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Genetic AlgorithmsGenetic Algorithms

AnAn algorithmalgorithmis a set of instructions thatis a set of instructions thatis repeated to solve a problem.is repeated to solve a problem.

AA genetic algorithmgenetic algorithmconceptually followsconceptually followssteps inspired by the biologicalsteps inspired by the biologicalprocesses of evolution.processes of evolution.

Genetic Algorithms follow the idea ofGenetic Algorithms follow the idea of

SURVIVAL OF THE FITTESTSURVIVAL OF THE FITTEST- Better- Betterand better solutions evolve fromand better solutions evolve fromprevious generations until a near optimalprevious generations until a near optimalsolution is obtained.solution is obtained.

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Also known asAlso known as evolutionary algorithmsevolutionary algorithms,,genetic algorithms demonstrate selfgenetic algorithms demonstrate selforganization and adaptation similar to theorganization and adaptation similar to the

way that the ttest biological organismway that the ttest biological organismsurvive and reproduce.survive and reproduce. A genetic algorithm is an iterative procedureA genetic algorithm is an iterative procedure

that represents its candidate solutions asthat represents its candidate solutions asstrings of genes called chromosomes.strings of genes called chromosomes.

Generally applied to spaces which aretoo large

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Genetic Algorithms are often used toGenetic Algorithms are often used toimprove the performance of other Aimprove the performance of other A

methods such as e!pert systems ormethods such as e!pert systems orneural networks.neural networks.

"he method learns by producing"he method learns by producingo#spring that are better and better aso#spring that are better and better as

measured by a tness function, which ismeasured by a tness function, which isa measure of the ob$ective to bea measure of the ob$ective to be

obtained %ma!imum or minimum&.obtained %ma!imum or minimum&.

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Simple GASimple GA initiali!e pop"lation#

e$al"ate pop"lation#

while Termination%riteria&otSatis'ed

select parents (or reprod"ction#

per(orm crosso$er and m"tation#

repair)*#

e$al"ate pop"lation# +

+

Every loop called

generation

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'oncepts'oncepts

Population(set of individuals eachrepresenting a possible solution to a givenproblem.

Gene(a solution to problem represented as

a set of parameters ,these parametersknown as genes.

Chromosome(genes $oined together toform a string of values called chromosome.

Fitness score(value)(every chromosomehas tness score can be inferred from thechromosome itself by using tness function.

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Recombination

)tochastic operators)tochastic operators

SelectionSelectionreplicates the most successfulreplicates the most successfulsolutions found in a population at a ratesolutions found in a population at a rate

proportional to their relative *ualityproportional to their relative *uality RecombinationRecombination)%rosso$er*)%rosso$er*

decomposes two distinct solutions anddecomposes two distinct solutions andthen randomly mi!es their parts to formthen randomly mi!es their parts to form

novel solutionsnovel solutions MutationMutationrandomly perturbs a candidaterandomly perturbs a candidate

solutionsolution

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)imulated +volution)imulated +volution

e need the followinge need the following epresentation of an individualepresentation of an individual

itness unctionitness unction eproduction /ethodeproduction /ethod

)election 'riteria)election 'riteria

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epresenting an ndividualepresenting an ndividual

An individual is data structureAn individual is data structure

representing the 0geneticrepresenting the 0genetic

structure1 of a possible solution.structure1 of a possible solution. Genetic structure consists of anGenetic structure consists of an

alphabet %usually 2,3&alphabet %usually 2,3&

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Binary +ncodingBinary +ncoding

/ost 'ommon 4 string of bits, 2 or 3./ost 'ommon 4 string of bits, 2 or 3.

'hrom( A 5 3 2 33 2 2 3 2 3 3'hrom( A 5 3 2 33 2 2 3 2 3 3

'hrom( B 5 3 3 3 3 3 3 2 2 2 2'hrom( B 5 3 3 3 3 3 3 2 2 2 2

Gives you many possibilitiesGives you many possibilities +!ample 6roblem( 7napsack problem+!ample 6roblem( 7napsack problem "he problem( there are things with given value"he problem( there are things with given value

and size. "he knapsack has given capacity.and size. "he knapsack has given capacity.)elect things to ma!imize the values.)elect things to ma!imize the values.

