Evolutionary Multi-Obje tive Optimization
Kalyanmoy Deband Sa hin Jain
Kanpur Geneti AlgorithmsLaboratory (KanGAL)
Indian Institute of Te hnology Kanpur
Kanpur, PIN 208016,India
fdeb,jsa hingiitk.a .in
KanGAL Report No. 2002004
Abstra t
It isnowwell establishedthatmorethan oneperforman emetri sare ne essaryforevalu-
atingamulti-obje tiveevolutionaryalgorithm(MOEA).Althoughthereexistanumberofper-
forman emetri sintheMOEAliterature,mostofthemareappliedtothe nalnon-dominated
set obtainedby anMOEA to evaluate its performan e. In thispaper, wesuggest a oupleof
running metri s{oneformeasuringthe onvergen etoareferen esetandotherformeasuring
thediversityinpopulationmembersateverygenerationofanMOEArun. Eitherusingaknown
Pareto-optimalfrontoranagglomerationofgeneration-wisepopulations,thesuggestedmetri s
revealimportantinsightsandinterestingdynami softheworkingofanMOEAorhelpprovide
a omparativeevaluationoftwoormoreMOEAs.
1 Introdu tion
Over the past ouple of years or so, more emphasis in the area of multi-obje tive evolutionary
algorithms (MOEAs) has been laid on the development of performan e metri s. The primary
reasonforits lukewarm interestinthepastisthatunlikeinthesingle-obje tive EAstudies,where
theperforman emetri isdire tlyrelatedtotheobje tivefun tion(bothbeings alarquantities),in
multi-obje tiveoptimizationtheperforman emetri mustassessanumberofsolutionsea hhaving
a ve tor of obje tive values. This immediately demands the need of more than one performan e
metri s. A re ent study (Zitzler et al., 2002) has shown that for an M-obje tive optimization
problem,atleastM performan emetri smustbeused. Inthe ontextofsingle-obje tiveEAs,the
re ordof a runningperforman emetri with generationprovideda plethora of informationabout
theworkingoftheEAontheproblemathand. Itisnotsurprisingthatifappropriateperforman e
measures of MOEA populations are also re orded and analyzed, salient information about their
working would beimminent. Inthispaper, we dis ussand suggestpotential runningperforman e
metri sformulti-obje tiveoptimization.
Althoughanumberofdi erentperforman emetri shave beensuggested,manyareonlyappli-
abletotwo-obje tiveproblemsandmostimportantlyitisnotobviouswhi hoftheseperforman e
metri sone mayuseinpra ti e. Knowlesand Corne(2002) omparedmostofthese metri sbased
ontheir ompatibilitywithoutperforman erelationsbetweentwosetsofsolutionsandtheirmono-
toni ityproperties. Themetri sare lassi edbasedonstrong, weakand ompleteoutperforman e
tentofoutperforman e,insteadofjustdeterminingwhetheroneoutperformestheotherornot. The
study also suggested the use of any of the three R-metri s introdu ed in Hansen and Jaszkiewi z
(1998). TheR-metri smeasuretheproportionofo asionsoneapproximationsetwillbepreferred
byapre-de ned setof utilityfun tions. Althoughsu h ametri is apragmati one,they maynot
be omputationally eÆ ient to beused asa runningperforman emetri .
Itisintuitivethattheuseofasetofmetri slessthanthenumberofobje tiveswouldmeanalose
of a dimensionand wouldimmediately make theapproa h theoreti allyina urate. However, one
ofthewaystoover omethedimensionalityproblempra ti allyistouseafun tionally independent
setofvariables(Goldberg,1993). Byrealizingthattherearetwomainfun tionalgoalsofMOEAs,
e orts an be made in devising two metri s: (i) one for measuring the onvergen e of solutions
to the Pareto-optimal front and (ii) the other formeasuring thediversityof solutions. Su h a set
of two metri s will enable two or more solution sets to be ompared among ea h other in terms
of their fun tional a hievements. Although the rst metri may simply be devised based on the
distan efrom areferen eset, these ondmetri isnotthatstraightforwardto explain,parti ularly
inthe ase of largenumberof obje tives.
In theremainderofthispaper,webrie y reviewtheexistingperforman emetri s. Thereafter,
we suggest a metri ea h forevaluating two fun tionalities des ribed above. Finally, the need of
using running metri s in MOEAs is amply demonstrated by applying them on a number of two
and three-obje tive problems.
2 Existing Performan e Metri s
Veldhuizen(1999),inhisdissertation,reportedanumberofperforman emetri sformulti-obje tive
optimization. Later, the rst author, in his book (Deb, 2001), lassi ed existing performan e
metri s into three lasses: (i) metri s for onvergen e, (ii) metri s for diversity estimation, and
(iii) metri s for both onvergen e and diversity. Although advantages and disadvantages of ea h
metri werequalitativelymentioned,Knowlesand Corne(2002)analyzedmostofthose metri son
thebasisof extentof outperforman erelations betweentwosets of non-dominatedsolutions. This
studysuggested theuseofanyofthethree performan emetri sofHansenandJaszkiewi z(1998).
We des ribethem brie y here.
