The University Timetabling Problem: Modeling and Solution Using Binary Integer Programming with Penalty Functions

Abstract

The University Course Timetabling Problem is a particular type of scheduling problems
known as a difficult problem arising in academic institutions, and an application of combinatorial
optimization. The problem consists of a coordination of lectures, students, teachers
and classrooms to avoid clashes between them. In this work, we address a course
timetabling problem encountered at Taibah University. A binary integer programming model
of the problem is proposed and a solution methodology based on an exterior penalty function
and two new penalty functions, called variance-penalty function and pseudo-convex
combination-penalty function, is developed. Solving this problem aims to minimize the waiting
time between lectures for students and teachers and preventing clashes of lectures and
classrooms.

Full text

The University Timetabling Problem: Modeling and Solution
Using Binary Integer Programming with Penalty Functions
Moustapha Abdellahi ∗1and Hussein Eledum 2
1Department of Mathematics
Taibah University, College of Science,
Madinah, Saudi Arabia
mouabdellahi@gmail.com
2Department of Statistics
University of Tabuk, College of Science,
Tabuk,Saudi Arabia
heledum@yahoo.com
Abstract
The University Course Timetabling Problem is a particular type of scheduling problems
known as a difficult problem arising in academic institutions, and an application of combi-
natorial optimization. The problem consists of a coordination of lectures, students, teach-
ers and classrooms to avoid clashes between them. In this work, we address a course
timetabling problem encountered at Taibah University. A binary integer programming model
of the problem is proposed and a solution methodology based on an exterior penalty func-
tion and two new penalty functions, called variance-penalty function and pseudo-convex
combination-penalty function, is developed. Solving this problem aims to minimize the wait-
ing time between lectures for students and teachers and preventing clashes of lectures and
classrooms.
Keywords:Timetabling, Integer programming, Decomposition, Penalty Functions.
Mathematics Subject Classifications: 90C10.
1 Introduction
The university course timetabling is a problem usually encountered in most academic institu-
tions. The problem is an assignment of a given number of teachers to a number of courses
in such a way that a teacher cannot have two courses at the same time. A special case of
assignment problems has been established and solved in (Guozhong and Xiao-Xiong, 2008).
The timetabling problem has received a great attention and still treated by researchers. The
∗Corresponding author.
International Journal of Applied Mathematics and Statistics,
Int. J. Appl. Math. Stat.; Vol. 56; Issue No. 6; Year 2017,
ISSN 0973-1377 (Print), ISSN 0973-7545 (Online)
Copyright © 2017 by International Journal of Applied Mathematics and Statistics
www.ceser.in/ceserp
www.ceserp.com/cp-jou
r

timetabling problems at educational institutions have been classified by (Schaerf, 1999) as fol-
lows: school timetabling, course timetabling, and exam timetabling problems. The timetabling
is a very difficult problem in scheduling. It belongs to the family of NP-complete problems (Bakir
and Akop, 2008). Some studies, such as (Grobner, Wilke and Buttcher, 2003) have tried to es-
tablish a common structure describing the timetabling problems, but no general model exists
which is applicable for all cases (Benli and Bostali, 2004). Many techniques have been used
to develop algorithms to solve timetabling problems such as graph coloring heuristics (Yanez
and Ramirez, 1999), tabu search (Azimi, 2005), genetic algorithms (Burke and Petrovic, 2002).
Another traditional solution approach for timetabling problems is mathematical programming
(Aizam and Liong, 2013), and (Daskalaki and Housos, 2004). The timetabling involves many
factors depending on students such as course clashes due to different levels or depending on
teachers such as teaching classrooms, which must be taken into account. We have developed
a mathematical model consisting of an objective function subject to some constraints for this
problem. The penalty function methods have successfully been used to solve different prob-
lems in various categories of optimization (Lucidi and Rinaldi, 2010). In this paper, we focus
on finding an optimal or suboptimal solution to the course timetabling problem using (0-1) inte-
ger programming with exterior penalty function method and two new penalty function methods,
called variance-penalty function and pseudo-convex combination-penalty function methods. A
decomposition of the problem into subproblems was necessary to deal with it.
2 The Math Department Description
The department of Math at Taibah University offers a program in Mathematics leading to a
bachelor degree. The courses of the program are organized in such a way that there is one
track for all students. Furthermore, these courses are classified into core and elective. Some
courses have practical part given in computer laboratory. A regular student has normally 18
hours per week, however, the pre-graduated student can take at most 21 hours. Any student
enrolled in the program is expected to graduate within eight semesters. Any core course must
be done after its prerequisite. Elective courses cannot be taken before fifth semester.
3 Problem Statement
3.1 Criteria for Constructing a Timetable
A suitable timetable for the department of Math has to meet the following rules:
Overlapping Conditions: The three main elements involved in any course timetable are (i)
the students, (ii) the teachers and (iii) the classrooms. Each element is associated to some
criterion. The first means that a students group (section) cannot be assigned more than one
course for a specific period. The second implies that, if a teacher gives more than one course,
they should not overlap. The last one means that there should not be more than one course in
a classroom at the same time.
International Journal of Applied Mathematics and Statistics
165

