Construct Examination Timetabling Using Graph Colouring in CAS UUM

8/13/2019 Construct Examination Timetabling Using Graph Colouring in CAS UUM

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CHAPTER ONE

INTRODUCTION

1.1BackgroundAccording to Hussin, Basari & Othman (2011), timetabling is at large covering many

different types of problems which have their own unique characteristics. There are three most

common academic timetabling problems which are school timetable, university timetable and

exam timetable (Hussin et. al, 2011). However, this study only covered on examination

timetabling. Examination timetabling is concerned with an assignment of exams into limited

number of timeslots subject to a set of hard constraint (Burke, Elliman, Ford & Weare, 1996

in Ayob, Abdullah & Abdul Malik, 2007). Basic problem in examination timetabling is to

assign examinations to a limited number of time periods in such a way that there are no

conflicts with some requirements (Carter, Laporte & Lee, 1996). A set of exams must be

scheduled to a set of timeslots such that every exam is located in exactly one timeslot within

the timetable, subject to certain constraints (Burke, Eckersley,McCollum, Patrovic & Qu,

2004).

There are two types of constraints known as hard constraints and soft constraints. Hicks et al.

(2006) defined hard constraints as those which definitely need to be satisfy. On the other

hand, soft constraint might be some requirements that are not essential but should be satisfied

as far as possible. Examination timetabling is concerned with an assignment of exams into a

limited number of timeslots subject to a set of hard constraints (Burke et al., 1996 in Ayob et

al., 2007). According to Ayob et al. (2007), common hard constraints for the examination

timetabling problem are: (i) no students should sit for two or more exams at the same timeslot

and (ii) the scheduled exams must not exceed the room capacity. In practical examination

timetabling problem, there are many other constraints and the constraints vary among

institutions (Ayob et al., 2007). Previous studies, problem statement and objective of the study

will be discussed in this chapter. Method that consist initial solution and improvement

solution will be discussed in chapter two. Chapter three will discussed about results,

contribution of the study, discussion and some recommendation for future works. However,


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this project will use Graph Coloring with Largest Degree approach and Tabu Search to find

best solution for examination timetabling at Universiti Utara Malaysia (UUM).

1.2Previous StudiesThere are lots of prior studies about examination timetabling in institutions. Rahim, Bargiela

& Qu (2013) introduced new optimization method for the examinations scheduling problem

by performing permutations of slots and assignments of exams upon the feasible schedules

obtained by the standard graph coloring methods with largest degree (LD) ordering. Hussin et

al. (2011) use graph coloring approach in order to guarantee that all exams are scheduled and

students can sit all the exams that they are required to do. After producing the examination

timetabling for all subjects, distribution of students among the rooms was done using

selection heuristic which equivalent to the knapsack filling problem. Cupic, Golub &

Jakobovic (2009) use genetic algorithm to produce best solution. The quality of the solution

will depend on how many students are scheduled to have more than one exam at the same day

and how many times this happens for those students. Quality solution also depends on how

many students have scheduled exams at adjacent days and how many times this happens for

those students.

Ayob et al., (2007) presented a real world examination timetabling problem proposed

objective function which called Penalty Cost. Penalty Cost attempt to spread out exams over

timeslots so students have larger gaps between exams. On the other hands, Malim, Khader &

Mustafa (2006) use three artificial immune algorithms and compare the effectiveness of the

algorithm on examination timetabling. This paper proved that the clonal selection and

negative selection algorithms are more effective than immune network algorithm in producing

good quality of examination timetabling. Hussin (2005); Gaspero & Schaerf (2001); White &

Xie (2001) used tabu search technique for examination timetabling. Burke et al. (2004)

suggest some methods which suitable to be used in examination timetabling problem which

are Simulated Annealing, Tabu Search and Great Deluge.

Carter (1996) focused on Graph Coloring Approach while Carter et al. (1996) focused on

comparing five Algorithmic Rules. Carter et al. (1996) considered five criteria for the list of


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processing scheme. There are five types of Algorithmic Rules which are Largest Degree (LD),

Saturation Degree (SD), Largest Weighted Degree (LWD), Largest Enrollment (LE) and

Random Ordering (RO). LD is that examinations conflicting with many others are hard to

schedule and should be assigned first. SD considered the examination that is selected next is

the one with the fewest number of feasible available periods remaining. In a sense, the most

difficult exam to schedule is the one that has the least flexibility in terms of choice of period.

