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.
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
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 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
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
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
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
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
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
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][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