evolutionary algorithms ga and de

7/25/2019 Evolutionary Algorithms GA and De

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EVOLUTIONARY ALGORITHMSEVOLUTIONARY ALGORITHMSEVOLUTIONARY ALGORITHMSEVOLUTIONARY ALGORITHMS

Our knowledge of the universe is always incomplete.Our knowledge of the universe is always incomplete.Our knowledge of the universe is always incomplete.Our knowledge of the universe is always incomplete.

New knowledge can, should and does alter current ideasNew knowledge can, should and does alter current ideasNew knowledge can, should and does alter current ideasNew knowledge can, should and does alter current ideas

and motivates new thinking. . . . . .and motivates new thinking. . . . . .and motivates new thinking. . . . . .and motivates new thinking. . . . . .

Dr. S. FarookDr. S. FarookDr. S. FarookDr. S. Farook

Associate Professor,Associate Professor,Associate Professor,Associate Professor,

Sree idyanikethan !ngineering "ollege,Sree idyanikethan !ngineering "ollege,Sree idyanikethan !ngineering "ollege,Sree idyanikethan !ngineering "ollege,

A. #angampetaA. #angampetaA. #angampetaA. #angampeta

Optimi$ationOptimi$ationOptimi$ationOptimi$ation isisisis anananan importantimportantimportantimportant tooltooltooltool inininin makingmakingmakingmaking decisionsdecisionsdecisionsdecisions andandandand inininin

analy$inganaly$inganaly$inganaly$ing physicalphysicalphysicalphysical systemssystemssystemssystems....

%n%n%n%n mathematicalmathematicalmathematicalmathematical terms,terms,terms,terms, ananananoptimi$ationoptimi$ationoptimi$ationoptimi$ation pro&lempro&lempro&lempro&lem isisisis thethethethe pro&lempro&lempro&lempro&lem

ofofofof findingfindingfindingfinding thethethethe &est&est&est&est solutionsolutionsolutionsolution fromfromfromfrom amongamongamongamong thethethethe set setsetset of ofofof allallallall feasi&lefeasi&lefeasi&lefeasi&le

solutionssolutionssolutionssolutions....

!ngineering!ngineering!ngineering!ngineering design,design,design,design, FinancialFinancialFinancialFinancial marketing,marketing,marketing,marketing, constructionconstructionconstructionconstruction andandandand

maintenancemaintenancemaintenancemaintenance ofofofof engineeringengineeringengineeringengineering systems,systems,systems,systems, travelling,travelling,travelling,travelling, inininin decisiondecisiondecisiondecision

makingmakingmakingmaking &oth&oth&oth&oth atatatat thethethethe managerialmanagerialmanagerialmanagerial andandandand thethethethe technologicaltechnologicaltechnologicaltechnological levellevellevellevel andandandand

manymanymanymany moremoremoremore....

'oals'oals'oals'oals ofofofof suchsuchsuchsuch decisionsdecisionsdecisionsdecisions ((((

)a*imi$ation)a*imi$ation)a*imi$ation)a*imi$ation

)inimi$ation)inimi$ation)inimi$ation)inimi$ation

Optimi$ationOptimi$ationOptimi$ationOptimi$ation

Dr. S.Dr. S.Dr. S.Dr. S. FarookFarookFarookFarook

!!! Dept.!!! Dept.!!! Dept.!!! Dept.

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FromFromFromFrom thethethethe viewviewviewview ofofofof optimi$ation,optimi$ation,optimi$ation,optimi$ation, thethethethe variousvariousvariousvarious techni+uestechni+uestechni+uestechni+ues includingincludingincludingincluding

traditionaltraditionaltraditionaltraditional andandandand modernmodernmodernmodern optimi$ationoptimi$ationoptimi$ationoptimi$ation methods,methods,methods,methods, whichwhichwhichwhich havehavehavehave &een&een&een&een

developeddevelopeddevelopeddeveloped totototo solvesolvesolvesolve thethethethe pro&lems,pro&lems,pro&lems,pro&lems, areareareare classifiedclassifiedclassifiedclassified intointointointo severalseveralseveralseveral

groupsgroupsgroupsgroups asasasas((((

"onventional"onventional"onventional"onventional optimi$ationoptimi$ationoptimi$ationoptimi$ation methodsmethodsmethodsmethods

inearinearinearinear programmingprogrammingprogrammingprogramming -P-P-P-P

/uadratic/uadratic/uadratic/uadratic programmingprogrammingprogrammingprogramming -/P-/P-/P-/P

