Download - Zehra KAMIŞLI ÖZTÜRK Anadolu University, TURKEY Müjgan SAĞIR Eskisehir Osmangazi University, TURKEY
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Zehra KAMIŞLI ÖZTÜRKAnadolu University, TURKEY
Müjgan SAĞIREskisehir Osmangazi University, TURKEY
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GOAL
Designing a flexible, computer
based and user interactive
system for the solution of the
Educational Timetabling
Problems (ETP).
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A general ETP includes;
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OutlineDifficulties on the solution of ETP.
the need for heuristics
Hyper heuristics on deciding the best
heuristic to solve the problem.
Small01 (a test problem from the literature)
Mathematical model
(course-room-time slot assignment)
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OutlineDimensional analysis
Investigating appropriate
heuristics from the literature
Evolutionary algorithms (GA)
A new genetic algorihmConstructing web interfaces
Problem solution
Comparison and conclusion
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DifficultiesNP-hard structureVaried natureConflicting objectivesSize
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SOLUTION METHODS
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Mathematical programmingMathematical programming HeuristicsHeuristics
Meta heuristicsMeta heuristics Hybrid meta heuristicsHybrid meta heuristics
Case Based ReasoningCase Based Reasoning Hyper heuristics …Hyper heuristics …
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CASE: Small01*
course-room-time slot assignment
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Small01
# of courses 100
# of rooms 5
# of features 5
# of students 80
Total time period(9 timeperiod/day* 5 days)
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*http://iridia.ulb.ac.be/supp/IridiaSupp2002-001/index.html
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Building the Mathematical Model
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HARD CONSTRAINTS
no student attends more than one event at the same time
the room is big enough for all the
attending students and satisfies all
the features required by the event only one event is in each room at any
timeslot
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no student has a event in the last slot of the day
no student has more than two different events consecutively
no student is allowed to have only one event on a day
SOFT CONSTRAINTS
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Objectives
To minimize soft constraint violations
Solution quality
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SMALL01 Parameters
Student Event Matrix (SE)
Room Feature Matrix (RF)
Event Feature Matrix (EF)
Room capasities
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Student Event MatrixS/E 1 2 3 4 5 6 7 8 9 10 … 88 89 90 98 99 100
1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0
2 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0
3 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0
4 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0
5 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0
6 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0
…
54 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0
55 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0
80 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0
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Room Feature Matrix
R/F 1 2 3 4 5
1 1 1 1 1 1
2 1 0 1 1 1
3 1 0 0 1 0
4 0 0 0 0 1
5 0 1 0 1 0
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Event Feature Matrix E/F 1 2 3 4 5
1 0 1 0 1 0
2 1 1 1 1 1
3 1 0 1 1 1
4 0 0 0 0 0
5 1 0 0 1 1
6 0 0 0 0 0
7 0 0 0 0 1
8 0 1 0 1 0
9 1 0 0 1 0
10 0 0 1 1 0
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Mathematical ModelDecision variables
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Mathematical Model (cont.)
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Mathematical Model (cont.)
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Dimension Analysis
Const. index # of total constraints
1 j,k,l j × k × l
2 j,k j × k
3 j,k j × k
4 j j
5 k,t k × t
6 j,k j × k
7 i, t i × t
8 j, t j × t
9 j, t j × t
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Dimension Analysis (cont.)
Goal no Index # of total goals
1 - 1
2 i,dsj × i
3 i i × 5
7×(3!) ×
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7×(3!) ×
variable index # of total variables
j,k,t j × k × t
j,k j × k
j,t j × t
- 2
i,j
× i × 2
i i × 5× 2
Dimension Analysis (cont.)
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3jk + 2jt + jkl +j + kt + it + 5i + 42i# of total constraints
# of total variables
jkt + jk + jt + 10i + 84i +2
for Small01
total constraints: 420525
total variables: 834702
Dimension Analysis (cont.)
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HYPERHEURISTICS
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Investigating appropriate heuristics from the literature
Burke et.al. (2003)Han and Kendall (2003)
Burke and Nevall (2004) …
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HYPERHEURISTICS
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performance of LLH
Hyper Heuristic
Heuristic selection
Low Level Heuristics
Problem
Solution quality
variability in the solution
Investigating appropriate heuristics from the literature
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Investigating appropriate heuristics from the literature
Year Study Authors
1994A Genetic Algorithm based University Timetabling System
Burke EK , Elliman DG and Weare RF
1992A genetic algorithm, to solve the timetable problem.
Colorni, A., Dorigo, M. and Maniezzo, V.
2002A genetic algorithm for a university weekly courses timetabling problem
Yu, E. and Sung K.S.