The essentialideaisthat foranytwo approximatesets(Aand B),these metri susea number
of utility fun tions and determine the expe ted number of o asions the solutions of one set is
better thantheother. Thesemetri s,in otherwords,de lare thatset asthewinnerwhi hwillbe
the hoi e ofmost de ision-makers. Forexample,the metri R 1isde nedas follows:
R 1(A;B;U;p)= Z
u2U
C(A;B;u)p(u)du; (1)
where U is the set of all utility fun tions onsidered, p(u) is the frequen y of o urren e of the
utilityfun tionu,and the out omefun tionC is de nedasfollows:
C(A;B;u)= 8
>
<
>
:
1; ifu
(A)>u
(B),
0:5; ifu
(A)=u
(B),
0; ifu
(A)<u
(B).
(2)
Here,u (A)is themaximumvalueof theutilityfun tionuon approximationset Aandis denoted
by u
(A) = max
z2A
fu(z)g (z is an obje tive ve tor from A). It is lear from equation 1 that
the R 1-metri measures the expe ted proportion of o asions the set A is better than set B.
The developers suggested to de lare A to be the winner if R 1(A;B;U;p)> 0:5 and B to be the
winner if R 1(A;B;U;p) < 0:5. If R 1(A;B;U;p) = 0:5, superiority of either A or B annot be
on luded from the study. However, the diÆ ulties with this metri are the requirement of a
numberofutilityfun tions,theirprobabilitiesofo urren e,andanumeri alintegrationte hnique
toa tuallyevaluatethemetri . Toavoidthenon-asso iativityprobleminvolvedin omparingmore
than two approximation sets, developers have also suggested omparing ea h set A with respe t
to a referen e set R , as follows: R 1
R
(A;U;p) = R 1(A;R ;U;p). In order to redu e the bias from
the hosen referen e set R , developers suggested using more than one referen e sets to on lude
thesuperiorityof one set overthe other. To quantifytheextent of superiorityof one set overthe
other,developershave suggestedtwomore metri s:
R 2(A;B;U;p) = Z
u2U (u
(A) u
(B))p(u)du; (3)
R 3(A;B;U;p) = Z
u2U u
(A) u
(B)
u
(A)
p(u)du: (4)
TheR 2-metri measurestheexpe teddegreeofsuperiorityandthetheR 3-metri measurestheex-
pe tedproportionofsuperiority. Thesetwometri s analsobemodi edtobeusedwithareferen e
set R , but need to be used as R 2
R
(A;U;p) = R 2(R ;A;U;p) and R 3
R
(A;U;p) = R 3(R ;A;U;p),
respe tively. As pointedoutearlier, these metri valueslargely dependonthe hosen utilityfun -
tions. Fortwo-obje tiveproblemswitha onvexPareto-optimalregion,thefollowingutilityfun tion
u
k
2U (where k 2[0;1℄) an be hosen:
u
k
=kf
1
+(1 k)f
2
: (5)
In general,however, weightedT heby he metri s an be usedasthe familyofutilityfun tions.
For measuring diversity and onvergen e of obtained solutions, Zitzler's hyper-volume metri
(also known as the S-metri ) an also be used. The metri omputes the hyper-volume of the
obje tive spa e dominated by an approximation set. Although a set with a good diversity of
solutionswouldmeanalargerhyper-volumemetri value(hen ebetter),theS-metri valuedepends
onthe hosenreferen epoint usedforthehyper-volume al ulationand demandsnormalizationof
theobje tivesbefore omputingthemetri .
Alltheseauthorsseemtohavemadeonepoint lear: the omparisonoftwonon-dominatedsetof
solutionsisnotastraightforwardmatter,be auseofthedimensionalityinvolvedinthesets. Itisalso
generallyagreedthatoneperforman emetri isnotenoughtojudgetheperforman eofanMOEA.
Re ently, Zitzleretal. (2002) have given a formal proof stating thatforan M-obje tive problem,
at least M performan e metri s are needed to ompare two or more set of solutions. Intuitively
thismakes sense, as otherwise thiswould suggest an ina uratejudgment made witha redu tion
in dimensionality. However, we argue that although mathemati allyin orre t, it may be possible
to ompare two or more approximation sets fun tionally, as is often followed in understanding
behaviorsof omplexsystems. Inmulti-obje tiveoptimization,therearetwoprimaryfun tionalities
that an MOEA must a hieve: (i) approa hing the Pareto-optimal front and (ii)maintenan e of a
fun tionalitiesofmulti-obje tiveoptimization,itmaybepossibletode netwoperforman emetri s,
one formeasuringea h fun tionalityex lusively, even forM >2 obje tives.
2.1 Need for Running Metri s
InmostMOEA studies,in ludingthose involving performan emetri s,two ormore MOEAshave
beenusually ompared purelyon thebasis of what havebeenobtained at theendof a simulation
run. Either a arefully updated ar hive or an EA population at the end of a simulation run
is evaluated for this purpose. However, interesting information about how an MOEA arrives at
the nal population has pra ti ally never been dis ussed in an MOEA study. On the ontrary,
mostsingleobje tiveEAstudiesanalyzesu hageneration-wiseperforman emeasureshowinghow
the average or best tness or some other performan e metri is varying with generation. Su h
an information provides a lot of insights to the working of an algorithm and helps de ide whi h
problemsarediÆ ultoreasy foranEA.Themainreasonfortheirabsen eintheMOEAliterature
is the ardinality of the ne essary performan e metri s to properly evaluate an MOEA and the
omplexitiesinvolvedin omputingthem.