'enerali$ed'enerali$ed'enerali$ed'enerali$ed gradientgradientgradientgradient methodmethodmethodmethod

NewtonNewtonNewtonNewton methodmethodmethodmethod

"lassification"lassification"lassification"lassification

0hese methods are suita&le for continuous and differentia&le0hese methods are suita&le for continuous and differentia&le0hese methods are suita&le for continuous and differentia&le0hese methods are suita&le for continuous and differentia&le

functions only, hence also known asfunctions only, hence also known asfunctions only, hence also known asfunctions only, hence also known as 1ard methods1ard methods1ard methods1ard methods



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2ased on physical phenomenon2ased on physical phenomenon2ased on physical phenomenon2ased on physical phenomenon

Simulated annealing algorithm

2ased on swarm &ehavior2ased on swarm &ehavior2ased on swarm &ehavior2ased on swarm &ehavior

Particle swarm algorithm

Firefly algorithm

2at algorithm

2ased on colony2ased on colony2ased on colony2ased on colony

Ant colony algorithm

Artificial 2ee colony algorithm

2ased on artificial intelligence2ased on artificial intelligence2ased on artificial intelligence2ased on artificial intelligence Neural network

Fu$$y logic

2ased on evolutionary strategy2ased on evolutionary strategy2ased on evolutionary strategy2ased on evolutionary strategy

'enetic algorithm'enetic algorithm'enetic algorithm'enetic algorithm

Differential evolution algorithmDifferential evolution algorithmDifferential evolution algorithmDifferential evolution algorithm

)odern Optimi$ation methods)odern Optimi$ation methods)odern Optimi$ation methods)odern Optimi$ation methods

0he0he0he0he conceptualconceptualconceptualconceptual ideaideaideaidea &ehind&ehind&ehind&ehind anyanyanyany algorithmsalgorithmsalgorithmsalgorithms waswaswaswas anananan attemptattemptattemptattempt totototo

mimicmimicmimicmimic somesomesomesome ofofofof thethethethe processesprocessesprocessesprocesses takingtakingtakingtaking placeplaceplaceplace inininin naturenaturenaturenature....



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!volution!volution!volution!volution



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0he &asic principle of evolutionary 'As has evolved from the

principlesprinciplesprinciplesprinciples ofofofof lifelifelifelifewhich were first proposed &yDarwinDarwinDarwinDarwin ----3456345634563456((((

!volutionary 'enetic Algorithm -'A!volutionary 'enetic Algorithm -'A!volutionary 'enetic Algorithm -'A!volutionary 'enetic Algorithm -'A

7Owing to this strugglestrugglestrugglestruggle forforforfor lifelifelifelife, variations, however slight and

from whatever cause proceeding, if they &e in any degree

profita&leprofita&leprofita&leprofita&le to the individuals of a species, in their infinitely

comple* relations to other organic &eings and to their physical

conditions of life, will tend to the preservation of such

individuals, and will generally &e inherited &y the offspring. 0he

offspring, also, will thus have a &etter chance ofsurvivingsurvivingsurvivingsurviving, for, of

the many individuals of any species which are periodically &orn,

&ut a small num&er can survive. % have called this principle, &ywhich each slight variation, if useful, ispreserved,preserved,preserved,preserved, &y the term

NaturalNaturalNaturalNatural SelectionSelectionSelectionSelection....8888

0he &asic principles of 'As were first laid down rigorously &y 1olland1olland1olland1olland

-Adaptation in natural and artificial systems( 3695


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!volutionary!volutionary!volutionary!volutionary algorithmsalgorithmsalgorithmsalgorithms areareareare stochasticstochasticstochasticstochastic :::: Pro&a&ilityPro&a&ilityPro&a&ilityPro&a&ility &ased&ased&ased&ased searchsearchsearchsearch

methodsmethodsmethodsmethods thatthatthatthat mimicmimicmimicmimic thethethethe naturalnaturalnaturalnatural processprocessprocessprocess ofofofof evolutionevolutionevolutionevolution....