LWD is done by selecting the exams with largest degree, which each edge is weighted by the

number of students in conflict. LE involved examinations with large enrollments are difficult

to schedule as they create more conflict will be assigned first. Lastly, RO is selecting the

exams randomly. This type of Algorithmic Rules is mainly considered for benchmark

comparisons. Carter et. al (1996) come out with LD strategy produces a better solution cost

most of the time. However, SD is better than LD on all measures: solution quality,

backtracking and CPU time.

1.3Problem StatementThis study is conducted to solve examination timetabling problem by using heuristic

techniques. There are 13 schools in UUM, but this study only focused on undergraduate

students examination data for School of Quantitative Science in Universiti Utara Malaysia

(SQS UUM) due to time restrictions. Sufficient time is needed to assign examination for all

undergraduate students in UUM. Particularly, the dataset use in this study is the real data for

undergraduate students examinations for second semester 2012/2013 session. The total

number of examinations is 33 exams with 582 students. Total of 350 capacities for 9 rooms

per timeslots are available in this study.

In SQS UUM there are two exams that taken by more than 350 students. In real situation,

these exams will be scheduled in huge halls rather than in halls SQS UUM due to capacity

constraint. Thus, these exams have to be excluded. Each exam must be scheduled in a time

slots and no students will be assigned in two or more exams in a same timeslots. This study

involved 7 consecutive days. This mean this exam will be conducted for whole 1 week

including weekend. Each day have 2 timeslots, so there are 14 timeslots in total. 3 hours are

provided for each timeslots. Morning session start from 9.00 a.m to 12.00 p.m while evening


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session start from 2.30 p.m to 5.00 p.m. In order to replicate the real-world timeslots model,

following vectors (Figure 1) which demonstrate the idea is produced.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Figur e 1 : Vector of Timeslots

In Figure 1, the timeslots are represented as indexes. Timeslots 0 and 1 are referring to day 1;

timeslots 2 and 3 are referring to day 2, etc. Room specifications are shown in Table 1. Each

examination should be assigned to a single room, unless this cannot be avoided. In

exceptional cases such as no room available to fit the exam, then the exam can be assigned to

multiple rooms but the room location should be close to each other. This constraint is

enforced due to the location practicality. Room location is shown in Figure 2.

Examination Room Capacity

BTB1 30

BTB3 30

BTB7 30

BTB9 30BTB11 30

DPB1 50

DPB2 50

DPB3 50

DPB4 50

Total 350

Table 1 : Examination room specif ications


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Figure 2 : Location for examination rooms

1.3 Objective of the Study

The objective of the examination timetabling is to minimize the cumulative inconvenience

implied by the proximity of consecutive exams taken by students. This measured by the cost

function originally proposed as in Formula 1.

Formula 1

where N is the number of exams,sijis the number of students enrolled in both exams i andj, tj

is the time slot where exam j is scheduled, tiis the time slot where exam i is scheduled, M is

the total number of students, w|tjti| is the weight that can be calculated as following formula

:

= Formula 2

The lower the cost obtained, the higher is the quality of the schedule, since the gap between

two consecutive exams allows students to have extra revision time. Hard constraints and soft

constraints involved in this study are stated in next section.


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CHAPTER TWO

METHODOLOGY

2.1 Introduction

This chapter will explain more on method in producing initial solution and improvement solution.

Graph Coloring approach are used in to initiate initial solution while Tabu Search are used to generate

best solution. However, assigning students into examination halls was done manually since several

exams were scheduled per timeslots. Figure 3 shows overall flow in this chapter.

Data Collection

and Analysis

Initial Solution

Improvement

Solution

Fitting into

Rooms

Start

FinalSolution

F igure 3 : Process in Examination Timetabli ng


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2.2 Data Collection

An interview with staffs in Academic Affair Department UUM was conducted in order to

identify soft and hard constraints involve for examination timetabling at UUM. Soft and hard

constraint involve in this study have been explained in Constraint section. However, there are

some soft constraints that excluded in this project such as weekend constraint since this study

just involve small data size and short duration which only 14 timeslots. Data for registered

students for each courses offered in SQS UUM was gathered from Computer Center UUM.

2.3 Data Analysis

Data gathered from Computer Center was analyzed. The data include all SQS UUM

undergraduate students matric numbers and courses taken. This study only considered

courses offered by SQS UUM thus other courses taken by these students but offered by other

schools are excluded. Specific codes are given to each students and courses. Table 2 shows

example of code given to each students.

