• No se han encontrado resultados

Evolutionary multiobjetive optimization in non-stationary environments

N/A
N/A
Protected

Academic year: 2017

Share "Evolutionary multiobjetive optimization in non-stationary environments"

Copied!
11
0
0

Texto completo

(1)

Evolutionary Multiobjetive Optimization in

Non-Stationary Environments

Vitoria Aragon, SusanaEsquivel

Lab. deInvestigaionyDesarrolloenInteligeniaComputaional(LIDIC).

Dpto. deInformatia-UniversidadNaionaldeSanLuis

EjeritodelosAndes950-5700-SanLuis-Argentina

fesquivel,vsaragongunsl.edu.ar

Carlos A. CoelloCoello

CINVESTAV-IPN(EvolutionaryComputationGroup)

EletrialEng. Department,ComputerSieneDept.

Av. IPNNo. 2508,Col. SanPedroZaateno

MexioD.F.07300,M

EXICO

oellos.investav.mx

Abstrat

This paper proposes an approah, alled

Multi-objetive Algorithm for Dynami Environments

(MADE), whihextendesFonseaandFleming's

MOGA (withanexternalarhive)sothat itan

deal with dynami environments. MADE

in-ludes two tehniquesto maintain diversity and

also uses speialized funtions that implements

the dynamism required. In order to validate

MADE,wedened adynami versionofastati

test problem (with3 objetives) previously

pro-posed in the speialized literature. The

prelimi-naryresultsobtainedindiate thattheproposed

approahprovidesan aeptableresponse tothe

typeofhangesstudied.

1 Introdution

In the ontext of single-objetive optimization,

manyreal-worldproblemsaredynamiinnature.

Ifthereisahangeovertimeinaertainproblem,

either beause theobjetivefuntion hanges or

beausesomeofitsonstrainthanges(orboth),

then evidently the global optimum hanges as

well. In order to deal with this type of

prob-lems, it is neessaryto haveheuristis that an

adaptquiklyenoughtoanyhanges. Giventhat

adaptationinnatureisaontinuousproess,the

use ofevolutionary algorithmsto dealwith

non-stationary environmentsseems anatural hoie.

However, note that in pratie evolutionary

al-gorithms tend to onvergeto a stationary point

(i.e., loal optimum) overtime,losingthe

diver-sity of the population neessary to explore the

searh spae. One this happens, an

evolution-aryalgorithmlosesitsapabilitytoadapttoany

hanges in the environment. Thus, the use of

goodmehanismstomaintaindiversityisritial

whendealingwithdynami funtions.

Theuseof evolutionaryalgorithmsfordealing

with non-stationary (or dynami) environments

hasreeivedinreasingattentionfromresearhers

[1,2,3℄. However,thedenionofdynami

multi-objetivetestfuntions andalgorithmshasbeen

the subjet of very little work in the

speial-izedliterature[4℄. Thispaperprovidesa

prelimi-narystudyregardingtheuseofamulti-objetive

evolutionaryalgorithmindynamienvironments.

UnliketheworkofFarinaetal. [4℄,inourase,we

don'tfousourresearhondynamiontrol

prob-lemsnoronthedesignofdynamitestfuntions.

Instead,thefousofourworkistostudythe

be-haviorofrelativelysimplemehanismstorespond

todynamihanges. Suhmehanismsare

inor-poratedinto awell-knownmulti-objetive

evolu-tionaryalgorithm(FonseaandFleming'sMOGA

[5℄)aiming to provide somepreliminary insights

regarding the possible hallenges that dynami

funtionspresentforurrentMOEAs.

Theremainderofthispaperisorganizedas

fol-lows. InSetion2,weprovideabriefintrodution

to optimization in non-stationary environments

both in the single-objetive and in the

multi-objetiveases. Setion3desribesourproposed

approah. InSetion4,weprovidethetest

fun-tion adopted for our study. Setion 5 desribes

theperformanemeasures adopted inour study.

(2)

a disussion of our ndings. Finally, Setion 8

providesouronlusionsandsomepossiblepaths

offuture researh.

2 Optimization in

Non-stationary Environments

Onepossibleapproahtodealwithdynami

fun-tionsistotreateahhangeasanewoptimization

problemthathastobesolvedfromsrath[6℄. It

should be obvious that this sort of approah is

impratial in many ases, beausedisregarding

previous information from the problem will

er-tainlyinreasetheomputationalosttosolveit.

Moreover, if the hange is small, one would

as-sumethat thenewsolutionwillbesimilartothe

previous one. Thus, it is desirable to have

op-timization algorithms apable of adapting

solu-tionstoadynami(i.e.,non-stationary)

environ-ment, reusing information obtained in the past.

In single-objetive dynami problems, as

evolu-tion progresses, dierent environments emerge,

whih must be optimized. Thus, the main goal

is to nd aset of pointssuh that eah of them

satises eah of the existing environments. The

samesituation ariseswhendealingwith

multiob-jetive optimization problems, only that in this

ase, the goalis notto nda singlesolutionfor

eah environment, but a set of them. Based on

the previous disussion, we will denote as P

(t)

and PF

(t) to the Pareto optimal set and the

Paretofront,respetively,bothdenedattimet.

