Stratified Random Sampling Technique for Integrated Two-stage Multineighbourhood Tabu Search for Examination Timetabling Problem Ariff Md Ab Malik
Masri Ayob
Abdul Razak Hamdan
Faculty of Office Management & Technology, Puncak Alam Campus, Universiti Teknologi MARA, 43200 Puncak Alam, Selangor, Malaysia. email:
[email protected]
Data Mining and Optimisation Research Group (DMO), Center of Artificial Intelligence (CAIT), Faculty of Information Science & Technology, Universiti Kebangsaan Malaysia 43600 Bangi, Selangor, Malaysia. email:
[email protected]
Data Mining and Optimisation Research Group (DMO), Center of Artificial Intelligence (CAIT), Faculty of Information Science & Technology, Universiti Kebangsaan Malaysia 43600 Bangi, Selangor, Malaysia. email:
[email protected]
Abstract –In this research, we introduce a stratified random sampling technique that guides the selection mechanism to select the events (exams) for the integrated two-stage multineighbourhood tabu search (ITMTS) in solving examination timetabling problem. This technique is used during the timetable improvement phase especially when dealing with the exhaustive search mechanism in order to reduce the possibilities of extensive neighbors evaluation in finding good neighbours, without scarifying (too much) on the performance of the ITMTS. The selection mechanism only selects a set of exams (that represents a whole exam population) for every exhaustive evaluation of ITMTS. Therefore, this strategy can speed up the searching process and might lead the ITMTS to search in more promising area. We test and evaluate this strategy on the uncapacitated Carter benchmark datasets by using the standard Carter’s proximity cost. Our results are comparable with other approaches that have been reported in the literatures subject to the Carter’s benchmark datasets. Keywords – meta-heuristic, iterated two-stage tabu search, multineighbourhood structures, stratified random sampling selection
I.
INTRODUCTION
The educational timetabling problem is among well studied optimization area, especially in university examination and course timetabling. The main objective of university examination timetabling problem is to produce a good quality timetable that could ease, at least, the minimum needs of the timetable’s stakeholders [1]. The compulsory requirement of this problem is known as hard constraints that need to be fulfilled in order to have a feasible timetable. For example, no student will sit more than one exam in any period of timeslot. Meanwhile the soft constraints are considered as highly desirable requirement to be fulfilled but can be relaxed if necessary. The examples of this constraint such as: (i) to increase the gap of conflicting exams within a given timeslots in order to give ample exam’s preparation time for the students [2] or (ii) to group the exams based on same duration length of exams in
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a single location or hall [3] and etc. In order to measure the quality of the produced solution, an objective function is used, where the violations of the timetable’s soft constraints can be evaluated [4]. Many techniques have been explored and introduced until today either single solution based approach or population based approach [5]. Tabu search is among popular techniques that widely studied in the educational timetabling problem and is viewed as a dynamic conversion of neighbourhood [5]. Tabu search components such as tabu list, aspiration criteria, and variation memories are among the strength that could lead the search mechanism particularly for intensification and diversification of the search exploration [9]. For further details about the techniques uses in solving the examination timetabling problem, please refer to [5, 6, 7, 8, 9]. II.
RELATED WORKS
Many selection strategies have been introduced and applied in examination timetabling problems (ETP) either in initial solution construction phase or improvement phase [5, 6]. [10] used the exhaustive and biased selection strategies in their work. These strategies were applied during selecting process of the exam moves based on two predefined violation lists, soft violation and hard-violation lists. A sample of exam candidates with high cost function will be selected through a dynamic random-variate probability distribution and will be evaluated exhaustively. Subsequently, in [11], they enhanced the work in [10] by applying a token ring mechanism to explore their neighbourhoods. Meanwhile, [12] used the constraint priority ordering strategy in enhancing the solution. The selection consideration will be made based on either one constraint (with highest priority first) at a time or all constraints at a time. On the other hand, [13] applied a tandem search strategy with different policy on neighbourhood and tabu
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list relaxation on every phase of OTTABU. The tabu list will be relaxed after no-improvement situation for certain period of time occurred. The random selection method had been applied by [14], where the selected exams will be examined temporarily and compared subject to the quality improvement of the moves and restriction of the tabu list. In this work, we introduce a stratified random sampling technique to speed up the search process particularly when evaluating a large number of exams. This guided selection will reduce indirectly the burden of the exhaustive search mechanism by sampling the exams based on certain criteria. This sampling technique has practically proven and widely applied in social research area especially when dealing with a large group of different background of respondents [16, 17]. III.
STRATIFIED RANDOM SELECTION (SRS) OF ITMTS
This work is an extension of ITMTS of [15] that based on two stages of the tabu search implementation iteratively. The technique is viewed as two dimentional structure of the solution (timetable), named as vertical neighbourhood search (VhS) and horizontal neighbourhood search (HhS). Both stages will work iteratively based on the improvement condition of the current generated solutions. In ITMTS, some neighbourhood structures, such as move, swap, and move-swap (variation of kemp-chain), will be explored in parallel and exhaustively at VhS stage until the search traps in a local optima. Then, the evaluation of large neighbourhood swapping options (based on the timeslots or group of exams) will be applied in HhS. The selection of the solutions in both stages is based on the best improvement selection. Some tabu list strategies and procedures are used in both stages. ITMTS uses the random descent searching method with exhaustive multi-neighbourhood exploration. This strategy has contributed to the inability of ITMTS to generate solutions faster and becomes too expensive, although the possibilities of generating good solutions based on these multi-neighbourhood options are high. Therefore, we propose a “random descent with stratified random selection” in the VhS in order to choose a group of exams (which may fairly represents the whole exams) to be evaluated. This selection technique is used with the objective to (semi) random choose the exams to be evaluated by using the allocated proportional fractions of the exam subgroups based on a predefined selection percentage. By limiting the number of selected exams to be evaluated, the search time can be reduced without affecting the neighbourhoods exploration. The selection will be implemented continuously until all the exams could be evaluated or the iteration limit is achieved. Figure 1 shows the pseudo code for ITMTS with Stratified Random Sampling (SRS) technique (SRS is in italic).
Construct initial solution, Sol; calculate the initial cost, f(Sol); Set best solution, SolVbest ← Sol; f( Solbest) ←f(Sol); Set Vertical tabu, SolVtabu; Set iteration=0; Counter=0; Construct conflict frequency table do while (iteration