However, some re ent studies (Abbass, 2002; Farhang-Mehr and Azarm, 2002; Lu and Yen,
2002)havejustbegunto showgeneration-wise dynami sofperforman emetri s. Inthispaper,we
demonstrate theuse of two su h runningperforman emetri s forMOEAsbyapplyingthem on a
numberoftest problemsand dis usssome interestingout omes.
3 Suggested Running Metri s
Parti ularlyfordesigningrunningmetri s,thefollowing propertiesare worth onsidering:
1. The metri shouldtake a value betweenzero and one inan absolute sense. Sin e themetri
isto be ompared generation-wise,anabsolute s aling ofarunningmetri betweenzero and
one willallowto assess the hangeof themetri valuefrom one generationto another.
2. The target (or desired) metri value ( al ulated foran ideally onverged and diversi ed set
of points)must be known.
3. Themetri shouldprovideamonotoni in reaseorde reaseinitsvalue,asthepopulationgets
improved ordeteriorated slightly. Thiswill also help in evaluating theextent of superiority
of oneapproximation setwith another.
4. Themetri shouldbes alabletoanynumberofobje tives. Althoughthisisnotanabsolutely
ne essary property, but if followed, it will ertainly be onvenient forevaluating s alability
issues ofMOEAsinterms of numberof obje tives.
5. The metri may be omputationally inexpensive, although this is not a stringent ondition
to befollowed.
Sin etwoindependentmetri saretobe hosen,ea hformeasuringoneofthetwofun tionalities
of multi-obje tive optimization, ea h metri may preferably evaluate the orresponding fun tion-
ality only, ignoringthe other fun tionality. Sin e two fun tionalities an not be measuredby one
suggested metri s,su h astheD1
R
metri (average distan e of referen e pointsfrom theapproxi-
mationset)suggestedbyCzyzakandJaszkiewi z(1998)for onvergen emeasureortheS-measure
(usedinZitzler(1999))fordiversitymeasure annotbeadequatelyused. Be auseof omputational
expensesinvolved in omputingtheR -metri s, they maynotbe suitable andidatesto be usedas
runningmetri s. In thefollowing two subse tions,we proposetwo di erent metri s forevaluating
MOEAsbykeepinginmindtheabove properties.
3.1 Metri for Convergen e
We usea simple metri forevaluating onvergen e towards a referen e set. A target set of points
P
(orreferen e set) an beeithera setof Pareto-optimal points(ifknown) orthenon-dominated
set of points ina ombined poolof all generation-wise populations obtained from an MOEA run.
Thereafter, for a population P (t)
at ea h generation, we ompute the onvergen e metri in the
followingmanner:
Step 1 Identifythenon-dominated setF (t)
of P (t)
.
Step 2 From ea h point i inF (t)
, al ulate the smallestnormalized Eu lidean distan e to P
as
follows:
d
i
= jP
j
min
j=1 v
u
u
t M
X
k=1 f
k (i) f
k (j)
f max
k f
min
k
!
2
: (6)
Here, f max
k
and f min
k
are the maximum and the minimum fun tion values of k-th obje tive
fun tioninP
.
Step 3 Cal ulate the onvergen e metri by averaging the normalized distan e for all points in
F (t)
:
C(P (t)
)= P
jF (t)
j
i=1 d
i
jF (t)
j
: (7)
In order to keep the onvergen e metri within[0;1℄, on e the above metri valuesare al ulated
for all generations, we normalize the C(P (t)
) values by its maximum value (usually C(P (0)
)):
C(P (t)
)=C(P (t)
)=C(P (0)
).
3.2 Metri for Diversity
In termsof measuring thediversityof solutions,a re ent study(Farhang-Mehr and Azarm, 2002)
suggested an entropy-based te hnique. Ea h obtained solution is proje ted on a suitable hyper-
plane 1
. Thereafter,a (M 1)-dimensional normallydistributedentropyfun tion isassigned with
its mean being on ea h proje ted point and with a user-de ned standard deviation. All su h
entropy fun tionsforallproje ted pointsareaddedtogether anda normalized entropyfun tionis
al ulated. Iftheproje ted pointsarewelldistributedon thehyper-planeand asuitablestandard
deviation of the entropy fun tion is hosen, the resulting normalized entropy fun tion will be a
atfun tion, thereby ausinga largevalue of theShannon'sentropy measure al ulatedusingthe
1
Aplanewithadire tionve torequaltotheunitve torofthelinejoiningtheidealpointandthepointwiththe
worstindividualobje tivevaluesofthereferen esetissuggestedintheoriginalstudy.
that points are rowded insome partsof theproje ted plane, the entropy measure willbe small,
thereby meaning a poor diversity among solutions. The idea of measuring diversity of solutions
usingasimulatedentropyfun tionismeaningful,buttheapproa hhasthefollowingshort omings:
1. The entropy measure largely depends on the hosen standard deviation, as the resulting
distributionbeingpeakyor atdependslargelyonthevarian eofthenormalentropyfun tion
used.
2. Sin ea ontinuousnormalentropyfun tionisused,ifaprobleminvolvesdis onne tedPareto-
optimalfronts, thismethodmaybeerroneous.
3. ForproblemswherethePareto-optimal front isadegenerated lesser-dimensional urve(su h
as DTLZ5 (Deb et al., 2001)), a sub-optimal front may produ e a higher entropy measure
than a set of Pareto-optimal points, thereby allo ating a large entropy value to an inferior
set.