+ncoding( +ach bit says, if the corresponding+ncoding( +ach bit says, if the correspondingthing is in the knapsackthing is in the knapsack

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6ermutation +ncoding6ermutation +ncoding

8sed in 0ordering problems18sed in 0ordering problems1 +very chromosome is a string of numbers,+very chromosome is a string of numbers,

which represents number is a se*uence.which represents number is a se*uence.'hrom A( 3 9 : ; < = > ? @'hrom A( 3 9 : ; < = > ? @'hrom B( @ 9 > > ; : 3 = ?'hrom B( @ 9 > > ; : 3 = ?

+!ample( "ravelling salesman problem+!ample( "ravelling salesman problem

"he problem( cities that must be visited."he problem( cities that must be visited. +ncoding says order of cities in which+ncoding says order of cities in which

salesman willl visit.salesman willl visit.

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Another +!ampleAnother +!ample

"o nd optimal *uantity of three"o nd optimal *uantity of three

ma$or ingredients %sugar, milk,ma$or ingredients %sugar, milk,

sesame oil& denoting ounces.sesame oil& denoting ounces. 8se an alphabet of 3-? denoting8se an alphabet of 3-? denoting

ounces..ounces..

)olutions might be 3-3-3, ;-3-=, :-:-3.)olutions might be 3-3-3, ;-3-=, :-:-3.

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alue +ncodingalue +ncoding

8sed for complicated values %real numbers&8sed for complicated values %real numbers&and when binary coding would be dicultand when binary coding would be dicult

+ach chromosome is a string of some values.+ach chromosome is a string of some values.'hrom A( 3.;:;: 9.:;=: 2.=99

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ule base systemule base system

Given a rule %if color5red and size5smallGiven a rule %if color5red and size5smalland shape5round then ob$ect5apple.and shape5round then ob$ect5apple.

Assume that each feature has nite set ofAssume that each feature has nite set ofvalues %e.g., size 5 small,large&values %e.g., size 5 small,large&

epresent the value as a substring of lengthepresent the value as a substring of lengthe*ul to the number of possible values. ore*ul to the number of possible values. ore!ample, small 5 32, large 5 23.e!ample, small 5 32, large 5 23.

"he entire rule would be 322 32 23 2322 4"he entire rule would be 322 32 23 2322 4set of rules concatenating the valuesset of rules concatenating the valuestogether.together.

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How o,spring areHow o,spring are

prod"ced -prod"ced -Reprod"ctionReprod"ction -eproduction-eproduction- "hrough- "hrough

reprod"ctionreprod"ction, genetic algorithms, genetic algorithms

produce new generations ofproduce new generations ofimproved solutions by selectingimproved solutions by selectingparents with higher tness ratingsparents with higher tness ratings

or by giving such parents a greateror by giving such parents a greaterprobability of being contributorsprobability of being contributorsand by using random selectionand by using random selection

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prod"cedprod"ced (Recombination)(Recombination)'rossover'rossover- /any- /any

genetic algorithms use strings of binarygenetic algorithms use strings of binary

symbols for chromosomes, as in oursymbols for chromosomes, as in our7napsack e!ample, to represent solutions.7napsack e!ample, to represent solutions.%rosso$er%rosso$ermeans choosing a randommeans choosing a randomposition in the string %say, after ; digits&position in the string %say, after ; digits&

and e!changing the segments either toand e!changing the segments either tothe right or to the left of this point withthe right or to the left of this point withanother string partitioned similarly toanother string partitioned similarly toproduce two new o# spring.produce two new o# spring.

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%rosso$er E.ample%rosso$er E.ample

6arent A 2332336arent A 233233

6arent B 3233226arent B 323322

00/ate the parents by splitting each/ate the parents by splitting eachnumber as shown between the secondnumber as shown between the secondand third digits %position is randomlyand third digits %position is randomly

selected&selected&

23C323323C3233 32C332232C3322

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'rossover +!ample'rossover +!ample

Dow combine the rst digits of A with the lastDow combine the rst digits of A with the lastdigits of B, and the rst digits of B with the lastdigits of B, and the rst digits of B with the lastdigits of Adigits of A

"his gives you two new o#spring"his gives you two new o#spring 233322233322 323233323233

f these new solutions, or o#spring, are betterf these new solutions, or o#spring, are better

solutions than the parent solutions, the system willsolutions than the parent solutions, the system willkeep these as more optimal solutions and they willkeep these as more optimal solutions and they willbecome parents. "his is repeated until somebecome parents. "his is repeated until somecondition %for e!ample number of populations orcondition %for e!ample number of populations orimprovement of the best solution& is satised.improvement of the best solution& is satised.

