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: christeoh.ck@gmail.com; ckteoh3@live.utm.my

A. Wibowoe-mail: antoni@utm.my

M. S. Ngadimane-mail: salihin@utm.my

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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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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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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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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)

123

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

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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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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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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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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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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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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-

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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.

References

Alvarez-Valdes R, Crespo E, Tamarit JM (2001) Design and implementation of a course scheduling systemusing Tabu search. Eur J Oper Res 137(3):512523

Asmuni H, Burke EK, Garibaldi JM, McCollum B (2005) Fuzzy multiple heuristic orderings for examinationtimetabling. Paper presented at the PATAT, LNCS

Aycan E, Ayav T (2009) Solving the course scheduling problem using simulate snnealing. Paper presented atthe IEEE international advance computing conference (IACC)

Baker KR (1974) Introduction to sequencing and scheduling. Wiley, New YorkBardadymVA (1996) Computer-aided school and university timetabling: the newwave. In: Practice and theory

of automated timetabling. Lecture notes in Computer Science, vol 1153. pp 2245Beligiannis GN, Moschopoulos CN, Kaperonis GP, Likothanassis SD (2008) Applying evolutionary compu-

tation to the school timetabling problem: the Greek case. Comput Oper Res 35(4):12651280Beligiannis GN, Moschopoulos CN, Likothanassis SD (2009) A genetic algorithm algorithm approach to

school timetabling. J Oper Res Soc 60(1):2342Blum C, Dorigo M (2002) On a particularity in model-based search. In: Paper presented at the genetic and

evolutionary computation conferenceBlum C, Dorigo M (2004) Theoretical and practical aspects of ant colony optimization. Theor Comput Sci

344(23):243278Brownlee J (2011) Clever algorithms: nature-inspired programming pecipes: Lulu EnterprisesBurke EK, Elliman DG, Weare RF (1994) A University timetabling system based on graph colouring and

constraint manipulation. J Res Comput Educ 27(1):118Burke EK, Elliman DG, Weare RF (1995) A hybrid genetic algorithm for highly constrained timetabling

problems. In: Proceedings of the 6th international conference on genetic algorithms, pp 605610Burke EK, Hart E, Kendall G, Newall J, Ross P, Schulenberg S (2003) Hyper-heuristics: an emerging direction

in modern search technology handbook of metaheuristics. In: International series in operations researchand management science, vol 57. Kluwer

Burke EK, Hyde M, Kendall G, Ochoa G, Ozcan E, Qu R (2010) Hyper-heuristics: a survey of the state of theart: School of Computer Science and Information Technology. University of Nottingham

Burke EK, McCollum B, Meisels A, Petrovic S, Qu R (2007) A graph-based hyper-heuristic for educationaltimetabling problems. Eur J Oper Res 176:177192

Casusmaecker PD, Demeester P, Berghe GV (2009) A decomposed metaheuristic approach for a real-worlduniversity timetabling problem. Eur J Oper Res 195:307318

Chakhlevitch K, Cowling P (2008) Hyperheuristics: recent developments. In: Cotta C, Sevaux M, SrensenK (eds) Adaptive and multilevel metaheuristics SE - 1, 136. Springer, Berlin, pp 329

Chaudhuri A, De K (2010) Fuzzy genetic heuristic for university course timetable problem. Int J Adv SoftComput Appl 2(1):100121

Cordon O, Viana IFD, Herrera F (2002) Analysis of the best-worst ant system and its variants on the QAP. In:Paper presented at the third international workshop on ant algorithms

Cordon O, Viana IFD, Herrera F, Moreno L (2000) A new ACO model integrating evolutionary computationconcepts: the best-worst ant system. In: Paper presented at the 2nd international workshop on ant algorithm.Universite Libre de Bruxelles, Belgium

Cupic M, Golub M, Jakobovic D (2009) Exam timetabling using genetic algorithm. In: Paper presented at theITI 31st international conference on information technology interfaces, Croatia

123

C. K. Teoh et al.

Denzinger J, Fuchs M, Fuchs M (1996) High performance ATP systems by combining several AI methods.University of Fachbereich Informatik, Berlin

Dorigo M, Birattari M, Stutzle T (2006) Ant colony optimization. Comput Intell Mag IEEE 1(4):2839Dorigo M, Blum C (2005) Ant colony optimization theory: a survey. Theor Comput Sci 344:243278Elmohamed MAS, Coddington P, Fox G (1998) A comparison of annealing techniques for academic course

scheduling. Springer, BerlinGhaemi S, Vakili MT (2006) Using a genetic algorithm optimizer tool to solve university timetable scheduling

problem. Faculty of Electrical and Computer Engineering, University of Tabriz, IranGhalia MB (2008) Particle swarm optimization with an improved exploration-exploitation balance. In: Paper

presented at the 51st IEEE international midwest symposium on circuits and systems.Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res

13(533):533549Glover F,McMillan C (1986) The general employee scheduling problem: an integration ofMS andAI. Comput

Oper Res 13(5):563573Goldberg DE (1989) Genetic algorithms in search optimization and machine learning. Addison-Wesley,

ReadingGonzalez TF (2007) Handbook of approximation algorithms and metaheuristics. CRC Press INCGuang-Feng D, Woo-Tsong L (2011) Ant colony optimization-based algorithm for airline crew scheduling

problem. Expert Syst Appl 38:57875793Gupta P, Bansal M, Prakash H (2006) Implementation of timetable problem using genetic algorithm. Depart-

ment of Computer Science Engineering, Indian Institute of Technology, Kanpur, Project ReportHaupt RL, Haupt SE (2004) Practical genetic algorithms. Wiley, Hoboken, New JerseyHolland JH (1975) Adaption in natural and artificial systems. University of Michigan Press, Ann HarborJohnson DS, McGeoch LA (1997) The travelling salesman problem: a case study in local optimization. Wiley,

