Advances in Operations Research

Volume 2016, Article ID 7597062, 9 pages

http://dx.doi.org/10.1155/2016/7597062

## Linear Integer Model for the Course Timetabling Problem of a Faculty in Rio de Janeiro

Department of Production Engineering, Fluminense Federal University, Rua Passo da Pátria 156, São Domingos, 24.210-240 Niterói, RJ, Brazil

Received 5 September 2015; Revised 28 December 2015; Accepted 30 December 2015

Academic Editor: Ahmed Ghoniem

Copyright © 2016 Valdecy Pereira and Helder Gomes Costa. This is an open access article 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

This work presents a linear integer programming model that solves a timetabling problem of a faculty in Rio de Janeiro, Brazil. The model was designed to generate solutions that meet the preferences of the faculty’s managers, namely, allocating the maximum number of lecturers with highest academic title and minimising costs by merging courses with equivalent syllabuses. The integer linear model also finds solutions that meet lecturers’ scheduling preferences, thereby generating more practical and comfortable schedules for these professionals. Preferences were represented in the objective function, each with a specific weight. The model outperformed manual solutions in terms of response time and quality. The model was also able to demonstrate that lecturers’ scheduling preferences are actually conflicting goals. The model was approved by the faculty’s managers and has been used since the second semester of 2011.

#### 1. Introduction

Tripathy [1] defines university course timetabling (UCT) as a programming problem that allocates a given number of meetings between students and lecturers for a given period. Each of these meetings is attended by specific groups of students and one lecturer. Each meeting may also require specific resources (projectors, computers, etc.). The allocation choices must take into consideration the availability of lecturers, resources, and, if necessary, other factors. Schaerf [2] defines UCT as scheduling a sequence of teaching sessions between lecturers and students in a predetermined period of time, normally within a week, while satisfying a set of constraints. The timetable may be created manually, often laboriously, and the solution thereby obtained is generally unsatisfactory in some sense. de Werra [3] believes that, because UCT problems can be very different, it seems unreasonable to develop universal programming models that might be used in all institutions.

The objective function of these problems usually consists of minimising costs and/or maximising preferences; that is, the objective function measures both the quality of solutions and the validity of the generated timetables [4]. The constraints of the UCT problems are the core of these models and are defined by the true characteristics of each institution.

Constraints are generally divided into two categories, hard and soft constraints [5]. A solution to timetable programming that violates even a single hard constraint is not feasible, whereas a solution that violates any number of soft constraints remains feasible. Soft constraints are used to improve the solution’s quality because satisfying one of them means the solution meets some particular preference; therefore, a feasible solution can be found satisfying or not a soft constraint. For instance, a lecturer is available to give lessons only on Mondays in the morning, afternoon, or night shifts, but he prefers the morning period. A schedule with lessons on any other given day, than Monday, is considered unfeasible, so we can conclude that this is a hard constraint. In the other hand a schedule with a lesson on Mondays in the afternoon or night shifts is considered feasible, and a schedule with a lesson on Mondays in the morning shift is considered feasible and satisfies the lecturer preference, therefore improving the quality of the schedule, so we can conclude that this is a soft constraint [6].

This work aims to propose a model that solves the UCT problem in a faculty of Rio de Janeiro. This paper is organized as follows. Section 2 provides a literature review to present the problem of using UCT programming for solving real-world cases. Section 3 details the characteristics of the educational institution in this study, and Section 4 details the modelling to solve the corresponding problems. Section 5 shows how the methodology was applied, and, lastly, Section 6 contains conclusions and a discussion of the main issues arising.

#### 2. Literature Review

The main goal of this literature review is to identify differences and similarities between the various models that have been developed. If each institution possesses unique characteristics, it is expected that their respective mathematical models will also possess unique characteristics that may be represented by hard or soft constraints or even by the objective function itself, when applicable.

