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An Approach to Accelerate Heuristic for Exam Timetabling

An Approach to Accelerate Heuristic for Exam Timetabling. Jiawei Li (Michael). Outlines. Introduction of exam timetabling Basic knowledge A simple example What this approach is Literature review Approach of combining exams Conditions for combining exams Compatibility measure

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An Approach to Accelerate Heuristic for Exam Timetabling

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  1. An Approach to Accelerate Heuristic for Exam Timetabling Jiawei Li (Michael)

  2. Outlines • Introduction of exam timetabling • Basic knowledge • A simple example • What this approach is • Literature review • Approach of combining exams • Conditions for combining exams • Compatibilitymeasure • Application (St.Andrews’83)

  3. Exam timetabling • Combinatorial optimization problem • NP-complete • Description of the problem: • A set of examinations (and students) • A limited number of time slots • Constraints • Standard constraints for University of Toronto Benchmarks

  4. An simple example • 8 students: • 6 exams: • Maths, English, Music, Painting, French, Physics • 4 time slots: • Monday, Wednesday, Friday, Next Monday

  5. An simple example • Maths: • English: • Painting: • Music: • French: • Physics: time slots: Monday Wednesday Friday Next Monday

  6. An sample example • How many computations does it need for a full-tree-search? 46=4096 possible solutions • How can we obtain the optimal solution in a minute? Possibility No.1: brain as fast as a computer Possibility No.2: make the problem simplified

  7. An simple example • English: • Painting: • Music: • French: • Maths: • Physics: • EPMF: time slots: Monday Wednesday Friday Next Monday

  8. Why these exams can be combined? Maths English Painting Music French Physics Maths ― 2 2 2 2 4 English 2 ― 0 0 0 1 Painting 2 0 ― 0 0 1 Music 2 0 0 ― 0 1 French 2 0 0 0 ― 1 Physics 4 1 1 1 1 ― Clash matrix (Clashes between exams are expressed by positive numbers).

  9. What is this approach? • To combine the exams that satisfy some conditions, which makes sure that the quality of solutions in the reduced search space does not become worse. • Objective of combining exams -To make the problem simplified • Relevant research

  10. Existing approaches for exam timetabling Carter and Laporte (1996,1998) categorized the existing approaches for exam timetabling into four types: • Sequential methods • Cluster methods • Constraint based methods • Meta-heuristic methods

  11. Cluster methods • Appeared in 1970s and has now been rare. • Cluster methods split the set of exams into groups which satisfy hard constraints and then assign the groups to time periods to fulfill the soft constraints. • A main drawback.

  12. Differences between Cluster methods and exam-combining approach • Different conditions • Different objectives -Cluster methods split the set of exams into groups so that every exam is assigned to a group; -Exam-combining approach only combines those exams that satisfy the conditions.

  13. What is the benefit of combining exams? • For the problem of exam timetabling with m exams and n timeslots, the size of the search space is nm. If two of the exams are combined, the size of the search space becomes nm-1. • Suppose that the quality of solutions in the reduced space does not become worse, then a previously used search method may either find feasible solutions in shorter time or reach a better solution. • Combining exams does not interfere with the applying of heuristic or meta-heuristic methods.

  14. Outlines • Introduction of exam timetabling • Basic knowledge • A simple example • What is our new approach • Literature review • Approach of combining exams • Conditions for combining exams • Compatibility measure • Application (St. Andrews’83)

  15. Conditions for combining exams • No clash between combined exams. (hard constraint) • They are equally clashed with other exams. (soft constraint)

  16. The second condition is too strict Maths English Painting Music French Physics Maths ― 2 2 2 2 4 English ― 0 0 0 1 Painting 0 ― 0 0 1 Music 0 0 ― 0 1 French 0 0 0 ― 1 Physics 4 1 1 1 1 ― 2 2 2 2 Clash matrix (Clashes between exams are expressed by positive numbers).