Courses Credit: Each course contains only 3 hours, and taught in two lectures with a period
of 1.5 hour. The lectures of each course must be distributed in such a way that there is one day
off between them. If the course consists of theory and practice, we treat them as two separate
courses, but the practice should be one period per week.
Slots Description: The number of slots per day is 8 and the period of each slot is 1.5 hours.
Furthermore, the fourth slot is considered as a break. No lecture can start before 8 a.m. and
no lecture can be taken after 8 p.m.
Classroom Occupancy: All lectures of a given course should be held in the same classroom.
Moreover, a lecture of any course consists of one period.
Theory-Practice Precedence: If a course is composed of theory and practice, then the prac-
tice must be held after theory; and they must be done in the same week.
Student’s Load: A regular student cannot take more than 18 hours per week for all courses. If
a given course is a prerequisite of another, their precedence must be respected.
Pre-Graduated Student: The pre-graduated student can have at most 21 hours for all courses.
If he failed a course of the precedent semester, he is allowed to take that course and its pre-
requisite together.
4 Binary Integer Linear Programming Formulation
Problem Sets: To construct a mathematical model of course university timetabling which rep-
resents the features of the problem, we define the following sets:
I={1, ..., ni}set of days in a week in which the courses are offered,
J={1, ..., nj}set of time slots in a day, K={1, ..., nk}set of courses, L={1, ..., nl}set of
student groups, M={1, ..., nm}set of teachers, N={1, ..., nn}set of classrooms.
Decision Variables of the Model:
To tackle the timetabling problem, we define four sets of variables. The first one is called the
set of basic variables xi,j,k,l,m,n, the second set of variables is called auxiliary variables yi,j,l,
the third set of variables zim represents the existence of lectures for teacher mon day iand the
fourth set of variables zil represents the existence of lectures for students group lon day ias
follows:
∀i∈I,∀j∈J, ∀k∈K, ∀l∈L, ∀m∈M,∀n∈N
xi,j,k,l,m,n =⎧
⎪
⎨
⎪
⎩
1if course ktaught by teacher mfor the group of students l
is assigned to jth time slot of day iin classroom n
0otherwise
∀i∈I,∀j∈J, ∀l∈L
yi,j,k,l =⎧
⎪
⎨
⎪
⎩
1if a course k+sfor group students loverlap with its prerequisite k
for the same group lin jth time slot of day i
0otherwise
for s∈{1, ..., nk−1},k+s≤nk∀m∈M
International Journal of Applied Mathematics and Statistics
166

