DECISION SUPPORT SYSTEM FOR EXAMINATION ...

2nd International Conference and Exhibition (OWSD-FUTA), November 1- 4, 2015, 439-444

DECISION SUPPORT SYSTEM FOR EXAMINATION TIMETABLING Akinrinwa, S. P., Olabode, O., Oluwadare, S. A., Adesuyi, A. T. Computer Science Department, Federal University of Technology, Akure, Ondo State, Nigeria. [email protected], [email protected], [email protected], [email protected] ABSTRACT Examination Timetabling poses a difficult to solve optimization problem. This paper presents a Decision Support System for Examination Timetabling called the Venue Based Examination Timetabling Model (VBET). The VBET model consists of a user interface, database and a Control System that houses a model base that uses Constraint Satisfaction Programming to generate examination timetables. The objective is to allow timeslot preassignments in generating a feasible timetable. This objective is formulated into the objective function while other major requirements for the exam timetable such as exam prioritization, room availability and exam collision are formulated into constraints. The VBET model was implemented with Hypertext Pre-processor (PHP) scripting language and My Structured Query Language (MySQL) on Apache Server used as the web server within Windows environment. The model was simulated using sample data collected from the Computer Resource Centre, FUTA. The result shows that the Decision Support System provides users with exam schedules that are conflict free and also satisfy the formulated constraints thereby assists in examination schedule generation. Keywords: Timetabling, Examination Scheduling, Constraint Programming, Decision Support Systems. 1.

INTRODUCTION

Academic resource planning is a difficult administrative practice that is based on broad analysis of data and proper allocation of resources associated with the educational structure. These resources which include people, space and time require proper allocation through workable and attractive schedules formed from decisions informed by accurate information. One of such resource allocation problems faced in academic administration is the University Timetabling Problem. The University Timetabling Problem (UTP) can be defined as the task of assigning a number of events, such as lectures, exams, meetings, and so on, to a limited set of timeslots and venues such as class rooms, exam rooms and laboratories, in accordance with a set of constraints [1]. The University Timetabling Problem (UTP) can be divided into Course and Examination Timetabling Problems [2]. In similarity to course timetabling, the Examination Timetable Problem is basically a problem of finding an exact time allocation within a slated time period to assign to some other resources in such a way as to satisfy some constraints, i.e. students, rooms and exams, to timeslots. The decisions involved are: room assignment and timeslot assignment. The focus of this study is on the Examination Timetable Problem. The timetabling methods in use fall into two basic categories, namely: Conventional Methods and Automated Methods. The conventional method of timetabling is difficult. Timetables in use in some tertiary institutions are put together almost completely manually and to avoid violating some hard constraints, the human scheduler spends substantial amount of energy and time. A human scheduler also does keep in mind the idea of creating a schedule that is of good quality and whenever possible 1 Corresponding Author: Akinrinwa, S. P.


will adjust the timetable in order to achieve a good quality schedule. Having a tool that would support the generation of quality exam timetables according to certain measuring standards would help avoid certain conflicts. Thus this study intends to formulate a model for a decision support system for examination timetabling which can be used to optimize the use of resources during educational timetabling. Examination timetabling is a well-researched and important topic in educational timetabling [3 - 5]. Although there are some novel research that focused on the modeling and formulation of Examination Timetabling Problem (ETP), [4,5] a large number of researchers are interested in the applications of heuristics to solve the problem [6,7], Shortcomings of research by [8] had limitations in their systems use of a large variable model that does not implement variable reduction and therefore requires large computation time, this research work attempts to develop a system with a reduced variable model. Shortcomings of research by [5], generated solutions that does not guarantee that no examination clashes occur and does not also incorporate constraints and requirements like scarce room resources or precedence constraints between exams. The objective of this research is to develop a Venue Based Decision Support System (DSS) for Examination Timetabling. 1.1