New YorkKanit R, Ozkan O, Gunduz M (2009) Effects of project size and resource constraints on project duration

through priority rule-base heuristics. Artif Intell Rev 32(14):115123Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Paper presented at the IEEE international

conference on neural networks, pp 19421948Kingston JH (2004)A tiling algorithm forHigh School timetabling. In: Paper presented at the fifth international

conference on practice and theory of automated timetablingKirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671680Kordalewski D, Liu C, Salvesen K (2009) Solving an exam scheduling problem using a genetic algorithm.

Department of Statistics, University of Toronto, Toronto, CanadaLewis R (2007) A survey of metaheuristic-based techniques for university timetabling problems. OR SpectR

30(1):167190Lewis R, Thompson J (2011) On the application of graph colouring techniques in round-robin sports schedul-

ing. Comput Oper Res 38:190204Lim HT, Razamin R (2010) Recent advancements of nurse scheduling models and a potential path. In: Paper

presented at the IMT-GT conference on mathematics, statistics and its applications (ICMSA2010), Uni-versiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia

Lutuksin T, Pongcharoen P (2010) Best-worst ant colony system parameter investigation by using experi-mental design and analysis for course timetabling problem. In: Paper presented at the second internationalconference on Computer and Network Technology

Mariott K, Stuckey PJ (1998) Programming with constraints: an introduction. MIT Press, CambridgeMd Sultan AB, Ramlan M (2008) Selecting quality initial random seed for metaheuristic approaches: a case

of timetabling problem. Int J Comput Internet Manag 16(1):8Moreira JJ (2008) A system for automatic construction of exam timetable using genetic algorithms. Tkhne-

Revista de Estudos Politchnicos (9):319336NuntasenN, Innet S (2007)A novel approach of genetic algorithm for solving university timetabling problems:

a case study of thai universities. In: Paper presented at the international conference on Applied ComputerScience

Omar M, Ainon RN, Zainuddin R (2003) Using a genetic algorithm optimizer tool to generate good qualitytimetables. In: Proceedings of the 10th IEEE international conference, electronics, circuits and systems,vol 3, pp 13001303

Papoutsis K, Valouxis C, Housos E (2003) A column generation approach for the timetabling problem ofGreek high schools. J Oper Res Soc 54(3):230238

Petrovic S, Patel V, Yang Y (2005) Examination timetabling with fuzzy constraints. In: Practice and theory ofautomated timetabling V. Lecture Notes in Computer Science, vol 3616

Pinedo ML (2012) Scheduling theory, algorithms and systems. Springer, Berlin

123

Review of state of the art

Pongcharoen P, Promtet W, Yenradee P, Hicks C (2007) Stochastic optimisation timetabling tool for universitycourse scheduling. Int J Prod Econ 112(2):903918

Qarouni-FardD,Najafi-ArdabliA,MoeinzadehM-H, (2007) Finding Feasible Timetableswith Particle SwarmOptimization. In: Proceedings of the 4th international conference on innovations in information technology,pp 387391

Qu R, Burke EK, Mccollum B, Merlot LT, Lee SY (2009) A survey of search methodologies and automatedsystem development for examination timetabling. J Sched 12(1):5589

Sabri MFM, Husin MH, Chai SK (2010) Development of a timetabling software using soft-computing tech-niques with a case study. IEEE 5:394397

Salman A, Ahmad I, Al-Madani S (2002) Particle swarm optimization for task assignment problem. Micro-process Microsyst 26(8):363371

Shu-Chuan C, Yi-Tin C (2006) Timetable scheduling using particle swarm optimization. In: Paper presentedat the first international conference on innovative computing, information and control

Singh E, Joshi VD, Gupta N (2008) Optimizing highly constrained examination timetable problems. J ApplMath Stat Inf 4(2):193197

Sivanandam SM, Deepa SN (2008) Introduction to genetic algorithms. Springer, BerlinSuyanto S (2010) An informed genetic algorithm for university course and student timetabling problems. In:

Proceedings of the 10th international conference on artifical intelligence and soft computing: Part II, Berlin,Heidelberg, pp 229236

Tahar M (2010) Universal tool for university course schedule using genetic algorithm. (IJCNS). Int J ComputNetw Secur 2(6):16

Tassopoulos IX, Beligiannis GN (2012) Solving effectively the school timetabling problem using particleswarm optimization. Expert Syst Appl 39:60296040

Terashima-Marin H, Ross P, Valenzuela-Rendon M (1999) Evolution of constraint satisfaction strategiesin examination timetabling. In: Paper presented at the genetic and evolutionary computation conference(GECCO-99)

Turabieh H, Abdullah S (2011) An integrated hybrid approach to the examination timetabling problem. Int JManag Sci 39:598607

Valouxis C, Housos E (2003) Constraint programming approach for school timetabling. Comput Oper Res30(10):15551572

Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338353ZhangD, Liu Y,MHallah R (2010) A simulated annealing with a new neighborhood structure based algorithm

for high school timetabling problems. Eur J Oper Res 203(3):550558Zhipeng L, Jin-Kao H (2010) Adaptive Tabu search for course imetabling. Eur J Oper Res 200:235244

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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