Course Opening, Assignment and Timetabling with Student Preferences

Sacha Varone and David Schindl

Economie d’Entreprise, Geneva School of Business Administration, Route de Drize 7, 1227, Carouge, Switzerland

Keywords:

Optimization, Scheduling, Application.

Abstract:

We consider the following problem of course scheduling and assignment of students. Students express their

preferences for each course from several sets of proposed courses and each student has to take a certain number

of courses from each set. A minimum number of students is required to open a course and a maximum number

of students is speciﬁed for each course. The courses have to be scheduled on a limited number of periods so

that simultaneous courses have no students in common. This problem can be seen as a generalization of the

Student Project Allocation problem. It consists in determining which courses to open, specifying the schedule

for these opened courses, and assigning students to them, so that their preferences are maximized. Our model

is an Integer Programming problem, which we solve with a common available solver using an iterative process.

1 INTRODUCTION

We present in this paper a real case from the Geneva

School of Business Administration (HEG), which

was modelled as an integer programming problem

and solved with available solvers using exact meth-

ods. The problem is the following: according to

the Bologna Declaration on the European space for

higher education , students may choose some of their

courses from a list of courses. The timetabling prob-

lem begins therefore with the following question:

which courses to open? The decision is based mainly

on the number of students and on their preferences for

each course.

In the literature, the most similar problem is the

Student-Project Allocation problem (SPA), in which

a list of projects is available for students who express

preferences over the projects. Methods for solving

the SPA problem are, among others, building an in-

teger program and solving it with a solver (Anwar

and Bahaj, 2003), deﬁning a linear program for some

particular cases (Saber and Ghosh, 2001), or develop-

ing speciﬁc heuristics like a genetic algorithm (Harper

et al., 2005). Manlove and O’Malley (Manlove and

O’Malley, 2008) show the complexity and give an ap-

proximation algorithm for a generalized SPA problem

in which both the students and the lecturers have pref-

erences over the projects. The main difference be-

tween the SPA problem and our case study is that in

the SPA problem there is no restriction on the min-

imum number of students assigned to a project. On

the contrary, our case requires a minimum number of

students to be assigned to a course in order to open it.

In other words, the SPA problem has a lower bound of

L = 0 and may have an upper bound of U > 0 on the

number of students, for each project assignment. Our

case study has a lower bound of L ≥ 0 and an upper

bound of U ≥ L. In that sense, the SPA problem may

be seen as a particular case of the problem we present

in this paper. More generally, our problem belongs to

the so called timetabling problems, which is an active

research area (e.g., (Rudov´a et al., 2011)).

2 PROBLEM DESCRIPTION

During the last year of bachelor education at the

Geneva School of Business Administration, students

have to take, apart from the required courses, one ma-

jor course and one or two minors courses, selected

from a list of courses. A major course is a full-day

course for the whole academic year, whereas a minor

course is a four-hour evening course for one semester.

Since the number of minor courses exceeds the num-

ber of available evenings, some of them have to be

scheduled at the same evening.

Selecting a major course and some minor courses

can also be done through a predeﬁned portfolio of

courses. It consists of one major course and two mi-

nor courses, one in the fall semester and one in the

spring semester. Students may or may not choose a

portfolio. If they do, then only one portfolio can be

chosen from a set of portfolios. Those having cho-

sen a portfolio are assigned to the courses contained

in the portfolio, with a priority over other students’

course assignment. This priority has been stated by

Varone S. and Schindl D..

Course Opening, Assignment and Timetabling with Student Preferences.

DOI: 10.5220/0004189901530158

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 153-158

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

the management team of the school.

The capacity of each course, as well as the capac-

ity of each portfolio, is limited, and therefore some

students will not be assigned to their preferred course.

There is also a minimum number of students required

to open a course or a portfolio.

The Geneva School of Business Administration

has two types of students: full-time students and

part-time students. The former have to be assigned

to one major course, two minor courses during the

fall semester and one minor course during the spring

semester. The latter, referring to students working

part-time and studying part-time, have to be assigned

to one major course, one minor course during the

Fall semester and one minor course during the Spring

semester.

Course Selection. Prior to 2010, students expressed

their preferences by ﬁlling out a paper questionnaire,

in which they speciﬁed their preferences for three ma-

jor courses, and three minor courses each semester.

Since the data treatment and the optimization were

done manually, it was tedious and time consuming to

do it each year for about 150 students. Such problems

often occur in universities and their solution involves

building an automated process, as it was done in

(Hinkin and Thompson, 2002) with their SchedulEx-

pert software, or using goal programming to build a

course schedule with preferences (Badri et al., 1998;

Schniederjans and Kim, 1987). It was decided that

each student will express his own preferences for each

available course, major and minor, for each semester,

using a web form.

