Journal of Optimization

Volume 2017 (2017), Article ID 5723239, 11 pages

https://doi.org/10.1155/2017/5723239

## A NNIA Scheme for Timetabling Problems

School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Yu Lei

Received 29 December 2016; Revised 6 March 2017; Accepted 16 March 2017; Published 30 May 2017

Academic Editor: Linqiang Pan

Copyright © 2017 Yu Lei and Jiao Shi. 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 paper presents a memetic multiobjective optimization algorithm based on NNIA for examination timetabling problems. In this paper, the examination timetabling problem is considered as a two-objective optimization problem while it is modeled as a single-objective optimization problem generally. Within the NNIA framework, the special crossover operator is utilized to search in the solution space; two local search techniques are employed to optimize these two objectives and a diversity-keeping strategy which consists of an elitism group operator and an extension optimization operator to ensure a sufficient number of solutions in the pareto front. The proposed algorithm was tested on the most widely used uncapacitated Carter benchmarks. Experimental results prove that the proposed algorithm is a competitive algorithm.

#### 1. Introduction

The examination timetabling problem has long been a challenging area for researchers in the fields of operational research and artificial intelligence, especially at the time that the Toronto benchmark dataset was stated by Carter and Laporte (1996) [1]. The problem has been more difficult because universities are recruiting more students into a larger variety of courses with a growing number of combined degree courses [2] (Merlot et al. 2002). In the past 40 years there are many methods that have been applied to this problem. The represented techniques include constraint-based techniques [3], population-based techniques including genetic algorithms [4], graph coloring techniques [5, 6], ant colony optimization [7], scatter search [8], local search methods including tabu search [9] and simulated annealing [10, 11], variable neighborhood search [12], and hybrid and hyperheuristic approaches [5]. Generally, this problem is modeled as a single-objective optimization problem; only the number of clashes is considered by researchers.

To minimize the number of clashes in an exam timetable, Burke and Newall (2005) [13] stated that the clashes can be eliminated if a large number of periods were allocated. Burke et al. (1998) [14] also stated that longer timetables are needed to decrease the number of clashes. It is obvious that the ETTP is definitely a two-objective optimization problem: the number of clashes and the number of periods. Within the reasonable scope of the number of periods, the number of clashes must be minimized as much as possible. Hence, it is needed to minimize multiple conflicting cost functions, which can be best solved through the method of multiobjective optimization [15] that imported several features from the research on the graph coloring problem and used a variable-length chromosome representation that this paper also adopts.

Evolutionary multiobjective optimization (EMO), whose main goal is to handle multiobjective optimization problems (MOPs), has become a hot topic in the field of evolutionary computation. By simultaneously optimizing more than one objective, Multiobjective Optimization Evolutionary Algorithms (MOEAs) can acquire a set of solutions considering the influence of all the objective functions. Each of those solutions cannot be said better than the other and corresponds to the tradeoffs between those different objectives. Multiobjective examination timetabling problem as a MOP has two contradictory objectives. The optimization of one objective tends to minimize the number of clashes; the other objective tends to decrease the number of time periods. Many MOEAs have been proposed in recent years. Malim et al. (2006) [16] studied three different Artificial Immune systems and indicated that the algorithms can be appropriate for both course and exam timetabling problems. However, after published they were found to represent a mistake in the code, and it is invalid [17].

Many of the existing methods for exam timetabling problems are applicable to single-objective exam timetabling problems. By calculation, these single-objective optimization algorithms only can obtain one result, and the computational efficiency is poor. In this paper, we proposed a novel MOEA-based approach for multiobjective examination timetabling problem. By calculation, multiple results can be obtained by our proposed algorithms. In order to simultaneously optimize the two objectives, we adopt the framework of multiobjective immune algorithm NNIA [18] with some modifications. The NNIA simulates the phenomenon of the multifarious antibodies symbiosis and a small number of antibody’s activation in the immune response according to a method of selecting the nondominated neighborhood individuals. It chooses the small number of relatively isolated individuals as active antibodies and clones according to the crowding-distance value and then applies on the operators of recombination and mutation to strengthen the searching of the sparse area of the pareto front. The reasons we adopt the frame of the NNIA are that NNIA is proposed by ourselves and the clone strategy can make the pareto front uniform and get the satisfied solutions. The main contributions are that we adopt elitism group strategy to keep the diversity of the group and the two vertical local search operators to get the optimized solutions. Experiments show that the proposed algorithm is able to find a set of tradeoff solutions between the two objectives.

The paper is organized as follows. Section 2 describes the background information with problem formulation and related works. Section 3 gives a description of the proposed algorithm in detail. Section 4 presents the experimental study. Finally, conclusions are given in Section 5.

#### 2. Background

##### 2.1. Mathematical Model

As previously mentioned, the format of examination timetabling problems described in this paper was first formulated by Carter and Laporte [1] in 1996. In this problem, a set of exams need to be scheduled into a set of periods with each period having a seating capacity . There are three periods per weekday and a Saturday morning period. No exam is held on Sundays. It is assumed that the exam period starts on a Monday. The problem can be formally specified by first defining the following:where is one if exam is allocated to period ; otherwise, equals zero. is the number of students registered for exams and .

Equations (1) and (2) are the two objectives of minimizing the number of clashes and timetable length, respectively. Equation (3) is the constraint that no student is to be scheduled to take two exams at any one time, while (4) states that every exam can only be scheduled once in any timetable.

