Applied Mathematical Sciences, Vol. 9, 2015, no. 130, 6449 - 6460

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.53292

Quadratic Assignment Approach for

Optimization of Examination Scheduling

Fadhilah Ahmad*, Zarina Mohammad, Hasni Hassan,

Ahmad Nazari Mohd Rose and Danlami Muktar

Faculty of Informatics and Computing

Sultan Zainal Abidin University (UniSZA)

Tembila Campus, 22200 Besut, Terengganu, Malaysia *Corresponding author

Copyright © 2015 Fadhilah Ahmad et al. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

The increase in student enrolment into institution of higher learning, and the great

increase in flexibility in choice of courses has made examination scheduling very

difficult to handle in the traditional scheduling system. This paper investigates a

quadratic assignment approach in examination scheduling by presenting a

quadratic assignment approach to eliminate or reduce the level of conflicts in the

examination timetabling. The examination with the highest number of students

will be prioritized across all faculty courses, followed by other criteria such as

courses with the highest number of students from different faculties. A framework

for optimizing and handling the conflicts at all constraint levels is proposed to

solve the examination scheduling problem.

Keywords: Optimization, Quadratic Assignment Problem, Examination scheduling

1. Introduction

Examination scheduling is the allocation of examinations subject while

considering the constraints of such as examination period, time slots, and rooms

sizes to entities being placed in space time when fulfilling a set of desirable

purposes or objectives. Examination scheduling problems involves time-related

scheduling and combinatorial optimization that tends to be unraveled with

combination of search and heuristics, which usually lead to satisfactory but

solutions that are not fully optimal.

6450 Fadhilah Ahmad et al.

In timetable scheduling problems, events must be slotted to certain times which

satisfy several constraints. It is difficult to design knowledge-based and operation

research based algorithms to solve such problems due to the complicity of the

system. On the other hand, constraint satisfaction techniques have been used to

solve hard constraints; however, it is more difficult to handle soft constraints such

as that of the examination scheduling problem.

These sets of objective demand an automated system built around mathematical

models or algorithm solutions for constraints satisfaction. Such constraints may

involve students having more than one examination at the same time or students

taking more than one course within a day, under-utilization of room facilities and

ensure capacity of individual rooms is not exceeded at any time throughout the

examination session, exceeding duration of examination periods and inadequate

availability of lecturers.

The Quadratic Assignment Problem (QAP) is a well-known combinatorial

optimization problem with a wide variety of practical applications. Many

heuristics solution has been proposed for QAP. This paper presents an

improvement of the quadratic assignment problem initiated in [1].

2. Problem definition and related works

An improper examination timetable may hinder the students for better preparation

for the examination. In addition, there are lots of subjects involved in the

examinations to be scheduled, a large number of students who are registering

different subjects, the limited number of rooms or exam halls that are available,

and some other constraints related to individual student’s examination timetable.

That is, a student is not scheduled to sit for more than one subjects at the same

time. The scheduling problem created by these set of circumstances clearly poses

an interesting intellectual challenge. Therefore, this study is to explore this

challenge and hence, will propose a solution to this problem.

3. Optimization approaches for examination scheduling

Examination timetabling is a well-researched and important area [3]. There is

some novel research that focused on the modeling and formulation of

Examination Timetabling Problem (ETP), and a number of research in the

applications of heuristics to solve the problem as shown in Table 1.

A new binary model for examination time tabling problem has been investigated

to deal with same causes that are scheduled in different time slots. It is also

possible to schedule two or more courses in a room simultaneously. The model

also considers some hard and soft constraints, where the objective function is set

in such a way that soft constraints are satisfied as much as possible [2].

Quadratic assignment approach for optimization 6451

Likewise in a research by [10], an integer programming (IP) model was adopted

to address certain constraints of examination scheduling problem such as

restricted availability of classrooms and the increased flexibility of the student’s

choices of courses. It assigns courses to the available time slots and rooms. The

quality of the schedule produced depends on the relative position of the courses

assigned to the available time periods, a condition that the IP model attempts to

satisfy by constructing groups of courses that are assigned to groups of time

periods. The system was adopted by the Athens University of Economics and

Business for use and with success. The approach is said to be theoretical

development and has not analyze an actual computer-based implementation

The issue of assigning invigilators to examination which are faculty lecturers or

graduate students in educational institutions using a computer based system has

been addressed by [11]. They have adopted Analytic Network Process model for

estimating exam objective weights. The exam weights used arbitrary assignment

ratio scale numbers to assign invigilators for certain examination slot.