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prod"cedprod"ced /utation/utation-- /"tation/"tationis an arbitraryis an arbitrary

change in a situation. )ometimes itchange in a situation. )ometimes it

is used to prevent the algorithmis used to prevent the algorithmfrom getting stuck. "he procedurefrom getting stuck. "he procedurechanges a 3 to a 2 to a 3 instead ofchanges a 3 to a 2 to a 3 instead of

duplicating them. "his changeduplicating them. "his changeoccurs with a very low probabilityoccurs with a very low probability%say 3 in 3222&%say 3 in 3222&

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1 0 1 0 1 1 1

1 1 0 0 0 1 1

Parent 1

Parent 2

1 0 1 0 0 1 1

1 1 0 0 1 1 0

Child 1

Child 2 Mutation

Genetic AlgorithmGenetic AlgorithmOperatorsOperators

/"tation and %rosso$er/"tation and %rosso$er

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'rossover Eperators'rossover Eperators )inglepoint crossover()inglepoint crossover( 6arent A(6arent A( 0 1 1 0 12 0 0 0 1 00 1 1 0 12 0 0 0 1 0 6arent B( 2 3 2 3 3 F3 2 3 3 26arent B( 2 3 2 3 3 F3 2 3 3 2

'hild AB('hild AB( 0 1 1 0 10 1 1 0 13 2 3 3 23 2 3 3 2 'hild BA( 2 3 2 3 3'hild BA( 2 3 2 3 3 0 0 0 1 00 0 0 1 0

"wopoint crossover("wopoint crossover( 6arent A(6arent A( 0 1 120 1 0 02 0 1 00 1 120 1 0 02 0 1 0

6arent B( 2 3 26arent B( 2 3 2CC3 3 3 23 3 3 2CC3 3 23 3 2

'hild AB('hild AB( 0 1 1 00 1 1 03 3 23 3 2 0 1 00 1 0 'hild BA( 2 3 2 3'hild BA( 2 3 2 3 1 0 01 0 03 3 23 3 2

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8niform 'rossover and8niform 'rossover and- A random mask is generated- A random mask is generated

- "he mask determines which bits are copied from- "he mask determines which bits are copied fromone parent and which from the other parentone parent and which from the other parent

- Bit density in mask determines how much material- Bit density in mask determines how much materialis taken from the other parent.is taken from the other parent.

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+!amples+!amples

/utation(/utation(

"he recipe e!ample("he recipe e!ample(

3-;-: may be changed to 3-:-: or :-;-3-;-: may be changed to 3-:-: or :-;-:, giving two new o#spring. ow:, giving two new o#spring. owoftenH ow many digits changeH owoftenH ow many digits changeH ow

bigH %parameters to ad$ust&bigH %parameters to ad$ust&

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/ore e!amples(/ore e!amples(

'rossover'rossover

ecipe (ecipe (

6arents 3-:-: I :-;-:. 'rossover6arents 3-:-: I :-;-:. 'rossoverpoint after the rst digit. Generatepoint after the rst digit. Generatetwo o#spring( :-:-: and 3-;-:.two o#spring( :-:-: and 3-;-:.

'an have one or two point crossover.'an have one or two point crossover.