!volutionary!volutionary!volutionary!volutionary algorithmsalgorithmsalgorithmsalgorithms operateoperateoperateoperate onononon aaaa populationpopulationpopulationpopulation ofofofof potentialpotentialpotentialpotential

solutionssolutionssolutionssolutions applyingapplyingapplyingapplying thethethethe principleprincipleprincipleprinciple ofofofofsurvivalsurvivalsurvivalsurvival ofofofof thethethethe fittestfittestfittestfittesttotototo produceproduceproduceproduce

&etter&etter&etter&etter andandandand &etter&etter&etter&etter appro*imationsappro*imationsappro*imationsappro*imations totototo aaaa solutionsolutionsolutionsolution....

AtAtAtAt eacheacheacheach generation,generation,generation,generation, aaaa newnewnewnew setsetsetset ofofofof appro*imationsappro*imationsappro*imationsappro*imations isisisis createdcreatedcreatedcreated &y&y&y&y thethethethe

processprocessprocessprocess ofofofofselectingselectingselectingselecting individualsindividualsindividualsindividuals accordingaccordingaccordingaccording totototo theirtheirtheirtheir levellevellevellevel ofofofoffitnessfitnessfitnessfitness

andandandand &reeding&reeding&reeding&reeding themthemthemthem togethertogethertogethertogether usingusingusingusing operatorsoperatorsoperatorsoperators &orrowed&orrowed&orrowed&orrowed fromfromfromfrom

naturalnaturalnaturalnatural geneticsgeneticsgeneticsgenetics....

0his0his0his0his processprocessprocessprocess leadsleadsleadsleads totototo thethethethe evolutionevolutionevolutionevolution ofofofof populationspopulationspopulationspopulations ofofofof individualsindividualsindividualsindividuals

thatthatthatthat areareareare &etter&etter&etter&etter suitedsuitedsuitedsuited totototo theirtheirtheirtheir environmentenvironmentenvironmentenvironment thanthanthanthan thethethethe individualsindividualsindividualsindividuals

thatthatthatthat theytheytheythey werewerewerewere createdcreatedcreatedcreated from,from,from,from, ;ust;ust;ust;ust asasasas ininininnaturalnaturalnaturalnatural adaptationadaptationadaptationadaptation....

!volutionary algorithms!volutionary algorithms!volutionary algorithms!volutionary algorithms



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Natural adaptationNatural adaptationNatural adaptationNatural adaptation



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'enetic manipulation'enetic manipulation'enetic manipulation'enetic manipulation

ProsProsProsPros "ons"ons"ons"ons



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Survival of the fittestSurvival of the fittestSurvival of the fittestSurvival of the fittest



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)odeling)odeling)odeling)odeling is the process of identifying and e*pressing in

mathematical terms the o&;ectiveo&;ectiveo&;ectiveo&;ective, the varia&lesvaria&lesvaria&lesvaria&les, and the

constraintsconstraintsconstraintsconstraints of the pro&lem.

Ano&;ectiveo&;ectiveo&;ectiveo&;ective is a +uantitative measure of the performance of the

system that we want to minimi$e or ma*imi$e.

0hevaria&lesvaria&lesvaria&lesvaria&lesor theunknownsare the components of the system

for which we want to find values.

0heconstraintsconstraintsconstraintsconstraints are the functions that descri&e the relationships

among the varia&les and that define the allowa&le values for the

varia&les.

