11

IOE/MFG 543IOE/MFG 543

Chapter 14: General purpose Chapter 14: General purpose procedures for scheduling in procedures for scheduling in practicepracticeSection 14.5: Local search – Section 14.5: Local search – Genetic AlgorithmsGenetic Algorithms

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Genetic AlgorithmsGenetic Algorithms

Keep a Keep a population population (or a (or a generation) of solutionsgeneration) of solutions

Selected members of the Selected members of the population population survivesurvive and and reproduce reproduce offspringoffspring– The offspring inherit characteristics The offspring inherit characteristics

from both parentsfrom both parents– A A mutationmutation can occur such that the can occur such that the

offspring has a feature that neither offspring has a feature that neither parent hasparent has

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Minimization of xMinimization of x2 2 via a via a simple GAsimple GA Gene representation of an Gene representation of an

individualindividual– Use binary encoding, e.g., Use binary encoding, e.g.,

I=1000100111=2I=1000100111=299+2+255+2+222+2+211+2+200=551=551– 10 binary digits => I 10 binary digits => I {0,1,…,1023} {0,1,…,1023} – Suppose we know xSuppose we know x[-2,2][-2,2]

Let x=(I-512)/256 => xLet x=(I-512)/256 => x[-2,1.996][-2,1.996]

Fitness of an individualFitness of an individualF(x)=1/(1+xF(x)=1/(1+x22))

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Simple GA (1)Simple GA (1)Initial populationInitial population Use a population size of 10Use a population size of 10 Randomly generate 10 individualsRandomly generate 10 individuals

Bit10 9 8 7 6 5 4 3 2 1 I x F(x)

1 0 0 1 0 1 0 0 1 1 0 166 -1.35 0.352 0 0 0 1 1 1 0 1 1 1 119 -1.54 0.303 0 0 1 1 1 0 1 0 0 1 233 -1.09 0.464 1 0 0 1 0 1 1 1 1 1 607 0.37 0.885 1 0 1 1 0 0 0 0 1 1 707 0.76 0.636 1 1 0 1 1 1 1 0 0 0 888 1.47 0.327 1 0 1 0 0 0 0 0 1 0 642 0.51 0.798 0 1 0 0 0 0 0 1 1 0 262 -0.98 0.519 0 1 0 0 1 0 1 0 1 0 298 -0.84 0.59

10 1 1 1 0 1 0 1 1 1 1 943 1.68 0.265.09

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Simple GA (2)Simple GA (2)Choosing the parentsChoosing the parents

F(x) P P U Pick

1 0.35 0.07 0.069 0.192 3

2 0.30 0.06 0.128 0.654 7

3 0.46 0.09 0.218 0.264 4

4 0.88 0.17 0.390 0.384 4

5 0.63 0.12 0.514 0.853 9

6 0.32 0.06 0.577 0.957 10

7 0.79 0.16 0.733 0.886 9

8 0.51 0.10 0.833 0.540 6

9 0.59 0.12 0.949 0.043 1

0 0.26 0.05 1.000 0.092 2

5.09

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Simple GA (3)Simple GA (3)CrossoverCrossover

ParenParentt Bits3 0 0 1 1 1 0 1 0 0 1 17 1 0 1 0 0 0 0 0 1 04 1 0 0 1 0 1 1 1 1 1 24 1 0 0 1 0 1 1 1 1 19 0 1 0 0 1 0 1 0 1 0 2

10 1 1 1 0 1 0 1 1 1 19 0 1 0 0 1 0 1 0 1 0 66 1 1 0 1 1 1 1 0 0 01 0 0 1 0 1 0 0 1 1 0 52 0 0 0 1 1 1 0 1 1 1

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Simple GA (4)Simple GA (4)OffspringOffspring

1 1 0 1 1 1 0 1 0 0 12 0 0 1 0 0 0 0 0 1 03 1 0 0 1 0 1 1 1 1 14 1 0 0 1 0 1 1 1 1 15 1 1 0 0 1 0 1 0 1 06 0 1 1 0 1 0 1 1 1 17 1 1 0 1 1 1 1 0 1 08 0 1 0 0 1 0 1 0 0 09 0 0 0 1 1 0 0 1 1 0

10 0 0 1 0 1 1 0 1 1 1

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Simple GA (5)Simple GA (5)MutationMutation Change each bit with probability Change each bit with probability

ppmm=0.05 =0.05 1 1 0 1 1 1 0 1 0 0 12 0 0 1 0 0 0 0 0 1 03 1 0 0 1 0 1 1 1 1 14 1 0 0 0 0 1 1 1 1 15 1 1 0 0 1 0 1 0 1 06 0 1 1 0 1 0 1 1 1 17 1 1 0 1 0 1 1 1 1 08 0 1 0 0 1 0 1 0 0 09 0 0 0 1 1 0 0 1 1 1

10 0 0 1 0 1 1 0 1 1 1

99

Simple GA (6)Simple GA (6)Second generationSecond generation

I x F(x)1 1 0 1 1 1 0 1 0 0 1 745 0.91 0.552 0 0 1 0 0 0 0 0 1 0 130 -1.49 0.313 1 0 0 1 0 1 1 1 1 1 607 0.37 0.884 1 0 0 0 0 1 1 1 1 1 543 0.12 0.995 1 1 0 0 1 0 1 0 1 0 810 1.16 0.426 0 1 1 0 1 0 1 1 1 1 431 -0.32 0.917 1 1 0 1 0 1 1 1 1 0 862 1.37 0.358 0 1 0 0 1 0 1 0 0 0 296 -0.84 0.589 0 0 0 1 1 0 0 1 1 1 103 -1.60 0.28