Students have to rank all the major courses by

preference. They also rank all the minor courses in

each semester. A strict ordering is therefore expressed

for each type of course (major and minor in the fall

and minor in the spring). Then, they may indicate if

they are willing to take one of the predeﬁned portfo-

lios.

Data Process. To avoid the previously used paper

form, we designed a web form linked to a database

to collect the data. The web form uses an internal

authentication via the Lightweight Directory Access

Protocol (LDAP

). Once the data has been collected,

a model is created using a mathematical modeling

language, for example AMPL. The model is then sent

with the data and the commands to a solver, for ex-

ample Gurobi, which returns the solution ﬁles. The

ultimate goal is to produce a worksheet so that the

project leader may easily use the result.

“an Internet protocol for accessing distributed directory

services”

3 MODEL FORMULATION

We built an integer programming model, with binary

variables in order to deﬁne the assignments of stu-

dents to portfolios (s) and to courses (x). Two types

of binary variables refer to the decision of whether

or not to open a course (z

) and a portfolio (z

). The

binary variables (y) deﬁne the assignments of courses

to periods.

First, we introduce the relevant notation.

J Set of courses

I Set of students

H Types of courses: major or minor

K Set of periods

P Set of portfolios

T = (t

) Selection of portfolio

Q = (q

) Deﬁnition of portfolios

A = (a

) Matrix of preferences

More precisely,



1 if portfolio p is selected by student i

0 otherwise



1 if course j ∈ portfolio p

0 otherwise

Student i ranks a

the course j, starting with 1

as the preferred course.

Moreover, in order to balance the weight between

a major course and a minor course, we introduce a

cost matrix C = (c

) such that



if course j is a major course

otherwise

This means that the utility of assigning a major

course with rank 1 and two minor courses with rank 2

is the same as the utility of assigning a major course

with rank 2 and two minor courses with rank 1.

The binary variables are of three types: those

related to assignment, those related to scheduling,

and those related to opening.



1 if student i is assigned to portfolio p

0 otherwise



1 if student i is assigned to course j

0 otherwise



1 if course j is scheduled in period k

0 otherwise



1 if course j is opened

0 otherwise



1 if portfolio p is opened

0 otherwise

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

3.1 Portfolio Problem

The opening of a portfolio only depends on the num-

ber of students assigned to it. If this number is smaller

than the minimum number needed to viably sustain

the portfolio, then the portfolio is not opened. The

minimum number of students is given by the aca-

demic department, which is a common practice (see

(Saber and Ghosh, 2001) for example).

If the number of students who wish to take a port-

folio exceeds its capacity, then some of them can not

be assigned to it. The students selected to attend the

portfolio are those with the highest sum of prefer-

ences for the courses in the portfolio. This has been

deﬁned during the ranking procedure, in which all

courses have to be ranked by the students. In other

words, thanks to the complete ordering deﬁned for

each set of courses (majors and minors during the fall

semester and minors during the spring semester), it

is possible to give a score to the portfolio choice of

each student. This selection process is expressed by

the objective function (1). In case of a similar portfo-

lio choice, with equal score for several students, and a

number of students exceeding the upper limit, the se-

lection may be done at random or by the solver itself.

max

∑

p∈P

∑

i∈I

(|J| −

∑

′

∈P

′

∑

j∈J

′

) s

(1)

≤ t

∀ i ∈ I, p ∈ P (2)

∑

i∈I

≥ 15z

∀ p ∈ P (3)

∑

i∈I

≤ 30z

∀ p ∈ P (4)

The objective function (1) indicates that the pref-

erences of the students have to be maximized. Con-

straint (2) ensures that portfolio p is assigned only to

students who have selected it. The minimum num-

ber of students necessary to open a portfolio p is

expressed by constraint (3), whereas the capacity of

portfolio p is deﬁned by constraint (4). The values of

these parameters in our school are 15, respectively 30.

3.2 Course Problem

The objective is to maximize the preferences of the

students, or equivalently, to minimize the dissatisfac-

tion of the students (5).

Each student should be assigned to the right num-

ber of courses of each type. This number of courses

is deﬁned by a function n(i, h), where i represents stu-

dent i ∈ I and h ∈ H represents the type of course.

The values of the function n, considered as a parame-

ter, are necessary as some students (full-time or part-

time) may already have successfully passed a major

or a minor course. The number of courses to be taken

by each student is expressed by equation (6). The

bounds on the capacity of each course also have to

be respected (7)(8). Each course may be opened and

in that case a single period has to be assigned to it (9).

