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