Development of a linear programming model for the University Course Timetabling Problem

Bucco, Guilherme Brandelli; Bornia-Poulsen, Camilo José; Bandeira, Denise Lindstrom

doi:10.1590/0104-530X2133-15

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Articles • Gest. Prod. 24 (1) • Jan-Apr 2017 • https://doi.org/10.1590/0104-530X2133-15 copy

Development of a linear programming model for the University Course Timetabling Problem

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract:

Creating timetables for courses is a problem that universities face at the beginning of every semester. This activity represents an important administrative task because it consumes significant amount of resources. In terms of computational complexity, this is classified as NP-hard, as it demands a huge amount of processing capacity. Timetabling is modeled in a number of different ways, aiming to fit the country’s educational context, meet specific rules of institutions of higher education or specific goals of managers, among others. In this paper, we propose a mathematical model to solve the University Course Timetabling Problem and optimize classroom utilization. To solve the proposed model, an algorithm that divides the problem was developed, solving it with mixed integer linear programming tools. Computational experiments applied to a real database of a Brazilian public university confirmed the good performance of the proposed approach, which greatly reduces the amount of assigned classrooms.

Keywords:
University Timetabling Problem; Integer programming; Mathematical programming; Decomposition; Timetable

Shift	$\| D \|$	$\| P \|$	$\| C \|$	$\| S \|$	$\| E \|$	$\| T \|$	$\| R \|$	$\| U \|$	$\| x_{e d p} \|$	$\| k_{r e d p} \|$	$\| m_{u} \|$
Morning	5	5	120	686	1,003	565	258	48	25,075	6,469,350	48
Afternoon	5	6	120	746	1,011	616	258	48	30,330	7,825,140	48
Evening	5	4	120	538	654	392	258	48	13,080	3,374,640	48
Total	5	15	120	1,688	2,688	1,083	258	48	201,600	52,012,800	48

Shifts		Running time (minutes)	Objective function value ( $Z_{1}$ )
Morning	( $h = 0$ )	4.68	101
Afternoon	( $h = 1$ )	16.14	88
Evening	( $h = 2$ )	3.31	95

Shifts		Running time (minutes)	Objective function value ( $Z_{2}$ )	Assigned units
Morning	( $h = 0$ )	2.13	730	0,1,2,3,4
Afternoon	( $h = 1$ )	3.82	490	0,1,2,3
Evening	( $h = 2$ )	1.27	536	0,1,2,4

Unit	Cost	# of classrooms	Unit	Cost	# of classrooms	Unit	Cost	# of classrooms
^* * Necessary and sufficient units to allocate the meetings, according to the model. 0	50	43	16	714	4	32	5,490	8
^* * Necessary and sufficient units to allocate the meetings, according to the model. 1	99	24	17	747	3	33	5,585	6
^* * Necessary and sufficient units to allocate the meetings, according to the model. 2	147	11	18	779	3	34	5,679	5
^* * Necessary and sufficient units to allocate the meetings, according to the model. 3	194	11	19	810	3	35	5,772	5
^* * Necessary and sufficient units to allocate the meetings, according to the model. 4	240	18	20	840	3	36	5,864	3
5	285	12	21	869	2	37	5,955	2
6	329	8	22	897	2	38	6,045	2
7	372	7	23	924	2	39	6,134	1
8	414	7	24	950	2	40	6,222	1
9	455	6	25	975	2	41	6,309	1
10	495	6	26	999	2	42	6,395	1
11	534	5	27	5,000	2	43	6,480	1
12	572	4	28	5,100	1	44	6,564	1
13	609	4	29	5,199	1	45	6,647	1
14	645	4	30	5,297	1	46	6,729	1
15	680	4	31	5,394	11	47	6,810	1

Universidade Federal de São Carlos Departamento de Engenharia de Produção , Caixa Postal 676 , 13.565-905 São Carlos SP Brazil, Tel.: +55 16 3351 8471 - São Carlos - SP - Brazil
E-mail: gp@dep.ufscar.br

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Necessary and sufficient units to allocate the meetings, according to the model.