Twoissuesare of partiular importane when

dealingwithdynamienvironments: (1)the

abil-ity of an approah to detet that a hange has

ourredand(2)theproperreation(i.e.,the

ve-loity of the response) to those hanges. Inthe

ontext of dynami multiobjetive optimization,

wewillallenvironmentbothtodeisionvariable

spae andto objetivefuntion spae. So,when

werefertohangesintheenvironment,thisould

beineither ofthesespaesorinboth. Farina et

al. [4℄proposedseveraltypesofhangesthatan

beproduedindynamimultiobjetive

optimiza-tion:

TypeI:TheParetooptimalsetP

hanges,

whiletheParetofrontPF

doesnot.

TypeII:BothP

andPF

hange.

TypeIII:P

doesnothangebutPF

does.

Type IV: The problem dynamially

hanges,butneitherP

norPF

hange.

The hange of Type IV is not of interest for

usforobviousreasons. Forthework reportedin

this paper, we onsidered only hanges of Type

III, beause what hanges is the loation of the

truePareto front. Although the Pareto optimal

set does not need to be hangedfor suh

prob-lemstoremainonthenewPF

,theremaybean

eet on the distributionof the solution on the

newPF

for theoldsolutions[4℄. Thus, we

im-plementedrandomhangesasdened in[1℄(i.e.,

eah hange does not depend from the previous

hange nor from time). Note that in this ase,

if the hange is too large, the new problem to

be optimized will be ompletely dierentto the

previousone. It isworthnotiingthat wewon't

dealwiththeautomatidetetionofthehanges,

but only the reation of the algorithm to suh

hanges. Thus, weassumein thisworkthatit is

knownthat ahangein theenvironmenthas

o-urred,sinetheyaresystemati(i.e.,thehanges

areperformedatertainintervalsdenedinterms

ofanumberofgenerations).

3 Proposed Approah

The approah proposed in this paper is alled

MultiobjetiveAlgorithm forDynami

Environ-ments(MADE),anditonsistsofanextensionof

FonseaandFleming'sMOGA [5℄withan

exter-nalarhive. Themain fous ofthis work wasto

experimentwith arelativelyonventional

multi-objetive evolutionary algorithm extended with

speial diversity maintenane mehanisms that

allowittoadapt tohangesin theenviroment.

MADE keeps the basi harateristis from

MOGA but adds speialized funtions that

im-plementthedynamism required. Oneof thefew

hangestoMOGAisthat wehaveeliminatedits

matingrestritions. Thisismainlyduetothefat

thatthereisnolearonsensusregardingthe

use-fulness of mating restritions [7℄. Furthermore,

MADEinludestwotehniquestomaintain

diver-sity andweonsidered unneessaryto introdue

this additional mehanism. MADE uses

real-numbers enoding, proportional seletion,

one-pointrossoverand uniformmutation.

Weusetwomehanismsto maintaindiversity:

Rerudesene: This approah was

pro-posed in [8℄ and it onsists of

maromuta-tions. Theapproahinreasesboth

reombi-nationand mutation probabilities of a

por-tionof the population. The operator is

ap-pliedateahgenerationwithaertain

prob-ability (p

reru

) and produes aradial

(3)

1. t=0

2. Initialize(P(0))andEmptyExternalFile()

3. Evaluate(P(0);F(0))

4. while(t<Nummaxgen)do

5. t=t+1

6. NewGeneration(P(t);P 0

(t))

7. P(t)=P 0

(t)

8. Evaluate(P(t);F(t))

9. Elistism()

10. if(ChangeFuntion(t))

11. StatistialReport()

12. Elitist SettoExternalFile()

13. Clean ElitistSet()

14. Funtion goto Change(F(t);F 0

(t))

15. F(t)=F 0

(t)

16. Evaluate(P(t);F(t))

17. Elistim()

18. InsertRandom Inmigrants()

19. Evaluate(P(t);F(t))

20. Elitism()

21. end if

22. end while

Figure1: GeneraloutlineofourMADEapproah

whih it operates. Individuals to whih the

operator is applied are randomly seleted

(adoptingauniformdistribution).

RandomImmigrants: Thisideawas

pro-posed in [9, 10℄. The approah onsists of

replaing aperentageof the population by

randomly generated individuals. The

teh-niqueisapplied onlywhenthere isahange

in theenvironment.

3.1 Pseudoode of our MADE

Oneweinitializethemainandseondary

popu-lations(line2), themainpopulationisevaluated

with the base funtion F(0) (line 3). The

algo-rithmentersaloop(line4)thatisexeuted

dur-ingaertainnumberofgenerations. Suha

num-berisdeterminedbasedonthenumberofhanges

that theenvironment experiments,and the

gen-erational interval between them. In the

proe-dure NewGeneration, for eah pair of

individu-alsseletedasparents,thereombination

opera-torisinvoked. Suhareombinationoperator

in-ludesthemutationoperator,whihisusedwith

alowprobabilityifthererudeseneoperatoris

notapplied. Otherwise,thererudesene

opera-tor(maromutation)isinvokedandthemutation

andreombinationprobabilitiesareinremented.

