thesearch space with minimal
objective function value.
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3 Approaches in the academic scheduling problem
Current survey indicates that all metaheuristic techniques fall
under one of these categoriesOne-stage optimization algorithm,
Two-stage optimization algorithm and algorithms thatallow
relaxation (Lewis 2007). In a one-stage algorithm, satisfaction of
both the hard and softconstraints is being attempted simultaneously
as opposed to two-stage algorithm, where thesatisfaction of hard
constraints will be attempted first in order to obtain a feasible
timetablebefore satisfying the soft constraints. In algorithms that
allow relaxation, the first phasewill generate a population of
feasible and high quality solutions which are obtained basedon a
specific primary criterion (relaxes other criteria). In the second
phase, the algorithmsearches for a compromised solution which
satisfies as many soft constraints as possiblewithout violating the
solution obtained from the first phase. In measuring the
performanceof a timetable, the computational time required to
generate a feasible timetable and theeffectiveness (the degree of
usability) of the timetable are always being considered. It isvery
difficult to generate an optimal (or near optimal) timetable which
does not violate anyconstraints at all within the shortest period
of time.
As an NP-hard and NP-complete problem, conventional heuristics
such as Graph Colour-ing (Burke et al. 1994, 1995), usage of
mathematical models such as integer linear pro-gramming, dynamic
programming (Kanit et al. 2009) and manual timetabling are
ofteninefficient and ineffective for solving the
resource-constrained problem effectively. Instead,meta-heuristics
methods were used and have grown popular over the years in the
areaof optimization due to its robustness and capability of
modelling many real world prob-lems such as nurse scheduling,
airline crew scheduling, round-robin sports schedulingetc.
(Guang-Feng and Woo-Tsong 2011; Lewis and Thompson 2011; Lim and
Razamin2010). Moreover, meta-heuristics methods can greatly reduce
the usage of rigid mathe-matical models which requires a
substantial amount of precision, which often at timesare difficult
to model as they are unable to take preferences (soft constraints)
into account(Pinedo 2012).
There have been many papers which described the usage of
meta-heuristics method tosolve the Academic Scheduling Problems,
such as Tabu-Search (Alvarez-Valdes et al. 2001),Hyper-heuristics
(Burke et al. 2007), Genetic Algorithm (GA) (Pongcharoen et al.
2007),Simulated Annealing (SA) (Aycan and Ayav 2009), Tabu Search
(Casusmaecker et al. 2009),Ant Colony Optimization (Lutuksin and
Pongcharoen 2010), The Great Deluge (TurabiehandAbdullah 2011),
Particle SwarmOptimization (PSO) (Tassopoulos andBeligiannis
2012)and hybrid algorithms such as Fuzzy Genetic (Chaudhuri and De
2010), 2-Point HybridEvolutionary algorithm (Md Sultan et al. 2008)
and many more. Results from these workshave exhibited very
promising results and have motivated the development of many
newmeta-heuristics algorithm today.
3.1 Tabu search
Tabu-Search (TS) is a type of local search algorithmandwasfirst
introduced in 1986byGloverand McMillan (1986). The advantage to TS
is that it incorporates an adaptive memory anda responsive
exploration (Gonzalez 2007). It utilizes a temporary memory to keep
a tabulist which stores the most recent visited solution. These
solutions are of course marked astaboo and prevents re-evaluation
(also known as cycling) in the future (Glover 1986).With the list
containing all the taboo solutions, the algorithm then continues to
iterativelyevaluate the immediate neighbouring candidate solution
for a potentially better solution. TheTS algorithm is described in
the works of Brownlee (2011).
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Table 1 Results of the three strategies for the use of candidate
list Alvarez-Valdes et al. (2001)
With candidate list Without candidate list Temporal use
ofcandidate list
Recover Non-recover Recover Non-recover Recover Non-recover
Changeweights(oscillation)
63,213 65,246 57,150 54,801 58,551 57,180
Non-changeweights(withoutoscillation)
64,341 64,695 58,691 56,950 58,829 55,282
The TS algorithm was applied by Alvarez-Valdes et al. (2001) in
the academic schedulingproblem to generate a master timetable, a
timetable which does not concern the preferencesof students. The
authors employed 2 consecutive phases which comprised of a main
methodused to generate a clash-free feasible timetable and
subsequently a tabu-search algorithm toenhance the generated
timetable. Additionally, the authors added another phase to
enhancethe room assignment process towards the end of phase 2. The
purpose of enhancing the roomassignment was to minimize themovement
of students between classes which is an importantcriterion to
increase the quality of the timetable. In order to diversify the
neighbourhoodstructure, the authors employed 3 move methods namely
simple move, swap, multiswapand oscillation of weights which
allowed greater exploration and of the three methods,multiswap
proved to be the most reliable as it allowed major modifications to
be done untothe solution. Comparisons were also made with the
parameters of candidate list, tabu-list andsolution recovery and it
was found that the use of a candidate list reduces the search
spaceby concentrating the search around the potential candidate
solution neighbourhood. Recoveris an intensification strategy used
to recover the best candidate solution after the algorithmstalls
for a period of time. Using the best candidate solution, another
search process wascarried out around the region to see if a better
solution could be obtained. The results foreach strategy are
tabulated in Table 1 and it was found that simple move and swap
methodswere not appropriate in solving a relatively large sized
problem. The utilization rate of eachroom for the institution
reported an average of 83% which is considerably satisfactory.