"onstructing Optimi$ation )odel"onstructing Optimi$ation )odel"onstructing Optimi$ation )odel"onstructing Optimi$ation )odel



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)odelling of optimi$ation pro&lem and !ssential)odelling of optimi$ation pro&lem and !ssential)odelling of optimi$ation pro&lem and !ssential)odelling of optimi$ation pro&lem and !ssential

components of 'enetic Algorithmcomponents of 'enetic Algorithmcomponents of 'enetic Algorithmcomponents of 'enetic Algorithm

'A


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PopulationPopulationPopulationPopulation

A population is a collection of individualsindividualsindividualsindividuals. A population

consists of a num&er of individuals &eing tested

0he two important aspects of population in 'A are(

0he initial population.

0he population si$e.



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Fitness functionFitness functionFitness functionFitness function

A fitnessfitnessfitnessfitness function or the o&;ectiveo&;ectiveo&;ectiveo&;ective function or

PerformancePerformancePerformancePerformance inde*inde*inde*inde*, satisfying all the constraints must &e

formulated for the pro&lem to &e solved.

0he fitness function returns a single numerical value

7fitnessfitnessfitnessfitness8 or 7figurefigurefigurefigure ofofofof meritmeritmeritmerit8 which represents the a&ility :

+uality of each individual chromosome to minimi$e or

ma*imi$e the optimi$ation pro&lem.

0he fitness not only indicates howgoodgoodgoodgood thethethethe solutionsolutionsolutionsolution is,

&ut also corresponds to howhowhowhow closeclosecloseclose the chromosome is to

the optimal one.



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=hen the optimi$ation pro&lem involves more than one

o&;ective function, then the task of finding one or more

optimal solutions is known as multi>o&;ective optimi$ation.

A solution that is e*treme -&etter with respect to one

o&;ective function may &e worse with respect to the other

o&;ective function

A single o&;ective function, there e*ists only one single glo&al

optimal solution

Single o&;ective functionSingle o&;ective functionSingle o&;ective functionSingle o&;ective function

)ulti)ulti)ulti)ulti>>>>o&;ective optimi$ationo&;ective optimi$ationo&;ective optimi$ationo&;ective optimi$ation



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Some applications in power systemsSome applications in power systemsSome applications in power systemsSome applications in power systems

Selecting the &est location for FA"0S devices

)inimi$ation of transmission losses

)inimi$ation of A"! in F"

%mprovement of voltage profile

Optimal power flow

Optimal scheduling of generators

Designing of controller gains : parameters to meet the

specified criteria and many



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0he total real power losses and the voltage deviations from the

specified limits at various &uses is

0he two o&;ective functions is transformed into an aggregated

single o&;ective function for minimi$ation as(

)inimi$ation of 0ransmission losses)inimi$ation of 0ransmission losses)inimi$ation of 0ransmission losses)inimi$ation of 0ransmission losses



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%ntegral of time multiplied a&solute value of the !rror

-%0A!

=here e-t? error considered

Area "ontrol !rror@A!".

0he fitness function to &e minimi$ed is given &y(

%mproving F" #egulations%mproving F" #egulations%mproving F" #egulations%mproving F" #egulations



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#eproduction : 2reeding#eproduction : 2reeding#eproduction : 2reeding#eproduction : 2reeding

0he two important aspects of genetic algorithm are

intensificationintensificationintensificationintensification and diversificationdiversificationdiversificationdiversification.

%t must &e a&le to generate a diversediversediversediverse rangerangerangerange of solutions

including the potentiallypotentiallypotentiallypotentially optimaloptimaloptimaloptimal solutionssolutionssolutionssolutions so as to e*plore

the whole search space effectively

%t intensifies its search around the neigh&orhood of an

optimal or nearlynearlynearlynearly optimaloptimaloptimaloptimal solutionsolutionsolutionsolution

DiversificationDiversificationDiversificationDiversification

%ntensification%ntensification%ntensification%ntensification



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SelectionSelectionSelectionSelection

During the reproduction phase of the 'enetic algorithm,

individuals are selectedselectedselectedselected from the population and recom&ined

to produce new offspring


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0here are many methods how to select the &est chromosomes,

#oulette wheel selection

#ank selection

0ournament selection

0runcated selection and many others.