One the nextpopulationP 0

(t) hasbeen

gener-ated, itreplae totheurrentP(t) andis ev

alu-atedwiththeurrentF(t)(line8),andweapply

elitism (line 9). The proedure elitism takes

eah nondominated individual from the

popula-tion and veries if it is not dominated with

re-spettotheelististset(whihisthesetofallthe

solutionsthatare nondominatedwith respetto

allthesolutionsgeneratedsofar). Shouldthatbe

the ase, the individual is insertedin the elitist

set. If during thisheking,anindividual in the

elitistsetisdominatedbyanindividualfromthe

urrentpopulation, thenthe dominated

individ-ualisremovedfromtheelitistset.

The funtion ChangeFuntion (line 10)

de-termines if it is neessary to produe a hange

in the environment in the urrent generation.

This is done by heking if the urrent

genera-tion is a multiple of the number of generations

between hanges that wasprovided as an input

to the algorithm. The hanges in the

environ-mentareproduedatonstantintervals(dened

intermsofaertainnumberofgenerations)

dur-ingthe evolutionary proess. Eah time the

en-vironment is about to hange, the

orrespond-ingstatistisare reported(line 11). Suh

statis-tis inlude the number of nondominated

indi-viduals, ESS, et. The nondominated solutions

found sofar (and temporarily retained in

mem-ory) are dumped into an external arhive (line

12). Then,theelitistsetisemptied,sinethe

ob-jetivefuntionsstoredwithindonotorrespond

to the new funtion any more (line 13). The

funtion Funtiongoto Change is responsible

forintroduinghangesintheenvironment. The

hangesimplementedarebothasendingand

de-sendingdisplaementsinall theobjetive

fun-tions or some of them. This is determined by

theuser (line 14). Theold objetivefuntion is

replaedbythenewone(line15)andthe

popula-tionisevaluatedusingthenewobjetivefuntion

(line16). Wethenapplyelitism (line17)sothat

we anretain thenondominated vetorspresent

inthepopulation. Next,aperentageofthe

pop-ulationisreplaed byindividuals randomly

gen-erated (line 18). The individuals seleted to be

replaedarethosedominatedbysomeother

indi-vidual in thepopulation. In asethe number of

individuals to be replaed is lessthan the

num-ber of individuals that are dominated, then we

replaeasmanynondominatedindividualsas

ne-essaryuntilompletingthe(pre-dened)

perent-age. It is worthnotiing that it is irrelevantto

lose nondominated individuals from the

popula-tion, sinethey have alreadybeen storedin the

seondarypopulation (line 20). We then

evalu-ateagain the population (withthe newinserted

(4)

4 Test Funtion

In order to validate our proposed approah, we

introdueadynamiversionofawell-knowntest

problem(DTLZ2[11℄)whih,initsstativersion,

has beenused to validate multi-objetive

evolu-tionaryalgorithms. Thisfuntionwashosen

be-auseitissalablebothindeisionvariablespae

and in objetivefuntion spae. Suhsalability

failitatestostudytheapabilityofanalgorithm

to reat to hanges in both spaes. Although

other test funtions have been adopted to

vali-date ourapproah,wehose to inlude onlyone

to allowamoredetailed analysisofthebehavior

ofthemehanismsproposed.

DTLZ2: Min (f

1 (x);f 2 (x);f 3 (x)), where: f 1

(x) = (1 + g(x

3 ;x 4 ))os(x 1 =2)os(x 2 =2), f 2

(x)=(1+g(x

3 ;x 4 ))os(x 1 =2)sin(x 2 =2),and f 3

(x)=(1+g(x

3 ;x 4 ))sin(x 1 =2),

with 0 x

i

1;i =4 and g(x) = P xi2x (x i 0:5) 2 .

Onthisbasefuntion,allobjetivefuntionstake

non-negativevaluesand the desired front is the

rst quadrant of a sphere of radius one. The

dynamienvironmentisgeneratedbytranslating

the base funtion DTLZ2 along alineartra

je-toryaordingto[12℄:

DTLZ2 Dyn(x;t)=DTLZ2(x)+Æ(t)

where t2N

0

denotestime (generationnumber).

Thedisplaementofthefuntionisdeterminedby

funtion Æ(t)=(Æ

1 (t);Æ

2 (t);Æ

3

(t))anditdepends

ontheupdatefrequenyofthefuntion (i.e.,the

numberofgenerationsbetweenhanges)andthe

severitys(afatorthatdeterminesthelengthof

thefuntion displaement).