The author concluded that the ideal combinations for the TS
algorithm should consist ofthe following:
i. The move is multiswap (The most critical parameter and is
ideal for exploring complexneighbourhood).
ii. Temporal use of candidate list (To improve the objective
function and reduce the numberof search move).
iii. Tabu list with dynamically changing length (Significantly
enhances the robustness of thealgorithm).
iv. Strategic oscillation of weights (To diversify the
exploration of the search space).v. Recovering (Recovery) the best
known solution after a given number of iterations without
improvement (An intensification process to obtain better
solutions).
The TS algorithm is heavily dependent on the neighbourhood
structure in locating theglobal optimum value as clearly
demonstrated in the works of Casusmaecker et al. (2009)who proposed
four different techniques to diversify the neighbourhood structure
namelySwap move, Time-Swap neighbourhood, Room-Swap neighbourhood
and Time-Room Swap
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neighbourhood.These techniqueswere also referred to as
thehorizontal swap as they involvedcontents swapping within the
same candidate solution. In tackling the issue of modularcourses, a
grouping technique adopted from the works of Kingston (2004) was
used as partof the solutionmodel to avoid generating numerous
independent timetables. In this algorithm,a grouping technique is
used to form a group between lecturers and lab sessions that
takeplace in the same room in order to reduce the complexity of the
combinatorial problem.
3.2 Genetic algorithm
Inspired by the process of natural selection and genetics, the
GA is an optimization andpopulation-based search technique based on
the aforementioned principles. It was first intro-duced byHolland
(1975) andwas later on diversified tomany other fields of discrete
optimiza-tion after the works authored by Goldberg (1989).
According to Haupt and Haupt (2004), GAis ideal at solving complex
problems as it possesses great variable optimization technique.One
of it is observed in the nature of the algorithm where the encoding
is performed directlyonto the variables as a set of candidate
solutions. A typical GA consists of a representationof potential
solutions known as population, genetic operators, fitness function,
a selectionscheme and stopping criteria. The GA algorithm is
described in Brownlee (2011).
Conventional GA often encodes the candidate solutions in binary
strings. However, in thearea of academic scheduling, the candidate
solutions are usually encoded in sets because itadds to the
robustness of the genetic operators such as the crossover operation
(Sabri et al.2010). These sets would contain the parameters to be
optimized as noted in the works ofNuntasen and Innet (2007), Tahar
(2010) and the encoding style may vary slightly dependingon the
requirement of the problem instance. For instance, in the works of
Nuntasen andInnet (2007), the candidate solutions (chromosomes)
were encoded in the form of set with 3parameters, namely Lecturer
(L), Subject (S) and Room (R) given as follow:
Chromosome Encoding,E = {L, S, R}.where E is the chromosome, L
is the lecturer, S is the subject and R is the room.
Each of the parameters contain a subset of their own, which can
be described as L ={L1, L2, . . . , Ln}, S = {S1, S2, . . . , Sm}
and R = {R1, R2, . . . , Rn}.
On the contrary, in the works proposed by Tahar (2010), the
parameters that were takeninto account during the encoding of the
chromosome were course (c), day (d), time (t),professor (p),
classroom (r), level (l) and a list of students (s).
Chromosome Encoding : {(ci , di , ti , pi , ri , li )/i = 1,2, .
. . , N }where N is the number of courses.
The encoded chromosomes will then undergo the selection process,
where they are eval-uated by a fitness function and the higher
quality chromosomes are carried to the next round.Thewill then
undergo the crossover operator, where the exchange of information
is performedbetween the two parent chromosomes, followed by the
mutation operator, which prevents thealgorithm to be stuck in a
local minimum (Sivanandam and Deepa 2008). Nevertheless, it
ispossible to omit the crossover operation (since it adds to the
complexity and computationaltime) as demonstrated in the works of
Beligiannis et al. (2008) and Suyanto (2010). Theresult of the
experiment by the latter is shown in Table 2.
Based on Table 2, it can be observed that the crossover
percentage affects the fitnessfunction value. A higher percentage
of crossover enhances the fitness function value withan increased
computational time. On the contrary, a higher percentage of
mutation enables afeasible solution to be obtained in lesser
generations, but with a higher fitness function value.
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Review of state of the art
Table 2 Ratio of GA operator,best generation and fitnessfunction
value (Suyanto 2010)
Ratio of GA operator Best generation Fitness function value
Crossover 80%:Mutation 20% 423 2,070
Crossover 50%:Mutation 50% 197 1,520
Crossover 20%:Mutation 80% 103 2,318
Table 3 Costs and execution times with three neighbourhood
search algorithms (Aycan and Ayav 2009)
SSN SWN S3WN
Cost CPU (s) Cost CPU (s) Cost CPU (s)
3,900 29 9,300 40 4,300 34
3.3 Simulated annealing
Simulated Annealing is a local search method and was first
introduced by Kirkpatrick et al.(1983) which mimics the principles
of metallurgy of metals boiling and cooling to achieve astable
crystal lattice structure with minimal energy state. The pseudocode
for a standard SAis described in the works of Gonzalez (2007).