10 0 0 1 0 1 1 0 1 1 1 183 -1.29 0.385.69

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GA and GA and schedulingscheduling Encoding of a sequenceEncoding of a sequence

– Easier to use job numbers than bitsEasier to use job numbers than bits Mutation can be done by an Mutation can be done by an

interchange of two jobsinterchange of two jobs A crossover can lead to an infeasible A crossover can lead to an infeasible

sequencesequence1-2-3-4-5 gives1-2-3-4-5 gives 1-2-3-2-1 and1-2-3-2-1 and5-4-3-2-1 5-4-3-2-1 5-4-3-4-55-4-3-4-5– repairrepair– random keysrandom keys– other methodsother methods

1111

Random keys Random keys (see Bean, 1994)(see Bean, 1994) Generate n U(0,1) random numbers for Generate n U(0,1) random numbers for

each parenteach parent ExampleExample

– Generate .29, .96, .17, .84 and .49Generate .29, .96, .17, .84 and .49

– PP11=(.29,.96,.17,.84,.49)=(4,1,5,2,3)=(.29,.96,.17,.84,.49)=(4,1,5,2,3)

– PP22=(.73,.14,.43,.54,.38)=(5,1,3,2,4)=(.73,.14,.43,.54,.38)=(5,1,3,2,4)

(.29,.96,.17,.84,.49) gives(.29,.96,.17,.84,.49) gives(.29,.96,.17,.54,.38)(.29,.96,.17,.54,.38)

(.73,.14,.43,.54,.38) (.73,.14,.43,.54,.38) (.73,.14,.43,.84,.49)(.73,.14,.43,.84,.49)

– CC11=(4,1,5,2,4) C=(4,1,5,2,4) C22=(2,5,4,1,3)=(2,5,4,1,3)

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GA example:GA example:1||1||wwjjTTjj

Jobs 1 2 3 4

wj 4 5 3 5

pj 12 8 15 9

dj 16 26 25 27

Use random keys and population size 6Use random keys and population size 6 Fitness = 1/wFitness = 1/wjjTTjj

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GA exampleGA exampleInitial populationInitial population

Random keysRandom keys SequenceSequence wjTj

1 0.206 0.781 0.32 0.864 4 2 3 1 1332 0.646 0.901 0.218 0.749 3 1 4 2 1793 0.682 0.688 0.089 0.893 3 2 4 1 1374 0.26 0.258 0.279 0.835 3 4 2 1 1425 0.653 0.828 0.281 0.24 2 1 3 4 1316 0.661 0.528 0.75 0.311 2 3 1 4 161

Fitness P CC U SelectSelect1 0.008 0.182 0.182 0.195 22 0.006 0.135 0.317 0.350 33 0.007 0.177 0.494 0.719 54 0.007 0.171 0.665 0.309 25 0.008 0.185 0.850 0.808 56 0.006 0.150 1.000 0.360 3

0.041

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GA exampleGA exampleCrossover and Crossover and mutationmutation

Parents Swap at Mutation2 0.646 0.901 0.218 0.749 3 0 0 0 0

3 0.682 0.688 0.089 0.893 0 0 1 15 0.653 0.828 0.281 0.24 4 0 0 0 0

2 0.646 0.901 0.218 0.749 0 0 0 05 0.653 0.828 0.281 0.24 4 0 0 0 0

3 0.682 0.688 0.089 0.893 0 0 0 0

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GA exampleGA exampleGeneration 2Generation 2

Very little improvement and the Very little improvement and the population is converging!?population is converging!?

Instead of mutation do Instead of mutation do immigrationimmigration– Replace an individual with a new randomly Replace an individual with a new randomly

generated individualgenerated individual

Random keysRandom keys SequenceSequence wjTj Fitness1 0.646 0.901 0.218 0.893 3 1 4 2 179 0.0062 0.682 0.688 0.439 0.317 2 1 3 4 131 0.0083 0.653 0.828 0.281 0.24 2 1 3 4 131 0.0084 0.646 0.901 0.218 0.749 3 1 4 2 179 0.0065 0.653 0.828 0.281 0.24 2 1 3 4 131 0.0086 0.682 0.688 0.089 0.893 3 2 4 1 137 0.007

0.041

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GA algorithmGA algorithm

1.1. Generate an initial populationGenerate an initial population2.2. Evaluate the fitness and select Evaluate the fitness and select

individuals for reproductionindividuals for reproduction Randomly, always keep the best few?Randomly, always keep the best few?

3.3. Generate new individualsGenerate new individuals Crossover, mutation, immigrationCrossover, mutation, immigration

4.4. Stop of stopping criteria are Stop of stopping criteria are satisfied. Otherwise go to 2satisfied. Otherwise go to 2

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GA design criteriaGA design criteria

Schedule representationSchedule representation Population sizePopulation size Fitness functionFitness function Selection procedureSelection procedure

– Probability based on fitness or rankingProbability based on fitness or ranking– TournamentTournament

Crossover, mutation and immigration Crossover, mutation and immigration probabilityprobability

Stopping criteriaStopping criteria