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'rossover 4 6ermutation'rossover 4 6ermutation

+ncoding+ncodingSingle point crosso$erSingle point crosso$er- one crossover point is selected,- one crossover point is selected,

till this point the permutation is copied from the rsttill this point the permutation is copied from the rstparent, then the second parent is scanned and if theparent, then the second parent is scanned and if thenumber is not yet in the o#spring it is addednumber is not yet in the o#spring it is added

%%0 3 4 5 620 3 4 5 62< > @ ?& J %< > @ ?& J %5 6 4 7 825 6 4 7 82? > ; 3& 5 %? > ; 3& 5 %0 3 4 5 60 3 4 5 6< @ ? >&< @ ? >&

/"tation/"tation

Order changingOrder changing- two numbers are selected and- two numbers are selected ande!changede!changed

%3%3 33: = 9 & 5K %3? >& 5K %3 88: = 9 &>&

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'rossover 4 alue'rossover 4 alue

+ncoding+ncoding%rosso$er%rosso$er

All crossovers fromAll crossovers from 9inary encoding9inary encodingcancan

be usedbe used

/"tation/"tationAddingAddinga small number %for real value encoding&a small number %for real value encoding&

- to selected values is added %or subtracted& a- to selected values is added %or subtracted& asmall numbersmall number

%%3.;? 9.

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)election 'riteria)election 'riteria

itness proportionate selection, rankitness proportionate selection, rankselection methods.selection methods. itness proportionate 4 each individual,itness proportionate 4 each individual, ,,hashas

thethe probabilityprobabilitytness%&MsumNoverNallNindividualN$ itness%$tness%&MsumNoverNallNindividualN$ itness%$&,&,wherewhere itness%&itness%&is the tness function value foris the tness function value for

individualindividual ..

ank selection 4 sorts individual by tness andank selection 4 sorts individual by tness andthe probability that an individual will bethe probability that an individual will beselected is proportional to its rank in this sortedselected is proportional to its rank in this sorted

list.list.

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itness unctionitness unction

epresents a rank of the 0representation1epresents a rank of the 0representation1 t is usually a real number.t is usually a real number.

"he function usually has a value between 2"he function usually has a value between 2and 3 and is monotonically increasing.and 3 and is monotonically increasing. Ene can have a sub$ective $udgment %e.g.Ene can have a sub$ective $udgment %e.g.

3-9 for recipe ;-3-=.&3-9 for recipe ;-3-=.&

)imilarly the length of the route in the)imilarly the length of the route in thetraveling salesperson problem is a goodtraveling salesperson problem is a goodmeasure, because the shorter the route,measure, because the shorter the route,the better the solution.the better the solution.

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O"tline o( the =&ew pop"lation> 'reate a new'reate a newpopulation by repeating followingpopulation by repeating followingsteps until the new population issteps until the new population iscompletecomplete

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O"tline o( the =/"tation>ith a mutation probability mutateith a mutation probability mutate

new o#spring at each locus %position innew o#spring at each locus %position inchromosome&.chromosome&.

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O"tline o( the

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Flow @iagram o( theFlow @iagram o( theGenetic AlgorithmGenetic Algorithm

rocessrocessDescribe

Problem

Generate

Initial

Solutions

Test: is initial

solution good enough?Stop

Select parentsto reproduce

ppl! crosso"er process

and create a set o# o##spring

ppl! random mutation

Step 1

Step 2

Step $

Step %

Step &

'es

(o

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'omponents of a GA'omponents of a GA

A problem denition as input, andA problem denition as input, and

+ncoding principles %gene, chromosome&+ncoding principles %gene, chromosome&

nitialization procedure %creation&nitialization procedure %creation& )election of parents %reproduction&)election of parents %reproduction&

Genetic operators %mutation, recombination&Genetic operators %mutation, recombination&

+valuation function %environment&+valuation function %environment&

"ermination condition"ermination condition

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epresentation %encoding&epresentation %encoding&

6ossible individualOs encoding6ossible individualOs encoding Bit strings %2323 ...Bit strings %2323 ...

3322&3322&

eal numbers %=:.; -::.3 ... 2.2eal numbers %=:.; -::.3 ... 2.2@?.;&@?.;&

6ermutations of element %+33 +: +> ... +3 +39&6ermutations of element %+33 +: +> ... +3 +39&

Pists of rules %3 ; : ... ;; ;:&Pists of rules %3 ; : ... ;; ;:&

6rogram elements %genetic6rogram elements %genetic

programming&programming&

... any data structure ...... any data structure ...

