1. 1 Approximate Solution to an Exam Timetabling Problem Adam White Dept of Computing Science University of Alberta Adam White Dept of Computing Science University of Alberta

2. 2 Combining Constraint Programming and Simulated Annealing on University ExamTimetableing Tuan-Anh Duong and Kim-Hoa Lam International Conference for frenchspeaking and vietnemesse computer scientisits - 2004 Combining Constraint Programming and Simulated Annealing on University ExamTimetableing Tuan-Anh Duong and Kim-Hoa Lam International Conference for frenchspeaking and vietnemesse computer scientisits - 2004

3. 3 Outline  Introduction  Problem  Motivation  Related Work  Two Phase Algorithm  Constraint Programming  Simulated Annealing  Results  Contribution  Future Work  Introduction  Problem  Motivation  Related Work  Two Phase Algorithm  Constraint Programming  Simulated Annealing  Results  Contribution  Future Work

4. 4 Problem  Shedule a number of exams in a given set of time slots  Students are divided into subgroups  Hard Constraints  No student has 2 exams at one time  No more than 2 exams in one room  Worst possible solution  But still acceptable  Shedule a number of exams in a given set of time slots  Students are divided into subgroups  Hard Constraints  No student has 2 exams at one time  No more than 2 exams in one room  Worst possible solution  But still acceptable

5. 5 Problem…  Soft Constraints  Student’s exams spread  Student subgroups have exams in near rooms  Some exams start after & end before regular  Room utilization is maximized  Some exams should be in special rooms  Improve the quality of the solution  Soft Constraints  Student’s exams spread  Student subgroups have exams in near rooms  Some exams start after & end before regular  Room utilization is maximized  Some exams should be in special rooms  Improve the quality of the solution

6. 6 Motivation  Exam scheduling is NP-Complete  Have a variety of courses and fields  Problem solved:  Chi Minh City University of Technology(China)  100,000 students  320 exams  2.5 week period  More close to home  University of Alberta University of Alberta  > 35,000 students  > 90 buildings  Number of exams???  Exam scheduling is NP-Complete  Have a variety of courses and fields  Problem solved:  Chi Minh City University of Technology(China)  100,000 students  320 exams  2.5 week period  More close to home  University of Alberta University of Alberta  > 35,000 students  > 90 buildings  Number of exams???

7. 7 Solution Methods  Constraint programming  Constraint Logic programming - CHIP  Constraint Logic programming - ECLiPSe  Local repair - Constraint Satisfaction Problem  Tabu Search  2 lists - 1 normal, 1 for most moved exams  Single list, exams stay in for random time  Genetic Algorithms  Automated timetabling  Constraint programming  Constraint Logic programming - CHIP  Constraint Logic programming - ECLiPSe  Local repair - Constraint Satisfaction Problem  Tabu Search  2 lists - 1 normal, 1 for most moved exams  Single list, exams stay in for random time  Genetic Algorithms  Automated timetabling

8. 8 Approach  Two Phase algorithm  Solve hard constraints using Constraint Programming(CP)  Satisfy as much as possible soft constraints using Simulated Annealing(SA)  Optimization problem  Solution from CP is used as an initial solution for SA algorithm  In practice…very important  Solved - assigning sessions to exams  Not Solved - assigning student groups to exams  Two Phase algorithm  Solve hard constraints using Constraint Programming(CP)  Satisfy as much as possible soft constraints using Simulated Annealing(SA)  Optimization problem  Solution from CP is used as an initial solution for SA algorithm  In practice…very important  Solved - assigning sessions to exams  Not Solved - assigning student groups to exams

9. 9 Phase I: Constraint Programming  Backtracking with forward checking(BTFC)  Consistency technique & chronological backtracking  Consistency: Arch-consistency  Pairs of yet initialized variables & instantiated vairables  Value assigned to current variable  Q: Any variable in domain of future variable conflicts?  A: remove from domain  Backtracking with forward checking(BTFC)  Consistency technique & chronological backtracking  Consistency: Arch-consistency  Pairs of yet initialized variables & instantiated vairables  Value assigned to current variable  Q: Any variable in domain of future variable conflicts?  A: remove from domain

10. 10 CP…  Chronological:  Variable ordering heuristic or Value ordering heuristic  Value order: selection of next variable  Dynamic variable ordering  Select smallest number of values in current domain  Variable ordering: order of exams scheduled  Priorety scores for exams  Order based on remaining domain size(# students)  Chronological:  Variable ordering heuristic or Value ordering heuristic  Value order: selection of next variable  Dynamic variable ordering  Select smallest number of values in current domain  Variable ordering: order of exams scheduled  Priorety scores for exams  Order based on remaining domain size(# students)

11. 11 Simulated Annealing  Analogy: metal cools and freezes into a minimum energy crystalline structure  search for a minimum in a system  Analogy: metal cools and freezes into a minimum energy crystalline structure  search for a minimum in a system

12. 12 Phase II: SA s 0, t 0 > 0,  loop until select s in N(s 0 ) ramdomly  = cost(s) - cost(s 0 ) if  < 0 then s 0 = s else x = rand(0,1) if x < e (-  /t) then s 0 = s end if until count = max t =  (t) until stop criteria s 0, t 0 > 0,  loop until select s in N(s 0 ) ramdomly  = cost(s) - cost(s 0 ) if  < 0 then s 0 = s else x = rand(0,1) if x < e (-  /t) then s 0 = s end if until count = max t =  (t) until stop criteria