An Ant colony optimization algorithms was adopted by [18] in solving

examination scheduling problem which was described as combinatorial

optimization problems derived from the field of swarm intelligence. The research

described how real ant colonies find shortest paths between food sources and their

nest. Ant Systems (AS) was implemented in the algorithms for solving the

traveling salesperson problem. The solution approach consists of n cycles of

examination construction process. Each cycle consists of m ants where each m ant

constructs a viable solution. Each ant builds a complete tour that visits all nodes.

Honey-bee mating optimization algorithm approach was pioneered by [19] to

address examination timetabling. This process was described as a swarm-based

approach to optimization inspired by real honey-bee mating. The algorithm

mimics the mating flight of a queen bee. The mating process (i.e. generating a

new solution) of the queen (current best solution) begins when the queen leaves

the nest to perform a mating flight during which time the drones (trial solutions)

follow the queen and mate with her. The algorithms start by initializing the

population and the best solution is selected as the queen.

A quadratic assignment was adopted in 2014 by [1] to address examination

timetabling problem by developing a mathematical model. The model will

consider E exams needed to be scheduled in a limited number of T, time slots and

limited number of R, room and capacity, taking into account all examination must

be scheduled within the time frame. The room capacities and the restrictions that

students cannot take more than one examinations at the same time are among

other constraints. The heuristic solution is formed to identify possible constraints

6452 Fadhilah Ahmad et al.

and groups them as hard and soft constraints. The heuristic algorithm ensure the

solution is improved if needed, until an optimal solution is achieved and all

constraints are satisfied.

[21] Has adopted a transformation process to solve examination time tabling as

quadratic assignment problem. The objective was to separate lists of high

interactions as far apart as possible in other to obtain a well-spaced schedule with

the use of activity and traffic cost transformation. A super imposition was

considered to spread courses from the same faculty by interweaving with courses

from different faculties.

Table 1: Optimization algorithms for examination scheduling problems

Bil. Authors date Model Strength Weaknesses

1 Thin-Yin

Leong, Wee-

Yong Yeong

1990[16]

Hierarchical

approach based

on Quadratic

Assignment

problem

Feasible solution, less

computational effort, easy to

understand, implement, and

interact.

Heuristic technique is used

in each stage causing

problems when the size

constraints increase.

2 Thin-Yin

Leong, Wee-

Yong Yeong

2012[21]

Quadratic

Assignment

Problem

Reduce the assignment cost

by

Adoption of monotonic

examination assignment.

Results were not

spectacular, due to the

complicity of the grouping

process.

3 Nasser R.

Sabar, Masri

Ayob Rong

Qu, Graham

Kendall

2011[19]

Honey-Bee

Mating

Optimization

Produce good quality

solutions for benchmarks

ETP.

This solution is yet to

receive wide acceptance.

4 Micheal Eley,

2006 [18]

Ant Systems

Optimization

The solution is capable of

solving large real world

exam timetabling problems.

This approach requires

proper adjusting parameters

to achieve an optimal

solution

5 Dimopoulou

and Miliotis

2001[10]

Integer

Programming

Contributed numerous

improvements to the

decision-making process

Schedule outcome is

parameters dependent.

Quadratic assignment approach for optimization 6453

Table 1: (Continued): Optimization algorithms for examination scheduling

problems

4. Proposed framework

This paper proposes a framework for examination scheduling based on QAP. The

framework consists of five phases (refer to Fig. 1) in which three phases applied

QAP approach. In Phase 1, all parameters for examination scheduling are

determined. Constraint tables are generated in Phase 2. Assignment of parameters

is done in Phase 3. The other two phases are for solving the constraints and

applying the solution. The whole phases are run consecutively, one after the other

from Phase 1 until 5 to obtain an optimized solution for examination scheduling.

The solution can be used to generate examination time table at the end of each

academic semester or whenever it is required.

4.1 Quadratic Assignment Problem

The Quadratic Assignment Problem (QAP) was introduced by Koopmans and

Beckman in 1957 in the context of locating "indivisible economic activities". The

objective of the problem is to assign a set of facilities to a set of locations that will

minimize the total assignment costs. The assignment cost for a pair of facilities is

a function of the flow between the facilities and the distance between the locations

of the facilities [13]. Due to wide assortment of applications, QAP has been

extensively used to formulate other practical problems such as economic problems

[22], Scheduling problems [24], Timetabling problems [21], and Facility layout

problems [23].

6 Alireza

Rashidi

Komijan and

Mehrdad

Nouri Koupaei

2012[2]

Binary Model Eliminates all conflicts in

individual student timetable.

The model dose not satisfy

all soft constraints such as

consecutiveness, and

mixed duration.