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epresentation %cont&epresentation %cont&

When choosing an encoding method rely on thefollowing key ideas

8se a data structure as close as possible to8se a data structure as close as possible tothe natural representationthe natural representation

rite appropriate genetic operators as neededrite appropriate genetic operators as needed f possible, ensure that all genotypesf possible, ensure that all genotypes

correspond to feasible solutionscorrespond to feasible solutions f possible, ensure that genetic operatorsf possible, ensure that genetic operators

preserve feasibilitypreserve feasibility

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nitializationnitialization

Start with a population of randomly

generated individuals, or use

- A previously saved population

- A set of solutions provided by

a human expert

- A set of solutions provided by

another heuristic algorithm

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)election)election

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itness 6roportionate )electionitness 6roportionate )election

Qerived by olland as the optimal trade-Qerived by olland as the optimal trade-

o# between e!ploration and e!ploitationo# between e!ploration and e!ploitation

QrawbacksQrawbacks Qi#erent selection forQi#erent selection for ff11(x)(x) andand ff22(x) = f(x) = f11(x) + c(x) + c

SuperindividualsSuperindividualscause convergence %thatcause convergence %that

may be premature&may be premature&

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Pinear anking )electionPinear anking )election

Based on sorting of individuals by decreasing fitness

The probability to be extracted for the ithindividual

in the ranking is defined as

21,1

1)1(2

1)(

=

n

i

nip

wherecan be interpreted as

the expected sampling rate ofthe best individual

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"ournament )election"ournament )election

Tournament Selection:

randomly select two individuals and the one

with the highest rank goes on and reproduces

cares only about the one with the higher rank,not the spread between the two fitness scores

puts an upper and lower bound on the chances

that any individual to reproduce for the next

generation equal to )2s 2r + 1*/ s2

! s is the si"e of the population

! r is the rank of the #winning# individual

can be generali"ed to select best of n individuals

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ecombination %'rossover&ecombination %'rossover&

!nables the evolutionary process

to move toward promising

regions of the search space "atches good parents# sub-solutions

to construct better offspring

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/utation/utation

"rposeB"rposeBto simulate the e#ect of errors thatto simulate the e#ect of errors thathappen with low probability during duplicationhappen with low probability during duplication

Res"ltsBRes"ltsB

- /ovement in the search space- /ovement in the search space

- estoration of lost information to the- estoration of lost information to the

populationpopulation

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+valuation %tness function&+valuation %tness function&

)olution is only as good as the)olution is only as good as the

evaluation functionR choosing a goodevaluation functionR choosing a good

one is often the hardest partone is often the hardest part )imilar-encoded solutions should have a)imilar-encoded solutions should have a

similar tnesssimilar tness

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"ermination condition"ermination condition

+!amples(+!amples(

A pre-determined number of generations or timeA pre-determined number of generations or time

has elapsedhas elapsed A satisfactory solution has been achievedA satisfactory solution has been achieved

Do improvement in solution *uality has takenDo improvement in solution *uality has taken

place for a pre-determined number ofplace for a pre-determined number of

generationsgenerations

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+!ample(+!ample(the /ASED+ problemthe /ASED+ problem

Suppose we want to maximi$e the number of

ones in a string of lbinary digits

%s it a trivial problem&

%t may seem so because we know the answer in

advance

'owever, we can think of it as maximi$ing the

number of correct answers, each encoded by (,

to l yes)no difficult *uestions+

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+!ample %cont&+!ample %cont&

An individual is encoded %naturally& as aAn individual is encoded %naturally& as astring ofstring of llbinary digitsbinary digits

"he tness"he tness ffof a candidate solution toof a candidate solution tothe /ASED+ problem is the number ofthe /ASED+ problem is the number ofones in its genetic codeones in its genetic code

e start with a population ofe start with a population of nnrandomrandom

strings. )uppose thatstrings. )uppose that ll5 32 and5 32 and nn5

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+!ample %initialization&+!ample %initialization&

We toss a fair coin times and get the

following initial population.

s1$ 1111%1%1%1 f (s1) $ &

s2$ %111%%%1%1 f (s2) $ '

s$ 111%11%1%1 f (s) $ &

s$ %1%%%1%%11 f (s) $

s'$ 111%1111%1 f (s') $ *

s+$ %1%%11%%%% f (s+) $

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+!ample %selection3&+!ample %selection3&

/ext we apply fitness proportionate selection with the

roulette wheel method.