The algorithm initializes by generating an initial random
solution. After that, adjacentsolution is being generated and these
two solutions will be evaluated by an objective function.If the
cost of the neighbour is lower than the cost of the initial
solution and lowers the energyof the system, the neighbourwill be
accepted as an improved solution. As for a non-improvingsolution,
it will gradually be acceptedwith a probability value given by a
probability function.
In SA, the performance of the algorithm is highly dependent on
its parameters, such ashow meticulous the neighbourhood is being
explored, update moves and cooling rate. Well-explored
neighbourhood provides the opportunity for quality solutions to be
obtained asdemonstrated in the works of Aycan and Ayav (2009). In
their work, 3 neighbourhood searchmethods were proposed, which were
Simple-Searching Neighbourhoods (SSN), SwappingNeighbourhoods (SWN)
and Simple-Searching and Swapping Neighbourhoods (S3W N ),each with
the ability to explore the search space region distinctly. Tables
3, 4, 5 give theresults of the search methods.
Additionally, high-quality solution is achievable if the update
moves at each temperaturestage is set to be proportional to the
neighbourhood size (Johnson and McGeoch 1997). Theworks of
Elmohamed et al. (1998) demonstrate how the different types of
cooling schedulecan add to the solution quality. The cooling
schedules consist of the typical geometric coolingschedule,
adaptive cooling schedule and adaptive cooling schedule with
reheating function.Geometric cooling schedule is themostwidely used
annealing schedule and has the advantageof being well understood.
In Adaptive cooling schedule, a new temperature is computedbased on
the existing temperature with slight deviation so as to maintain
the system closeto equilibrium. Reheating allows the algorithm to
escape the local minima by reheating thesystem temperature above
transition phase which in turn allows the algorithm to exploreother
optima.
It was found that the adaptive cooling schedule, when used
together with a pre-processor(to yield a good starting point),
produced the best result. To further enhance the solution, thefinal
solution can be reheated further as well. The results are presented
in Table 6.
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Table 4 Costs and execution times with the combinations of SN,
SWN and S3WN (Aycan and Ayav 2009)
SSN and SWN SSN and S3WN SWN and S3WN
Cost CPU (s) Cost CPU (s) Cost CPU (s)
3,900 28 4,900 27 3700 31
Table 5 Costs and executiontimes when SSN, SWN andS3WN are used
altogether(Aycan and Ayav 2009)
Case A (sequentially) Case B (in turn)
Cost CPU (s) Cost CPU (s)
4,100 87 3,600 28
Table 6 Percentage of scheduled classes, averaged over 10 runs
of the same initial temperature and otherparameters, for three
terms using simulated annealing with an expert system as
pre-processor (Elmohamed etal. 1998)
Academic time period Algorithm Scheduled(average) %
Highestscheduled %
Lowestscheduled %
First semester SA (geometric) 93.90 95.12 85.20
SA (adaptive) 98.80 99.20 95.00
SA (cost-based) 100.0 100.0 100.0
Second semester SA (geometric) 95.00 98.95 89.40
SA (adaptive) 99.00 99.50 98.50
SA (cost-based) 100.0 100.0 100.0
Third semester SA (geometric) 97.60 98.88 90.90
SA (adaptive) 100.0 100.0 100.0
SA (cost-based) 100.0 100.0 100.0
A comparison between the SSN, SWN and S3W N shows that the
Simple-Searching Neigh-bourhood structure yields the best result of
all the 3 neighbourhood search methods. Uponhybridizing SWN and S3W
N , a combination of all 3 neighbourhood search algorithm
yieldedthe best result with the cost of 3,700 within 31s. The
effectiveness of the hybrid algorithmwas tested and evaluated based
on 2 casesCase A and B. In Case A, the algorithm wasexecuted
sequentially and in Case B, the algorithm was executed in turn
basis. From theexperiment, it was found that the algorithm performs
more effectively when executed in turnbasis.
From Table 6, it is evident that the adaptive cooling schedule
outperforms the geometriccooling schedule. The cost-based SA can
only be used upon obtaining the best solution fromthe adaptive SA.
In the experiment, it was used with a pre-processor, reheating
function andwill always return a valid solution.
3.4 Particle swarm optimization
Particle swarm optimization was first proposed by Kennedy and
Eberhart (1995). The algo-rithm mimics the social behaviour of
collective species such as a school of fishes, a flockof birds and
a group of humans. It is also closely associated with GA due to
their many
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Review of state of the art
similarities (Shu-Chuan and Yi-Tin 2006). The PSO algorithm is
particularly concerned withthe exploration and exploitation of the
search space. Essentially, exploration is the ability toexplore the
different regions of a search space in order to locate the global
optimum valuewhereas exploitation is the ability to concentrate the
search around a promising region inhope to refine an existing
candidate solution (Ghalia 2008). It is reported by Salman et
al.(2002) that PSO excels GA in terms of solutions quality and
computational speed. In PSO, apopulation of candidate solutions
known as particles is initialized over random positions in asearch
space. As the iterations increase, each of the particles with their
individual and globalexperience will share information with one
another and converge to a global optimum.