7 Sagir and

Ozturk,

2010[11]

Analytic

Network Process

Model

Considers system dynamics;

people's expectations, and

system parameters.

Focus more on the

Prioritization of invigilator

assignment.

6454 Fadhilah Ahmad et al.

This work enhances the QAP to emulate the scheduling of examination time table

such that the objective function will eliminate or reduce the rate of conflict at each

assignment. The enhanced QAP considers the complicity of constraints; conflict,

capacity of individual rooms, capacity of the whole rooms, and consecutiveness of

scheduled exams. The original QAP was implemented by considering only a

limited number of constraints. In order for the QAP to be applied for examination

scheduling, its formula needs to be adjusted so that all considered constraints can

be taken into account. The mathematical formula for this purpose is describe in

the following sections.

4.2 Mathematical Formulation

An E number of exams need to be scheduled in a limited number of T time slots

and limited number of R room and capacity, taking into account all examination

must be scheduled within the time frame and the rooms capacities and the

restrictions that students cannot take more than one examination at the same time

as other constraints.

N= {1, 2, 3 … n}

:n T E R , the set of all permutations.

ijt T , is the time slot for an exam to be assigned to a room

1 1

minn n

ij ij

i j

t

i = 1, 2, 3 … n

j = 1, 2, 3 ... (n-1)

Quadratic assignment approach for optimization 6455

Fig. 1: Proposed framework for Examination scheduling based on Enhanced QAP

Represents the assignment of exams to specific time slot and different rooms,

to prevent conflicts. The conflicts degree of each examination is minimized when

exams i are assigned to room j at a specific time and is denoted by ij ijt where

1 if exam i is assigned to room r

0 otherwise ij

6456 Fadhilah Ahmad et al.

Room Assignment

An examination can be assign to one or more room depending on the number of

the student to take the examination or otherwise, ie two or more examination in a

room. But for all examination assigned to the room/s the number of students to

seat for the examination most be less or equal the seat capacity/ies of the room/s.

this is mathematically express in the figure below.

1

n

i j i j

j

e r se sr

ie E ∀ i = 1, 2 … n

jr R ∀ j = 1, 2 … (n-1)

Sei= Size of student in the exam e

i

Srj = Size of room

jr

Course Assignment to time slot

For each course assignment to time slot it ensure that no one student have more

than one examination for a particular time slot ie for all examination assigned to a

particular time slot, it ensure that all the examination have no common student.

This can be express mathematically in the figure below.

1 2 ...i i ij ne E e t e e e

1

n

i i

i

e e

i ije t

i = 1,2,…n

lei = student in examination i

In examination scheduling, Quadratic assignment reduces the rate of conflicts

during the scheduling of the examination timetable. The examination for a course

taken by students from different faculties is being prioritized. Next, the same

courses with students from the same faculty, but different program is considered

and finally the highest number of repeating student from the previous semesters.

At each stage, course with the highest number of registered students is given

prioritization.

Quadratic assignment approach for optimization 6457

The following algorithm and flowchart describe the scheduling process:

Initialize the parameters

Get the total number of available seats, tot_seats and total number of students per time

slot, tot_students

If tot_seats ≥ tot_students

Begin assignment

{Set the current solution, current_sol = initial_sol

Apply a sequential search to list of examination E

Select an examination E in ascending order of conflicts where E1 > E2 >........En

}

If current solution is best solution, current_sol = best_sol

Apply current_sol to E1

If the result is feasible

Apply current_sol to E1………En

Else {

Improve current_sol

Update the best solution, best _sol = current_sol

}

End

The framework defines all possible conflicts and groups them as hard and soft

constraints. A number of algorithms for solving the conflicts have been proposed

to eliminate the possible constraints in the system.

5. Conclusion and recommendation

In this paper, we highlighted some weaknesses and strength of the related studies

on examination scheduling. The proposed examination timetabling problem was

formulated based a Quadratic Assignment Problem (QAP). Since the number of

examinations can be greater than the number of time periods and in a quest to

minimize all possible conflicts, the objective function is set in such a way that,

hard constraint satisfaction is maximized. The future work of this research is to

implement an examination scheduling system which will allow simulation and

evaluation of various examination data including the benchmark data. The result

of the implementation will provide the planning of examination scheduling

becomes easier and fulfil the user expectation.

Acknowledgements. The author would like to extend appreciation of the support

given by Universiti Sultan Zainal Abidin (UniSZA) and the sponsorship by Kano

State Government.

6458 Fadhilah Ahmad et al.

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Received: April 15, 2015; Published: November 2, 2015