For an asending linear displaement, we have:

Æ

1 (0)=Æ

2 (0)=Æ

3 (0)=0

Æ(t+1)=

Æ

i (t)+s

i

if(t+1)modinterval=0

Æ

i

(t) otherwise

(1)

where i is the objetive funtion number to be

hanged,intervalis thenumberofgenerations

betweenhanges ands

i

is theseveritydegreeof

thedisplaementoff

i .

Foralineardesendingdisplaement,wehave:

Æ

1

(0)=amountof hanges

1

Æ

2

(0)=amountof hanges

2

Æ

3

(0)=amountof hanges

3 Æ(t+1)= Æ i (t) s i

if(t+1)modinterval=0

Æ

i

(t) otherwise

(2)

where i is index of the objetive funtion to

be hanged, interval is the number of

gener-ations allowed between hanges, s

i

is the

de-gree of severity of the displaement of f

i and

amountof hangeisthenumberofhangesthat

theenvironmentwill experiment.

Iftheseverityistoohigh,thenthesequeneof

problems to be optimized won't share anything

in ommon. This would be similar to solving

ompletely dierent problems by separate. On

theontrary,if theseverityis small, there ould

benopereptibledierenebetweentwo

onseu-tivehangesandthisanbeonsideredasa

non-dynami problem (i.e., it ould be treated as a

stati problem and one ould build robust

solu-tions for suh problem [1℄). As a onsequene,

the severity of the hanges produed in our

ex-perimentsissuhthatthesetoffeasiblesolutions

betweenhangesgetspartially overlapped.

5 Performane Measures

Itis obviouslydesirablethat ourmulti-objetive

evolutionaryalgorithm(MOEA) is abletoreah

(either in stati or dynami environments) the

trueParetofrontofaproblemwithagoodspread

of points. In order to evaluate the performane

ofourapproah,weadoptedthefollowing

perfor-manemeasures:

1. UnsuessfulCounting(USCC):We

de-ne this measure based on the idea of the

ErrorRatiometriproposedin[13℄whih

in-diatestheperentageofsolutions(fromthe

nondominatedvetorsfoundsofar)that are

notmembersofthetrueParetooptimalset.

Inthisase,weountthenumberofvetors

(inthe urrent set ofnondominated vetors

available)thatarenotmembersofthePareto

optimal set: USCC = P

n

i=1 u

i

; where n is

the numberof vetors in the urrent set of

nondominated vetors available; u

i

= 1 if

vetoriisnotamemberofthePareto

opti-malset,andu

i

=0otherwise. Itshouldthen

be lear that USCC =0 indiates an ideal

behavior, sine it would mean that all the

vetors generated by our algorithm belong

to the true Pareto optimal set of the

prob-lem. Forafairomparison,whenweusethis

measure,allthealgorithmsshouldlimittheir

nal number of non-dominated solutionsto

thesamevalue.

2. InvertedGenerationalDistane(IGD):

Theoneptofgenerationaldistanewas

(5)

ofestimatinghowfararetheelementsinthe

Paretofrontproduedbyouralgorithmfrom

thoseinthetrueParetofrontoftheproblem.

Thismeasureisdenedas: GD= pP

n

i=1 d

2

i

n

wherenisthenumberofnondominated

ve-tors found bythe algorithm beinganalyzed

and d

i

is the Eulidean distane (measured

in objetive spae) between eah of these

and the nearest memberof thetrue Pareto

front. It should be lear that a value of

GD=0indiates thatall theelements

gen-erated are in the true Pareto front of the

problem. Therefore, any other value will

indiate how \far" we are from the global

Pareto front of our problem. In our ase,

weimplemented an \inverted"generational

distane measure (IGD) in whih weuse as

a referene the true Pareto front, and we

ompareeahofitselementswithrespetto

thefrontprodued byanalgorithm. Inthis

way,weare alulatinghowfar arethe

ele-mentsofthetrueParetofront,fromthosein

theParetofrontproduedbyouralgorithm.

Computing this\inverted"generational

dis-tane value redues the bias that an arise

whenanalgorithmdidn'tfullyoverthetrue

Paretofront.

3. EÆiently Spaed Set (ESS): Here, one

desiresto measure thespread(distribution)

ofvetorsthroughoutthenondominated

ve-tors found so far. Sine the \beginning"

and\end" oftheurrentPareto frontfound

are known, asuitably dened metrijudges

howwell thesolutionsin suhfront are

dis-tributed. Shott [14℄ proposed suh a

met-ri measuring the range (distane) variane

of neighboring vetorsin thenondominated

vetorsfoundso far. This metriis dened

as:

S= v

u

u

t 1

n 1 n

X

i=1 (d d

i )

2

; (3)

whered

i =min

j (jf

i

1 (~x ) f

j

1

(~x )j+jf i

2 (~x )

f j

2

(~x )j)+:::jf i

k (~x f

j

k

(~x )j,i;j=1;:::;n,

d is the mean of all d

i

, and n is the

num-ber of nondominated vetors found so far.

A valueof zero for this metri indiates all

membersoftheParetofronturrently

avail-ableareequidistantlyspaed.