The encoding of the particles is similar to that of GA. In the
works of Tassopoulos andBeligiannis (2012), the encoded solution
model proposed satisfied 3 hard constraints duringthe
initialization phase in contrast to the work proposed by
Qarouni-Fard et al. (2007) whereit satisfied only one constraint.
It is desirable for the solution model to satisfy as many
hardconstraints as possible as this reduces the stress on the
algorithm which in turn, decreasesthe computational time. The
satisfied hard constraints guarantee that all lecturers cannotbe
lecturing 2 different subjects in a class at all time, can only
teach up to a maximumnumber of hours as stipulated by the school
and the total hours assigned to each class shouldequal to the total
hours permitted to teach. Additionally, the timetabling system
possesses aninherent adaptive behaviour, which added to the quality
of the solutions produced. The authorincorporated a feature where
lecturers can specify their preferences by attaching priorities
inthe form of (adjustable) weights to the selected constraints. The
results of the experiment aredescribed in Tables 7, 8 and 9. In
Table 7, the algorithm records 13/18 cases in outperformingother
methods, 3/18 cases with similar results and 2/18 cases with
unsatisfactory results. InTable 8, 8/12 cases for outstanding
performance, 3/12 cases with similar results and 1/12case with
unsatisfactory results. In Table 9, 6/9 cases for outperforming
other methods, 2/9cases with similar results and 1/9 case for
unsatisfactory result.
One of the noted soft constraints which add to the solution
quality is to minimize theteachers idle time as much as possible.
In order to achieve this, a local search methodknown as the
Refining procedure was incorporated after obtaining a fairly good
solution inthe first phase. In the distribution teachers column in
Table 7, the first number indicatesthe number of teachers whose
teaching hours are not evenly distributed while the numberin the
parentheses indicates the frequency of the uneven distribution. The
first number inthe distribution courses indicates the classes in
which the same course is being taughtrepeatedly while the number in
the parentheses indicates the total number of classes in whichthe
incident occurs. In the teachers idle periods column, the first
number indicates theteacher who has idle hours while the number in
the parentheses indicates the total hour idlesfor all teachers.
3.5 Fuzzy logic algorithm
Fuzzy logic (FL) is a form of probabilistic logic which provides
a greater range of optionswhen it comes to making decisions and was
proposed by Zadeh (1965). In conventionaldecision making option, an
algorithm can only evaluate a condition using binary logic, whichis
either true or false. When FL was introduced, it expands the
capability of evaluationby the introduction of linguistic
variables, enhancing the ability to evaluate constraints witha
certain degree of truth. In other words, FL is able to provide good
reasoning even undervague conditions and uncertainties.
The algorithm was adopted by Petrovic et al. (2005) and Asmuni
et al. (2005) in the fieldof academic scheduling. In the works of
Petrovic et al. (2005), flexible constraints pertaining
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C. K. Teoh et al.
Tabl
e7
Com
paring
timetablesconstructedby
theproposed
PSOalgorithm
with
real-w
orld
timetablesused
atschoolsandtim
etablescreatedin
Beligiannisetal.(20
08,200
9)
Test
dataset
Tim
etablesused
atschools
Geneticalgorithm
(Beligiannisetal.2
009)
Evolutio
nary
algorithm
(Beligiannisetal.2
008)
PSOalgorithm
Distribution
teachers
Distribution
courses
Teachers
idleperiods
Distribution
teachers
Distribution
courses
Teachers
idleperiods
Distribution
teachers
Distribution
courses
Teachers
idleperiods
Distribution
teachers
Distribution
courses
Teachers
idleperiods
121(48)
1(1)
25(34)
18(40)
1(1)
25(29)
18(40)
1(1)
25(29)
1(2)
2(2)
11(15)
215(34)
0(0)
30(52)
15(34)
0(0)
26(42)
15(34)
0(0)
26(42)
1(2)
2(2)
12(19)
39(23)
3(3)
8(24)
4(8)
3(3)
9(9)
4(8)
3(3)
9(9)
2(4)
3(3)
0(0)
46(14)
0(0)
15(31)
5(12)
0(0)
17(29)
5(12)
0(0)
17(29)
2(3)
0(0)
9(10)
56(17)
0(0)
15(39)
1(2)
0(0)
8(8)
1(2)
0(0)
8(8)
0(0)
0(0)
0(0)
724(56)
13(36)
24(33)
24(50)
13(27)
24(32)
24(50)
13(27)
24(32)
0(0)
0(0)
20(43)
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Review of state of the art
Tabl
e8
Com
paring
timetablesconstructedby
theproposed
PSOalgorithm
with
timetablescreatedin
Beligiannisetal.(20
08,200
9)andPapoutsisetal.(20
03)
Test
dataset
Colum
ngeneratio
napproach
(Papou
tsisetal.2
003)
Geneticalgorithm
(Beligiann
isetal.2
009)
Evolutio
nary
algo
rithm
(Beligiann
isetal.2
008)
PSOAlgorith
m
Distribution
teachers
Distribution
courses
Teachers
idleperiod
sDistribution
teachers
Distribution
courses
Teachers
idleperiod
sDistribution
teachers
Distribution
courses
Teachers
idleperiod
sDistribution
teachers
Distribution
courses
Teachers
idleperiod
s
85(10
)5(15
)0(0)
4(8)
5(15
)0(0)
3(6)
5(15
)0(0)
0(0)
5(11
)0(0)
96(12
)6(22
)0(0)
7(14
)6(21
)0(0)
7(14
)6(21
)0(0)
0(0)
0(0)
2(2)
106(16
)6(16
)0(0)
4(6)
7(22
)0(0)
2(4)
7(22
)0(0)
0(0)
6(10
)0(0)
116(16
)8(29
)0(0)
6(14
)9(29
)0(0)
6(13
)9(29
)0(0)
0(0)
8(18
)0(0)