4. A performan e metri measuring the diversity of solutions annot adequately measure the
onvergingabilityof an MOEA,and vi eversa.
The diversity metri suggested next is similar in on ept to the above metri , ex ept that it
attempts to alleviate most of the above diÆ ulties. Moreover, this and the onvergen e metri
suggestedearliertogether anbeusedto systemati allyevaluateMOEAsforboth onvergen eand
diversityproperties. Thesuggested metri is also omputationally fast.
The essential idea is that the obtained non-dominated points at ea h generation is proje ted
on a suitable hyper-plane, thereby losing a dimension of the points. The plane is divided into a
numberof smallgrids (or(M 1) dimensional boxes). Dependingon whether ea h grid ontains
an obtained non-dominated pointor not, a diversitymetri isde ned. If allgrids arerepresented
with at least one point, the best possible (with respe t to the hosen numberof grids) diversity
measure is a hieved. If some gridsare notrepresented by a non-dominated point, thediversityis
poor. The parameters required from the user are the dire tion osine of the referen e plane, the
numberofgrids(G
i
)in ea h of (M 1) dimension,andthe target (orreferen e) set ofpointsP
.
Here isthepro edure:
Step 1 From P (t)
,determinetheset F (t)
whi h arenon-dominated to P
.
Step 2 Forea h grid indexedby(i;j;:::), al ulate followingtwo arrays:
H(i;j;:::) = (
1; ifthegrid hasa representative point inP
;
0; otherwise.
(8)
h(i;j;:::) = (
1; ifH(i;j;:::)=1and thegrid hasa representative point inF (t)
;
0; otherwise.
(9)
Step 3 Assignavaluem(h(i;j;:::))toea hgriddependingonitsanditsneighbor'sh(). Similarly,
al ulatem(H(i;j;:::))usingH()forreferen e points.
to thatforH():
D(P (t)
)= P
i;j;:::
H(i;j;:::)6=0
m(h(i;j;:::))
P
i;j;:::
H(i;j;:::)6=0
m(H(i;j;:::))
: (10)
In the simple ase, the value fun tion m() for a grid an be omputed by using its h() and two
neighboringh()dimension-wise. Withasetofthree onse utivebinaryh()values,thereareatotal
of 8possibilities. Anyvalue fun tionmaybe assignedbykeepinginmindthefollowing:
A 111isthebest distributionand a000 istheworst.
A 010or a 101meansa periodi pattern with agood spreadand may bevalued more than
a 110ora 011. For example,the above valuation willmake an approximation setwith 50%
overage of grids but having a wider spread (su h as 1010101010) better than another set
havingthesame overage butwitha smallerspread(su has1111100000).
A 110ora011 maybe valued more thana 001ora 100,be ause of more overed grids.
Basedonaboveobservations, thefollowingm()valuesforh()aresuggestedandusedinthisstudy:
h(:::j 1:::) h(:::j::: ) h(:::j+1:::) m(h(:::j::: ))
0 0 0 0.00
0 0 1 0.50
1 0 0 0.50
0 1 1 0.67
1 1 0 0.67
0 1 0 0.75
1 0 1 0.75
1 1 1 1.00
Identi al value are usedfor H(). In the urrent study,two ormore dimensional hyper-planesare
handledby al ulating the above metri dimension-wise,whereas a higher-dimensional version of
theabovevaluefun tion analso be arefullydesignedby onsideringamovingsetofhyper-boxes.
One su h onsideration foratwo-dimensionalset of9 boxes is shownbelow:
is better than
Obviously,withmorenumberofobje tives,thevaluefun tionwillbediÆ ulttode ne. Itremains
as an interesting future study to ndif su h higher-dimensional hyper-boxes are really ne essary
ompared to thedimension-wise al ulationof theproposedone-dimensionalmetri .
As an illustration to the above al ulation pro edure, Figure 1 shows a set of target points
(marked as lled ir les)and a setof population points(marked as shaded and open boxes) fora
two-obje tive minimizationproblem. Thepointsmarkedwithshadedboxesarethenon-dominated
pointswithrespe tto thetarget pointsandareusedforthediversity al ulation (thisisStep1 of
thepro edure). Thef
2
=0 plane isusedas thereferen eplane here andthe ompleterangeof f
1
valuesaredividedinto G
1
=10 grids. In thenext step, forea h grid bothh() and H()valuesare
al ulated. Fortheboundarygrids(extremegrids andgrids (:::;j;:::) withH(:::;j 1;:::)=0
f
f
2
1
1 1 0 0 1 0 1 1 0 0
h() 1 1
0.67 0.67 0.50 0.75 0.75
0.50 0.67 0.50
1.00 0.50
m() = 6.50
1 1 1 1 1 1 1 1 1 1
H()
1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00
1.00
m() 1.00
1 1
= 10.00
Diversity Measure = 6.50/10.00 = 0.65
Figure 1: Thediversitymetri omputationis illustrated.
or H(:::;j+1;:::)=0), an imaginary neighboringgrid with a h() orH() value of one is always
assumed. Inthe gure,these gridsare shown indashed boxes. The h()valuesare one inthe rst
two grids, fth, seventh and eighth grids. Noti ethat althoughmore thanone point mayexist in
a grid,theH() orh() value is stillone forthat grid. Based on a movingwindow ontaining three
onse utivegrids, them() valuesare omputedinthe gureand thediversitymetri is al ulated.