4. Number of Nondominated Individuals

perEnvironment(NNIE):Thismeasureis

self-explanatory.

Furthermore,wealso onsidered thefollowing

performanemeasures:

1. Average Unsuessful Counting

(AUSCC): Average of the unsuessful

ounting values of the population in the

generation just before the hange. It is

denedas:

AUSCC=(1=k) P

k

j=1 USCC

j

wherekisthenumberofhangesinthe

envi-ronment, USCC

j

is theunsuessful

ount-ingintheenvironmentj.

2. Average Inverted Generational

Dis-tane (AIGD): The average of the

genera-tional distane values of the population at

thegeneration just before thehange. It is

denedby:

AIGD=(1=k) P

k

j=1 IGD

j

wherekisthenumberofhangesinthe

envi-ronment, IGD

j

is thegenerationaldistane

intheenvironmentj.

3. Average EÆientlySpaed Set(AESS):

TheaverageoftheESSvaluesofthe

popula-tionatthegenerationjustbeforethehange.

Itisdenedby:

AESS =(1=k) P

k

j=1 ESS

j

wherekisthenumberofhangesinthe

envi-ronment,ESS

j

istheEÆientlySpaedSet

intheenvironmentj.

4. AverageNumberofNondominated

In-dividuals (ANNIE): Average of the NNIE

values of the population in the generation

justbeforethehange. Itisdened by:

ANNIE=(1=k) P

k

j=1 NNIE

j

wherekisthenumberofhangesinthe

envi-ronment,NNIE

j

isthenumberof

individu-alsnondominatedin theenvironmentj.

6 Desription of the

Experi-ments

This setion aims to desribe the experiments

performed to validate our proposed approah.

Obviously, the aim is to evaluate the apability

of ourapproahto trakdown thenew loation

ofthe truePareto front, one thealgorithm has

deteted a hange in the environment. The

ex-perimentstookplaeondierentsenarios.Eah

of these senarios represents a dierent type of

(6)

1. Senario 1: PositiveLinearDisplaementof

f

1

(x)only,

2. Senario 2: PositiveLinearDisplaementof

f

2

(x)only,

3. Senario 3: PositiveLinearDisplaementof

f

3

(x)only,

4. Senario 4: PositiveLinearDisplaementof

bothf

1

(x)andf

2

(x)only,

5. Senario 5: PositiveLinearDisplaementof

bothf

1

(x)andf

3

(x)only,

6. Senario 6: PositiveLinearDisplaementof

bothf

2

(x)andf

3

(x)only,

7. Senario7: NegativeLinearDisplaementof

f

1

(x)only,

8. Senario8: NegativeLinearDisplaementof

f

2

(x)only,

9. Senario9: NegativeLinearDisplaementof

f

3

(x)only,

10. Senario 10: NegativeLinear Displaement

ofbothf

1

(x)andf

2

(x)only,

11. Senario 11: NegativeLinear Displaement

ofbothf

1

(x)andf

3

(x)only,

12. Senario 12: NegativeLinear Displaement

ofbothf

2

(x)andf

3

(x)only,

13. Senario13: PositiveLinearDisplaementof

f

1 (x),f

2

(x)andf

3

(x)simultaneously,and

14. Senario 14: NegativeLinear Displaement

off

1 (x), f

2

(x)andf

3

(x)simultaneously.

The parametersrequired byour approah are

thefollowing:

1. Probabilities for the operators:

reombina-tion, mutation, rerudesene, inrease for

thereombination,inreaseforthemutation.

2. PerentageofRandomImmigrants,

3. Numberof GenerationsbetweenChangesof

theEnvironment,

4. NumberofChangesoftheFuntion,

5. ObjetiveFuntionsthatwewishtomodify,

6. Degreeofseverityforf

1 ,f

2 andf

3 ,

7. TypeofDisplaement(positiveornegative).

Table1: Valuesadopted fortheparametersused

byourapproah

Populationsize 200

Sizeoftheseondarypopulation 250

InitializationoftheSeondaryPopulation Random

Prossover 0.65

P

mutation

0.5

P

reombination inrease

0.8

P

mutation inrease

0.8

P

rerudesene

0.2

PerentageofRandomImmigrants 30individuals

NumberofChangesoftheFuntion 9

s1forDTLZ2Dyn 0.5

s

2

forDTLZ2Dyn 0.5

s

3

forDTLZ2Dyn 0.5

Eah experimentwasrepeated10 times (with

10 dierent random seeds), and we olleted

statistisfrom theorresponding runs. Foreah

senario, the hanges in the environment were

produedatevery5and10generations. The

val-uesadopted for theparameters of ourapproah

areshownin Table1.