123
-
C. K. Teoh et al.
Tabl
e9
Com
paring
timetablesconstructedby
theproposed
PSOalgorithm
with
timetablescreatedin
Beligiannisetal.(20
08)andValou
xisandHou
sos(200
3)
Test
dataset
Con
straintp
rogram
mingapproach
(Valou
xisandHou
sos20
03)
Evolutio
nary
algo
rithm
(Beligiann
isetal.2
008)
PSOalgorithm
Distribution
teachers
Distribution
courses
Teachers
idle
period
sDistribution
teachers
Distribution
courses
Teachers
idle
period
sDistribution
teachers
Distribution
courses
Teachers
idle
period
s
85(10
)5(15
)0(0)
3(6)
5(15
)0(0)
0(0)
5(11
)0(0)
95(10
)6(22
)0(0)
7(14
)6(21
)0(0)
0(0)
0(0)
2(2)
106(16
)6(16
)0(0)
2(4)
7(22
)0(0)
0(0)
6(10
)0(0)
123
-
Review of state of the art
Table 10 Comparison betweenmanual, GA and FGH
Solutions(Chaudhuri and De 2010)
Feature Manualsolution
GA solution(Gupta et al. 2006)
FGHsolution
Fitness 2, 286 2, 599 2, 809
Objective value 2, 286 1, 107 809
Penalty value 0 2, 000 2, 000
Number of hardconstraints violated
0 0 0
Number of softconstraints violated
0 0 0
Classroom hour gaps 90 7 5
Teacher hour gaps 98 2 1
to exam timetabling were introduced to loosen the evaluation
process as opposed to a rigidevaluation. For example, in
determining the size of an exam, 2 extra linguistic variableswere
introduced which are size and time period defined by small, medium,
large and early,middle, late respectively. 9 different combinations
of possibilities were then derived based onthe linguistic
constraints to determine the degree of constraint satisfaction.
Similarly, in thework proposed by Asmuni et al. (2005), exam
ranking was taken into consideration by takinginto account multiple
heuristics methods such as largest degree (LD) first, largest
enrolment(LE) first and least saturation degree (LSD) first with
largest enrolment (LE) first excellingthe rest.
Fuzzy logic can easily be hybridizedwith other algorithms as
demonstrated in theworks ofChaudhuri and De (2010)where it was
hybridized with GA denoted as Fuzzy Genetic Heuris-tic (FGH). In
the works proposed, the timetable generation was constructed in 2
phasesthefirst phasewas to obtain a feasible timetablewith
theGAoperator and the second phasewas tominimize the violation of
soft constraints as much as possible using the notation sets of
FuzzyLogic. During the first phase, whenever a feasible timetable
(satisfies all hard constraints anda certain amount of soft
constraints) is produced, the solution was often impractical to
imple-ment as it possessed some invalid solutions. Hence, the
author introduced a direct and indirectencoding method into the
construction of the timetable. In direct encoding, all the
parame-ters in GA were encoded into the chromosome while indirect
encoding invoked a timetablebuilder which allowed FL to solve the
soft constraints violated. A comparison between themanual solution,
GA solution and FGH solution is given in Table 10 while the
execution timeof fuzzy genetic against other instances of GA-based
heuristic is shown in Table 11.
In Table 10, the results of three methods which comprised of
manual solution, GA andFGH were tabulated and FGH was able to
achieve a reasonably good solution. The FGHalgorithm scored the
least in the objective function value, obtained a minimum
classroomhour gaps of 5h andwas able tominimize the teachers idle
time to 1h, increasing the resourceutility rate. Moreover, the
algorithm was reported to satisfy both the hard and soft
constraintsin a balanced manner rather than satisfying only the
hard constraints as solved by the manualmethod. In Table 11, the
execution time of the FGH algorithm was compared against otherGA
methods on different problem instances and FGH exhibited almost
similar results withthe other algorithms.