zim =⎧
⎪
⎨
⎪
⎩
1if j∈Jxi,j,k,l,m,n =0
0otherwise
∀l∈L
zil =⎧
⎪
⎨
⎪
⎩
1if j∈Jxi,j,k,l,m,n =0
0otherwise
The Model Constraints: As we defined the above rules for course university timetabling, now
we build the corresponding model in which the courses of students groups cannot overlap.
Courses overlapping: The following conditions ensures that there is no overlapping for courses.
k∈Km∈Mn∈Nxi,j,k,l,m,n ≤1,∀i∈I, ∀j∈J, ∀l∈L
Teachers overlapping: Each teacher cannot be assigned to more than one course for any
given period.
l∈Lk∈Kn∈Nxi,j,k,l,m,n ≤1,∀i∈I, ∀j∈J, ∀m∈M
Classrooms overlapping: Each classroom cannot hold more than one course for any given
period.
l∈Lm∈Mk∈Kxi,j,k,l,m,n ≤1,∀i∈I, ∀j∈J, ∀n∈N
Student Load: A student has some courses amount to 18 hours per week. Each course
consists of 3 hours and taught in 2 periods. We can express these requirements in terms of
number of slots worked per day.
j∈Jk∈Km∈Mn∈Nxi,j,k,l,m,n ≤nj,∀i∈I,∀l∈L
In order to guarantee that each course occupies only one slot per day, the following constraints
must be satisfied.
j∈Jxi,j,k,l,m,n ≤1,∀i∈I, ∀k∈K, ∀l∈L, ∀m∈M, ∀n∈N
Course Distribution: To make sure that the lectures of each course are distributed in such a
way that there is one day off between them, the following constraint has been designed.
j∈J(xi,j,k,l,m,n +xi+2,j,k,l,m,n )=2,i+2≤5,∀i∈I,∀k∈K, ∀l∈L, ∀m∈M, ∀n∈N
Classroom Occupancy: All lectures of a given course in a week must be held in the same
classroom. This can be described by the constraint below.
i∈Ij∈Jl∈Lm∈Mxi,j,k,l,m,n +i∈Ij∈Jl∈Lkm∈Mxi+2,j,k,l,m,n =2nl,i+2≤5
∀k∈K, ∀n∈N
Pre-Graduated Student: The overlap of courses with their prerequisites for the same group of
students lis permitted.
International Journal of Applied Mathematics and Statistics
167

xi,j,k,l,m,n +xi,j,k+s,l,m,n ≤1+yi,j,k,l ,∀i∈I, ∀j∈J, ∀k∈K, ∀l∈L, ∀m∈M, ∀n∈N
s∈{1, ..., nk−1},k+s≤nk
Objective Function: This section presents the objective function of the model which represent
the satisfaction of teachers and students. It consists of six terms to be minimized, where the
first two terms are associated to teachers, while the third and fourth terms are related to regular
students and the last two terms concern predicted graduated students.
This is equivalent to the objective function consisting of maximizing the number of lectures per
day, which implies decreasing the waiting time between lectures.
(Total dissatisfaction)= Teacher number of lectures per day +
Regular student number of lectures per day +
Predicted graduated student number of lectures per day 
Since each course consists of two periods of one and half an hour, it is necessary to multiply the
number of courses by 1.5 hours. In this case the objective function can be written as follows:
1.5j∈Jk∈Kl∈Lm∈Mn∈Nxi,j,k,l,m,n +m∈Mzim +l∈Lzil +
j∈Jl∈Lk∈Kyi,j,k,l,∀i∈I
5 Reduction of the Model
The model we obtained is very large because it contains several millions decision variables.
This model is not tractable, so it needs to be reduced in order to find an acceptable solution.
For this purpose we neglect the classroom index. To prevent classrooms overlapping, we will
add the following constraint to the model:
k∈Kl∈Lm∈Mxi,j,k,l,m ≤nn,∀i∈I, ∀j∈J
As the number of classrooms is limited, removing the index representing the classroom must
be compensated by the above constraint which means that the number of courses cannot ex-
ceed the number of classrooms in a time slot jin a given day i. Also, to avoid an intractable
model with strong complexity, we have chosen to discard the term j∈Jl∈Lk∈Kyi,j,k,l cor-
responding to overlapping of courses and their prerequisites from objective function, and the
related constraint xi,j,k,l,m +xi,j,k+s,l,m ≤1+yi,j,k,l.
Reduced Model: After reduction we get the following model:
max 1.5j∈Jk∈Kl∈Lm∈Mxi,j,k,l,m +m∈Mzim +l∈Lzil
subject to k∈Km∈Mxi,j,k,l,m ≤1∀i∈I, ∀j∈J, ∀l∈L
l∈Lk∈Kxi,j,k,l,m ≤1∀i∈I, ∀j∈J, ∀m∈M
j∈Jk∈Km∈Mxi,j,k,l,m ≤nj∀i∈I, ∀l∈L
International Journal of Applied Mathematics and Statistics
168