Daskalaki et al. [7] developed a binary linear integer programming model to which several constraints can be added to incorporate a large number of operational requirements and rules, as occurs in most academic institutions. The model was used in the Department of Electrical and Computer Engineering of the University of Patras in Greece. The objective of this model is to minimise a linear cost function while satisfying preferences related to the time, day of week, or even classroom for a specified course. The cost function consists of two terms: the first represents the cost of allocating a module (lecture) to a given time of a teaching session; the second represents the cost incurred in allocating these modules to a given day of the week. Hard constraints are as follows: modules are not allowed to overlap; all planned modules for a given semester should be offered; and the teaching session duration should meet lecturers’ demands and should occur in the required rooms. Soft constraints are as follows: collisions in lecturers’ availability schedules are not permitted; the timetable should be as compact as possible; and modules should be restricted to a single classroom. The model was used in a case study, and the authors reported that a feasible solution was generated.

Thompson [8] investigated the construction of a UCT with information on students’ preferences on course allocation. The objective function measures the solution’s quality based on how it satisfies students’ preferences. The model was tested with 3 data sets of the School of Hotel Administration of Cornell University in the USA. Its objective function consisted of the following: minimising the number of modules allocated to lecturers; minimising the number of modules not in the timetable; minimising collisions of modules in time; minimising violations regarding modules that should be scheduled in the same days; and minimising cases in which modules should be scheduled for the same time but are not. Its hard constraints are the following: lecturers schedules are not allowed to collide; the distance between the location of one class and the next should be as small as possible; a lecturer must have his or her work hours fully allocated; classroom schedules are not allowed to collide; and module schedules are not allowed to collide. Soft constraints are as follows: some modules should not be offered on the same days as some other modules, while some modules should be offered on the same day as some other modules. The author believes that his model is applicable to other institutions.

Schimmelpfeng and Helber [9] described a binary integer programming approach that was used in the School of Economics and Administration of the University of Hannover in Germany. The model solves the UCT problem and allocates classrooms. Its objective function consists of minimising violations of each one of the problem’s soft constraints. Hard constraints are as follows: lecturers’ availability schedules must be respected; collisions in lecturers’ availability schedules are not allowed, nor are collisions between modules; and modules’ sequences must be respected. Soft constraints are as follows: some classes’ scheduling preferences must be respected, and some lecturers’ scheduling preferences must be respected. After applying the model, the authors conducted a survey to assess the model’s results from the lecturers’ perspective. Only one of the 100 respondents was dissatisfied with the new system.

Bakir and Aksop [10] formulated a binary integer model for timetable programming in the Department of Statistics of Gazi University in Turkey. The model aims to minimise the dissatisfaction of students and lecturers, and thus the objective function assesses these quantities. Hard constraints are as follows: modules and classrooms are not allowed to collide; students may only enrol in courses corresponding to their current semester; some courses must obey a logical sequence in time; and teaching sessions allocated to modules must respect the required duration of each module. Soft constraints are as follows: the timetable should be as compact as possible, and failing students may only enrol in courses from their current and previous semesters, if possible. The authors claim the optimal solution was found; however, the size of the problem required a very long time to find this solution.

Adewumi et al. [11] developed a genetic algorithm to address a UCT in an unidentified university in Nigeria. The hard constraints are that lecturers’ and classrooms’ schedules are not allowed to collide and classrooms’ capacities must be respected. The authors claim that the heuristic found good results and that their model may easily be adapted to satisfy soft constraints, if necessary.

Razak et al. [12] present an approach that uses a bipartite graph and graph colouring to solve a UCT problem for the Department of Information Technology of a private university. The hard constraints are that modules and lecturers’ availabilities are not allowed to collide. The soft constraints are as follows: the timetable should accommodate lunch breaks if possible; teaching sessions should have a limited maximum duration; the total number of modules should not exceed the lecturers’ available working hours; lecturers should not teach courses for more than three consecutive hours; students should not attend classes for more than four consecutive hours; older students should have preferential schedules; a class of students should not have only one course in any given day; and a module should not be taught for more than two consecutive hours. The authors reported that the results of experimental studies demonstrated that the proposed model solves UCT problems more easily than other methods.