7 Results

TheperformaneofMADEusingthebenhmark

funtion previously desribed (with hanges at

every5and10generations,andwithboth

asend-ing desending displaements) varies from good

to relatively good on the dierent senarios. In

thefollowingTables,thelabelsof theolumns2

to 4 and 2 to 5 are interpreted as follows: the

AorDmeansAsendingorDesending

displae-ment,respetively,andthenumberfollowingthe

undersoreindiates thesenarionumber. First,

weanalyzetheresultsforthesenarios13and14

where we apply positive and negative

displae-ments over the three objetives, simultaneously.

Inthisase,regardlessofthesize of theinterval

between hanges and regardless of the

displae-menttypes,theParetofrontsproduedbyMADE

are very lose to the true Pareto fronts with a

gooddistributionofpointsinbothases. Thisis

showninFigures2to 5fortheaseinwhihthe

hangesareproduedatevery5generations.

Analogously, when analyzing the results from

Table2,weanseethatweobtainlittle

variabil-ity of results for all metris,exept for ANNIE.

Intheaseofthismetri(averagenumberof

non-dominated individuals),weobtain alowervalue

fordesendingdisplaements. Betweentwo

on-seutivehanges, regardlessofthedisplaement,

the old and new feasible zone share a set of

(7)

de-sending: the old PF

is ontained in the new

feasible zone and 2) asending: the new PF

is

ontained in the old feasible zone. The reason

oftheMADE'spoorperformane,ondesending

displaementsould be beause ofthe newPF

is not ontained in the oldfeasible zone, soany

nondominated individuals from the old feasible

zone belong tothenewPF

. ThenewPF

has

to befoundfrom srathbut theold PF

ould

be used likeabase to nd the newPF

.

How-ever,onasendingdisplaements,thenewPF

is

ontainedintheoldfeasiblezone,sothesearhof

thenewPF

ouldnotstartfromsrathifsome

nondominatedindividualsbelongtothenewPF

ortheyarelosefromit.

For senarios4 to 6 and 10 to 12, where the

displaementsareappliedsimultaneouslytopairs

of objetives (see Tables 3to 6), wean see in

Figures6to 11thatourapproahanobtain

rea-sonablygoodapproximationsandagood spread

ofsolutions.

Finally, for the senarios 1 to 3 and 7 to 9,

where only oneof the objetivefuntions is

dis-plaed,MADEproduesgood approximationsof

the truePareto front, witha good spreadof

so-lutions. However,inallases,afewsolutionsare

produed away from the true Pareto front (see

Figures 12to 17).

When looking at the numerial resultsin

Ta-bles 7, 8, 9and10, weobservethat whenonly

f

1

isdisplaed,regardlessoftheintervalbetween

hanges,asendingdisplaementsseemtobethe

mostdiÆulttohandlebyMADE.

It is worthemphasizingthat, despitethehigh

valuesobtainedfortheAUSCCmetriinallthe

senarios studied, our approah systematially

onvergedverylosetothetrueParetofront(this

anbeappreiatedbylookingatthevaluesofthe

IGD metri). However,sinetheexatfrontwas

not reahed, the AUSCC metri provided poor

results. So, the reahability problem seems to

be more an aurany problem whih may be

related to the small number of generations

be-tween hanges (i.e., the algorithm doesn't have

enoughtimetoprodueaner approximationof

theParetofront).

Figure2: FrontgeneratedbyMADEwhenf

1 ,f

2

andf

3

aredisplaedinanasendingway

Figure3: PF

true

orrespondingtothehange

in-diatedin Figure 2

Figure4: FrontgeneratedbyMADEwhenf

1 ,f

2

andf

3

aredisplaedinadesendingway

Figure5: PF

true

orrespondingtothehange

(8)

Table2: DTLZ2 Dyn,modiationsinf

1 ,f

2 and

f

3

, valuestaken from the runin themedian

re-spetofIGD

Interval=5 Interval=10

Displaement A13 D14 A13 D14

AESS 0.04 0.05 0.04 0.05

IGD 0.000022 0.000025 0.000023 0.000024

ANNIE 114 97 113 98

AUSCC 249.9 246.1 250.0 249.5

Table 3: DTLZ2 Dyn, modiations in f

1 f

2 ;

f

1 f

3 ; f

2 f

3

; values of the run in the median

re-spetofIGD

Interval=5

Displaement A4 A5 A6

AESS 0.05 0.05 0.05

AIGD 0.000023 0.000024 0.000024

ANNIE 100 104 97

AUSCC 248.5 249.8 247.6

Table 4: DTLZ2 Dyn, modiations in f

1 f

2 ;

f

1 f

3 ; f

2 f

3

; values of the run in the median

re-spetofIGD

Interval=5

Displaement D10 D11 D12

AESS 0.04 0.05 0.04

AIGD 0.000023 0.000023 0.000023

ANNIE 113 105 108

AUSCC 249.6 249.6 248.8

Table 5: DTLZ2 Dyn, modiations in f

1 f

2 ;

f

1 f

3 ; f

2 f

3

; values of the run in the median

re-spetofIGD

Interval=10

Displaement A4 A5 A6

AESS 0.04 0.05 0.04

AIGD 0.000023 0.000024 0.000024

ANNIE 113 95 99

AUSCC 249.9 249.1 247.1

Table 6: DTLZ2 Dyn, modiations in f

1 f

2 ;