j∈Jxi,j,k,l,m ≤1∀i∈I, ∀k∈K, ∀l∈L, ∀m∈M
j∈J(xi,j,k,l,m +xi+2,j,k,l,m )=2,i+2≤5∀i∈I,∀k∈K, ∀l∈L, ∀m∈M
k∈Kl∈Lm∈Mxi,j,k,l,m ≤nn∀i∈I, ∀j∈J
s∈{1, ..., nk−1},k+s≤nk
m∈Mzim ≤1∀i∈I
l∈Lzil ≤nl∀i∈I
xi,j,k,l,m,y
i,j,k,l,z
im,z
il ≥0
5.1 Decomposition of the Model
Reduced Student Model: For the sake of simplicity, we drop the index corresponding to the
teacher, but since the role of the teacher is important in the model the constraint below has
been considered to express that each course is taught by only one teacher for several groups.
xi,j,k,l +xi,j,k,s ≤1∀i∈I, ∀j∈J, ∀l=s∈L, ∀k∈K
Considering all courses given in all slots on a given day, the above constraint becomes
k∈Kj∈J(xi,j,k,l +xi,j,k,s )≤2nj∀i∈I,∀l=s∈L
After reduction we get the following model:
max 1.5j∈Jk∈Kl∈Lxi,j,k,l +l∈Lzil
subject to k∈Kxi,j,k,l ≤1∀i∈I,∀j∈J, ∀l∈L
k∈Kl∈Lxi,j,k,l ≤nl∀i∈I,∀j∈J
j∈Jk∈Kxi,j,k,l ≤nj∀i∈I,∀l∈L
j∈Jxi,j,k,l ≤1∀i∈I,∀k∈K, ∀l∈L
j∈J(xi,j,k,l +xi+2,j,k,l )=2,i+2≤5∀i∈I,∀k∈K, ∀l∈L
k∈Kj∈J(xi,j,k,l +xi,j,k,s )≤2nj∀i∈I,∀l=s∈L
s∈{1, ..., nk−1},k+s≤nk
l∈Lzil ≤nl∀i∈I
xi,j,k,l,z
il ≥0
Reduced Teacher Model : To obtain a simplified model, we have eliminated the index corre-
sponding to the students groups, and replaced it by the constraint below, ensuring that each
student group has at most one course at a time.
k∈Kxi,j,k,m ≤1∀i∈I,∀j∈J, ∀m∈M
This results in the following reduced teacher model:
max 1.5j∈Jk∈Km∈Mxi,j,k,m +m∈Mzim
subject to k∈Kxi,j,k,m ≤1∀i∈I,∀j∈J, ∀m∈M
k∈Km∈Mxi,j,k,m ≤nk∀i∈I,∀j∈J
j∈Jk∈Kxi,j,k,m ≤nj∀i∈I,∀m∈M
j∈Jxi,j,k,m ≤nl∀i∈I,∀k∈K, ∀m∈M
k∈Km∈Mxi,j,k,m ≤nn∀i∈I,∀j∈J
International Journal of Applied Mathematics and Statistics
169

