A Multiobjective Optimisation Technique for Exam Timetabling Problems Based on the Defined Trajectory Sanja Petrovic, Yuri Bykov University of Nottingham, School of Computer Science & IT, Wollaton Road Nottingham NG8 1BB, UK {sxp, yxb}@cs.nott.ac.uk University examination timetabling problems comprise arranging exams in the given number of timeslots through the examination sessions. The primary objective of this process is avoiding of students' clashes (i.e. no one student should take two exams simultaneously). This requirement is generally considered as a hard constraint and should be obligatory satisfied in a feasible timetable. However, the number of other restrictions and regulations, which depend on a particular institution, should be also taken into account while solving examination timetabling problems. Some of them are also considered as hard constraints, while other constraints are soft, i.e. they cannot be completely satisfied, and therefore their violations should be minimised. The soft constraints usually have different importance for the timetable officer (decision maker). They are generally incompatible and often conflicting with each other. Timetabling problems with various soft constraints typically have no single supreme solution. Formally, a number of solutions can be considered as optimal. The decision maker’s goal is to obtain a desirable one in accordance to his/her preferences. This can be done by grouping the soft constraints into several objectives, where objectives measure the number of violations of the soft constraints, and applying an appropriate multiobjective optimisation technique. Those techniques show different performance and require different strategies for their application. The multiobjective techniques are traditionally divided into two groups. The first group is known as “Decide-then-Search” (a priori). In these methods the decision maker specifies his/her preferences regarding the solution before starting a search. Generally, such approach involves the aggregation of problem’s objectives into one objective function in order to apply some singleobjective searching algorithm. The examples of application of this approach to examination timetabling can be found in [1], [2]. The second group of the approaches is called “Search-thenDecide” (a posteriori). Those methods produce a number of solutions among which the decision maker chooses the preferable one. To our knowledge, there are no publications about the use of these methods for exam timetabling. However several authors (ex. [4], [5]) have applied aposteriori approach to the class/teacher timetabling, i.e. to the problem, which is similar to examination timetabling. In this paper we introduce a new a priori multiobjective algorithm, which drives a search procedure through a predefined trajectory. The proposed algorithm is based on the local search algorithm, which we named degraded ceiling algorithm [3]. In the degraded ceiling algorithm a new candidate solution is accepted if its objective function is either better than a current one or does not exceed a value of the ceiling. The value of the ceiling reduces gradually throughout the search taking into consideration a predefined searching time of the algorithm. This algorithm was initially tested with a weighted sum as its objective function. It was shown that a longer search usually yielded a better final result. The algorithm can produce the results which outperform other techniques applied to exam timetabling problems at the price of relatively longer search period (time is usually not an issue in the exam timetabling problems).

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In the developed multiobjective approach to degraded ceiling algorithm, in order to direct the search to a predefined trajectory, the weights of the objectives are changed dynamically, one at a time. The proposed algorithm was experimentally checked on the real-world large scale examination timetabling data. The experiments confirmed the ability of the algorithm to drive the search close to the chosen trajectory. The developed approach is compared with other multiobjective optimisation techniques. References [1] E. K. Burke J. P. Newall, “A Multi-Stage Evolutionary Algorithm for the Timetabling Problem”. The IEEE Transactions of Evolutionary Computation 3.1, 1999, 63-74. [2] E. K. Burke, Y. Bykov, S. Petrovic. “A Multicriteria Approach to Examination Timetabling”. E. Burke, W. Erben, eds. The Practice and Theory of Automated Timetabling III: Selected Papers (PATAT 2000). Lecture Notes in Computer Science 2079. Springer-Verlag, Berlin Heidelberg, New York, 2001. 118-131. [3] E. K. Burke, Y. Bykov, J. P. Newall, S. Petrovic “A Time-Predefined Local Search Approach to Exam Timetabling Problems”, Computer Science Technical Report No. NOTTCS-TR-2001-6, University of Nottingham, 2001. [4] M. P. Carrasco, M. V. Pato, “A Multiobjective Genetic Algorithm for the Class/Teacher Timetabling Problem”, E. Burke, W. Erben, eds. The Practice and Theory of Automated Timetabling III: Selected Papers (PATAT 2000). Lecture Notes in Computer Science 2079. Springer-Verlag, Berlin Heidelberg, New York, 2001. 3-17. [5] M. Tanaka, S. Adachi, “Request-based timetabling by Genetic Algorithm with Tabu Search”, The Third International Workshop on Frontiers in Evolutionary Algorithms, 999-1002, 2000.

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In the developed multiobjective approach to degraded ceiling algorithm, in order to direct the search to a predefined trajectory, the weights of the objectives are changed dynamically, one at a time. The proposed algorithm was experimentally checked on the real-world large scale examination timetabling data. The experiments confirmed the ability of the algorithm to drive the search close to the chosen trajectory. The developed approach is compared with other multiobjective optimisation techniques. References [1] E. K. Burke J. P. Newall, “A Multi-Stage Evolutionary Algorithm for the Timetabling Problem”. The IEEE Transactions of Evolutionary Computation 3.1, 1999, 63-74. [2] E. K. Burke, Y. Bykov, S. Petrovic. “A Multicriteria Approach to Examination Timetabling”. E. Burke, W. Erben, eds. The Practice and Theory of Automated Timetabling III: Selected Papers (PATAT 2000). Lecture Notes in Computer Science 2079. Springer-Verlag, Berlin Heidelberg, New York, 2001. 118-131. [3] E. K. Burke, Y. Bykov, J. P. Newall, S. Petrovic “A Time-Predefined Local Search Approach to Exam Timetabling Problems”, Computer Science Technical Report No. NOTTCS-TR-2001-6, University of Nottingham, 2001. [4] M. P. Carrasco, M. V. Pato, “A Multiobjective Genetic Algorithm for the Class/Teacher Timetabling Problem”, E. Burke, W. Erben, eds. The Practice and Theory of Automated Timetabling III: Selected Papers (PATAT 2000). Lecture Notes in Computer Science 2079. Springer-Verlag, Berlin Heidelberg, New York, 2001. 3-17. [5] M. Tanaka, S. Adachi, “Request-based timetabling by Genetic Algorithm with Tabu Search”, The Third International Workshop on Frontiers in Evolutionary Algorithms, 999-1002, 2000.

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