2001A Constructive Evolutionary Approach to School Timetabling
Filho, G.R. and Lorena, L.A.N.
2002School Tımetable Generating Using Genetic Algorithm
Voráč, J, I. Vondrák, and K. Vlček
1994 Fast Practical Evolutionary TimetablingCorne, D. Ross, P. and Fang, .L.
1995A Genetic Algorithm Solving a Weekly Course-Timetabling Problem
Erben, W. and Keppler, J.
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Search Techniqes
Calculus
Base Techniques
Guided random search techniqes
Enumerative Techniques
BFSDFS Dynamic Programming
Tabu Search Hill Climbing
Simulated Anealing
Evolutionary Algorithms
Genetic Algorithms
Fibonacci Sort
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Building the Genetic Algorithm
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Basic Operators in GA’s
population
parents
offsprings
selectionselection
mutationmutation
crossovercrossover
selectionselection
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Basic StepsDefinition of encoding principles
(gene, chromosome)Definition initialization procedure
(creation)Selection of parents
(reproduction)Genetic operators
(mutation, recombination)Evaluation function
(environment)Termination condition
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MATRIX Representation
Mon1 Mon2 … Fri5 Fri6
Place 1Event 1-
1Event 2-2 … … …
… … … … … …
Place i … Event i-2 … Event i-5 …
Lab 1 … … … … …
… … … … … …
Lab j … … … … Event j-6
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Abramson, 1991
MATRIX Representation
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PERMUTATION Representation
Course no: 1 2 3 4 … 50
Time period: 5 3 20 45 … 14
Course no: 1 2 3 4 … 50
classroom: 1 2 5 4 … 6
Course no: 1 1 2 2 3 3 4 4 . . . 49 49 50 50
5 1 3 2 20 5 45 4 10 3 14 6
timeperiod classroom
Chromosome length :50
Chromosome length :50
Chromosome length :100
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Restrictions for different representations
1.Matrix representation
needs some special genetic operators
(PMX, imitation etc.)
can not handle all resources.
does not guarantee feasible solution.
2. Binary and permutation representation
needs some special genetic operators
takes too much space
can not handle all resources.
does not guarantee feasible solution.
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New cromosomes
Course: 1 2 3 4 5
Time perio
d1 2 3 4 5
Course: 1 2 3 4 5
Time perio
d5 3 4 2 1
Course: 1 2 3 4 5
Time period 5 3 3 4 5
Course: 1 2 3 4 5
Time period 1 2 4 2 2
Restrictions for different representations
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Create initial population
14689587…7708929858643513
EE-1…4321
small01.tim
İnclude parameters # of events, rooms, features, students and capasities
Calculate total num.of students for each event
Construct correlated events matrix
Decode cromosome as constructing feasible solutions and evaluate them.
Reproduction, crossover, Mutation and Elitist operators
1 2 3 4 E-1 E
3513586492987708 95871468
1 2 1 1 1 33 3 5 2 54 3
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AR 3 2 2 5 1 2
R 3 3 5 4 1 3
(35*3)/100 +1=2
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Solution 1
Derslik 2 Derslik 3
S2 G1G2G3G4 G5
G
1
G2G3G2G3G4 G5
Z1 44 31 Z1 35 Z2 18 10 Z2 Z3 Z3 14 Z4 2332 Z4 34Z5 Z5 1 Z6 Z6 Z7 Z7 Z8 Z8 9Z9 Z9
Derslik 4 Derlsik 5S4 G1 G2 G3 G4 G5 S5 G1 G2 G3 G4 G5
Z1 9 14 Z1 25 19 Z2 3 14 2 Z2 12 20 Z3 13 39 Z3 25 17 27 Z4 39 Z4 24 3 14 35 Z5 25 Z5 18 6Z6 42 36 Z6 34 21
Z7 21 16 Z7 10 7 18 6 36
Z8 41 2 Z8 15 14 Z9 13 25 Z9 29 21 31
Room 1
R1 D1 D2 D3 D4 D5
T1 43 26
T2 41 14 43
T3 33 8
T4 4 28 1 13
T5 39 8 39 18 2
T6 0 27 6 7
T7 12 31 2 19
T8 42 27 4
T9 23 25 21
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Solution 2
Trial no:solution(fitness
function)
1 149
2 154
3 127
4 90
5 147
6 125
7 116
8 92
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Case: GA based HHHLH: GA
LLHs:
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Ongoing studies …
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Ongoing studies …
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Conclusion Feasible solutions without
hard constraint violations
A general solution
methodolgy by HHs
Hybrid methodolgies for future work…
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Future work
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