21n

rea is

-roportional

to fitness

value

%ndividual iwill have a

probability to be chosen

i

if

if

)(

)(

4

We repeat the extraction

as many times as the

number of individuals we

need to have the sameparent population si$e

0 in our case1

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+!ample %selection;&+!ample %selection;&

Suppose that, after performing selection, we get

the following population.

s1. $ 1111%1%1%1 (s1)

s2. $ 111%11%1%1 (s)

s. $ 111%1111%1 (s')

s

. $ %111%%%1%1 (s2

)

s'. $ %1%%%1%%11 (s)

s+. $ 111%1111%1 (s')

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+!ample %crossover3&+!ample %crossover3&

/ext we mate strings for crossover2 3or each

couple we decide according to crossover

probability 0for instance 21 whether to actually

perform crossover or notSuppose that we decide to actually perform

crossover only for couples 0s(+, s4+1 and 0s5+, s+12

3or each couple, we randomly extract a

crossover point, for instance 4 for the first and 5

for the second

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+!ample %crossover;&+!ample %crossover;&

s1. $ 1111%1%1%1

s2. $ 111%11%1%1

s'. $ %1%%%1%%11

s+. $ 111%1111%1

Before crossover.

After crossover.

s1.. $ 111%11%1%1

s2.. $ 1111%1%1%1

s'.. $ %1%%%111%1

s+.. $ 111%11%%11

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+!ample %mutation3&+!ample %mutation3&

The final step is to apply random mutation. for each bit that

we are to copy to the new population we allow a smallprobability of error 0for instance 2(1

Before applying mutation.

s1.. $ 111%11%1%1

s2.. $ 1111%1%1%1

s.. $ 111%1111%1

s.. $ %111%%%1%1

s'.. $ %1%%%111%1

s+.. $ 111%11%%11

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+!ample %mutation;&+!ample %mutation;&

After applying mutation.

s1... $ 111%1%%1%1 f(s1... ) $ +

s2... $ 111111%1%% f(s2... ) $ &

s... $ 111%1%1111 f(s... ) $ *

s... $ %111%%%1%1 f(s... ) $ '

s'... $ %1%%%111%1 f(s'... ) $ '

s+... $ 111%11%%%1 f(s+... ) $ +

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+!ample %end&+!ample %end&

%n one generation, the total population fitness

changed from 67 to 68, thus improved by 9:;

At this point, we go through the same process

all over again, until a stopping criterion is met

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E.ample BE.ample B S"ppose a Genetic Algorithm "ses chromosomes o( the (ormS"ppose a Genetic Algorithm "ses chromosomes o( the (orm

.Ca9cde(gh with a '.ed length o( eight genes : Each gene.Ca9cde(gh with a '.ed length o( eight genes : Each genecan 9e any digit 9etween 1 and ? : Let the 'tness o(can 9e any digit 9etween 1 and ? : Let the 'tness o(indi$id"al . 9e calc"lated as Bindi$id"al . 9e calc"lated as B

().* C)aD9*-)cDd*D)eD(*- ) gDh*().* C)aD9*-)cDd*D)eD(*- ) gDh*And let the initial pop"lation consist o( (o"r indi$id"als .0 :::And let the initial pop"lation consist o( (o"r indi$id"als .0 :::

.5 with the (ollowing chromosomes B.5 with the (ollowing chromosomes B0 C 7 6 5 0 4 6 4 30 C 7 6 5 0 4 6 4 3F).0* C)7D6*-)5D0*D)4D6*-)4D3* C ?F).0* C)7D6*-)5D0*D)4D6*-)4D3* C ?3 C 8 ; 0 3 7 7 1 03 C 8 ; 0 3 7 7 1 0F).3* C )8D;*-)0D3*D)7D7*-)1D0* C 34F).3* C )8D;*-)0D3*D)7D7*-)1D0* C 34