Current Code New Code

Matric Number Course Matric Number Course

122826 BJTM2033

s1

excluded

122826 SBLF1053 excluded

122826 SQIT3033 e4

122826 SQPX3908 excluded122826 SQQS3033 e28

122826 SQQS3073 e32

129184 BPMM1013 s2 excluded

129825 BWFN3013excluded

129825 BWRR3023

129831 SQQM2043s3

e12

129831 SQQS2043 e26

129873 SQQM2043 s4 e12

129874 SQQM2043s5

e12

129874 SQQP3043 e21

Table 2: Code Given for Each Students and Courses

Based on Table 2, each students are assign assnwhich n = {1,2,582}while each courses are

assign as ei which i = {1,2,33}. There are some courses that we exclude, such as courses


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BJTM2033 since this course are offered by other school. We also exclude some case such as

student 129825 since this student only take subjects offered by other schools. This process

continues for all 4153 enrollments. However, there was only 1789 enrollment that included in

this study. Table 3 shows the 33 courses and new code given for each of them.

NO. COURSES Current Code New Code

1 PENGATURCARAAN DALAM APLIKASI PERNIAGAAN SQIT1013 e1

2 KOMPUTER DALAM PEMUTUSAN PERNIAGAAN SQIT3013 e2

3 SISTEM MAKLUMAT DAN PEMBUATAN KEPUTUSAN SQIT3023 e3

4 PEROLEHAN PENGETAHUAN DALAM PEMBUATAN SQIT3033 e4

5 KALKULUS I SQQM1034 e5

6 KALKULUS I SQQM1043 e6

7 PERISIAN MATEMATIK DAN PENGGUNAANNYA SQQM1053 e7

8 MATEMATIK DISKRET SQQM1063 e89 ALGEBRA LINEAR SQQM2023 e9

10 KALKULUS LANJUTAN SQQM2033 e10

11 KALKULUS II SQQM2034 e11

12 KALKULUS II SQQM2043 e12

13 PERSAMAAN PEMBEZAAN SQQM2053 e13

14 PERMODELAN MATEMATIK SQQM3023 e14

15 MATEMATIK PERNIAGAAN SQQM3063 e15

16 TEKNIK PEMBUATAN KEPUTUSAN I SQQP1013 e16

17 TEKNIK PEMBUATAN KEPUTUSAN II SQQP2013 e17

18 TEKNIK PEMBUATAN KEPUTUSAN III SQQP3013 e18

19 PEMODELAN PEMUTUSAN SQQP3023 e19

20 PEMODELAN BERKOMPUTER DALAM PERNIAGAAN SQQP3033 e20

21 TEKNIK-TEKNIK HEURISTIK SQQP3043 e21

22 PEMODELAN SISTEM DINAMIK SQQP3063 e22

23 PENJELAJAHAN DAN PENGITLAKAN DATA SQQS1033 e23

24 KEBARANGKALIAN DAN STATISTIK SQQS1043 e24

25 STATISTIK PERNIAGAAN DAN PENTADBIRAN SQQS2023 e25

26 ANALISIS REGRESI BERGANDA DALAM PERNIAGAAN SQQS2043 e26

27 PEMUTUSAN MELALUI KAEDAH TIDAK BERPARAMETER SQQS3023 e27

28 RAMALAN PERNIAGAAN SQQS3033 e28

29 ANALISIS MULTIVARIATE BAGI DATA PERNIAGAAN SQQS3043 e2930 REKABENTUK UJIKAJI DALAM PERNIAGAAN SQQS3053 e30

31 PENINGKATAN KUALITI BERSTATISTIK SQQS3063 e31

32 KAEDAH PENYELIDIKAN SQQS3073 e32

33 PERSAMPELAN UNTUK PEMBUATAN KEPUTUSAN SQQS3083 e33

Table 3: Cour ses with Code


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2.4 Initial Solution

This study used Use Graph Coloring technique to assign the examination to timeslot. First

step involve construction of conflict matrix. The conflict matrix is one of the most important

aspects in exam timetabling problem representing hard constraint or a pair of clashing exams.

The construction of the conflict matrix helps in determines the constraints that no student

must attend more one exam at the same time. Two subject conflict with each other if there are

at least one student take both subject. Conflict matrix in Table 5 is developed based on

students courses registration. Based on conflict matrix, 33 examinations (e1,e2,,e33) is

assigned into 14 timeslots (t1,t2,,t14) using LD Algorithmic rules. As mention earlier in

previous studies section, LD is that examinations conflicting with many others are hard to

schedule and should be assigned first. Initial solution obtained is shown in Table 4. Objective

function,z(x)is calculated asFormula 1.