To evaluate the quality of one feasible timetable, a function evaluating the average cost for per student based on soft constraints has been proposed. It can be presented as follows:where is the weight that represents the importance of scheduling exams with common students either 4, 3, 2, 1, or 0 timeslots away in one timetable and is the number of students involved in the violation of the soft constraint. is the total number of students in the problem. For this reason that (5) emphasizes the most important indicators, that is, whether the exams in the timetable are allocated throughout the timetable equally, we use this function as one of the objectives in our algorithm. The two objectives of our algorithm optimized are described as follows:

##### 2.2. Related Works

The ETTP is a semiannual or annual problem for colleges and is studied by many operational researches widely due to its complexity and utility. There have been proposed a large range of approaches to solve the problem, discussed in the existing literature. These approaches can be classified into the following broad categories [19]: graph-based sequential techniques, local search-based techniques, population-based techniques, and hyperheuristics.

The graph coloring heuristics are one of the earliest algorithms. Welsh and Powell [20] in 1967 proposed a bridge that is built between graph coloring and timetabling and made a great contribution to the field of the timetabling. The five ordering strategies which extended from graph coloring heuristics on examination timetabling problems and a series of examination timetabling problems were introduced by Carter, Laporte, and Lee in 1996, called University of Toronto Benchmark Data. By developing two variants of selection strategies, Burke et al. [21] studied the influence of bringing a random element into the employment of graph heuristics in 1998. These simple strategies showed improved pure graph heuristics on the sides of both the quality and diversity of the solutions when tested on three of the Toronto datasets. Asmuni et al. [22] in 2005 employed fuzzy logic to order the exams to be scheduled on account of graph coloring heuristics on the Toronto datasets and indicated that it is an appropriate evaluation for arranging the exams. Corr et al. [23] investigated a neural network, the objective of which is to arrange the most difficult exams at an early stage of solution construction. The work has showed the feasibility of using neural network as a generally adaptive applicable technique on timetabling problems.

The local search-based techniques represent a large portion of the work which has appeared in the last decade [1]. Mainly because various constraints can be handled relatively easily, they have been applied on a variety of timetabling problems. Di Gaspero and Schaerf (2001) [24] carried out a valuable investigation on a family of tabu search based techniques whose neighborhoods concerned those which contributed to the violations of hard and soft constraints. Burke et al. [5] investigated variants of Variable Neighbourhood Search and obtained the best results in the literature across some of the problems in the Toronto datasets. Caramia et al. [25, 26] developed a fine-tuned local search method where a greedy scheduler assigned examinations into the least timeslots and a penalty decrease improved the timetable without increasing the number of timeslots.

The genetic algorithm is one of the most typical representatives of the population-based techniques. It is noticed that the algorithm has a good performance in the literatures. Particularly, the hybridizations of genetic algorithms with local search methods, memetic algorithms, have an excellent performance in this area. In 1994, Corne et al. [27] introduced genetic algorithms to solve general educational timetabling problems. The function of this work is that certain problem structures in some particularly generated graph coloring problems cannot be handled by obtaining direct representation in the genetic algorithms. Ross et al. [28] in 1996 indicated that by testing on specially generated graph coloring problems of different homogeneity and connectivity the transition regions were existent in solvable timetabling problems. The study can make researchers understand how different algorithms perform on complex timetabling problems. Terashima-Marin et al. [29] in 1999 indicated a clique-based crossover on the timetabling problems which was turned into graph problems. Erben [30] (2001) indicated a grouping genetic algorithm with appropriate encoding, crossover and mutation operators, and fitness functions studied. This method requires less computational time than some of the methods in the literature. Burke and Landa Silva [31] discussed some issues concerning the design of memetic algorithms for scheduling and timetabling problems. Burke et al. [21] developed a memetic algorithm to reassign single exams and sets of exams and employed light and heavy mutation operators. However, neither of these mutations on their own improved the solution quality. Malim et al. [16] developed three variants of Artificial Immune systems and indicated that the algorithms can be suitable for course and exam timetabling problems. However, there was a problem in the results; after publication they were showed to represent an error in the code and invalidness.

More and more researchers pay attention to the hyperheuristics approach. In 2003 Ahmadi et al. [32] investigated a variable neighborhood search, aiming to find good combinations of heuristics for different examination timetabling problem. Kendall and Hussin [33] in 2005 developed a tabu search hyperheuristic; they adopted moving strategies and constructive graph heuristics to be the low level heuristics. In 2007 Burke et al. [5] researched obtaining tabu search to find sequences of graph heuristics to construct solutions for timetabling problems and considered the effects of various numbers of low level graph heuristics on the examination timetabling problems. By conducting an empirical study on both benchmark functions and exam timetabling problems, Bilgin et al. [34] (2007) studied 7 heuristic selection methods and 5 acceptance criteria within a hyperheuristic. The memetic algorithm hyperheuristic with a single hill climber at a time showed that it performed better on approaches tested. For the interested readers, more details can be referred from [5].

In summary, during the recent years, there are an increasing number of excellent algorithms; almost all of these algorithms were tested on either benchmark datasets or in real applications, which had made quite good achievements. In this paper, we also proposed a multiobjective optimization algorithm, called Nondominated Neighbor Immune Algorithm (NNIA) in [18]. NNIA adopts an immune inspired operator, a nondominated neighbor-based selection technique, two heuristic search operators, and elitism. It indicates that NNIA is an effective method for solving MOPs by a number of experiments. Due to its good performance, we will adopt the framework of NNIA with some modifications, which will be described in the following section. The contribution of this paper is that we solve this task by using multiobjective optimization technique.

#### 3. The Proposed Algorithm

##### 3.1. Algorithmic Flow of MOEA Based on NNIA

The algorithmic flow of our algorithm is presented in Figure 1. At the start of the algorithm, a conflict matrix was created according to Burke and Newall [13], which has dimensions by with the definition from Section 2.1 being its th element. This matrix can check and eliminate the conflicts in the timetables efficiently.