m∈Mzim ≤1∀i∈I
xi,j,k,m,z
im ≥0
5.2 Comparison between Reduced Student Model and Reduced Teacher Model
To point out the similarities and differences between the reduced student model and the re-
duced teacher model, we expose the following facts:
•Objective Functions
Both objective functions have two terms. In a given day, the first one in the reduced
student model means the number of lectures for student groups, and the second term
represents all lectures for all student groups. While the first term in the reduced teacher
model refers to the number of lectures for teachers and the second term represents all
lectures for all teachers in the same day. One can notice that these objectives functions
are equivalent because their terms lead to similar results.
•Constraints
Examining the constraints of the two models, it turns out that they are close to each other.
These constraints almost stand for the same things because they are associated to same
courses. Therefore, a strong relationship exists between them.
Consequently, we think that it is not necessary to study both models, and then we deal only
with the reduced student model for the formulation based on exterior penalty function method.
Proposition 1. The teacher problem (TPB) has a suboptimal solution if and only if the student
problem (SPB) has a suboptimal solution.
Proof.(By equivalence between objective functions and constraints.)
Referring to the objective functions of the reduced student model and the reduced teacher
model, one can observe that they are equivalent. In fact, the first term in the objective function
of the reduced student model j∈Jk∈Kl∈Lxi,j,k,l means that the number of all lectures
for all groups in a given day must be equal to the number of all lectures offered by all teachers
in the same day, that is j∈Jk∈Km∈Mxi,j,k,m . The second term in the reduced student
model l∈Lzil signifies that in a given day there is a ceratin number of lectures for all student
groups. In fact, the second term in the reduced teacher model l∈Mzim corresponds to the
same number of lectures given by all teachers in that day.
Comparing the constraints of the two reduced models, it can be seen that these constraints
achieve similar goals. In other words, these constraints reflect somewhat the same features
in different manners. For instance, the constraint k∈Kl∈Lxi,j,k,l ≤nlof the reduced stu-
dent model, expresses that the number of all lectures for all student groups do not exceed the
number of student groups. meanwhile, the constraint k∈Km∈Mxi,j,k,m ≤nkof the reduced
teacher model, pertains to the total number of lectures taught by all teachers and must be at
most equal to the total number of lectures in specific slot. In the same way, the remaining con-
straints can be compared.
International Journal of Applied Mathematics and Statistics
170

Proposition 2. If the student problem (SPB) has a suboptimal solution or the teacher prob-
lem (TPB) has a suboptimal solution then the original problem (OP) has a suboptimal solution.
Proof.
Based on the above proposition and the decomposition of the original problem, it seems rea-
sonable to get this result. Considering the objective functions of the three models, the original
problem, the reduced student model and the reduced teacher model, it is easy to check that
the objective function of the original problem is equivalent to the two other objective functions.
All terms in the objective function of the original problem j∈Jk∈Kl∈Lm∈Mxi,j,k,l,m;
m∈Mzim;l∈Lzil are minimized if the corresponding terms in the reduced student model
and the reduced teacher model are minimized.
6 Solution of the Problem based on Exterior Penalty Function Method
The exterior penalty function method is a well-known technique used to solve nonlinear op-
timization problem. Also, it would be useful to tackle large-scale integer programming after
relaxation. The method consists to approximate a constrained optimization problem with an un-
constrained problem, in order to obtain a solution using Sequential Unconstrained Minimization
Techniques (SUMT). The exterior penalty function method transforms the optimization problem:
min f(x1,x
2,...,x
n)
subject to hk(x1,x
2,...,x
n)=0,k=1,...,l
k
gj(x1,x
2,...,x
n)≤0,j=1,...,l
j
xl
i≤xi≤xu
i,i=1,...,n
to an unconstrained problem, whose formulation is the following:
min F(X, rh,r
g)=f(X)+P(X, rh,r
g)
xl
i≤xi≤xu
i,i=1,...,n
where P(X, rh,r
g)is the penalty function, rhand rgare penalty constants (also called multipli-
ers). The penalty function is expressed as:
P(X, rh,r
g)=rhl
i=1 hi(X)2+rgm
q=1(max{0,g
q(X)})2
It can be shown that the transformed unconstrained problem solves the original constrained
problem as the multipliers rhand rgapproach ∞.
In computer implementation,this limit is replaced by a large value instead of ∞. If the original
objective function f(x)seeks to maximize, then the augmented objective function is
max F(X, rh,r
g)=f(X)−P(X, rh,r
g)
Actually, the exterior penalty function method provides a real solution to the unconstrained opti-
mization problem, then it is convenient to approximate such solution to a binary one. To achieve
this end, we have designed the following algorithm.
International Journal of Applied Mathematics and Statistics
171