4 C 3 4 ? 3 0 3 8 64 C 3 4 ? 3 0 3 8 6F).4* C )3D4*-)?D3*D)0D3*-)8D6* C -07F).4* C )3D4*-)?D3*D)0D3*-)8D6* C -075C 5 0 8 6 3 1 ? 55C 5 0 8 6 3 1 ? 5F).5* C )5D0*-)8D6*D)3D1*-)?D5* C -0?F).5* C )5D0*-)8D6*D)3D1*-)?D5* C -0?The arrangement is ) ass"me ma.imi!ation *The arrangement is ) ass"me ma.imi!ation *33 .0.0 .4.4 .5.5) the 'ttest indi$id"al *) the 'ttest indi$id"al * ) least 't indi$id"al *) least 't indi$id"al *

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6ut the calculations in6ut the calculations in

table for simplicitytable for simplicityIndi$id"alIndi$id"al

ss

StringStringRepresentatioRepresentatio

nn

FitnesFitnesss

Arrangement Ass"meArrangement Ass"mema.imi!ationma.imi!ation

S3S3

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3 C 83 C 8 ;; 00 33 77 77 11 000 C 70 C 7 66 55 00 44 66 44 33

O,spring 0 C 8O,spring 0 C 8 ;; 00 33 44 66 44 33

O,spring 3 C 7O,spring 3 C 7 66 55 00 77 77 11 00

0 C 70 C 7 66 55 00 44 66 44 334 C 34 C 3 44 ?? 33 00 33 88 66

O,spring 4 C 7O,spring 4 C 7 66 ?? 33 00 33 44 33O,spring 5 C 3O,spring 5 C 3 44 55 00 44 66 88 66

Middlecrossover

crossover crossover

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S; 5 @S; 5 @ >> 33 ;;

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E#spring 3 5 @E#spring 3 5 @ >> 33 ;; :: 99 :: ;;

F )O,spring 0* C)8D;*-)0D3*D)4D6*-)4D3* C 06F )O,spring 0* C)8D;*-)0D3*D)4D6*-)4D3* C 06

E#spring ; 5

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Indi$id"alsIndi$id"als String RepresentationString Representation FitnessFitness

E#spring 3E#spring 3 @>3;:9:;@>3;:9:; 3939

E#springE#spring;;

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+!ample( "he 7napsack+!ample( "he 7napsack

6roblem6roblemTou are going on an overnight hike andTou are going on an overnight hike and

have a number of items that you could takehave a number of items that you could takealong. +ach item has a weight %in pounds&along. +ach item has a weight %in pounds&

and a benet or value to you on the hike%forand a benet or value to you on the hike%formeasurements sake letOs say, in 8) dollars&,measurements sake letOs say, in 8) dollars&,and you can take one of each item at most.and you can take one of each item at most."here is a capacity limit on the weight you"here is a capacity limit on the weight youcan carry %constraint&. "his problem onlycan carry %constraint&. "his problem onlyillustrates one constraint, but in realityillustrates one constraint, but in realitythere could be many constraints includingthere could be many constraints includingvolume, time, etc.volume, time, etc.

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Item:Item: 1 2 $ % & + ,1 2 $ % & + ,

-ene#it:-ene#it: & . $ 2 , / %& . $ 2 , / %

eight:eight: , . %, . % 10 % + %10 % + %

napsac holds a ma3imum o# 22 poundsnapsac holds a ma3imum o# 22 pounds 4ill it to get the ma3imum bene#it4ill it to get the ma3imum bene#it

Solutions tae the #orm o# a string o# 15s and 05sSolutions tae the #orm o# a string o# 15s and 05s

Solutions: lso no6n as strings o# genes called ChromosomesSolutions: lso no6n as strings o# genes called Chromosomes

3. 23232323. 2323232 ;. 3322322;. 3322322 :. 2322333:. 2322333

GA E.ampleB The napsacJGA E.ampleB The napsacJro9lemro9lem

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7napsack +!ample7napsack +!ample

"ypically, a string of 3s and 2s can"ypically, a string of 3s and 2s canrepresent a solution.represent a solution.