Timeslot Exams

1 e18 e11 e16

2 e3 e19 e7

3 e2 e1 e15

4 e9 e27

5 e12 e5 e33

6 e20 e8

7 e4 e14 e24

8 e22 e10 e23

9 e28 e17

10 e21 e6

11 e26 e32

12 e31 e13

13 e29

14 e30 e25

Table 4: I ni tial solution based on Largest Degree (LD)


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Table 5: Conflict M atrix


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2.5 Improvement Solution

Tabu Search (TS) is used in getting best solution. At each state ti, TS explores a subset V of the

current neighborhood N (ti). Among the elements in V, the one that gives the minimum value of

the cost function becomes the new current state ti+1, independently of the fact whether z(ti+1) is

less or greater than f(ti).Such a choice allows the algorithm to escape from local minima, but

creates the risk of cycling among a set of states. In order to prevent cycling, tabu list is used,

which determines the forbidden moves. This list stores the most recently accepted moves. The

inverses of the moves in the list are forbidden. The simplest way to run the tabu list is as a queue

of fixed size k. That is, when a new move is added to the list, the oldest one is discarded. The

stop criterion is based on the so-called idle iterations: The search terminates when it reaches a

given number of iterations elapsed from the last improvement of the current best state. However,

iteration=12 and tabu tenure = 5 were fixed in this project.

TS involve three steps which are initialization, choice and termination, and update. We begin

with the same initialization used in neighbourhood search. After determine the neighborhood,

candidate solution from the set that minimize solution that minimize cost was chosen in choice

and termination step. Then, we perform update for the search method. Appendix A shows

calculation for best solution with fixed iteration=12.

2.6 Assign Exams into Rooms

After producing the exam timetable for all the subjects, distribution of students among the room

will be done manually since there are 9 rooms available with 350 capacities per timeslots. There

are only one to three subject scheduled per timeslots due to clashes constraint. The objective is to

assign every exam to single rooms. Rooms capacities are shown in Table 1. However, in some

cases, exams may be scheduled to multiple rooms that close to each other based on room

location inFigure 2. Table 6show total students enrollment for each exam.


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Exams Number of Students Exams Number of Students

e1 22 e18 102

e2 60 e19 34

e3 52 e20 92

e4 39 e21 20

e5 72 e22 58

e6 12 e23 35

e7 1 e24 146

e8 32 e25 36

e9 187 e26 78

e10 27 e27 50

e11 136 e28 83

e12 56 e29 24

e13 13 e30 14

e14 30 e31 97e15 3 e32 101

e16 31 e33 25

e17 1

Table 6: Total Students for Each Exam


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CHAPTER THREE

FINDINGS

3.1 Results

Table 7 shows list of iteration

Iteration nTabu Tenure Criteria

(swap)z(x)