Since several minor courses may be taken by the

same student, a non-simultaneous course constraint

has to be valid (10). To understand this constraint,

suppose that a student i is assigned to courses j and

′

. This would mean that x

= x

′

= 1. The simul-

taneity of courses j and j

′

would be expressed by as-

signing the period k to both j and j

′

, that is to say

= y

′

= 1. Therefore, in such a case, the left-hand

side of inequality (10) would be equal to 4, which

would violate the constraint. Suppose now that either

student i is not assigned to j or j

′

, or courses j and

′

are not simultaneously given. Then the maximum

value of the left-hand side of inequality (10) would be

3, which satisﬁes the constraint.

Finally, the students assigned to a portfolio have

to take the courses in it (11). This last constraint (11)

is linear if the values of z

and s

are ﬁxed. Notice

that the portfolio capacity should not be set to a value

larger than the courses’ capacities, otherwise the stu-

dent to course assignment may produce an infeasible

solution. This will indeed be the case since, as ex-

plained in section 4, the portfolio problem is solved

before the course problem.

The model is presented below:

min

∑

j∈J

∑

i∈I

(5)

∑

j∈h

= n(i, h) ∀i ∈ I, h ∈ H (6)

∑

i∈I

≤ 30z

∀ j ∈ J (7)

∑

i∈I

≥ 15z

∀ j ∈ J (8)

∑

k∈K

= z

∀ j ∈ J (9)

+ x

′

+ y

′

≤ 3 ∀i ∈ I, k ∈ K,

j, j

′

∈ J (10)

≤ x

∀i ∈ I (11)

4 SOLUTION METHODOLOGY

We ﬁrst solve the portfolio allocation problem, as it

has priority over the course allocation. Once a solu-

tion is found, constraint (11) ensures that students en-

CourseOpening,AssignmentandTimetablingwithStudentPreferences

rolled in a portfolio are assigned to the corresponding

major and minor courses. Then we solve the remain-

ing assignment and scheduling problem, which con-

sists in optimizing the students assignment to courses

along with the courses’ schedule.

4.1 Portfolio Allocation

Since portfolio allocation is given priority over course

allocation, this sub-problem can be solved ﬁrst. Its

goal is to deﬁne which portfolio to open and which

not, and which students to assign to each opened port-

folio. In our case, even though we have a binary pro-

gram, the size of the instance is small so that usage

of common solvers allows to easily ﬁnd an optimal

solution within a few seconds.

4.2 Course Allocation

Once the portfolio sub-problem is solved, it is neces-

sary to assign students enrolled in a portfolio to the

corresponding courses. This will ﬁx some of the vari-

ables and hence reduce the size of the instance.

A major difﬁculty in solving “difﬁcult” binary

programs, in the sense of NP-hard problem, is that the

time to solve them optimally increases exponentially

with the size of the instance. One approach is to build

a speciﬁc heuristic which solves the problem but of-

ten without a guarantee of optimality, or in some cases

with a bound on the gap between the solution and the

optimal solution. This approach is described in sec-

tion 4.3. Another approach, described by Algorithm

2, is to arbitrarily ﬁx some variables to values they

are likely to take at optimality, so as to reduce the size

of the problem in the hope that it will become small

enough to be tractable by available solvers.

Instead of considering all available courses, a sub-

set of them may be selected with the aim of solving

the subproblem optimally (Algorithm 2). The selec-

tion of the subset of courses is based on the expressed

preferences of the students: a maximum rank is de-

ﬁned as an upper bound on the possible assignment

of a student to a course. Then, for each course, if

the number of students with a preference lower than

or equal to this maximum rank is lower than the re-

quired lower bound, then the course is not opened

(Algorithm 1). Since there is a limited capacity for

each course, it may imply that the problem is infeasi-

ble. In that case the maximum rank is increased by

one unit and this process is run iteratively.

4.3 Further Ideas

A direct attempt to solve the course problem pre-

Algorithm 1: Course elimination.

Require: MaxRank

1: for all j ∈ J do

2: if

∑

δ(a

≤ MaxRank) < 15 then

3: J = J\{ j}

4: end if

5: end for

Algorithm 2: Direct method : IterateElimination procedure.

1: Presolve {Course assignment with respect to

portfolio assignment}

2: MaxOrder = 2 {Initialization of maximal prefer-

ence}

3: repeat

4: MaxOrder = MaxOrder + 1

5: Call algorithm 1(MaxOrder)

6: Solve the courses alloc. problem (5)-(11)

7: until Problem solved or MaxOrder= min{|{ j :

j ∈ h}| : h ∈ H}

sented in Section 3.2 may not result in a solution,

due to the intrinsic complexity of this binary problem.

Then an heuristic procedure would be necessary.