Literature Reviews

In literature, many approaches have been used to solve the timetabling problem. [9] divided timetabling approaches into approaches based on operation research, meta-heuristic and modern intelligent methods. The solution to an exam timetable problem would be to assign all exams to timeslots without violating any constraint thereby generating a feasible timetable. The task is usually to determine whether a solution exists and to find the solution. In cases where a partial solution exists, they are to be extended to full solution, and also find optimal solutions relative to a given cost function [10]. Constraints can be described by explicitly presenting the consistent or inconsistent value combinations by mathematical expressions or computable procedures that specify these combinations. In general, tasks posed in the constraint satisfaction problem paradigm are usually computationally difficult i.e. NP-hard [11]. Over the last two decades, a great deal of theoretical and experimental research has focused on developing algorithms for solving constraint satisfaction problems. [12] combined a genetic algorithm with constraint based reasoning and they presented a possible and near to optimum solution to the lecture timetabling problem. [13] used a timetabling planning problem as formulated by using constraint based reasoning technique in an object oriented approach. [10] presented the creation of University timetabling by using an approach based on constraint satisfaction programming. 1.2

Description of the Timetabling Problem

The examination timetabling problem consists of the following entities: a) A set of timeslots: These are separate examination periods have date, start and end time. b) A set of examination venues. Each examination venue has an examination seating capacity, venue owner and an availability status defined. c) A set of examinations. Each exam has a length, type of venue required, and a list of students enrolled in the exam. d) A set of distribution preferences – A distribution preference is set between exams common to a certain group of students. Example of distribution preferences are: 2 Corresponding Author: Akinrinwa, S. P.


i. Date and time preferences – this specifies a particular time of a particular date for an exam. ii. Day preferences – this specifies a particular day for an exam iii. Time preference – this specifies a particular time on any day for an exam iv. Room preferences –each department has a certain room for their exams. Exams written by the students in that department are held in their room except the exams with need for special venues. v. Timeslot distribution - exams that have no students/room in common can be assigned to the same timeslot. vi. Exam distribution preferences – exams are distributed in a particular order. For instance exams can be assigned to timeslots in descending order of the number of students involved. In education, decisions made may be highly consequential. The Venue Based DSS for examination timetabling developed in this research is to support the administrative task of planning the University’s examination under the specified resource constraints. 2.

MATERIALS AND METHODS

The Venue-Based Examination Timetable (VBET) model is based on the concept of a fixed exam venue allocation to departments for the exam duration; to the best of our knowledge, no other set of constructive heuristics proposed for solving the examination timetabling problem uses the same assumption. The VBET consists of five modules which are the user interface, Data entry, Database, Constraint programming module and report generation module. Constraint Programming Module The constraint programming module houses the constraint mapping engine which maps the chosen constraints to the timetable. The venue and course input are fed to the constraint mapping engine which then enforces the constraints on the venue and courses. If there is a constraint on a venue, take for example, a laboratory may require only practical exams to be allocated, the constraint mapping engine enforces the constraints. This mapping engine first takes into consideration the objective function in equation (1) which is to allow for pretimetable generation course allocations before the whole timetable is generated. In examination timetabling, exam schedules must satisfy a set of constraints which are divided into hard and soft constraints. [14-17] Hard constraints must be respected (must be attained) while soft constraints are to be satisfied as much as possible. Generally, a timetable is said to be feasible if all the hard constraints of the problem are satisfied, the quality of that feasible timetable depends on how well they satisfy the soft constraints. Constraints are categorized into two namely: objective function (sometimes called soft constraint) and hard constraints. a) Constraint Based on the Objective function 𝑓(𝑥)are further divided into the following: i. Slots only constraints: These are constraints that are placed on courses to restrict them to a particular slot in any of the exam days. ii. Date only constraints: These are constraints that are placed on courses to restrict them to any slot in a particular date iii. Date and slots constraints: These are constraints that are placed on courses to restrict them to a particular slot on a particular date b) Hard constraints 𝑔(𝑥) considered include: Resource Assignment, Resource Availability, Time Assignment, Venue Capacities, Room Feature, Exam collision 3 Corresponding Author: Akinrinwa, S. P.


