ALTA-MONTAÑA

This study discusses an approach based on adaptive strategies that decomposes the examinations in a given problem into two sets: difficult to schedule examinations and easy to schedule examinations. This decomposition is performed automatically at each iteration, and is augmented with a suitable ordering of examinations within each set. In this study, it is observed that by merging or swapping the boundary set with the difficult set, solution quality could be improved. A stochastic component based on roulette wheel selection is embedded into the approach in order to shuffle the order of examinations. This mechanism provides a higher chance that an examination with a higher score will be selected for timetabling. Different parameters are tested on the boundary size and roulette wheel selection size and the parameter setting is undertaken based on the statistical analysis. It is observed that, using saturation degree heuristic, the possibility of creating infeasible solutions could be decreased and that dynamic ordering achieves better ordering of examinations in the list. This study shows that the proposed approach is simple to implement, yet it is competitive with previously published constructive and improvement approaches. In this study, the same ordering heuristics are used for reordering the examinations in the difficult and easy sets. The proposed framework allows the use of different strategies. The next chapter will discuss the improvement strategy for enhancing solutions obtained from the proposed constructive approaches.

te r 5. A C o n st ru ct io n A p p ro a ch fo r E xa m in a tio n T im et a bl in g ba se d o n A d a p tive m po si tio n a n d O rd er in g 148

Table 5.6: _{Comparison of different approaches for the ITC2007 benchmark datasets (The bold entries indicate the best results)}

Problem [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] ADO Exam 1 4370 5905 8006 6670 12035 4370 4633 4699 8559 6235 4775 4128 4368 11378 Exam 2 400 1008 3470 623 3074 385 405 385 830 2974 385 380 390 3243 Exam 3 10049 13862 18622 - 15917 9378 9064 8500 11576 15832 8996 7769 9830 20403 Exam 4 18141 18674 22559 - 23582 15368 15663 14879 21901 35106 16204 13103 17251 25449 Exam 5 2988 4139 4714 3847 6860 2988 3042 2795 3969 4873 2929 2513 3022 8199 Exam 6 26950 27640 29155 27815 32250 26365 25880 25410 28340 31756 25740 25330 25995 28745 Exam 7 4213 6683 10473 5420 17666 4138 4037 3884 8167 11562 4087 3537 4067 16381 Exam 8 7861 10521 14317 - 16184 7516 7461 7440 12658 20994 7777 7087 7519 20308 Exam 9 1047 1159 1737 1288 2055 1014 1071 - - - 964 913 - 2171 Exam 10 16682 - 15085 14778 17724 14555 14374 - - - 13203 13053 - 16876 Exam 11 34129 43888 - - 40535 31425 29180 - - - 28704 24369 - 46649 Exam 12 5535 - 5264 - 6310 5357 5693 - - - 5197 5095 - 7445

[1] M¨uller (2008), [2] Gogos et al. (2008), [3] Atsuta et al. (2008), [4] De Smet (2008), [5] Pillay (2008), [6] M¨uller (2009), [7] McCollum et al. (2009), [8] Gogos et al. (2009), [9] Pillay (2010a), [10] Burke et al. (2010f), [11] Gogos et al. (2010a), [12] Gogos et al. (2010b), [13] Turabieh and Abdullah (2011))

A Variable Neighbourhood Search

- Great Deluge for Examination

Timetabling Problem

Most successful approaches to examination timetabling, including that adopted by the winner of the ITC2007 competition (M¨uller, 2009), consist of multiple stages, in which a constructive approach is used in finding a good initial solution, and then one or more im- provement approaches are employed to further improve the quality of solution obtained in the previous stage. The main objectives of this chapter are to show how previously constructed solutions are improved and to investigate the influence of various initialisa- tion methods and neighbourhood orderings on the performance of the search algorithm. The chapter presents a variable neighbourhood search approach combined with a great deluge acceptance method, allowing the acceptance of some worsening solutions for solv- ing the examination timetabling problem. A range of neighbourhood structures were tested within this approach for diversification purposes. The neighbourhood orderings in VNS and the effect of the initialisation methods on the solution quality of the improve- ment approach were investigated in relation to two well-known examination timetabling benchmarks. The results illustrate the success of the variable neighbourhood search - great deluge (VNS-GD) approach in solving examination timetabling problems. This study has shown that initialisation is crucial for the success of the improvement ap- proach used in a multistage setting, particularly for VNS. The present chapter describes variable neighbourhood search in Section 6.1. The details of the variable neighbourhood search - great deluge approach are discussed in Section 6.2, focusing on the algorithmic components, initialisations, neighbourhood structures and acceptance criteria. Section 6.3 provides the experimental results and comments. Finally, the conclusion is presented in Section 6.4.