6ossible solutions generated by the6ossible solutions generated by thesystem usingsystem using -eproduction,-eproduction,'rossover'rossover, or, or /utations/utations 3. 23232323. 2323232 ;. 3322322;. 3322322 :. 2322333:. 2322333

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)olution 3)olution 3

Benet @ J ; J ? 5 3?Benet @ J ; J ? 5 3? eight @ J 32 J < 5 ;=eight @ J 32 J < 5 ;=

temtem 33 ;; :: == 99 >

Sol"tioSol"tionn 11 00 11 00 11 00 11

BenetBenet 99 @@ :: ;; >> ?? ==

eighteight >> @@ == 3232 ==

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)olution ;)olution ;

Benet 9 J @ J > 5 ;2Benet 9 J @ J > 5 ;2

eight > J @ J = 5 3?eight > J @ J = 5 3?

temtem 33 ;; :: == 99 >

Sol"tioSol"tio

nn

00 00 11 11 00 11 11

BenetBenet 99 @@ :: ;; >> ?? ==

eighteight >> @@ == 3232 ==

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)olution :)olution :

Benet @ J > J ? J = 5 ;@Benet @ J > J ? J = 5 ;@

eight @ J = J < J = 5 ;;eight @ J = J < J = 5 ;;

temtem 33 ;; :: == 99 >

Sol"tioSol"tio

nn

11 00 11 11 00 00 00

BenetBenet 99 @@ :: ;; >> ?? ==

eighteight >> @@ == 3232 ==

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)olution : is clearly the best solution)olution : is clearly the best solutionand has met our conditions, therefore,and has met our conditions, therefore,item number ;, 9, will be takenitem number ;, 9, will be takenon the hiking trip. e will be able to geton the hiking trip. e will be able to getthe most benet out of these itemsthe most benet out of these itemswhile still having weight e*ual to ;;while still having weight e*ual to ;;

pounds.pounds."his is a simple e!ample illustrating a"his is a simple e!ample illustrating a

genetic algorithm approach.genetic algorithm approach.

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7napsack 6roblem7napsack 6roblem"o understand GA must work with the following

problem((Knap Sack Problem) "hief wants to steal gold store.

"hief has a bag%the bag can hold a specicweight&. +very piece of gold has a specic weight and price. "hief wants to steal gold with high price but the

weight must e*ual or less than the weight that bagcan hold it.

f we gave every gold piece a specic number3,;,:,U,n%suggest n5@ in this e!ample&.

1 2 3 4 5 6 7 8

weight 5 3 10 6 5 5 4 4

price 100 30 250 150 50 75 50 50

0-Encoding )representation*

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0-Encoding )representation*)genechromosome*

%hromosome co"ld 9eB Bit strings %323323232322&. eal numbers %=:.3,=9.;,

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3-Initiali!e pop"lation

Implementers speci(y pop"lation si!e : To initiali!e pop"lation create

chromosomes randomly and store them in

list o( length the pop"lation si!e: In o"r pro9lem lets taJe pop"lation si!e 7

chromosomes: e can initiali!e pop"lation a (ollowingB

1 "10100001"

2 "01110011"

3 "11110000"

4 "00110011"

5 "01010101"

6 "00001111"

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4-E$al"ation o( pop"lation:

Giving every chromosome in population

'tness $al"eby using 'tness ("nction. Fitness ("nctionswill di#er according to the

problem and encoding techni*ue. Fitness ("nctionreturns a single

numerical%'tness& which reVe! the "tilityorthe a9ilityof the individual which thatchromosome represents.

itness function can calculate ( strength,weight, width, ma!imum load,cost,construction

time or combination of all these. itness value well be stored with chromosome.

chromosome /itness function /itness value

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-

n our e!ample we will make tness function as

1 "10100001" 225

2 "01110011" 305

3 "11110000" 225

4 "00110011" 225

5 "01010101" 450

6 "00001111" 530

Chromosome #itness "alue

5-Selection o( new

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5 Selection o( newparents)reproduction*

ndividuals are selected from population randomly orby using any selection method to improve thepopulation itself.

Good individuals will probably be selected severaltimes in a generation ,poor ones may not be at all.

Methods of selectionRandom ,est, !ournament, Roulette "heel,

!runcation, Rank, #$ponential, olt%man,Stead& state, 'nteractive and binar&tournament selection

n our e!ample we will use Tr"ncationselectionwith parameter :.

search best : chromosomes and then select

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