avoidance, and Room Continuity. The mathematical programming formulations for this problem are of the form: 𝐹𝑖𝑛𝑑: 𝑓(𝑥) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑔(𝑥) ≤ 0 (1) where x is the solution vector and 𝑓(𝑥) is the objective function and 𝑔(𝑥) contains all hard constraints. The examination timetable consists of a set of m timeslots 𝑇 {𝑘1 , … , 𝑘𝑚 } An examination schedule that lasts for 2 weeks, where there are 6 examination days in a week and 3 timeslots per day will have 2×6×3=36 timeslots, a set of rooms 𝑅 {𝑟1 , … , 𝑟𝑗 } in which examinations 𝐶 {𝑖1 , … , 𝑖𝑛 } can take place , 𝑖, 𝑘, 𝑟 represents an exam, a timeslot and a room respectively, 𝑆𝑖,𝑗 represents the number of students for an exam 𝑖, 𝑅𝑟 refers to the size of room r, 𝑅̂ is the maximum exams per room given relative to each examination room and 𝑆𝑖 is the number of departments involved in an examination 𝑖, 𝑃𝑖,𝑗 represents a function to determine exam with the larger student size, 𝑝𝑖,𝑘 represents the pre-assignment of a course i to period k. The conflict matrix 𝑀𝑖1 , 𝑖𝑛 is a binary matrix such that 𝑀𝑖1 , 𝑖2 = 1 if courses i1 and i2 have students in common, and 𝑀𝑖1 , 𝑖2 = 0 otherwise. All courses have students. 1, 𝑖𝑓 𝑒𝑥𝑎𝑚 𝑖 𝑖𝑠 𝑡𝑎𝑘𝑒𝑛 𝑏𝑦 𝑗 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 ,𝑗 > 0 𝐹𝑖,𝑗 = { (2) 0, 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Conflict matrix: The matrix 𝑀 is a used to detect collisions between courses [18]. 1, 𝑖𝑓 𝑒𝑥𝑎𝑚 𝑖1 𝑐𝑜𝑙𝑙𝑖𝑑𝑒𝑠 𝑤𝑖𝑡ℎ 𝑒𝑥𝑎𝑚 𝑖2 𝑀𝑖1 , 𝑖2 = { (3) 0, 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Note that, two exams collide if they have students in common. Objective Function: The objective function f(x) is to enforce a pre-timetable generation allocation of courses to timeslots on the timetable. The formulation for the objective function is the following: 𝑛

𝑚

𝑓(𝑥) = ∑ ∑ 𝑝𝑖,𝑘 𝑥𝑖,𝑘,𝑟

(4)

𝑖=1 𝑘=1

Hard constraints: The formulations for the hard constraints are as follows: Resource Assignment: Ensures that each exam is assigned at least a venue and a timeslot. 1, 𝑖𝑓 𝑒𝑥𝑎𝑚 𝑖 𝑖𝑠 𝑠𝑙𝑜𝑡𝑡𝑒𝑑 𝑖𝑛 𝑡𝑖𝑚𝑒𝑠𝑙𝑜𝑡 𝑘 𝑎𝑛𝑑 𝑟𝑜𝑜𝑚 𝑟 𝑥𝑖,𝑘,𝑟 = { (5) 0, 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑇𝑚

∑

𝐽

∑ 𝑘=1

𝑥𝑖,𝑘,𝑟 = 1

∀𝑖 = 1 … 𝑛

𝑟=1

(6) 𝑥𝑖,𝑘,𝑟 is assigned 1 if an exam 𝑖 is assigned a room 𝑟 and a timeslot 𝑘. Exam Collision: Ensures that courses with common students are not scheduled together. 𝑀𝑖1 ,𝑖2 𝑥𝑖1 ,𝑘,𝑟1 + 𝑀𝑖1 ,𝑖2 𝑥𝑖1 ,𝑘,𝑟1 ≤ 1, ∀𝑖1 ≠ 𝑖2 , ∀𝑟, ∀𝑘 (7) This enforces that for any pair of exam (𝑖1 , 𝑖2 ) where 𝑖1 ≠ 𝑖2 , if there is a collision, then the two exams cannot be scheduled in the same time slot k. (adapted from [18]) Room Availability and Capacity: Ensures that there are no exam clashes in an exam room. ∑ 𝑥𝑖,𝑘,𝑟 Si,j ≤ 𝑅𝑟

∀𝑖1 , 𝑖2 … 𝑖𝑛 , 𝑘 𝑎𝑛𝑑 𝑟

(8)

𝑖

Where Rr refers to the size of room r and Si,j refers to the number of students sitting for the exam i. Thus, if any of the courses 𝑖1 , 𝑖2 … 𝑖𝑛 are assigned to time slot k at room r, then sum of all students taking courses 𝑖1 , 𝑖2 … 𝑖𝑛 must not exceed room capacity. 4 Corresponding Author: Akinrinwa, S. P.


Room Collision: Not more than m mutually exclusive exams can hold at once in one accommodative exam room. 𝑖1 , 𝑖2 , …, 𝑖𝑛 (𝑖1 ≠ 𝑖2 ≠ … ≠ 𝑖 𝑅̂ ) can be allocated in the same room r at the same timeslot k, where 𝑅̂ is the maximum exams per room given relative to a particular exam room, which exists for all rooms. ∑ 𝑥𝑖,𝑘,𝑟 ≤ 𝑅̂ ∀𝑖1 ≠ 𝑖2 ≠ ⋯ ≠ 𝑖𝑛 , ∀𝑘, ∀𝑟 𝑤ℎ𝑒𝑟𝑒 ∀𝑟 ∃ 𝑅̂ (9) 𝑖

Exam Prioritization: This constraint ensures that large exams are scheduled as early as possible to allow enough time for marking. 𝑖1 , 𝑖𝑓𝑓 𝑆𝑖1 ,𝑗 ≥ 𝑆𝑖2 ,𝑗 𝑃𝑖,𝑗 = { (10) 𝑖2 , 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 The constraint satisfaction model described in equations (1 – 10) form the basis of the control system of the venue based examination timetabling system. 3.

RESULTS AND DISCUSSION

This following shows the performance of the VBET techniques for ETP on experimental data. The proposed VBET technique was tested on data instances available on [20]. The datasets are of varying sizes. Comparisons of the mean proximity cost of the VBET technique as well as some other techniques in [19] are given in Tables 1. Table 1. Comparison of mean Proximity Cost of Examination Timetabling Techniques Datasets M1 M2 M3 M4 M5 M6 M7 VBET CAR–F–92 6.25 4.36 4.86 4.42 4.6 4.38 4.32 8.25 CAR–S–91 7.07 5.1 5.82 5.3 5.37 5.18 5.3 6.96 EAR–F–83 42.24 37.22 36.86 36.98 37.18 38.45 37.79 48.8 HEC–S–92 12.96 11.85 11.97 12.31 12.8 11.97 11.87 18.2 KFU–S–93 18.5 14.62 15.04 14.67 15.84 14.7 14.99 22.6 LSE–F–91 15.64 11.14 12.12 11.21 13.47 11.7 12.86 16.17 60 50

M1

40

M2 M3

30

M4

20

M5

10

M6

0

M7 VBET

Fig. 2. Comparison of mean Proximity Cost of Examination Timetabling Techniques The comparisons above show that the proposed VBET model does well in the production of examination timetables although the VBET model may not outperform the other techniques in literature. 5 Corresponding Author: Akinrinwa, S. P.


4. CONCLUSION In this research, the University Examination Timetabling problem is analyzed and the Venue Based Examination Timetabling System is proposed with an objective of allowing pretimetable generation allocation of courses to timeslots thus providing a Decision Support System for solving this problem. The results from the analysis and comparisons with other existing models for exam timetabling show that the proposed VBET model provides students with quality exam schedules. A limitation however is that the system is not self configuring, a user should be able to express his constraints and these constraints should be taken into considerations in an interactive manner when generating a timetable. In pursuing this, a future research might be to create a self configuring timetabling system. 5. REFERENCES [1] Cambazard, H., Demazeau, F., Jussien, N., and David, P. (2004). Interactively Solving School Timetabling Problems Using Extensions of Constraint Programming. In Practice and Theory of Automated Timetabling 5(3616): 190–207. Berlin: Springer. [2] Schaerf A. (1999). A survey of automated timetabling. Artificial Intelligence Review, 13: 303–316. [3] Ayob M., Ab Malik A. M., Abdullah S., Hamdan A. R., Kendall G., and Qu R. (2007). Solving a Practical Examination Timetabling Problem: A Case Study. ICCSA, LNCS, 4707(3): 611–624. [4] Burke E. K. and Bykov Y. (2006), Solving exam timetabling problems with the flex-deluge algorithm. PATAT, 370-372. [5] Eley M. (2006). Ant algorithms for the exam timetabling problem. Practice and Theory of Automated Timetabling (PATAT), 5:167–180. [6] Malek A. and Salwani A. (2011). Artificial bee colony search algorithm for examination timetabling problems. International Journal of the Physical Sciences, 6(17): 4264-4272. [7] Ozcan E, Misir M, Ochoa G & Burke E (2010). A Reinforcement Learning - Great-Deluge Hyper-Heuristic for Examination Timetabling. International Journal of Applied Metaheuristic Computing, 1(1): 39-59. [8] Benli O. S. and Botsah A. R. (2004). An Optimization-Based Decision Support System for a University Timetabling Problem: An Integrated Constraint and Binary Integer Programming Approach. http://web.csulb.edu/~obenli/Research/benli-botsali.pdf [9] Feizi-Derakhshi M., Babaei H. and Heidarzadeh J. (2012). A Survey of Approaches for University Course Timetabling Problem. IMS: 307-321. [10] Zhang L. and Lau S. K. (2005). Constructing University timetable using constraint satisfaction programming approach. International Conference on Intelligent Agents, Web Technologies and Internet Commerce, 28-30(2): 55-60. [11] Ionita M., Breaban M. and Croitoru C. (2010). Evolutionary Computation in Constraint Satisfaction. New Achievement in Evolutionary computation, ISBN 978-953-053-7: 318 – 330. [12] Deris S., Omatu S. and Ohta H. (2000) Timetable Planning Using the Constraint-Based Reasoning. Computers and Operations Research, 27: 819-840. [13] Deris S., Omatu S., Ohta H. & Saad P. (1999). Incorporating constraint propagation in genetic algorithm for university timetable planning. Engineering Applications of Artificial Intelligence, 12:241-253. [14] Abdullah S. (2006). Heuristic Approaches for University Timetabling Problems. PhD thesis, School of Computer Science and Information Technology, The University of Nottingham, United Kingdom. [15] Shahvali M., and et al.(2011). A fuzzy genetic algorithm with local search for university course timetabling. Proc. of ICMI, 250-254. [16] Yang S, Jat S. N. (2011). Genetic algorithms with guided and local search strategies for university course timetabling. IEEE TSMC, 41(1). [17] Abdullah S. and et al, (2007). Using a randomized iterative improvement algorithm with composite neighborhood structures. Proc. of 6th ICMH, 153–169. [18] Chacha S. (2012), “Mathematical programming formulations for optimization of University course Timetabling problem” University of Dar es Salaam, pages 1 – 36. [19] Chaudhari A. and De K.(2010). Fuzzy Integer Linear Programming Mathematical Models for Examination Timetable Problem. http://arxiv.org/ftp/arxiv/papers/1307/1307.1900.pdf retrieved 10 October, 2015. [20] http://www.cs.nott.ac.uk/~rxq/data.htm retrieved 10 October, 2015.

6 Corresponding Author: Akinrinwa, S. P.