Approximate Solution to an Exam Timetabling Problem

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

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

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

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

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

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 • > 35,000 students • > 90 buildings • Number of exams???

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

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

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

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)

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

12. Phase II: SA s0, t0 > 0,  loop until loop until select s in N(s0) ramdomly  = cost(s) - cost(s0) if  < 0 then s0 = s else x = rand(0,1) if x < e(-  /t) then s0 = s end if until count = max t = (t) until stop criteria

13. SA … • s0 - initial solution • t0 - 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. 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

15. Cost Function • Time distance (Fc) • ti, tj sessions for exami and examj • Summed distance between each pair of exams • Distance significant iff 1|ti-tj| 5 • Then penalty = 26 - |ti-tj| - shorter distance higher penalty • Same day (Fl) • Summed sessions that are adjacent and on same day • If |ti-tj| = 1 then pen = 1 • Both cases penalty = 0 otherwise • Minimize Fc + Fl

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 T0 --> Tf •  = 1 - (ln(T0 ) - ln(Tf ))/N • Where N is the desired number of SA steps

17. Parameter Initialization • Nrep - steps where T constant • Tested 1, 4, 6, 10, 50, 5000 • T0 - starting temp • Find T0 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 • Tf - final temp • Tested 0.5, 0.05, 0.005, 0.0005, 0.00005

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

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

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

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

22. Questions