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 twoobjective optimization problem while it is modeled as a singleobjective 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 diversitykeeping 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.
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) [
To minimize the number of clashes in an exam timetable, Burke and Newall (2005) [
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) [
Many of the existing methods for exam timetabling problems are applicable to singleobjective exam timetabling problems. By calculation, these singleobjective optimization algorithms only can obtain one result, and the computational efficiency is poor. In this paper, we proposed a novel MOEAbased 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 [
The paper is organized as follows. Section
As previously mentioned, the format of examination timetabling problems described in this paper was first formulated by Carter and Laporte [
Equations (
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:
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 [
The graph coloring heuristics are one of the earliest algorithms. Welsh and Powell [
The local searchbased techniques represent a large portion of the work which has appeared in the last decade [
The genetic algorithm is one of the most typical representatives of the populationbased 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. [
More and more researchers pay attention to the hyperheuristics approach. In 2003 Ahmadi et al. [
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 [
The algorithmic flow of our algorithm is presented in Figure
The flow of algorithm.
Because of the largescale individual of the population, it is inadvisable to search in the normal population. Elitism strategy and crowded selection optimization mechanism are put forward. In our algorithm we adopt elitism strategy two times to reduce the computation burden and extend the range of nondominated solutions in elitism group. In the normal population, the children population after crossover and mutation is mixed with parent population. Then the nondomined solutions of this new population are put into the elitism group. The purpose of the strategy is to offer more nondominated solutions to the elitism group.
As a common step, initialization is to produce an initial population. In our algorithm, there are a large number of feasible solutions that need to be optimized, but the process of generating the feasible solutions is hard for most of the examination timetabling problems. The difficulty level that feasible solutions generated for different examination timetabling problems is different. It is hard to make some unfeasible solutions be the feasible ones with some conventional ways in the subsequent operation. The result of the algorithm is influenced by the number of feasible solutions in the initial population for some issues. The initialization in our algorithm produces a set of solutions randomly and updates the random solutions to be the feasible ones with the simple genetic algorithm. The details are shown in the following.
The hyperheuristic initialization is that the exams are selected for insertion with the help of some heuristic information when used in the graph coloring problem [
Largest degree (LD): exams with the largest number of conflicts with other exams are inserted first.
Largest weighted degree (LWD): it is the same as LD but weighted by the number of students involved.
Saturation degree (SD): exams with the fewest valid timeslots, in terms of satisfying the hard constraints, remaining in the timetable are inserted first.
There are two terminated conditions, maximum iteration number and the maximum number of feasible solutions, which can make the whole algorithm achieve the stable result for most examination timetabling problems. But our algorithm has obvious superiority for the problems that the feasible solutions are hardly generated, because of this strategy of the initialization. The advantage of the hyperheuristic initialization is that we can get the timetables with the lengths being close to the demands of the users. The result can be seen in the next section.
Some researchers indicated that adopting local search within evolutionary algorithms is an much effective approach for finding high quality exam timetables which can also contribute to the intensification of the optimization results [
The first kind is to minimize the timetable length as far as possible without concerning the conflict number, aiming at the operator in the local search to minimize the time periods among the nondominated elitism group.
The selected individual is
Set the original mutation probability
Set search depth variable sign
Randomly select
Rearrange the exams in
If the number of
If
If
The second kind aims at minimizing the number of conflicts without concerning the number of timeslots. The details are described below.
The selected probability is
Set the original examination selected probability
Set search depth variable sign
Randomly select
Rearrange exams in
Insert the exams in
Compare the number of conflicts in
If
If
Although the local search can intensify the optimization results, the discrete optimization is different from the continuous optimization that a small disturbance in decision domain may probably let individuals transform irregularly and even result in deterioration. As such, in order to avoid this phenomenon we put forward a novel local search exploitation with an extra elitism group to save nondominated solutions in every generation. However, normal population is just offering a space of updating the nondominated solutions. In our algorithm we also introduce a corresponding elitism strategy and a crowded selection optimization mechanism; the details will be introduced in the next part.
The local search operators are applied after the strategy of extension and optimization of the elitism group. The operators avoid an objective in an individual deterioration and then minimize the other objective and get the new elitist solutions mixed with original elitist solutions to conduct a nondominated sorting. The frontier may extend to the two different directions as far as possible according to the operators. The local search is searching vertically between two objectives in order which is shown as Figure
Elitism group local search.
Due to the normal selection and mutation operators making a little contribution to the nondominated elitism group, we present a strategy to extend and optimize the elitism group based on the congestion degree which is shown in Figure
The computation of congestion degree.
Our algorithm is programmed in Matlab and simulations are performed on the 2.8 GHz Core Personal Computer. We use 9 uncapacitated benchmark examinations timetabling datasets proposed by Carter and Laporte [
Characteristics of datasets.
Dataset  Timeslots  Exams  Students  Conflict density 

Car 91  35  682  16925  0.13 
Car 92  32  543  18419  0.14 
Ear 83  24  190  1125  0.27 
Hec 92  18  81  2823  0.42 
Kfu 93  20  461  5349  0.6 
Lse 91  18  381  2726  0.6 
Rye 92  23  486  11483  0.08 
Sta 83  13  139  611  0.14 
Tre 92  23  261  4360  0.18 
Ute 92  10  184  2750  0.08 
York 83  3521  181  941  0.29 
Parameter setting for simulation study.
Parameter  Value 

Population size  100 

2 

0.8 

0.2 
SD: the search depth of the local search  10 
Itermax: the maximum iteration number  500 
In the following sections, we will study our algorithm in two sides. One is to make analysis on the contribution of diversitykeeping local search operators searching in two different directions orderly which is shown with four comparative experiments. The other one is to discuss the contribution of elitism group strategy applied on our algorithm two times with four experiments presented.
This section presents the performance of the diversitykeeping operators. To assess the effectiveness of the strategies, a comparison was conducted as in Figure
Performance comparison for MOEA with and without diversitykeeping strategy.
In this section, we use the hypervolume as an indicator to estimate the effectiveness of the algorithm; in our comparable experiments the indicator
To prove the efficiency of the proposed two local search operators, this section shows the performance of the algorithms with and without local search operators. As is shown in Figure
Performance comparison for MOEA with different local search settings.
From the statistic boxplots we can see that our algorithm is robust to the indicator of the students conflict numbers. The outliers are few and the differences between the highest and the lowest values are small, which also demonstrate the robustness of our algorithm.
This section presents the multiobjective optimization performance of the algorithm based on NNIA. On top of showing the advantages of our algorithm, the role of the two objectives will be validated as follows. The experiment was conducted running ten times independently.
The boxplots in Figure
Number of pareto optimal solutions for the datasets.
Experiments were conducted to further show the results. From Figure
Pareto optimal solutions for the datasets.
In this paper, the exam timetabling problem has been regarded as a multiobjective optimization problem which involves the minimizing of the number of clashes and number of periods in a timetable. A multiobjective evolutionary algorithm, based on NNIA, featured with elitism group strategy, congestion degree based on extension optimization strategy, and two local search operators, has been presented.
The proposed MOEA is different from most existing singleobjectivebased methods in the fact that it optimizes two objectives concurrently and get a set of solutions reasonable instead of producing singlelength timetables. It has been proved that such an approach is more universal and would be able to function effectively. The results also show that the algorithm can generate relatively short clashfree timetables and various solutions which are convenient for deciders to choose on their own preference.
The work we do in this paper focuses on the temporal aspect of the ETTP, which has solved the problem well in a sense. However, it still has some shortcomings, how to balance the diversity and approximation, which can be subjected for future study.
The authors declare that they have no conflicts of interest.