The idea expressed by Algorithm 3 is the follow-

ing: a ﬁrst difﬁculty in solving the courses allocation

problem (5)-(11) is the assignment of courses to pe-

riods, so that constraint (10) is satisﬁed. In our ap-

plication, the set of periods associated to each type of

courses (major, minor in spring, minor in autumn) are

disjoint: i.e., the set of available periods for a major

course is different from the one for a minor course,

and the set of available periods for a minor course in

spring is different from the one for a minor course in

autumn. This assumption is made as a requirement

for Algorithm 3. The ﬁrst step is therefore to choose

the assignment of periods to courses (line 1). To pro-

ceed, one should avoid two courses with a lot of high

preferences to be scheduled at the same period.

Algorithm 3: Heuristic for the courses allocation problem.

Require: Non overlapping periods between types of

courses

1: Assign periods to courses

2: Build the associated ﬂow network

3: Solve the associated min cost max ﬂow problem

4: while feasible solution is not found do

5: Eliminate 1 course with insufﬁcient number of

students

6: Solve the associated min cost max ﬂow prob-

lem

7: end while

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

A second difﬁculty lies in the minimal number

of students necessary to open a course. This con-

straint (8) is temporarily relaxed. The second step

(line 2) consists of building a transshipment network

G = (V, A, c, w) with capacities c and weights w on

the arcs. The set of nodes V contains a source and a

sink, a node (i, h) for each student with each type of

course, a node (i, k) for each student with each period,

and the courses with their already associated periods

( j, k). The set of arcs is (source, (i, h)), ((i, h), (i, k)),

((i, k), ( j, k)), (( j, k), sink). The capacity c and weight

w functions are deﬁned as follows:

Arcs A Capacity c Weight w

(source,(i, h)) n(i, h) 0

((i, h), (i, k)) 1 0

((i, k), ( j, k)) 1 a

(( j, k), sink) 30 0

Since the capacities and the weights are all inte-

gers, this transshipment problem can be solved opti-

mally using continuous variables, as stated by the well

known integrality theorem; this problem is known un-

der the name of min cost max ﬂow problem and is

polynomially solvable: For every network ﬂow prob-

lem with integer data, every basic feasible solution

and, in particular, every basic optimal solution as-

signs integer ﬂow to every arc.

Once the problem solved, if some courses do not

contain enough students, one of these is eliminated

from the set of courses. The choice of the course to be

eliminated may depend on the preferences expressed

by the students, as stated by Algorithm 1. Technically,

the elimination of a course may be done by either re-

moving the course from the network model, or by set-

ting the capacity of the course to 0. And this process

iterates until all courses have the minimal number of

students (while loop) required to be opened.

Of course, in order to get a feasible solution, prior

to running Algorithm 3, one has to check that the

number of students in each type of course (major, mi-

nor in spring, minor in autumn) is less than or equal

to the sum of the capacities.

One could also extend the model by changing

some unused arc costs or capacities. For instance,

one could take into account students preferences over

days with costs on the arcs ((i, h), ( j, k)), or even by

putting a zero capacity if a student is not available a

given day. Moreover, if a course has a cost which

is linear with the number of students (for instance if

some material has to be bought for each student), one

could put a cost, say b on each arc (( j, k), sink), ex-

pressing that each student following the course costs

an additional amount of b.

5 APPLICATION

The student assignment problem to major and mi-

nor courses was successfully applied at the Geneva

School of Business Administration. In 2011, it in-

volved 146 students, of which 98 full-time students

and 48 part-time students, 14 major courses, 10 minor

courses during the fall semester and 9 minor courses

during the spring semester. The minimum number of

students necessary to open a portfolio or a course is

15, and the capacity of each course is 30.

The generated model contains 5904 variables and

66541 constraints. It is solved optimally within a

few seconds on a Intel Core 2 Quad PC with 4 Gb

memory, using the AMPL modeling language and the

Gurobi 4.5 solver. It was therefore, with our speciﬁc

data set, not necessary to apply the heuristic described

by Algorithm 3.

6 CONCLUSIONS

In this article, we described a successful application

of operations research tools. We created a convenient

way of data collection, modelled an assignment prob-

lem as a binary problem and ﬁnally solved it using

available solvers.

Solving an NP-hard problem, or at least one

viewed as ”difﬁcult” from a practical point of view,

requires several steps: data collection, modeling,

solving and presenting the results. It is often agreed

that the development of a speciﬁc heuristic is required

even for middle-size “difﬁcult” problems. We learned

from our experience in the ﬁeld of practical opti-

mization of “difﬁcult” problems that before develop-

ing home-made heuristics, it makes sense to try solv-

ing the problem with available commercial or open-

source solvers.

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CourseOpening,AssignmentandTimetablingwithStudentPreferences

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ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems