Hybrid heuristics for Examination Timetabling problem

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Abstract

Simulated Annealing (SA), Tabu Search (TS), Genetic Algorithm (GA), and Ant Colony System (ACS) are four of the main algorithms for solving challenging problems of intelligent systems. In this paper, we apply these four techniques and three novel hybrid combinations of them to a classical Examination Timetabling problem (ETP), an NP complete problem. The novel hybrid algorithms consist of a Sequential TS–ACS, a Hybrid ACS/TS, and a Sequential ACS–TS algorithms. These various hybrid combinations are then tested on 10 different scenarios of the classical ETP. Statistical comparative analysis conclude that all of the three proposed novel techniques are significantly better than each of their non-hybrid competitors, and furthermore the Sequential ACS–TS provides the superior solution of all.

Introduction

Proper scheduling of exams is a common problem for all universities and institutions of higher education. Quite often it is done by hand or with the limited help of a simple administration system and usually involves taking the previous year's timetable and modifying it so it will work for the new year. But changing the number of students, variety of the courses which are offered and student's freedom in selecting them requires significant alteration of previous year's timetable. The Examination Timetabling problem regards the scheduling for the exams of a set of university courses, avoiding overlap of exams of courses having common students and spreading the exams for the students as much as possible.

The process of finding a period for each exam so that no two conflict has been shown to be equivalent to assigning colours to vertices in graph so that adjacent vertices always have different colours [1]. This in turn has been proved to lie in the set of NP complete problems [2] which means that carrying out an exhaustive search for the timetable is not possible in a reasonable time. There are many heuristic methods which have been offered for solving this problem based on graph colouring as well as many metaheuristic methods such as SA, TS, GA, and ACS [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].

In this paper we consider the above metaheuristics and introduce three hybrid heuristic methods for solving the ETP, and compare their results. In the following sections, we first begin with describing the Examination Timetabling problem. In Section 3, a common structure of the four metaheuristics are provided. The four basic metaheuristics and their structure as used in this paper is discussed in Section 4. In Section 5, the structure of the input data set is discussed, and the results of application of the pure metaheuristics is analysed in Section 6. The proposed hybrid methods are explained in Section 7. Section 8 includes the final analysis of the hybrid methods and comparison with pure methods.

Section snippets

Problem description

Given is a set of examinations, a set of (contiguous) time slots, a set of students, and a set of student enrollments to examinations. The problem is to assign examinations to time slots satisfying a set of constraints [17]. Many different constraint types have been proposed in the literature. In this work, we consider the version proposed by Carter et al. [4], which is based on the so-called first-order and second-order conflicts.

First-order conflicts arise when a student has to take two exams

Common structure

In this section we define a common framework for all non-hybrid metaheuristics that are used for solving the ETP here.

Description of the heuristics

In this section, the four basic heuristic methods are separately but briefly described.

Problem datasets

We produce several problems in different size in order to apply these algorithms for different ones. In these problems the number of exams varies from 40 to 200 in line with the number of students and number of periods. The elements of conflict matrix of student Aij (that shows the common students in both i and j exams) has been produced randomly. You can see the information about these problems in Table 9.

Heuristic analysis

Due to the fact that the stopping criterion of the metaheuristics are not similar, a simple comparison of only the final solution values of the four metaheuristics is not appropriate. Furthermore, the computing time of heuristics highly depends on the value assigned to the parameters. Also it is difficult to estimate the processing time of heuristics. Moreover, the probability of finding a better final solution increases with the run time. Therefore a simple comparison of the final solution of

Proposed hybrid methods

By studying the behavior of each of the above pure methods, a properly chosen combination may be expected to yield improved performance. Here, we introduce three hybrid methods that combine the best metaheuristics of above-discussed methods, ACS and TS, in various ways.

Hybrid heuristic analysis

Note that in all of these methods, the parameters have been used that have provided the best performance in previous simulations when we introduced them separately.

The path of the objective function versus computing time for one of the runs is shown in Fig. 5.

The cumulative result of these three hybrid methods is shown in Table 16. The framework of this table is as same as Table 10 and the notation for bold faces and symbol `*' are same. Also the same computer has been used for all experiments.

Conclusions

In this paper we considered four conventional metaheuristic methods (Simulated Annealing, Tabu Search, Genetic Algorithm and Ant Colony System) and proposed three new hybrid methods for solving ETP and compared the results of them on 10 datasets. When comparing the results of the conventional methods, ACS showed better ability to find best solutions while TS algorithm worked better terms of improving its performance over time.

In comparison, all of the three hybrid methods demonstrate better

Acknowledgements

I thank Dr. Mohamad-Reza Akbarzadeh-Totonchi for his review on this paper and editing it and his good comments.

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