Zehra KAMIŞLI ÖZTÜRKAnadolu University, TURKEY

Müjgan SAĞIREskisehir Osmangazi University, TURKEY

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;

3

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)

4

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

6

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 …

CASE: Small01*

course-room-time slot assignment

8

  Small01

# of courses 100

# of rooms 5

# of features 5

# of students 80

Total time period(9 timeperiod/day* 5 days)

45

*http://iridia.ulb.ac.be/supp/IridiaSupp2002-001/index.html

Building the Mathematical Model

9

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) …

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

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:

41

Ongoing studies …

42

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

45