3.6 Ant colony optimization
Ant colony optimization (ACO) was first proposed by Dorigo et
al. (2006) and is a relativelynew algorithm. It is inspired by the
foraging behaviour of ants through their deposit ofpheromone where
they are able to identify the shortest path to transport their
food. Despite
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C. K. Teoh et al.
Table 11 Execution time (in minutes) of various GA instances
against FGH (Chaudhuri and De 2010)
Datasets GA1(Cupicet al. 2009)
GA2(Ghaemiand Vakili2006)
GA3(Qu et al.2009)
GA4(Moreira2008)
GA5(Singhet al. 2008)
GA6(Guptaet al. 2006)
GA7(Kordalewskiet al. 2009)
FG H
Small1 11.55 15.79 16.60 17.45 14.56 16.62 18.32 19.76
Small2 11.59 15.86 16.64 17.46 14.57 16.64 18.33 19.90
Small3 11.62 15.96 16.66 17.47 14.59 16.66 18.34 19.86
Small4 11.64 15.98 16.69 17.50 14.60 16.67 18.35 19.89
Small5 11.69 15.99 16.86 17.52 14.69 16.69 18.37 19.89
Medium1 109.96 106.90 116.99 115.90 111.30 112.30 112.28
119.07
Medium2 104.86 109.50 107.84 107.86 86.32 80.32 79.86 119.56
Medium3 110.99 118.30 117.99 114.37 112.37 112.16 118.66Medium4
100.79 104.56 115.57 114.54 112.50 112.37 117.96Medium5 105.99
115.96 85.69 75.69 69.86 118.98Large 116.32 119.30 119.55
119.99
the fact that it is a stochastic and multi-directional search
algorithm, it does not guaranteethe discovery of an optimal
solution (Lutuksin and Pongcharoen 2010). The algorithm isdeveloped
based on a parameterized probabilistic model known as the pheromone
modelwith various pheromone values. A pheromone value is associated
to each pheromone trailand is updated during every runtime in order
to obtain a bias towards high quality solutions(Dorigo and Blum
2005). However, it was also reported by the same author that the
originalACO suffers frombias deception known as the first order and
second order deceptionwherebysome solution components are updated
more frequently than the others on the average, whichin turn may
not guarantee an optimum solution at all (Blum and Dorigo 2002,
2004). Thealgorithm for ACO is given in Fig. 1.
Recent works in the Ant ColonyOptimization algorithm, ACO have
resulted inmany vari-ants of the algorithm. For example, the works
of Cordon et al. (2002) have led to algorithmssuch as the Ant
System (AS), Ant Colony System (ACS), Best-Worst Ant System (BWAS)
andBest-Worst Ant Colony System (BWACS).
In his study, 3 parameters namely Restart (Rs), Mutation(M) and
Worst Ant Update (W )were taken into account. Rs is a mechanism
that enables the algorithm to escape from itslocal optima should it
get stuck, M mutates the pheromone trails to enhance the
explorationof the search space and W is an updating mechanism of
the worst ant. Among the proposedalgorithms, the BWAS and BWACS
models performed excellently in obtaining a high qualitysolution.
It was found that the W parameter would remove irrelevant search
spaces whileboth mutation and restart would avoid the algorithm to
be trapped in local optima. A priorityorder was also established to
denote the importance of each parameter: Restart, Mutation andWorst
Ant Trail Update. The proposed algorithm was applied by Lutuksin
and Pongcharoen(2010) in solving the academic scheduling problem
using 6 problem instances adopted fromthe literature review of
Cordon et al. (2000).
The computational results for MMAS, ACS and BWACS are given in
Table 12, and fromthe results, theMMAS outperformed theACS
andBWACSmethod for problems 1 and 2whichwere relatively small.
However, the ACS method excelled in solving problem 3. Problems 4,5
and 6 were relatively large and they were effectively solved by the
BWACS method. The
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Review of state of the art
Fig. 1 Algorithm for Ant Colony Optimization (Brownlee 2011)
proposed methods were found to be effective for certain
problems, hence it can be concludedthat the methods available are
unique based on the attributes of the problem.
3.7 Hyper-heuristics
Hyper-heuristics first surfaced when there was a need for a
robust algorithm which couldeasily generalize and extend to a new
but yet similar problem. Proposed by Denzinger et al.(1996), the
idea is very similar to that of hybridized algorithms, except that
hyper-heuristicsis a high level algorithm which consists of a huge
amount of lower level heuristics algorithm(Burke et al. 2010. The
algorithm is designed such that it is able to generate a solutionof
acceptable quality within the shortest period of time (Chakhlevitch
and Cowling 2008).While conventional meta-heuristics method
searches for a possible solution in the searchspace,
hyper-heuristics differ to meta-heuristics by searching for
combinations of lower-heuristics techniques in a space of
heuristics than a space of solutions (Burke et al. 2010).
Itoperates on a higher level of meta-heuristics to select an ideal
combinations of lower levelheuristics method solve the problem
based on the assignment of weights, rather than solvingthe problem
directly. The fundamental algorithm of hyper-heuristics is
described in Fig. 2.