f

1 f

3 ; f

2 f

3

; values of the run in the median

re-spetofIGD

Interval=10

Displaement D10 D11 D12

AESS 0.04 0.05 0.05

AIGD 0.000022 0.000023 0.000023

ANNIE 111 108 110

AUSCC 249.9 249.6 250.0

Figure 6: Front generated by MADE when f

1 ,

andf

2

aredisplaedinanasendingway

Figure7: PF

true

orrespondingtothehange

in-diatedin Figure 6

Figure8:FrontgeneratedbyMADEwhenf

1 and

f

3

aredisplaedinadesendingway

Figure9: PF

true

orrespondingtothehange

(9)

Table7: DTLZ2Dyn,modiationsinf

1 ;f

2 ;f

3 ;

valuesoftherunin themedianrespetofIGD

Interval=5

Displaement A1 A2 A3

AESS 0.05 0.04 0.05

AIGD 0.000024 0.000024 0.000024

ANNIE 94 101 97

AUSCC 247.0 248.3 249.3

Table8: DTLZ2Dyn,modiationsinf

1 ;f

2 ;f

3 ;

valuesoftherunin themedianrespetofIGD

Interval=5

Displaement D7 D8 D9

AESS 0.05 0.04 0.04

AIGD 0.000023 0.000023 0.000023

ANNIE 104 111 110

AUSCC 249.3 250.0 250.0

Table9: DTLZ2Dyn,modiationsin f

1 ;f

2 ;f

3 ;

valuesoftherunin themedianrespetofGDM

Interval=10

Displaement A1 A2 A3

AESS 0.054 0.05 0.05

AIGD 0.000025 0.000024 0.000024

ANNIE 98 99 101

AUSCC 246.1 248.1 249.8

Table10: DTLZ2 Dyn,modiationsinf

1 ;f

2 ;f

3 ;

valuesoftherunin themedianrespetofGDM

Interval=10

Displaement D7 D8 D9

AESS 0.04 0.04 0.04

AIGD 0.000023 0.000022 0.000023

ANNIE 113 111 108

AUSCC 250.0 249.8 249.8

Figure 10: Frontgenerated by MADE when f

2 ,

andf

3

aredisplaedinanasendingway

Figure 11: PF

true

orresponding to the hange

indiatedin Figure 10

Figure12: FrontgeneratedbyMADEwhenf

1 is

displaedinanasendingway

Figure 13: PF

true

orresponding to the hange

(10)

Figure14: FrontgeneratedbyMADEwhenf

2 is

displaedin adesendingway

Figure 15: PF

true

orresponding to the hange

indiatedinFigure 14

Figure16: FrontgeneratedbyMADEwhenf

3 is

displaedin anasending way

Figure 17: PF

true

orresponding to the hange

indiatedinFigure 16

8 Conlusions and Future

Work

Theperformaneoftheproposedapproahinthe

test ase adopted turned out to be satisfatory

in the sense that the aim was not to generate

theompletetrueParetofronts,buttodetermine

ifthe proposed approah wasable to adapt fast

enoughtothehangesintheloationofthetrue

Paretofront.

Aspartofourfuturework,weintendtoexplore

the use of other (more sophistiated) operators

to handle dynami environments [15℄. We will

alsovalidatetheperformaneofourMADEwhen

saling the test funtion both onthe number of

deisionvariables andobjetive funtions. Also,

wewanttoexploretheimpatof-dominane[16℄

intheperformaneofourapproah.

Another aspet that deserves more attention

is the hoie of performane measures adopted

tovalidatetheperformaneofourapproah.

Fi-nally,wealsoaimtoevaluateothersearhengines

dierentfromMOGA (e.g.,weintendto usethe

NSGA-II[17℄whihisahighlyompetitive

multi-objetiveevolutionaryalgorithm).

9 Aknowledgements

Thersttwoauthors aknowledgesupportfrom

the Universidad Naional de San Luis and the

ANPCYT. The third author aknowledges

sup-port from the Consejo Naional de Cienia y

Tenologa(CONACyT)throughprojetnumber

42435-Y.

Referenes

[1℄ Branke,J.: Evolutionaryoptimizationin

dy-namienvironments.KluwerAademi

Pub-lishers(2002)

[2℄ Yamasaki, K.: Dynami pareto optimum

GA against the hangingenvironments. In

Branke,J.,Bak,T.,eds.: Evolutionary

Al-gorithms for Dynami Optimization

Prob-lems, SanFraniso,California,USA(2001)

47{50

[3℄ Jin,Y.,Sendho,B.: Construtingdynami

optimization test problems using

multi-objetiveoptimization onept(2004)InG.