To avoid the boundary e e ts (the e e t of using the imaginary grids), we normalize the above
metri asfollows:
D(P (t)
)= P
i;j;:::
H(i;j;:::)6=0
m(h(i;j;:::)) P
i;j;:::
H(i;j;:::)6=0 m(0 )
P
i;j;:::
H(i;j;:::)6=0
m(H(i;j;:::)) P
i;j;:::
H(i;j;:::)6=0 m(0 )
;
where0 isa zero-valuedarray. A arefulthought willrevealthattheH(i;j;:::)6=0 onsideration
in omputingtheD(P (t)
)termandtheboundarygridadjustmentsuggested above allowageneri
wayto handleproblemswithdis onne tedPareto-optimal fronts. The metri doesnotin ludethe
value fun tion for a grid on whi h there exists no referen e solution. We have in luded one su h
test problem(ZDT3)inoursimulationstudies.
IfthePareto-optimalfrontisnotknown(parti ularlytoreal-worldproblems),thetargetsetmay
bedetermined inthefollowingway. First,an MOEAis runforT generations and thegeneration-
wisepopulations(P (t)
,t=0;1;:::;T) arestored. Thereafter,thenon-dominatedmembersF (t)
of
ea h population are ombined together and the target set is de ned as the non-dominated set of
the ombinedpopulation: P
=Non-dominated([
T
t=0 F
(t)
).
Convergence
Diversity
0 0.2 0.4 0.6 0.8 1
0 10 20 30 40 50 60 70 80 90 100
Metrics
Generation Number ZDT1
Figure 2: Two runningmetri sforZDT1 usinga setof Pareto-optimal points.
4 Simulation Results
Inthisse tion,we applytheabove metri sonsolutionsobtainedusingNSGA-II(Debetal.,2000)
onanumberoftestproblemssuggestedintheliteratureandonareal-worldtwo-obje tive gearbox
designproblem. In ea h ase, the metri sare appliedon a typi al runof NSGA-II. However, it is
advisedto applythemetri sonmultiplerunsofanMOEAandto reportaverage metri values. In
all ases with NSGA-II, we have used theSBX re ombination operator(Deb, 2001) withp
=1:0
and
=10,and thepolynomialmutationoperatorwith p
m
=1=nand
m
=20.
First four test problems are two-obje tive test problems hosen from Zitzler, Deb and Thiele
(2000). Sin e they are test problems and the exa t lo ationof the Pareto-optimal front is known
in ea h ase, we apply the suggested metri s using a set of 200 uniformly set of points on the
Pareto-optimal front (intheobje tive spa e).
Figure 2 shows the two metri s on simulation results obtained usingNSGA-II using N = 100
populationsizerunningtill100generations. Inthisandotherproblemsofthisstudy,wesetnumber
of gridsinea h dimension asG
i
=N
1=(M 1)
. Thus,fortwo-obje tive problems, we have usedthe
number of grids equal to the population size, as if ea h population member is expe ted on ea h
grid. Wealsousef
2
=0planetoproje tthepoints,unlessotherwisestated. The gureshowsthat
the onvergen e metri qui kly moves to zero, thereby implying that NSGA-II solutions starting
from a random set of solutions qui kly approa h the Pareto-optimal front. A value of zero of
the onvergen e metri impliesthat allnon-dominated solutions mat h the hosen Pareto-optimal
points. Afterabout30generations,theNSGA-IIpopulation omesvery losetothePareto-optimal
front. Similarly,thediversitymetri shows thattillaboutthe rst26generations,nosolutionnon-
dominatedtothePareto-optimalset wasfound. Butfrom thisgenerationonwards,NSGA-II nds
more andmore pointsnon-dominatedwiththe hosen Pareto-optimal points. The hoi e ofpoints
aresu hthatthediversitymetri in reasesexponentiallytillabout70generations, afterwhi hthe
diversityremains more or less the same. Although the obtained pointsare very lose the hosen
Pareto-optimalpoints, diversitymetri os illatesnear astablevalue,a matterwhi his inherentto
Convergence
Diversity
0 0.2 0.4 0.6 0.8 1
0 10 20 30 40 50 60 70 80 90 100
Metrics
Generation Number ZDT2
Figure 3: Two runningmetri sforZDT2 usinga setof Pareto-optimal points.
NSGA-II and other MOEAs and wasalso dis ussed for NSGA-II and SPEA2 in an earlier study
(Laumanns etal.,2001).
It isinterestingto notethatthediversitymeasuresuggested hereusesonlysolutionswhi hare
non-dominated with thereferen e set. This maymean to suggest that onvergen e properties are
somehow in-built into thisdiversitymeasure, but thisis not quite true. Noti e how the diversity
measure is onsistentlyequal to zero tillabout 26 generations, untilwhi h time no solutionsnon-
dominatedto the referen e set is found. On the other hand, the onvergen e metri duringthese
early generations shows how the population has been onsistently approa hing the referen e set.
Thus,a ombination of thesetwo metri s providea lear insight into theworkingof an MOEA.
Next, we hoosethetest problemZDT2. Thisproblemhasanon- onvex Pareto-optimal front.