Algorithm
Initialization:
i=1;
X=[x1,...,x
n]T,xiis the ith component of vector X, actual solution of the
exterior penalty function method.
xbin =[];is the (0-1)-binary approximation of the actual solution X.
For t∈{0,1},t=[t1,t
2,...,t
n];
Step1:
xi=t;
Step2:
Evaluate each constraint gi(x)for xi=t;
If gi(x)≤biis satisfied, then:
•Eliminate xiand put bi=bi−t;
•xbin =[xbin,x
i]
Else go to Step1
i=i+1;
6.1 Problem Formulation for Reduced Student Model
The reduced student model can be formulated as an integer program, and then transformed into
an unconstrained optimization problem. That is, the maximization of an augmented objective
function including the original objective function and an exterior penalty function.
max F(X, rh,r
g)=f(X)−P(X, rh,r
g)
f(X)=max1.5j∈Jk∈Kl∈Lxi,j,k ,l +l∈Lziland
P(X, rh,r
g)=rg14
q=1(max{0,g
q(X)})2
g1(x)=k∈Kxi,j,k,l ≤1∀i∈I,∀j∈J, ∀l∈L
g2(x)=l∈Lk∈Kxi,j,k,l ≤nl∀i∈I,∀j∈J
g3(x)=j∈Jk∈Kxi,j,k,l ≤nj∀i∈I,∀l∈L
g4(x)=j∈Jxi,j,k,l ≤1∀i∈I,∀k∈K, ∀l∈L
g5(x)=j∈J(xi,j,k,l +xi+2,j,k,l )=2,i+2≤5∀i∈I,∀k∈K, ∀l∈L
g6(x)=k∈Kj∈J(xi,j,k,l +xi,j,k,s )≤2nj∀i∈I,∀l=s∈L
s∈{1, ..., nk−1},k+s≤nk
g7(x)=l∈Lzil ≤nl∀i∈I
xi,j,k,l,z
il ≥0
International Journal of Applied Mathematics and Statistics
172

6.2 Numerical Example for Reduced Student Model
To illustrate our approach of solution of the university course timetabling problem and as the
reduced student model remains including great number of variables and constraints, we assign
small values to the parameters of the model as follows:
ni=5is the number workable days per week, nj=2is the number of slots per day,
nk=3is the number of courses to be assigned, nl=2is the number of students groups,
nn=2is the number of classrooms.
max F(X, rh,r
g)=f(X)−P(X, rh,r
g)
f(X)=max1.5j∈Jk∈Kl∈Lxi,j,k ,l +l∈Lzil
and P(X, rh,r
g)=rg14
j=1(max{0,g
q(X)})2
g1(x)=k∈Kxi,j,k,l ≤1∀i∈I,∀j∈J, ∀l∈L
g2(x)=l∈Lk∈Kxi,j,k,l ≤2∀i∈I,∀j∈J
g3(x)=j∈Jk∈Kxi,j,k,l ≤2∀i∈I,∀l∈L
g4(x)=j∈Jxi,j,k,l ≤1∀i∈I,∀k∈K, ∀l∈L
g5(x)=l∈Lzil ≤2∀i∈I
xi,j,k,l,z
il ≥0
The constraint of the above system can be written in matrix form as follows:
AX ≤b
where A=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
10101000000000
00000010101000
11111100000000
00000011111100
10101010101000
01010101010100
10000010000000
01000001000000
00100000100000
00010000010000
00001000001000
00000100000100
00000000000010
00000000000001
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
;X=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x111
x112
x121
x122
x131
x132
x211
x212
x221
x222
x231
x232
z11
z12
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
;b=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1
1
2
2
2
2
1
1
1
1
1
1
1
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
In order to solve the optimization problem of the reduced student model with exterior penalty
function formulation, we have tried different values for the penalty parameter rg.Wehave
observed that rgmust be greater than one hundred so that the objective function converges to
its optimal value. The results obtained are presented in the table 1.
International Journal of Applied Mathematics and Statistics
173