Selection methodsSelection methodsSelection methodsSelection methods



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Parents are selected according

to their fitness. 0he &etter the

chromosomes are, the more

chances to &e selected they

have. %magine arouletterouletterouletteroulette wheelwheelwheelwheel

where are placed all

chromosomes in the

population, every has its place

accordingly to its fitness

function, like on the following

picture.

#oulette wheel selection#oulette wheel selection#oulette wheel selection#oulette wheel selection



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#ank Selection#ank Selection#ank Selection#ank Selection

#ank selection first ranks the population and then every

chromosome receives fitness from this ranking. 0he worst will

have fitness3333, second worstCCCCetc. and the &est will have fitness

NNNN-num&er of chromosomes in population.



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%n tournament selection, n individuals are selected

randomly from the larger population, and the selected

individuals compete against each other.

0he individual with the highest fitness wins and will &e

included as one of the ne*t generation population.

0ournament selection0ournament selection0ournament selection0ournament selection

0ournament selection also gives a chance to all

individuals to &e selected and thus it preserves diversity

of the algorithm



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0runcation selection0runcation selection0runcation selection0runcation selection

%n truncation selection individuals are sorted according to their

fitness.OnlyOnlyOnlyOnly thethethethe &est&est&est&est individualsindividualsindividualsindividuals areareareare selectedselectedselectedselected forforforfor parentsparentsparentsparents....

0he parameter for truncation selection is the truncation

threshold0runc0runc0runc0runc.

0runc0runc0runc0runc indicates the proportion of the population to &e

selected as parents and takes values ranging from 5E>3E.

%ndividuals%ndividuals%ndividuals%ndividuals &elow&elow&elow&elow thethethethe truncationtruncationtruncationtruncation thresholdthresholdthresholdthreshold dodododo notnotnotnot produceproduceproduceproduce

offspringoffspringoffspringoffspring.



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"rossover"rossover"rossover"rossover

"rossover operator is an important operator of 'A as it

increases the diversitydiversitydiversitydiversity of the population and evolvesevolvesevolvesevolves newnewnewnew

solutionsolutionsolutionsolutionwhich may potentially optimi$e the pro&lem.

%n crossover generally two chromosomes called parentsparentsparentsparents are

selected among the population with preference towards the

fitness value and forms new chromosomes called offspringoffspringoffspringoffspring

-"hildren.

0he offspring shares the &est features of the parent

chromosomes and evolvesevolvesevolvesevolves asasasas dominantdominantdominantdominant solutionsolutionsolutionsolution in the

population, eventually leading to convergence to an overall

good solution.



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"rossover pro&a&ility says how often will &e crossover

performed.

%f the crossover pro&a&ility is low then there will &e

few offspring in the population

%f the crossover pro&a&ility is high then there will &e

large num&er of offspring which promisingly

converges to optimal solution.

"rossover pro&a&ility"rossover pro&a&ility"rossover pro&a&ility"rossover pro&a&ility

0ypically the crossover pro&a&ility of....9999 totototo ....6666 is selected

for optimum crossover operation.



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inear recom&inationinear recom&inationinear recom&inationinear recom&ination

%ntermediate recom&ination is a method only applica&le to real

varia&les -and not &inary varia&les. 1ere the varia&le values of

the offspring are chosen somewhere around and &etween the

varia&le values of the parents.



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2inary valued recom&ination -crossover2inary valued recom&ination -crossover2inary valued recom&ination -crossover2inary valued recom&ination -crossover

During the recom&ination of &inary varia&les onlypartspartspartsparts ofofofof

thethethethe individualsindividualsindividualsindividuals areareareare e*changede*changede*changede*changed &etween&etween&etween&etween thethethethe individualsindividualsindividualsindividuals.

Depending on the num&er of parts, the individuals are

divided &efore the e*change of varia&les -the num&er of

crosscrosscrosscross pointspointspointspoints. 0he num&er of cross points distinguish the

methods.