R. Raidl, editor, Apliations of

Evolution-aryComputing,volume3005ofLNCS,pages

[image:10.595.85.232.279.371.2]
(11)

[4℄ Farina, M., Deb, K., Amato, P.: Dynami

MultiobjetiveOptimizationProblems: Test

Cases,Approximation,andAppliations. In

Fonsea, C.M., Fleming, P.J., Zitzler, E.,

Deb, K., Thiele, L., eds.: Evolutionary

Multi-CriterionOptimization.Seond

Inter-national Conferene,EMO2003, Faro,

Por-tugal, Springer.Leture Notes in Computer

Siene. Volume 2632(2003)311{326

[5℄ Fonsea,C.M.,Fleming,P.J.: Geneti

Algo-rithmsforMultiobjetiveOptimization:

For-mulation,DisussionandGeneralization. In

Forrest, S., ed.: Proeedings of the Fifth

International Conferene on Geneti

Algo-rithms, San Mateo, California, University

of Illinois at Urbana-Champaign, Morgan

KaumanPublishers(1993)416{423

[6℄ Raman,N.,Talbot,F.B.: Thejobshop

tar-dinessproblem: Adesompositionapproah.

European Journal of Operational Researh

69(1993)187{199

[7℄ CoelloCoello, C.A., Van Veldhuizen, D.A.,

Lamont,G.B.: EvolutionaryAlgorithmsfor

Solving Multi-Objetive Problems. Kluwer

Aademi Publishers, New York (2002)

ISBN0-3064-6762-3.

[8℄ Kwasnika, H.: Redundany of genotypes

as the way for some advaned operators in

evolutionaryalgorithms -SimulationStudy.

VIVEK, A Quarterly in Artiial

Intelli-gene10(1997)2{11

[9℄ Cobb, H., Grefenstette, J.: Geneti

algo-rithms for traking hanging environments.

In: Proeedingofthe5thIEEEInternational

Conferene onGeneti Algorithms, Morgan

Kauman(1993)523{530

[10℄ Grefenstette, J.: Optimization of ontrol

parameters for geneti algorithms. IEEE

Transation on Systems, Man and

Cyber-neti16(1986)122{128

[11℄ Deb, K., Thiele, L.,Laumanns, M., Zitzler,

E.: Salable Multi-Objetive Optimization

Test Problems. In: Congress on

Evolution-ary Computation (CEC'2002). Volume 1.,

Pisataway,NewJersey, IEEEServie

Cen-ter(2002)825{830

[12℄ Bak, T.: On the behaviorof evolutionary

algorithms in dynami environnments. In:

Proeedings of International Conferene on

EvolutionaryComputation,Pisataway,NJ,

IEEEPress(1998)446{451

[13℄ Veldhuizen, D.A.V.: Multiobjetive

Evolu-tionary Algorithms: Classiations,

Analy-ses, and New Innovations. PhD thesis,

De-partment of Eletrial and Computer

En-gineering. Graduate Shool of Engineering.

Air Fore Institute of Tehnology,

Wright-PattersonAFB,Ohio(1999)

[14℄ Shott, J.R.: Fault tolerant design

us-ing single and multiriteria geneti

algo-rith optimization. Master's thesis,

Depart-mentofAeronautisandAstronautis,

Mas-sahusetts Institute of Tehnology,

Cam-bridge,Massahussetts(1995)

[15℄ Morrison, R.W.: Designing

Evolution-ary Algorithms for Dynami Environments.

Springer-Verlag,Berlin(2004)

[16℄ Laumanns, M., Thiele, L.,Deb, K., Zitzler,

E.: CombiningConvergeneandDiversityin

Evolutionary Multi-objetive Optimization.

Evolutionary Computation 10 (2002) 263{

282

[17℄ Deb,K.,Pratap,A.,Agarwal,S.,Meyarivan,

T.: A Fast and Elitist Multiobjetive

Ge-neti Algorithm: NSGA{II. IEEE

Transa-tionsonEvolutionaryComputation6(2002)

182{197

Figure

Figure16:displa
edFrontgeneratedbyMADEwhenf3isinanas
endingway

Referencias

Documento similar

We obtained a corpus to contrast the terms used in the articles written in English and in the articles written in Spanish in order to study what types of translations are made

The Dwellers in the Garden of Allah 109... The Dwellers in the Garden of Allah

The power processing circuit (PPC) created by the evolutionary algorithm is connected on one side to an input voltage source V in , and on the other side to an output circuitry made

Elaborar el Diagrama de Pareto a partir de estudios de casos aplicando el aprender haciendo para resolver problemas de la vida diaria y la gestión

Government policy varies between nations and this guidance sets out the need for balanced decision-making about ways of working, and the ongoing safety considerations

It is interesting to remark that, if we compare the evolutionary results with the Bertrand game, we see how in the evolutionary model, from both the point of view of firms (which

Por supuesto que esta teoría ha quedado desfasada con el tiempo y sólo tendrá vigencia en aquellos países en los que prevalezcan pautas de compor- tamiento como las de principios

Keywords: acoustic sensing; sound location; sensor synchronization; sensor network planning; derivative-free optimization; Pareto