Figure3 showsthetwo runningmetri swiththesameGA parametersasbefore. The onvergen e
metri showsinabout30generations,NSGA-IIpopulationmovesvery losetothePareto-optimal
front. Itisalso interestingtonotethatNSGA-IItakesabout10moregenerationsto ndasolution
non-dominated with thereferen e set in ZDT2 than thatrequired inZDT1. On e some solutions
very losetothePareto-optimalfrontarefound,moreandmorepointsnearthefrontaredis overed
and the diversityamong solutionsin reases rapidly,indi atingthat NSGA-II nds solutions lose
to thePareto-optimal front witha good diversityamong them.
The test problemZDT3 hasa disjointedset of Pareto-optimal fronts. NSGA-II with identi al
parameter setting are applied to thisproblemfor 100 generations and the orresponding running
metri s are shown in Figure 4. The gure shows a similar behavior as that in ZDT1. In this
problem, we have also omputed thediversitymetri byproje tingthe pointson an in linedline,
as suggested in the footnote in se tion 3.2. The orresponding diversity metri is plotted with a
solidline. The gureshowsthatalthoughthereissomedi eren einmagnitudeofthetwodiversity
metri omputations(proje tedonf
2
=0planeandproje tedonthein linedline),theirbehaviors
arevery similar.
Next, we hoose the test problemZDT6, whi h has a non-uniform density of solutions along
thePareto-optimal front. This time, theNSGA-II isrunfor300 generations. Figure5 shows that
Diversity (projected) Diversity
(on f_1)
Convergence
0 0.2 0.4 0.6 0.8 1
0 10 20 30 40 50 60 70 80 90 100
Metrics
Generation Number ZDT3
Figure 4: Two runningmetri s forZDT3 usinga set of Pareto-optimal points al ulated usingf
1
axisand proje tedon an in linedplane.
Convergence
Diversity 0
0.2 0.4 0.6 0.8 1
0 50 100 150 200 250 300
Metrics
Generation Number ZDT6
Figure 5: Tworunningmetri s forZDT6 using
a setof Pareto-optimal points.
Convergence
Diversity
0 0.2 0.4 0.6 0.8 1
0 50 100 150 200 250 300
Metrics
Generation Number ZDT6
Figure 6: Two running metri s for ZDT6 us-
ing an agglomeration of generation-wise popu-
lations.
thepopulationsteadilyapproa hesthePareto-optimalfront,butalwaysmaintainsa nitedistan e
from the true Pareto-optimal front. Sin e almost all obtained solutions are not non-dominated
withthe hosen Pareto-optimal points,thediversitymetri is al ulatedwithonlyafewsolutions,
thereby having a very smallvalue. For problemswhi hare diÆ ultto solve to Pareto-optimality,
wemayuseanagglomerationofpopulations(dis ussedinthelastparagraphofse tion3.2),instead
of Pareto-optimal points,to ompute theabove metri s. Figure6 showsthat NSGA-IIpopulation
rea hes its nal non-dominated front after around 165 generations. After this generation, the
populationqui kly onverges to the nalnon-dominatedfront andthepopulationdiversityresorts
into a noisy steady-state value. This study shows that when it is diÆ ult to obtainsolutions on
Diversity
Convergence 0
0.2 0.4 0.6 0.8 1
0 50 100 150 200 250 300 350 400 450 500
Metrics
Generation Number Gearbox Design Problem
Figure 7: Two runningmetri sshowing the onvergen eand diversityinsolutionsof NSGA-II.
the true Pareto-optimal front, it is better to use the population-agglomeration te hnique. This
te hnique an alsobe usedfor omparingtwo ormore MOEAs.
Next, we use the population-agglomeration te hnique to evaluate the performan e of NSGA-
II to a two-obje tive gearbox design problem dis ussed elsewhere (Deb and Jain, 2002). In this
problem,thePareto-optimalfrontisnotknown. With100gridpointsonf
1
,themetri sareplotted
inFigure7forNSGA-IIrunusing100populationmembers. The gureshowsthatthe onvergen e
near the obtained non-dominated front is fast and the rst point near this front was found after
about65generations. Itisinterestingtonotethatatthe nalgenerationthediversitymetri value
is less thanone. This meansthat NSGA-II does not ontain all 100 solutions used as a referen e
set at theendofthesimulationrun. Some better distributedsolutionsarelostduringtherun.
The naltwotest problemsarethree-obje tivetest problemsborrowed fromDebetal. (2001).
TheproblemDTLZ2hasaspheri alPareto-optimalsurfa e. Here,weuse100populationmembers
and NSGA-II and SPEA2 (Zitzler, Laumanns,and Thiele, 2001) arerun for300 generations. For
omputingthemetri s,wehave hosen10 10or100gridsintheinterval[0;1℄off
1 andf
2
axes. The
f
3
=0planeis hosentoproje tthepoints. Figure8showsthe orrespondingmetri s,astheyvary
withthe generationnumber. The onvergen e metri valuesforbothMOEAssuggestthat notall
non-dominatedpointseven atthe nalgenerationhave rea hed losetothe hosenPareto-optimal
points. The diversity metri values show that both NSGA-II and SPEA2 have found a solution
non-dominated to the hosen Pareto-optimal pointsat around 20th generation. Thereafter, more
and more pointswith in reasing diversityamong them have beenfound. Thereafter, like in ZDT
problems, the populationmaintains a parti ular levelof diversitywith some variations. However,
thediversityofsolutionsobtainedbySPEA2ismu hbetter thanthatofNSGA-IIinthisproblem.