1 2 3 4 5

0

1 e4 & e12 15.8959

2 e2 & e22 16.3785

3

e18 &

e22 16.9743

4 e20&e29 16.0922

5 e2 & e29 16.883

1

1 e4 & e12 e33&e26 16.2754

2 e4 & e12 e20&e29 15.7842

3 e4 & e12 e31&e2 17.7497

4 e4 & e12 e7&e14 16.3088

5 e4 & e12 e3&e21 17.1312

2

1 e4 & e12 e20&e29 e5&e11 17.4025

2 e4 & e12 e20&e29 e3&e26 19.0072

3 e4 & e12 e20&e29 e5&e25 14.2779

4 e4 & e12 e20&e29 e2&e4 15.9765

5 e4 & e12 e20&e29 e4&e30 15.7788

3

1 e4 & e12 e20&e29 e5&e25 e7&e14 14.7344

2 e4 & e12 e20&e29 e5&e25 e4&e30 14.2725

3 e4 & e12 e20&e29 e5&e25 e15&e13 14.3894

4 e4 & e12 e20&e29 e5&e25 e30&e31 13.4179

5 e4 & e12 e20&e29 e5&e25 e27&e30 13.6527

4

1 e4 & e12 e20&e29 e5&e25 e30&e31 e21&e28 13.5317

2 e4 & e12 e20&e29 e5&e25 e30&e31 e28&e29 13.4813

3 e4 & e12 e20&e29 e5&e25 e30&e31 e3&e22 13.5844

4 e4 & e12 e20&e29 e5&e25 e30&e31 e27&e30 12.9117

5 e4 & e12 e20&e29 e5&e25 e30&e31 e13&e16 13.4100


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5

1 e4 & e12 e20&e29 e5&e25 e30&e31 e27&e30 e3&e22 12.9587

2 e4 & e12 e20&e29 e5&e25 e30&e31 e27&e30 e18&e22 13.6140

3 e4 & e12 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 12.9186

4 e4 & e12 e20&e29 e5&e25 e30&e31 e27&e30 e7&e14 13.3682

5 e4 & e12 e20&e29 e5&e25 e30&e31 e27&e30 e2&e22 13.4747

6

1 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 e3&e26 16.4625

2 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 12.8371

3 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 e18&e22 13.6209

4 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 e2&e20 13.9227

5 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 e2&e31 14.0099

7

1 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 12.8976

2 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 e7&e16 12.9005

3 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 e13&e15 12.9211

4 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 e2&e20 13.2187

5 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 e12&e27 13.9936

8

1 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 e7&e15 12.8748

2 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 e7&e16 12.9609

3 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 12.8495

4 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 e11&e19 13.5488

5 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 e18&e22 13.1666

9

1 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 e6&e17 12.7738

2 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 e7&e15 12.8267

3 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 12.6825

4 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 e20&e29 13.44405 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 e20&e27 13.4985

10

1 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 e12&e21 13.2529

2 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 e6&e17 12.6377

3 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 e7&e15 12.6597

4 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 e5&e25 14.0510

5 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 12.5614

11

1 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 e5&e25 13.9608

2 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 e27&e30 13.2939

3 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 e10&e13 12.8300

4 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 e1&e6 12.4938

5 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 e10&e14 12.5606

Table 7: Tabu Search


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Iteration nTabu Tenure Criteria

(Swap)z(x)

1 2 3 4 5

0 1 e4&e12 15.8459

1 2 e4&e12 e20&e29 15.7842

2 3 e4&e12 e20&e29 e5&e25 14.2779

3 4 e4&e12 e20&e29 e5&e25 e30&e31 13.4179

4 4 e4&e12 e20&e29 e5&e25 e30&e31 e27&e30 12.9117

5 3 e4&e12 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 12.9186

6 2 e20&e29 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 12.8371

7 1 e5&e25 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 12.89768 3 e30&e31 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 12.8495

9 3 e27&e30 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 12.6825

10 5 e10&e14 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 12.5614

11 4 e4&e32 e3&e22 e16&e19 e12&e20 e2&e21 e1&e6 12.4938

Table 8: Tabu List

Timeslot Exams/Rooms Exams

0Exams e11 e18 e19

RoomsDPB1, DPB2,

DPB3

BTB3, BTB7,

BTB9, BTB11 DPB4

1Exams e7 e16 e22

Rooms BTB 1 DPB3 DPB1, DPB2

2Exams e6 e15 e21

Rooms BTB9 BTB7 BTB11

3

Exams e9 e30 -

RoomsDPB1, DPB2,

DPB3, DPB3 BTB1 -

4Exams e25 e32 e33

RoomsDPB4

BTB3, BTB7,

BTB9, BTB11 DPB3

5Exams e8 e29 -

Rooms DPB3 BTB1 -

6Exams e10 e20 e24

Rooms BTB1 BTB3, BTB7, DPB1, DPB2,


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BTB9, BTB11 DPB3

7Exams e3 e14 e23

Rooms DPB3, BTB1 BTB11 DPB4

8Exams e17 e28 -

Rooms BTB1 DPB1, DPB2 -

9Exams e1 e2 -

Rooms BTB1 DPB3. DPB4 -

10Exams e4 e26 -

Rooms DPB4 BTB1. DPB3 -

11Exams e13 e27 -

Rooms BTB1 DPB2 -

12Exams e12 - -

Rooms BTB9, BTB11 - -

13

Exams e5 e31 -

Rooms DPB1, DPB 2 DPB3, DPB 4 -

Table 9: Exams Timetable with Rooms

3.2 Contribution of the Study

solve the basic examination timetabling problem of assigning examinations totimeslots without violating a clash constraint.

Assign the exams while considering more spacing between exams to maximizestudents exams preparation time

3.3 Conclusion

Use a computerized system which the output can be more consistent and the experimentcan be applied for any problem size.

Use hybridization method to generate best solution. Use heuristic method in assigning exams into rooms.


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Documents

Construct Examination Timetabling Using Graph Colouring in CAS UUM