Tab l e 1: The reduced student model results with exterior penalty function.
Variables x1x2x3x4x5x6x7
Actual sol. 0.3669 0.3770 0.3669 0.3770 0.3669 0.3770 0.3669
Approx. sol. 1 0 0 1 0 0 0
Variables x8x9x10 x11 x12 x13 x14
Actual sol. 0.3770 0.3669 0.3770 0.3669 0.3770 0.6671 0.6671
Approx. sol. 1 1 0 0 0 1 1
f(x)Actual value 5.7975
Approximate value 6
where each variable xjkl =xr,r=1,...,14.
x111 x112 x121 x122 x131 x132 x211 x212 x221 x222 x231 x232 z11 z12
x1x2x3x4x5x6x7x8x9x10 x11 x12 x13 x14
The above solution could be interpreted in terms of timetable using the scheduling table 2,
below. Each shaded area xjkl corresponds to the existence of a lecture of course k, for student
group lin slot j.
Tab l e 2:The course scheduling
The shaded area in the first column in the previous table 2., signifies that the variable x111
equals 1, which means the slot 1 is occupied by course 1, and the second shaded area in the
second column, corresponds to x221 equals 1, which implies that a lecture of course 2 exists
for student group 1 in slot 2. Similarly, x122 equal to 1 and x212 equal to 1 indicate that group 2
has two different courses in different slots. One can notice that the proposed solution is a good
one because no clash between courses and no waiting time between lectures for any group.
International Journal of Applied Mathematics and Statistics
174

7 New Penalty Functions
To search a good solution for our problem, we have designed the two new penalty functions in
the following.
7.1 Variance-Penalty Function
This method aims to establish a relationship between the constraints and the weighted mean
of the variables ˜g(x). Obviously, this can be expressed in terms of variance of constraints gi(x)
with respect to ˜g(x). We call this type of penalty function, variance-penalty function. In this
case the augmented objective function becomes as follows:
max F(X, rv)=f(X)−P(X, rv)
xl
i≤xi≤xu
i,i=1,...,n
where P(X, rv)is the penalty function, rhand rgare penalty multipliers. The variance-penalty
function is given by:
P(X, rv)=rvl
i=1(gi(X)−˜g(x))2)
where ˜g(x)=x1+···+xn
nand rvis the total weight of the constraints. The partial weight of each
constraint equal to the number of variables in the constraint divided by the total number of
variables in the model.
7.2 Numerical Example for Reduced Student Model
Adopting the same approach to solve the optimization problem of the reduced student model
given above, we formulate it using variance-penalty function method which leads to the following
numerical results.
Tab l e 3:The reduced student model results with variance-penalty function
Variables x1x2x3x4x5x6x7
Actual sol. 0.4116 0.3934 0.4116 0.3934 0.4116 0.3934 0.3669
Approx. sol. 0 1 1 0 0 0 1
Variables x8x9x10 x11 x12 x13 x14
Actual sol. 0.3934 0.4116 0.3934 0.4116 0.3934 0.5050 0.5050
Approx. sol. 0 0 1 0 0 1 1
f(x)Actual value 5.8402
Approximate value 6
The solution in table 3 could be expressed in terms of timetable using the scheduling table 4,
below.
International Journal of Applied Mathematics and Statistics
175

Tab l e 4:The course scheduling
The shaded area in the first column in the precedent table 4., denotes that the variable x211
equals 1, that is the slot 2 is devoted to course 1, and the second shaded area in the second
column, indicates x121 equals 1, which means that a lecture of course 2 exists for student group
1 in slot 1. Likewise, x112 equal to 1 and x222 equal to 1 stand for that group 2 has two different
courses in different slots. It can be notice that the suggested solution is a suitable because no
overlap between courses and no waiting time between lectures for any group.
7.3 Pseudo-Convex Combination Penalty Function
The method states a pseudo-convex combination between the square of the objective function
and the constraints. The new penalty function is called pseudo-convex combination-penalty
function. There is a penalty pseudo-convex parameter for the objective function and each
constraint gi(x). The augmented objective function is defined by:
max F(X, rci)=rci{f(X)2+P(X, rci)2},x
l
i≤xi≤xu
i,i=1,...,n
where P(X, rci)is the pseudo-convex penalty function, rci=βi
Nare penalty multipliers, βi,i∈
{1,...,m+1}is the number of variables in the objective function and in constraint iand Nis
the total number of variables in the objective function and all constraints.
The pseudo-convex penalty function is given by:
P(X, rci)=rcim
i=1 gi(X)2
7.4 Numerical Example for Reduced Student Model
The table 5 below illustrate the pseudo-convex penalty function approach for solving the re-
duced student problem.
International Journal of Applied Mathematics and Statistics
176

