An integer programming formulation for a case study

in university timetabling

S. Daskalaki

b

a,*

, T. Birbas b, E. Housos

b

a

Department of Engineering Sciences, University of Patras, GR-26500 Rio Patras, Greece Department of Electrical and Computer Engineering, University of Patras, GR-26500 Rio Patras, Greece

Abstract

A novel 0–1 integer programming formulation of the university timetabling problem is presented. The model provides constraints for a great number of operational rules and requirements found in most academic institutions. Treated as an optimization problem, the objective is to minimize a linear cost function. With this objective, it is possible to consider the satisfaction of expressed preferences regarding teaching periods or days of the week or even classrooms for speciﬁed courses. Moreover, with suitable deﬁnition of the cost coeﬃcients in the objective function it is possible to reduce the solution space and make the problem tractable. The model is solvable by existing software tools with IP solvers, even for large departments. The case of a ﬁve-year Engineering Department with a large number of courses and teachers is presented along with its solution as resulted from the presented IP formulation. Ó 2003 Elsevier B.V. All rights reserved.

Keywords: Timetabling; Integer programming; University timetabling

1. Introduction

The construction of a timetable that satisﬁes all operational rules and needs in an academic institution, while at the same time fulﬁlls as many of the wishes and requirements of the staﬀ and the students is an important but extremely diﬃcult task for the staﬀ involved. In most institutions this task is left to administrative staﬀ and the current practice is to replicate the timetables of previous years with minor changes to accommodate newly developed situations. However, in recent years, changes occur more frequently and patching of what has been developed historically is not always the best policy. Under these circumstances, and in light of the progress achieved both in the hardware and software technologies, the scientiﬁc community continues to work on the problem in order to develop formal and automated procedures for constructing eﬃcient and desirable timetables.

Formally, the university timetabling problem is deﬁned as the process of assigning university courses to speciﬁc time periods throughout the ﬁve working days of the week and to speciﬁc classrooms suitable for

*

Corresponding author. Tel./fax: +30-2610-997810.

E-mail addresses: sdask@upatras.gr (S. Daskalaki), tbirbas@ee.upatras.gr (T. Birbas), housos@ee.upatras.gr (E. Housos).

0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(03)00103-6

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S. Daskalaki et al. / European Journal of Operational Research 153 (2004) 117–135

the number of students registered and the needs of each course. For the problem we are modeling in this paper, courses are oﬀered to well-deﬁned groups of students that follow a semi-structured schedule and in that sense the problem belongs to the category of the class/teacher timetabling problem for university environments [10]. For every educational institution the objective is always the construction of eﬀective and satisfactory weekly timetables. A timetable is considered to be eﬀective when it is feasible and may be realised by the institution, while it is considered to be satisfactory when it carries certain quality characteristics that keep its users satisﬁed at least to a certain degree. The university timetabling is the third stage of our eﬀort to solve the timetabling problem for all three levels of education in Greece. The ﬁrst step faced the problem of constructing timetables for the Greek high schools [7,8], the second solved the problem for the Greek Lyceums and this last eﬀort gives a thorough model and an...