A choice function whose preference can be easily specified is
used to guide the hyper-heuristics in selecting the best method to
solve a problem. Equation 3 describes a simplechoice function.
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C. K. Teoh et al.
Table 12 Computational results obtained from the MMAS, ACS and
BWACS methods (Lutuksin andPongcharoen 2010)
Problem Methods Best so far solutions
Minimum Maximum Average Standard deviation Time (h)
1 MMAS 62 78 72 6.52 1.08
ACS 105 117 110.8 5.36 1.05
BWACS 49 130 83.8 31.68 1.06
2 MMAS 22 31 26 3.24 2.25
ACS 52 59 56.2 3.03 2.30
BWACS 30 43 38 4.84 2.23
3 MMAS 520 566 539.4 18.08 3.66
ACS 515 541 522 10.98 3.86
BWACS 517 572 548.2 20.96 3.17
4 MMAS 406 458 426.4 23.69 4.09
ACS 349 385 359 15.54 4.33
BWACS 314 349 337.8 14.51 4.53
5 MMAS 339 369 355 11.11 5.91
ACS 263 286 274 8.89 5.92
BWACS 254 289 267.6 13.01 5.94
6 MMAS 405 441 424 15.33 5.54
ACS 331 341 336.6 4.39 5.57
BWACS 318 340 330.2 9.49 5.67
Fig. 2 Algorithm for hyper-heuristics (Burke et al. 2003)
G (Hk) = f1 (Hk)+ f2(Hj , Hk
)+ f3(HK ) (3)where:
Hk is the kth heuristic,, and are weights which reflect the
importance of each term. (Can be varied accordingto users
preference where + + = 1.0),f1(Hk) is the recent performance of
heuristic Hk.f2(Hj,Hk) is the recent performance of heuristic pair
Hj,Hk.f3(Hk) is a measure of the amount of time since heuristic Hk
was called.
In the algorithm applied by Terashima-Marin et al. (1999) in an
academic schedulingproblem, the algorithmwas tested on several
problem instances known as theToronto problem
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-
Review of state of the art
Table 13 Brelaz algorithm on Carters real-life exam timetable
(Toronto set) with edge, near-clash andcapacity constraints using
various heuristics (Terashima-Marin et al. 1999)
Problems Slots Maximumexam size
Seats Heuristics
1 2 3 4
HECS92 21 634 1, 250 0/318/0 0/302/0 0/322/0 0/1,112/0
STAF83 15 237 600 0/1,338/0 0/1,348/0 0/1,450/0 0/2,418/0
YORF83 21 175 500 0/790/0 0/865/0 0/783/0 0/1,171/0
UTES92 12 482 1, 250 0/816/0 0/870/ 0/1,593/0 0/2,643/0
EARF83 24 232 700 0/880/0 0/933/0 0/946/0 0/1,448/0
TRES92 27 407 655 0/613/0 0/645/0 0/716/0 0/1,239/0
LSEF91 21 382 900 0/421/0 0/428/0 0/302/0 0/1,309/0
KFUS93 24 1, 280 1, 955 0/951/0 0/957/0 0/996/0 0/2,484/0
RYES93 27 943 2, 500 0/1,471/0 0/1,045/0 0/1,451/0 0/4,676/0
CARF92 40 1, 566 2, 000 0/428/0 0/383/0 0/427/0 0/2,441/0
UTAS92 38 1, 314 2, 800 0/952/0 0/1,104/0 0/1,032/0
0/2,984/0
CARS91 51 1, 385 1, 550 0/342/0 0/230/ 0/356/0 0/2,217/0
set, which is a collection of real-world data. Whenever a
condition which involves a selectedamount of constraints is marked
X , the heuristics involved will be H1 and H2. After thefirst round
of evaluation, it will proceed to the second phase which deals with
the remainingset of constraints, which then involves heuristics H3
and H4. The higher level heuristics inthis instance is GA while the
lower level heuristics are composed by the variations of
Brelazalgorithms. The variations of Brelaz algorithmhandle the
clash and capacity constraintswhileGA was used to search for the
ideal heuristic combinations to solve the various constraintsfor
various datasets. The results are described in Tables 13 and 14
respectively.
In Table 13, the Slots column denotes the available timeslots
for the problem; the Max-imum Exam Size column denotes the size of
the exams with the most number of registeredstudents and the Seats
column denotes the capacity for any timeslot. The generated
sched-ules were all feasible because the number of seats was larger
than the exam size, fulfillingthe capacity constraint. The values
present in the Heuristics column refer to the edge (withexam being
the node, and the edge implicates that 2 nodes cannot be at the
same time),near clash and capacity constraints respectively (e.g.
0/318/0 implies 0 edge, 318 near clashinstances and 0 room whose
students exceeded the room capacity). It can be concluded thateach
variant of the heuristics was capable of solving a problem
effectively through the inter-action of heuristics. The results in
bold denote the best heuristic in solving the
particulardataset.
In Table 14, the GABest column is further divided into 3
sub-columns. The first sub-column describes the violation result in
the form of edge/ near clash/ capacity. The secondsub-column
describes the combination of the modified Brelaz algorithm together
with theheuristics strategies (values in parentheses) which
involved Brelaz (BR), Backtracking (BT)and Forward Checking (FC).
The third sub-column describes the rules that were used tochange
the strategies which involved With-Large (WL) and With- (W) and the
numberof events scheduled. The results in bold denote the best
heuristic in solving the particulardataset. From the table, it is
evident that the strategies obtained from GA yielded a more
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C. K. Teoh et al.
Table 14 Evolution of CSP strategies against best solution of
modified Brelaz algorithm on Carters real-lifeexam timetable
(Toronto set) problems (Terashima-Marin et al. 1999)
Problems Slots Seats Brelaz best GA average G ABest (best
strategy)
HECS92 21 1, 250 0/302/0 0/190/0 0/154/0 BR(7,1)-BT(0,1)
WL-24
STAF83 15 600 0/1,338/0 0/932/0 0/821/0 BR(8,2)-BT(3,0)
W-127
YORF83 21 500 0/783/0 0/764/0 0/708/0 BR(0,2)-FC(2,1) W-119
UTES92 12 1, 250 0/816/0 0/632/ 0/594/0 BR(2,0)-BT(1,1) W-16
EARF83 24 700 0/880/0 0/723/0 0/723/0 FC(4,0)
TRES92 27 655 0/613/0 0/599/0 0/586/0 FC(4,1)-BT(3,0) WL-25
LSEF91 21 900 0/302/0 0/247/0 0/221/0 BR(8,0)
KFUS93 24 1, 955 0/951/0 0/231/0 0/223/0 BR(1,0)-FC(3,0)
W-97
RYES93 27 2, 500 0/1,045/0 0/754/0 0/671/0 BR(8,1)
CARF92 40 2, 000 0/383/0 0/285/0 0/285/0 BR(2,0)
UTAS92 38 2, 800 0/952/0 0/936/0 0/902/0 BR(0,0)-BR(6,2)
W-262
CARS91 51 1, 550 0/230/0 0/170/0 0/130/0 BR(8,0) WL-24
effective overall result. It can be concluded that a better
overall performance and solutionquality can be achieved by
discreetly choosing and evolving a suitable pair or combinationsof
strategies rather than solving a problem with a single strategy. As
for GA, the authors alsosuggested to using a non-direct
representation of chromosomes in solving similar problemsbecause in
a large problem instance, long chromosomes are required to
represent the solutionwhich could lead to various failures.
4 Conclusion
In this review, the nature of the Academic Scheduling Problems
and the properties of thevarious meta-heuristics techniques used in
solving the academic scheduling problem havebeen surveyed. In
general, Academic Scheduling Problems encompass both course
schedul-ing and exam scheduling problem. The difference between
them is that the goal in coursescheduling is to minimize the time
gap for both lecturers and students while exam schedulingmaximizes
the time gap between each examination.
From the survey, it can be said that eachmetaheuristic technique
has the ability to yield fea-sible solutionswith certain
characteristics and tradeoffs. For example, theAcademic Schedul-ing
Problems have been successfully solved by SA, a method which
promises high qualitysolutions with the condition of an optimum
parameter tuning while GA promises greaterexploration of the search
space due to the algorithm operators but with a longer
compu-tational time. Primitive decision making skills which
involved only binary outcome havealso evolved into an algorithm
known as Fuzzy Logic which has the ability to offer mul-tiple
options which benefited the decision making process. In order to
effectively utilizethe strength of each algorithm, hybridization
techniques were proposed such as the FuzzyGenetic Heuristics which
employed the operators from GA and the enhanced decision mak-ing
ability by fuzzy logic and hyper-heuristics which functions as a
collection of algorithmstailored to specific problem types. It is
no doubt that meta-heuristics methods are capable ofproducing high
quality solutions with the development of hyper-heuristics which
involvedhybridization of more than one technique. However, it is
due to the various parameter set-
123
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Review of state of the art
tings and stochastic nature of the algorithm, that obtaining
high quality solutions becomesdifficult.
In conclusion, it can be said that there is no definite
algorithm which is more superiorto solving an academic scheduling
problem as each algorithm possesses a unique strength.It simply
depends on the difficulty of the problem which increases
proportionately with theproblem size. Additionally, the parameter
settings and the complexity of the algorithm arealso a key factor
in contributing to the quality of the solutions. On one hand, in
view ofaspects such as exploration of the search space, it can be
observed that GA and PSO seem toperform better. On the other, high
quality solutions seem to be achievable with SA and FGHwhich
satisfied a considerable amount of soft constraints.
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Review of state of the art for metaheuristic techniques in
Academic Scheduling ProblemsAbstract1 Introduction2 Problem
background2.1 The academic scheduling problem
3 Approaches in the academic scheduling problem3.1 Tabu
search3.2 Genetic algorithm3.3 Simulated annealing3.4 Particle
swarm optimization3.5 Fuzzy logic algorithm3.6 Ant colony
optimization3.7 Hyper-heuristics
4 ConclusionReferences