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

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

3. 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. 4 8 students: 6 exams: –Maths, English, Music, Painting, French, Physics 4 time slots: –Monday, Wednesday, Friday, Next Monday An simple example

5. 5 Maths: English: Painting: Music: French: Physics: time slots: Monday Wednesday Friday Next Monday

6. 6 An sample example How many computations does it need for a full-tree-search? 4 6 =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. 7 An simple example English: Painting: Music: French: time slots: Monday Wednesday Friday Next Monday Maths: Physics: EPMF:

8. 8 Why these exams can be combined? MathsEnglishPainting MusicFrenchPhysics 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. 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. 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. 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. 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. 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 n m. If two of the exams are combined, the size of the search space becomes n m-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. 14 Outlines Introduction of exam timetablingIntroduction of exam timetabling –Basic knowledge –A simple example –What is our new approach –Literature review Approach of combining examsApproach of combining exams –Conditions for combining exams –Compatibility measure Application (St. Andrews’83)Application (St. Andrews’83)

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

16. 16 The second condition is too strict MathsEnglishPainting MusicFrenchPhysics 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 ― Clash matrix (Clashes between exams are expressed by positive numbers). 2222

17. 17 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 Compatibility measure

18. 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 C AB =1 C CD =0 C EF =0.8 C GH =0.4 C IJ =0.2

19. 19 Compatibility matrix MathsEnglishPainting MusicFrenchPhysics 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. 20 Compatibility measure 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.

21. 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. 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. 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 13 139 to 13 65. Heuristic ordering method and meta- heuristic are applied.

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

25. 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. 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. 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