  17. Compatibility measure • Compatibility is defined to measure to what degree two exams are suitable to be combined. • where m denote the number of exams; whether there is clash between exams i and j; and

  18. Compatibility measure A B C D E F G H I J Exam A ― 0 1 0 1 1 0 1 0 1 Exam B 0 ― 1 0 1 1 0 1 0 1 Exam C 1 1 ― 1 1 1 1 0 1 0 Exam D 0 0 1 ― 1 1 1 0 1 0 Exam E 1 1 1 1 ― 0 1 0 1 0 Exam F 1 1 1 1 0 ― 1 0 0 1 Exam G 0 0 1 1 1 1 ― 0 1 0 Exam H 1 1 0 0 0 0 0 ― 1 0 Exam I 0 0 1 1 1 0 1 1 ― 0 Exam J 1 1 0 0 0 1 0 0 0 ― Simplified clash matrix CAB=1 CCD=0 CEF=0.8 CGH=0.4 CIJ=0.2

  19. Compatibility matrix Maths English Painting Music French Physics Maths ― 0 0 0 0 0 English 0 ― 1 1 1 0 Painting 0 1 ― 1 1 0 Music 0 1 1 ― 1 0 French 0 1 1 1 ― 0 Physics 0 0 0 0 0 ― Compatibility matrix for the example

  20. Values of are ranged in [0,1]. denotes perfect compatibility between two exams. Small values of denote unsuitability of combining these exams together. In applying the criteria of compatibility, we can set a value and combine those exams that satisfy . A trade-off. Compatibility measure

  21. Application (St.Andrews’83) • One instance of the University of Toronto benchmarks. • St.Andrews83 (sta83-I) has 139 exams, 611 students, 5751 enrolments, and 13 timeslots. • Soft constraint is to minimize an evaluation function which denotes the cost of timetables that are generated.

  22. Application (St.Andrews’83) Table 1. Compatibility between nine exams for sta83-I benchmark E1 E2 E3 E4 E5 E6 E7 E8 E9 E1 ---- 0.94 0.93 0.91 0.94 0.93 0.94 0.94 0.94 E2 0.94 ---- 0.93 0.91 0.94 0.93 0.94 0.94 0.94 E3 0.93 0.93 ---- 0.92 0.93 0.93 0.93 0.93 0.93 E4 0.91 0.91 0.92 ---- 0.91 0.91 0.91 0.91 0.91 E5 0.94 0.94 0.93 0.91 ---- 0.93 0.94 0.94 0.94 E6 0.93 0.93 0.93 0.91 0.93 ---- 0.93 0.93 0.93 E7 0.94 0.94 0.93 0.91 0.94 0.93 ---- 0.94 0.94 E8 0.94 0.94 0.93 0.91 0.94 0.93 0.94 ---- 0.94 E9 0.94 0.94 0.93 0.91 0.94 0.93 0.94 0.94 ----

  23. Application (St.Andrews’83) • With , a total of 10 groups of 84 exams was combined to form 10 new larger exams, and the number of exams decreased from 139 to 65. • The size of search space changes from 13139 to 1365. • Heuristic ordering method and meta-heuristic are applied.

  24. Application (St.Andrews’83) • Largest Degree (LD), Largest Colour degree (LC), Saturation Degree (SatD),Largest Enrolment (LE) and Random ordering.

  25. Application (St.Andrews’83) • The local search adopts a simple strategy that removes several exams from a solution and reschedules them. If a better solution is found, the local search will restart based on the new solution. Otherwise, several other exams will be tried. • Two-stage structure: it reschedules two exams in the first stage and four exams in the second stage.

  26. Conclusions and discussion • Exam-combining makes the problem of exam timetabling simplified. • This approach can be a supplement to the heuristic and meta-heuristic method. • Why St.Andrews83 only? • Other measures for combining exams.

  27. An Approach to Accelerate Heuristic for Exam Timetabling Jiawei (Michael) Li RA at ASAP group Email: jwl@cs.nott.ac.uk Room: C43 Ext: 6555

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