)ulti)ulti)ulti)ulti>>>>point crossoverpoint crossoverpoint crossoverpoint crossover

SingleSingleSingleSingle>>>>point crossoverpoint crossoverpoint crossoverpoint crossover



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2inary valued recom&ination -crossover2inary valued recom&ination -crossover2inary valued recom&ination -crossover2inary valued recom&ination -crossover



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)utation)utation)utation)utation

0he mutation operator according to

the mutationmutationmutationmutation pro&a&ilitypro&a&ilitypro&a&ilitypro&a&ility introduces

random change into the characteristic

of chromosome there&y reintroduces

geneticgeneticgeneticgenetic diversitydiversitydiversitydiversity into the population

and thus overcomes locallocallocallocal trapstrapstrapstraps &y

slightly pertur&ing current solutions.



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A randomly created values are added to the varia&les with a

low pro&a&ility.

)utation of real varia&les



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2inary2inary2inary2inary mutationmutationmutationmutation

For &inary valued individuals mutation means the flipping of

varia&le values, &ecause every varia&le has only two states.

During the mutation operation the modification of salient genes

may degradedegradedegradedegrade the solution, hence to carry out the continual

improvement in the search the pro&a&ilitypro&a&ilitypro&a&ilitypro&a&ility of mutation must &e

setlowlowlowlow usuallyusuallyusuallyusually ....3333 totototo ....3333.



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#einsertion#einsertion#einsertion#einsertion

Once the offspring have &een produced &y selection,

recom&ination and mutation of individuals from the old

population, the fitness of the offspring may &e determined.

0o maintain the si$e of the original population, the offspring

have to &e reinserted into the old population



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#andom #eplacement#andom #eplacement#andom #eplacement#andom #eplacement

0he children replace twotwotwotwo randomlyrandomlyrandomlyrandomlychosen individuals in the

population.

=eak Parent #eplacement=eak Parent #eplacement=eak Parent #eplacement=eak Parent #eplacement

Aweakerweakerweakerweaker parentparentparentparentis replaced &y astrongstrongstrongstrong childchildchildchild. =ith the four

individuals only the fit test two, parent or child, return to

population.

2oth Parents2oth Parents2oth Parents2oth Parents0he child replaces the parent. %n this case, each individual

only gets to &reed once.

#eplacement methods#eplacement methods#eplacement methods#eplacement methods



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Survival of the fittestSurvival of the fittestSurvival of the fittestSurvival of the fittest

Survival of the fittest is the driving force in evolutionary

algorithm in which the individual with high fitness has a

large chance of survival than the individuals with low fitness.

0he individuals with a higher fitness are guaranteed to

survive for the ne*t generation. 0his process of preserving

the &est solution is known as!litism!litism!litism!litism

0he elitism will increase the selection pressure &y

preventingpreventingpreventingpreventing thethethethe losslosslossloss ofofofof &est&est&est&est candidatecandidatecandidatecandidate solutionsolutionsolutionsolution



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0ermination criteria of 'enetic algorithm0ermination criteria of 'enetic algorithm0ermination criteria of 'enetic algorithm0ermination criteria of 'enetic algorithm

0he genetic algorithm should &e properly terminated so as

to prevent the needless computation and prevent premature

termination.

0erminate after aprepreprepre specifiedspecifiedspecifiedspecified &est&est&est&est num&ernum&ernum&ernum&er ofofofof iterationsiterationsiterationsiterations

: generations

%f the chromosome reaches aspecifiedspecifiedspecifiedspecified fitnessfitnessfitnessfitness levellevellevellevel

Succeeds in solving the pro&lem within a specifiedspecifiedspecifiedspecified

tolerancetolerancetolerancetolerance

%f there is nononono changechangechangechange to the population


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!*ample( )inimi$ation : )a*imation!*ample( )inimi$ation : )a*imation!*ample( )inimi$ation : )a*imation!*ample( )inimi$ation : )a*imation