Inorder to investigate thediversityobtainedinea h ase, we haveplotted the nalobtainednon-
dominated solutions in Figures 9 and 10 for NSGA-II and SPEA2, respe tively. It is lear from
these two plots that SPEA2has obtaineda better spread thanNSGA-II. The lustering operator
of SPEA2 is more omputationally expensive than the rowding operator used in NSGA-II, but
SPEA2isbetterabletodistributepoints. Figure8revealsthelatterfa tquantitativelybyshowing
Diversity (NSGA−II)
Convergence (SPEA2)
Diversity (SPEA2)
Convergence (NSGA−II)
0 0.2 0.4 0.6 0.8 1
0 50 100 150 200 250 300
Metrics
Generation Number DTLZ2 (Three Objectives)
Figure 8: Two runningmetri s for DTLZ2using a set of Pareto-optimal points for NSGA-II and
SPEA2.
0 0.25 0.5 0.75 1 1.25
f1 0
0.25 0.5
0.75 1
1.25 f2
0 0.25 0.5 0.75 1
f3
Figure9: Non-dominatedpointsobtainedusing
NSGA-II.
0 0.25 0.5 0.75 1 1.25
f1 0
0.25 0.5
0.75 1
1.25 f2
0 0.25 0.5 0.75 1
f3
Figure 10: Non-dominatedpoints obtained us-
ingSPEA2.
SPEA2having abetter diversitymetri value thanNSGA-II notonlyat the nal generation, but
in the entire run of the two algorithms; however the onvergen e levels of both algorithms are
more or less thesame. The u tuations inthe diversity metri bybothmethods arisedue to the
dynami loss and gain of riti al points on the non-dominated solutions, on e they ome loseto
the referen e set. In any ase, thisstudyshows how thetwo suggested metri s an bring outthe
di eren es intheworking of twoMOEAsquantitativelyover theentire runofthealgorithms.
Our next problemisDTLZ5, whi hhas aPareto-optimal urve,instead ofathree-dimensional
Pareto-optimal surfa e. Here, we run NSGA-II for 200 generations, whereas all other NSGA-II
parametersand parametersfor omputingthemetri sarethesameasinDTLZ2. Figure11shows
the onvergen eanddiversitymetri sofasinglerunofNSGA-II.The onvergen emetri depi tsa
good onvergen eofNSGA-IIonthePareto-optimal front. Sin ethenumberofgridsinvolvingthe
Diversity (10X10)
Diversity (50X50)
Convergence 0
0.2 0.4 0.6 0.8 1
0 20 40 60 80 100 120 140 160 180 200
Metrics
Generation Number DLTZ5
Figure 11: Two runningmetri sforDTLZ5usinga setof Pareto-optimal points.
Pareto-optimalfrontisfewinthisproblem omparedto thetotal numberofgridsbeing onsidered
inthe diversity omputation(due to the redu eddimensionalityof thePareto-optimal front), the
diversitymetri hangesin nitesteps. Theattainmentofadiversitymetri valueofunitysuggests
theperfe tdiversitybeinga hievedwith10gridsinea hobje tive. When50gridsonea hobje tive
(f
1 and f
2
) axis are hosen, thePareto-optimal front is represented by more grids. The diversity
metri forthis aseisalsoshowninthe gure. Ahyper-planewithmoregridsrequiremoreobtained
solutions to attain a good diversity measure. It is not surprisingthat the diversity metri values
degrade inthis ase from before. Thisillustrates thesensitivityof the suggested diversitymetri
on the hosennumberofgrids.
5 Dis ussions and Future Studies
Theneedofdevelopinggoodrunningmetri sandtheiruseinMOEAstudiesisnow learfromthe
above omputersimulations. Althoughtwometri sareusedinthisstudy,moremetri smeasuring
furtherdetailedpropertiesofanapproximationset analsobeusedand areamatterofimmediate
resear hinterests.
Although the suggested template-based diversity metri is a way to measure the extent of
diversity in an approximation set, there are at least a ouple of diÆ ulties with this metri : (i)
the metri value depends on the hosen template (the value fun tionm()) and (ii) as mentioned
earlier it is diÆ ult to assign a value fun tion for higher dimensions, although a dimension-wise
appli ationhasreasonablyworkedwellinthispaper. However, theoverallstrategyofproje tingan
approximationsetto asuitablegrid-edhyper-planeandthenapplyingadiversitymetri remainsa
goodapproa hfor ndingthediversityofanapproximationset. Insteadofthesuggestedtemplate-
based diversitymetri , thefollowingmetri s an also beused:
1. A grid- ount diversitymetri ounting the numberof o upied gridsin the proje ted plane
an simply be used. To normalizethe metri ,the ount an be dividedbythe total number
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 10 20 30 40 50 60 70 80 90 100
Different Diversity Metrics
Generation Number D2 D3
D1 ZDT1
Figure 12: Three diversity running metri s on
ZDT1 are shown. D1 is the metri de ned in
se tion3.2, D2isthegrid- ount metri andD3
isthe varian e-metri V.
f_1
Generation 49 (7 grids) Generation 50 (11 grids) Generation 51 (10 grids)
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Figure 13: The grids o upied by NSGA-II at
generations49,50,and51areshownforZDT1.
of o upied grids obtained using the referen e set. Although this metri does not tell the
sparseness of o upied grids, the sheer number of them may be an important fa tor for
omparing two or more MOEAs. Figure 12 plots this normalized grid- ount metri (shown
as D2) on thepopulations obtainedusing NSGA-II inproblem ZDT1. The plot shows that
NSGA-II solutionsshare about70%gridswiththe referen eset at theendof therun.