6.1

Variable Neighbourhood Search (VNS)

Most local search approaches could trap at a local minimum during the search process. As described by Mladenovi´c and Hansen (1997) and Hansen and Mladenovi´c (2001), different neighbourhood structures have their own local minimum. Motivated by this, the use of a set of neighbourhood structures was introduced by Mladenovi´c and Hansen (1997) so that the search process could continue by changing to one of the other neigh- bourhood structures, while at the same time the search could avoid being trapped at a local minimum. VNS is a descent-ascent approach that iteratively applies a shaking strategy and local search in order to find the best solution for a problem at hand. This algorithm can be considered as a single point-based selection hyper-heuristic (Burke et al., 2010b) which requires several neighbourhood structures with different natures. If one of these fails to improve the solution, the other may still have a chance. The VNS main framework consists of three main steps: ‘shaking’, ‘local search’ and ‘move’. Algorithm 13 below illustrates the steps of the basic VNS algorithm adopted by Mlade- novi´c and Hansen (1997). Nk is a set of neighbourhood structures that will be selected

during the search, where k = 1, ..., kmax. The foundation of basic VNS was a descent

approach that only accepts the improving move. The current solution is accepted if it is better than the incumbent solution. In this circumstance, the neighbourhood structures are alternated in order to avoid local optimum since different neighbourhood structures have their own local optimum. The local search procedure acts as an intensification strategy to converge to a good solution. At the shaking procedure, the point s′ is gen- erated randomly. The shaking procedure is a diversification strategy that avoids cycling during the search. There are several stopping conditions utilised in this approach, such as the maximum number of iterations, the number of non-improving iterations and the preset of CPU time.

Algorithm 13 Basic VNS algorithm

Initialisation: Select the set of neighbourhood structures Nk, k = 1, ..., kmax, to be

used in the search; find an initial solution s; choose stopping condition; Repeat until stopping criteria is satisfied:

1. Set k := 1;

2. Until k = kmax, repeat:

(a) Shaking: Generate a point s′ at random from the kth

neighbourhood of s(s′ ∈ Nk(s));

(b) Local Search: Apply a local search method with s′ as initial solution until local optimum s′′ is obtained.

(c) Move or not: Accept s′′_{(s ← s}′′_{) if it is better than incumbent solution and}

The VNS approach has been successfully implemented in the field of university timetabling. Wong et al. (2005), for example, implemented variable neighbourhood descent using mul- tiple neighbourhood structures for solving the un-capacitated problem of examination timetabling. Each neighbourhood structure was formulated into a different local search operator in order to explore and exploit the search space of solutions. The aim was to balance the intensification and diversification during the exploration of the search space. Furthermore, Ahmadi et al. (2003) employed the VNS as a high-level heuris- tic that chose low-level heuristics. Their study described the application of VNS as a perturbation-based algorithm to the examination timetabling problem that incorporated weight values to each low-level heuristic in order to find good quality solutions. The low- level heuristics used within this hyper-heuristic were based on a combined selection of an examination, time-slot and room.

A recent study has incorporated a genetic algorithm as neighbourhood selector within a VNS approach. Burke et al. (2010a) showed that VNS and hybridisation with a genetic algorithm could yield good quality solutions, a method which produced several best known results in the literature. In their study, the genetic algorithm imitated the concept of hyper-heuristics and case-based reasoning, where it was not directly applied to the problem but instead worked at a high-level. Since the solution quality was dependent on the selection of a neighbourhood, the genetic algorithm performed the search by selecting the list of neighbourhoods from the VNS framework.

The VNS approach was implemented for the course timetabling by Abdullah et al. (2005). The course timetabling problem required a search for the best assignment of lectures to time-slots and rooms, subject to constraints. The study employed a VNS approach using exponential Monte-Carlo as an acceptance criteria for worsening moves, and during the search, a tabu list was utilised to prohibit a non-improving neighbourhood being used for a certain period.

There have been further recent studies based on the VNS approach. For example, VNS was employed to solve a nurse rostering problem (Burke et al., 2003b), a graph colouring problem (Avanthay et al., 2003), a median cycle problem (Morena P´erez et al., 2003), project scheduling (Fleszar and Hindi, 2004) and an external graph problem (Caporossi and Hansen, 2000).