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)inimi$ation of the function f-*)inimi$ation of the function f-*)inimi$ation of the function f-*)inimi$ation of the function f-*

!*ample( )inimi$ation!*ample( )inimi$ation!*ample( )inimi$ation!*ample( )inimi$ation



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DIFFERENTIAL EVOLUTION ALGORITHMDIFFERENTIAL EVOLUTION ALGORITHMDIFFERENTIAL EVOLUTION ALGORITHMDIFFERENTIAL EVOLUTION ALGORITHM

0he differential evolution -D! algorithm, an evolutionary

algorithm introduced &y Price and Storn, is designed for

glo&al optimi$ation pro&lems over continuouscontinuouscontinuouscontinuous domains

which can also work with discretediscretediscretediscretevaria&les

D! has earned a reputation of a very effectiveeffectiveeffectiveeffective glo&alglo&alglo&alglo&al

optimi$eroptimi$eroptimi$eroptimi$er

Differential !volution uses mutationmutationmutationmutation as a search

mechanism and selectionselectionselectionselection to direct the search toward the

prospective regions of the search space.



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%n D!, each varia&le


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0he general convention used a&ove isD!D!D!D!:****:yyyy:$$$$.

D!D!D!D! stands for Differential !volution

**** represents a string denoting the vector to &e pertur&ed

yyyy is the num&er of difference vectors considered for

pertur&ation of *

$$$$ stands for the type of crossover &eing used

D!:&est:D!:&est:D!:&est:D!:&est:3333::::e*pe*pe*pe*p D!:rand: D!:rand:D!:rand:D!:rand:3333::::e*pe*pe*pe*p

D!:randD!:randD!:randD!:rand>>>>totototo>>>>&est:&est:&est:&est:3333::::e*pe*pe*pe*p D!:&est: D!:&est:D!:&est:D!:&est:CCCC::::e*pe*pe*pe*p

D!:rand:D!:rand:D!:rand:D!:rand:CCCC::::e*pe*pe*pe*p D!:&est: D!:&est:D!:&est:D!:&est:3333:&in:&in:&in:&in

D!:rand:D!:rand:D!:rand:D!:rand:3333:&in:&in:&in:&in D!:randD!:randD!:randD!:rand>>>>totototo>>>>&est:&est:&est:&est:3333:&in:&in:&in:&in

D!:&est:D!:&est:D!:&est:D!:&est:CCCC:&in:&in:&in:&in D!:rand:D!:rand:D!:rand:D!:rand:CCCC:&in:&in:&in:&in

ariants of D!ariants of D!ariants of D!ariants of D!



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!volutionary process in D!!volutionary process in D!!volutionary process in D!!volutionary process in D!


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2inomial : niform crossover2inomial : niform crossover2inomial : niform crossover2inomial : niform crossover

!*ponential crossover!*ponential crossover!*ponential crossover!*ponential crossover



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Selection process determines the ne*tne*tne*tne*t generationgenerationgenerationgeneration populationpopulationpopulationpopulation

which is likely the most promising feasi&le candidate solutions.

'reedy'reedy'reedy'reedy selectionselectionselectionselection

%f the trial vector produces a fitness value which is lesslesslessless than the

corresponding target vector, then the trialtrialtrialtrial vectorvectorvectorvector willwillwillwill replacereplacereplacereplace thethethethe

targettargettargettarget vectorvectorvectorvector and will &ecome the population of ne*t generation.

SSSSelectionelectionelectionelection



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n constrained optimi$ation Pro&lemn constrained optimi$ation Pro&lemn constrained optimi$ation Pro&lemn constrained optimi$ation Pro&lem



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Optimal SolutionOptimal SolutionOptimal SolutionOptimal Solution

0 10 20 30 40 50 60 70 80 90 100-0.05

0

0.05

0.1

0.15

0.2

Generations

Fitness

value

Convergence

"onvergence characteristics"onvergence characteristics"onvergence characteristics"onvergence characteristics



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evolutionary algorithms ga and de

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