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Approximate Solution to an Exam Timetabling Problem

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

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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

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OutlineOutline

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

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ProblemProblem

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

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Problem…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

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MotivationMotivation

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???

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???

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Solution MethodsSolution 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

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ApproachApproach

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

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Phase I: Constraint ProgrammingPhase 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

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CP…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)

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Simulated AnnealingSimulated 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

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Phase II: SAPhase II: SAs0, t0 > 0, loop until

loop until

select s in N(s0) ramdomly

= cost(s) - cost(s0)

if < 0 then s0 = selse

x = rand(0,1)

if x < e(- /t) then s0 = send if

until count = maxt = (t)

until stop criteria

s0, t0 > 0, loop until

loop until

select s in N(s0) ramdomly

= cost(s) - cost(s0)

if < 0 then s0 = selse

x = rand(0,1)

if x < e(- /t) then s0 = send if

until count = maxt = (t)

until stop criteria

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SA …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

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

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NeighborhoodNeighborhood

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 FG & 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 FG & 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

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Cost FunctionCost 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

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

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Cooling SchemeCooling 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

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

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Parameter InitializationParameter 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

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

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ResultsResults

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

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Results…Results…COST

6900000

.

.

6200000

6100000

6000000

5900000

SA steps

70000

.

.

30000

20000

10000

0

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ContributionContribution

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

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Future WorkFuture 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

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QuestionsQuestions