Al-Tarawneh and Ayob [13] applied a tabu search and a multineighbourhood structure to solve the UCT problem of the School of Engineering of Kebangsan University in Malaysia. The objective function sums the weights of penalties associated with the model’s soft constraints, which were determined by specialists. The hard constraints are as follows: collisions between classrooms and in lecturers’ availability schedules are not allowed; classroom capacities must be respected; and modules that take longer than two hours cannot be split into more than one teaching session. The soft constraints are as follows: each class of students must have one module in isolation every day; no class of students may have two classes in sequence; intervals between modules must not be long; no lecturer should teach two consecutive sessions; and no female lecturer should be allocated for modules in the evening. The authors claim that the results they obtained are better than those obtained manually.

This literature review shows that the core of hard constraints is quite coherent in the analysed works because these are, in fact, logical constraints that cannot be left unsatisfied. By contrast, the set of soft constraints may be viewed as a type of “institutional fingerprint” that makes the modelling extremely specific for each case and thus cannot be generalised.

#### 3. Description of the Institution to Be Analysed

The private university that is the object of study of this work is located in São Gonçalo, state of Rio de Janeiro, Brazil. The period of analysis is the year 2011 because this was the period for which data were fully available to the authors. Data gathering on the institution was all-encompassing, and the main sources were the institution page on Internet, institutional Intranet, documental research, and key staff interviews. In the first semester of 2011, the institution had a total of 1,297 students distributed over nine undergraduate courses and 92 lecturers. Its infrastructure consists of 47 classrooms and five laboratories, which seat a total of 1,857 students. There are two distinct shifts for classes, namely, morning and evening shifts. The morning shift is divided into two teaching sessions: the first session starts at 7:20 a.m. and ends at 9:00 a.m.; after a 20-minute interval, the second session starts at 9:20 a.m. and ends at 10:20 a.m. The evening shift also has two teaching sessions. The first starts at 7:00 p.m. and ends at 8:20 p.m.; after a 20-minute break, the second session starts at 8:40 p.m. and ends at 10:00 p.m.

Because each teaching shift is split into two teaching sessions, modules are classified into two categories: double-session (DS) and single-session (SS) modules. The first corresponds to modules that comprise two teaching sessions (from 7:20 a.m. to 10:20 a.m. or from 7:00 p.m. to 10:00 p.m.), and the second corresponds to modules that comprise only one teaching session (from 7:20 a.m. to 9:00 a.m.; from 9:20 a.m. to 10:20 a.m.; from 7:00 p.m. to 8:20 p.m.; or from 8:40 p.m. to 10 p.m.). Each module has strictly one class per week, meaning a regular student may be enrolled in five to 10 modules in a single shift.

Some modules may have joint classes; that is, students from different undergraduate programs may attend the same class because the modules have the same syllabuses; Foulds and Johnson [14] call this practice* piggybacking*.

The institution offers nine undergraduate programmes, with different durations each: Administration (ADM) in eight semesters, Law School (LAW) in ten semesters, Information Systems (INF) in eight semesters, Pedagogy (PED) in six semesters, Languages (LAN) in six semesters, Tourism (TOU) in seven semesters, Logistics Technology (LOG) in five semesters, Quality Management Technology (QUA) in five semesters, and Financial Management (FIN) Technology in five semesters.

The technology group is a new step in Brazil’s higher education system towards professional education. These undergraduate courses grant technology degrees to students that want to fulfil specific demands of the job market and can be rapidly allocated as specialised professionals. The “technologist” title is granted upon the completion of these courses, which are authorised by the Brazilian Ministry of Education.

Table 1 shows the total distribution of students among the 9 undergraduate programmes in the first semester of 2011.