2. A varian e in inter-member distan es an be another measure of diversityof solutions on a
proje ted plane. The al ulation pro edure is as follows. For a population P (t)
, rst the
non-dominated solutions F (t)
(with respe t to P
) are found and are proje ted to a hosen
hyper-plane. Ea h o upied grid on the hyper-plane is represented by its enter point ~x (i)
.
First, the entroid~g of all o upied gridsis al ulated. Thereafter, the ve tor ~v (i)
from the
grid~x (i)
to the entroid~g is omputed. Then,a diversitymetri V is al ulated asfollows:
V(P (t)
)= M 1
X
j=1 v
u
u
u
t jF
(t)
j
X
i=1
v (i)
j
2
: (11)
For ea h dimension j, the term inside the square-root is the sum of the square of the pro-
je tions of all solutions from the entroid along the j-th dimension. Thus, if the solutions
are sparsely pla ed, the above metri willprodu e a large value. Sin e the proje tions are
squared andadded, thepositive and negative proje tionsdo not an elea h other.
In order to normalize themetri , the above value an be dividedbythe metri value of the
referen e set: V =V(P (t)
)=V(P
). Figure 12 also shows aplot of thismetri (shown asD3)
on ZDT1. The omparison of these three plots shows the similarityin their variations with
intunewiththeundulationsinthenumberofo upiedgrids(D2). An interestingdi eren e
amongthesethree metri sis learduringthetransitionofthepopulationfromgeneration49
through51:
Generation Numberof Diversity Metri s
number o upiedgrids D1 D2 D3
49 7 8.754/100 7/100 0.192/2.887
50 11 14.899/100 11/100 0.405/2.887
51 10 13.468/100 10/100 0.891/2.887
Figure 13 also shows the o upied grids at generations 49, 50, and 51. A total of 100 grids
areassumed forf
1
2[0;1℄. The grid- ount (D2) metri forthese generationsdoesnotreveal
the fa t that a better diversity of solutions is obtained in generation51, although thegrid-
ount is less in generation 51 ompared to that in generation 50. Be ause of the use of
the three-grid template inD1, thedi eren e in diversity obtained from generation 50 to 51
is not apparent by this metri either. But the D3 metri shows a big jump in this metri
value during this transition, meaning that a set with a mu h better diversity among them
is obtained. However, the real di eren e among these metri s may be apparent in solving
higher-dimensional or more omplex problems and the hoi e of one metri over the other
mayalso dependon the omputational omplexityasso iatedwithea h metri .
3. Theentropy-basedmetri suggested byFarhang-Mehrand Azarm(2002) an be modi edto
handledis onne tedPareto-optimal sets. The holes ordis ontinuitiesintheproje ted plane
anbeeliminatedandallindividualregionsofPareto-optimalsets anbepla ednextto ea h
other,sothatinthemodi edplane,there isnoholeordis ontinuity. Althoughthe hoi e of
thestandarddeviation ofthe normaldistributionstillremainsasan importantfa tor, other
distributionsmayalso betried.
Adetailstudyapplyingthesemetri stothe hosentestproblemsand omparingthemwithexisting
metri ssu hasthe R -metri sortheS-metri also remainsanimportant futuretask.
6 Con lusions
In order to investigate the generation-wise dynami s of an MOEA, two metri s are suggested in
this paper. The rst metri is a distan e measure of a population from a referen e set and is
used to measure the onvergen e ability of an MOEA. The se ond metri uses a lo al template
based evaluation te hnique to estimate the diversityof one set omparedto a referen e set. Both
metri s are normalized so as to have their values bounded between zero and one. On six test
problems and one real-world engineering design problem, the appli ation of these two metri s
have showninterestingpropertiesof NSGA-II. The non-dominatedsolutions ontinually approa h
the referen e set in all ases, meaning thereby that the domination-based sele tion operator of
NSGA-II is apableof bringingthepopulationnear thereferen e setwithoutgetting stu kin any
intermediatelevel. On esomepointsnearthereferen esetaredis overed,thesear hoperatorsand
thediversity-preservingme hanismofNSGA-IIareableto ndmoreandmoresolutionswhi hare
test problem, NSGA-IIand SPEA2 populations are ompared generation-wise,revealing superior
performan eofSPEA2 inmaintaininga betterdiversityamong non-dominatedsolutions.
Forproblemswith unknownPareto-optimal front, thisstudyhas alsosuggested away to on-
stru t a referen e set based on an agglomeration of generation-wise non-dominated populations.
Su ha te hnique an alsobeusedto omputeother performan emetri srequiringareferen eset.
Itisalsoimportanttohighlightthatthe omputational omplexityofMOEAsisanotherimpor-
tantmatterandmustbetakeninto onsiderationwhile omparingtwoormoreMOEAs. Neverthe-
less,theresultsofthisstudyhasshowntheimportan eofusingrunningmetri sformulti-obje tive
evolutionary omputation andre ommend more su h studiesinthe nearfuture.
A knowledgement
Authorsthank Mar o Laumannsforsharing theSPEA2resultson test problemDTLZ2.
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