13. 13 SA …  s 0 - initial solution  t 0 - initial temperature   - temperature reduction function  N(s) - neighborhood of s  cost(s) - cost of a solution  What we are minimizing…related to soft constraints  N, cost and   Problem specific  s 0 - initial solution  t 0 - initial temperature   - temperature reduction function  N(s) - neighborhood of s  cost(s) - cost of a solution  What we are minimizing…related to soft constraints  N, cost and   Problem specific

14. 14 Neighborhood  Variant of Kempe Chains  Exam i allocated to session t and t’ and t’ != t  G - set of exams allocated to t  G’ - set of exams allocated to t’  Find unique minimum pair of G’s  F  G & F’  G’ s.t. I  F & (G\F)  F & (G’\F’)  F  Are conflict free  The timetable where we can find F and F’ s.t.  Reallocate all exams in F’ to session t  Reallocate all exams in F to session t’  Is a neighbor of current solution  Variant of Kempe Chains  Exam i allocated to session t and t’ and t’ != t  G - set of exams allocated to t  G’ - set of exams allocated to t’  Find unique minimum pair of G’s  F  G & F’  G’ s.t. I  F & (G\F)  F & (G’\F’)  F  Are conflict free  The timetable where we can find F and F’ s.t.  Reallocate all exams in F’ to session t  Reallocate all exams in F to session t’  Is a neighbor of current solution

15. 15 Cost Function  Time distance (F c )  t i, t j sessions for exam i and exam j  Summed distance between each pair of exams  Distance significant iff 1  |t i -t j |  5  Then penalty = 2 6 - |ti-tj| - shorter distance higher penalty  Same day (F l )  Summed sessions that are adjacent and on same day  If |t i -t j | = 1 then pen = 1  Both cases penalty = 0 otherwise  Minimize F c + F l  Time distance (F c )  t i, t j sessions for exam i and exam j  Summed distance between each pair of exams  Distance significant iff 1  |t i -t j |  5  Then penalty = 2 6 - |ti-tj| - shorter distance higher penalty  Same day (F l )  Summed sessions that are adjacent and on same day  If |t i -t j | = 1 then pen = 1  Both cases penalty = 0 otherwise  Minimize F c + F l

16. 16 Cooling Scheme  Lowering T  Size of T determines if higher cost neighbor solution is accepted x < e (-  /t)  Climb out of local minima  Geometric cooling  Every nrep steps T is multiplyed by    set s.t. takes N steps from T 0 --> T f   = 1 - (ln(T 0 ) - ln(T f ))/N  Where N is the desired number of SA steps  Lowering T  Size of T determines if higher cost neighbor solution is accepted x < e (-  /t)  Climb out of local minima  Geometric cooling  Every nrep steps T is multiplyed by    set s.t. takes N steps from T 0 --> T f   = 1 - (ln(T 0 ) - ln(T f ))/N  Where N is the desired number of SA steps

17. 17 Parameter Initialization  Nrep - steps where T constant  Tested 1, 4, 6, 10, 50, 5000  T 0 - starting temp  Find T 0 s.t. starting probability of neighbor solution acceptance is [70%,80%]  Algorithm test all n*(n-1)/2 neighbors - n exams  Dynamic - run everytime scheduler is  T f - final temp  Tested 0.5, 0.05, 0.005, 0.0005, 0.00005  Nrep - steps where T constant  Tested 1, 4, 6, 10, 50, 5000  T 0 - starting temp  Find T 0 s.t. starting probability of neighbor solution acceptance is [70%,80%]  Algorithm test all n*(n-1)/2 neighbors - n exams  Dynamic - run everytime scheduler is  T f - final temp  Tested 0.5, 0.05, 0.005, 0.0005, 0.00005

18. 18 Results  Microsoft Visual C++ 6.0  PII 450 MHz PC  On … HCMC University of Technology  30 exams sessions  324 exams  64,000 students  N - tested from 500 - 70,000  Compared against? NOTHING  Microsoft Visual C++ 6.0  PII 450 MHz PC  On … HCMC University of Technology  30 exams sessions  324 exams  64,000 students  N - tested from 500 - 70,000  Compared against? NOTHING

19. 19 Results… COST 6900000. 6200000 6100000 6000000 5900000 SA steps 70000. 30000 20000 10000 0

20. 20 Contribution  Impelemented on a real data set  Illustrated the importance of proper determination of SA parameters  Developed methods to select  Timetable cost decreases as N increases  Runtime, N = 70,000  SA - 50min CP - 2min  Acceptable - run once a term  Impelemented on a real data set  Illustrated the importance of proper determination of SA parameters  Developed methods to select  Timetable cost decreases as N increases  Runtime, N = 70,000  SA - 50min CP - 2min  Acceptable - run once a term

21. 21 Future Work  Implement in smodels  Representation??  Approximate soln to soft constraints??  Aquire SA code  Empirical results for varying sizes  Compare solution methods  Accuracy  Speed  If time allows compare with dlv  Implement in smodels  Representation??  Approximate soln to soft constraints??  Aquire SA code  Empirical results for varying sizes  Compare solution methods  Accuracy  Speed  If time allows compare with dlv

22. 22 Questions