Tab l e 5:The reduced student model results with pseudo-convex penalty function.
Variables x1x2x3x4x5x6x7
Actual sol. 0.4417 0.4420 0.4414 0.4420 0.4414 0.4420 0.4417
Approx. sol. 0 1 0 0 1 0 0
Variables x8x9x10 x11 x12 x13 x14
Actual sol. 0.4420 0.4414 0.4420 0.4414 0.4420 0.4470 0.4470
Approx. sol. 0 1 0 0 1 1 1
f(x)Actual value 6.1944
Approximate value 6
The solution in table 5 could be expressed in terms of timetable using the scheduling table 6,
below.
Tab l e 6:The course scheduling
The solution found in the precedent table 6., indicates that the variable x221 equals 1, namely
that slot 2 is occupied by course 2, and the second shaded area in the third column, denotes
that x131 equals 1, which means that a lecture of course 3 exists for student group 1 in slot 1.
Similarly, x112 equal to 1 and x232 equal to 1 signify that group 2 has two different courses in
different slots.
8 Conclusion
This work has dealt with the university course timetabling. Our case study has been the de-
partment of Math at Taibah University in Madinah, Saudi Arabia. First, we have developed a
binary integer linear programming model to capture the characteristics of the problem. Since
the model was very large with several million of variables, we have reduced the original model,
to get a tractable one. Although the reduction of the model, it was necessary to decompose
it into two sub-models, the reduced student model and the reduced teacher model. Second,
we provided an approach of solution based on exterior penalty function method to transform
International Journal of Applied Mathematics and Statistics
177

the original constrained discrete problem into unconstrained continuous problem. To obtain
the best possible solution, we have proposed two new penalty functions, respectively called
variance-penalty function and pseudo-convex penalty function. Third, we have designed an
algorithm to produce the integer approximation of the real solution given by the three prece-
dent penalty function methods. Referring to the numerical examples, we notice that all penalty
functions lead to good results but the solution given by the variance-penalty function is slightly
better. As the university course timetabling problem is very difficult, we intend to tackle it in
another approach based on penalty functions and inspired by probability distributions in future
works.
References
Aizam, N. and Liong, C. 2013. Mathematical modeling of university timetabling : A mathemat-
ical programming approach, International Journal of Applied Mathematics and Statistics
37(7): 110–122.
Azimi, Z. 2005. Hybrid heuristics for examination timetabling problem, ”Applied Mathematics
and Computation” 163: 705–733.
Bakir, M. A. and Akop, C. 2008. A 0-1 integer programming approach to a university timetabling
problem, Hacettepe Journal of Mathematics and Statistics 37: 41–55.
Benli, O. and Bostali, A. 2004. An optimization-based decision support system for a university
timetabling problem: An integrated constraint and binary integer programming approach,
California State University, Long Beach pp. 41–55.
Burke, E. K. and Petrovic, S. 2002. Recent research directions in automated timetabling, Eur J
Opnl Res 140: 266–280.
Daskalaki, S.and Birbas, T. and Housos, E. 2004. An integer programming formulation for a
case study in university timetabling, Eur J Opnl Res 153: 117–135.
Grobner, M., Wilke, P. and Buttcher, S. 2003. A standard framework for timetabling problems, In
E. Burke and P. De Causemacker, editors, Practice and Theory of Automated Timetabling
IV, Lecture Notes in Computer Science .
Guozhong, B. and Xiao-Xiong, G. 2008. The time limit assignment problems, International
Journal of Applied Mathematics and Statistics 13(S08): 24–38.
Lucidi, S. and Rinaldi, F. 2010. Exact penalty functions for nonlinear integer programming
problems, Journal of Optimization Theory and Applications 145: 479–488.
Schaerf, A. 1999. A survey of automated timetabling., Artificial Intelligence Review 13: 303–
316.
Yanez, J. and Ramirez, J. 1999. The robust coloring problem, Eur J Opnl Res 148: 546–558.
International Journal of Applied Mathematics and Statistics
178