Multiobjective Genetic Algorithms:Problem
Dif®culties and Construction of Test Problems
Kalyanmoy Deb
Kanpur Genetic Algorithms Laboratory (KanGAL)
Department of Mechanical Engineering
Indian Institute of Technology Kanpur
Kanpur,PIN208 016,India
deb@iitk.ac.in
Abstract
In this paper,we study the problem features that may cause a multiobjective genetic
algorithm (GA) dif®culty in converging to the true Paretooptimal front.Identi®cation
of such features helps us develop dif®cult test problems for multiobjective optimiza
tion.Multiobjective test problems are constructed from singleobjective optimization
problems,thereby allowing known dif®cult features of singleobjective problems (such as
multimodality,isolation,or deception) to be directly transferred to the corresponding
multiobjective problem.In addition,test problems having features speci®c to multi
objective optimization are also constructed.More importantly,these dif®cult test prob
lems will enable researchers to test their algorithms for speci®c aspects of multiobjective
optimization.
Keywords
Genetic algorithms,multiobjective optimization,niching,paretooptimality,problemdif
®culties,test problems.
1 Introduction
After a decade since the pioneering work by Schaffer (1984),a number of studies on multi
objective genetic algorithms (GAs) have emerged.Most of these studies were motivated
by a suggestion of a nondominated GA outlined in Goldberg (1989).The primary reason
for these studies is a unique feature of GAsÐa population approachÐthat is highly suitable
for use in multiobjective optimization.Since GAs work with a population of solutions,
multiple Paretooptimal solutions can be found in a GA population in a single simulation
run.During the years 199395,a number of independent GA implementations (Fonseca
and Fleming,1993;Horn et al.,1994;Srinivas and Deb,1995) emerged.Later,other
researchers successfully used these implementations in various multiobjective optimization
applications (Cunha et al.,1997;Eheart et al.,1993;Mitra et al.,1998;Parks and Miller,
1998;Weile et al.,1996).A number of studies have also concentrated on developing new
GA implementations (Kursawe,1990;Laumanns et al.,1998;Zitzler and Thiele,1998).
Fonseca and Fleming (1995) and Horn (1997) presented overviews of different multi
objective GA implementations,and Van Veldhuizen and Lamont (1998) made a survey of
test problems that exist in the literature.
Despite these interests,there seems to be a lack of studies discussing problem fea
tures that may cause dif®culty for multiobjective GAs.The literature also lacks a set of
c
1999 by the Massachusetts Institute of Technology Evolutionary Computation 7(3):205230
K.Deb
test problems with known and controlled dif®culty measure for systematically testing the
performance of an optimization algorithm.Studies seeking problem features that cause
dif®culty for an algorithmmay seema pessimist's job,but we feel that the true ef®ciency of
an algorithmis revealed when it is applied to challenging test problems,not easy ones.Such
studies in singleobjective GAs (studies ondeceptive test problems,NK`rugged'landscapes,
and others) have all enabled researchers to better understand the working of GAs.
In this paper,we attempt to highlight a number of problemfeatures that may cause a
dif®culty for a multiobjective GA.Keeping these properties in mind,we showprocedures
for constructing multiobjective test problems with controlled dif®culty.Speci®cally,there
exist some features shared by a multiobjective GA and a singleobjective GA.Our con
struction of multiobjective problems fromsingleobjective problems allowsuch dif®culties
to be directly transferred to an equivalent multiobjective GA.Some speci®c dif®culties of
multiobjective GAs are also discussed.
We also discuss and de®ne local and global Paretooptimal solutions.We show the
construction of a simple twovariable,twoobjective problemfrom singlevariable,single
objective problems and show how multimodal and deceptive multiobjective problems
may cause dif®culty for a multiobjective GA.We present a tunable twoobjective prob
lem of varying complexity constructed from three functionals.Speci®cally,a systematic
construction of multiobjective problems having convex,nonconvex,and discontinuous
Paretooptimal fronts is demonstrated.We then discuss the use of parameterspace versus
functionspace based niching and suggest which one to use when.Finally,future challenges
in the area of multiobjective optimization are discussed.
2 Paretooptimal Solutions
As the name suggests,Paretooptimal solutions are optimal in some sense.Therefore,like
singleobjective optimization problems,there exist possibilities of having both local and
global Paretooptimal solutions.Before we de®ne both these types of solutions,we discuss
dominated and nondominated solutions.
For a problem having more than one objective function (say,
,
and
),a solution
is said to dominate the other solution
if both the following
conditions are true (Steuer,1986):
1.The solution
is no worse (say the operator
denotes worse and
denotes better)
than
in all objectives,or
for all
objectives.
2.The solution
is strictly better than
in at least one objective,or
for at least one
.
If any of the above conditions is violated,the solution
does not dominate the
solution
.If
dominates the solution
,it is also customary to write
is
dominated by
,or
is nondominated by
.
The above concept can also be extended to ®nd a nondominated set of solutions in
a population of solutions.Consider a set of
solutions,each having
(
) objective
function values.The following procedure can be used to ®nd the nondominated set of
solutions:
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Evolutionary Computation Volume 7,Number 3
MultiObjective GAs
Step 0:Begin with
.
Step 1:For all
,compare solutions
and
for domination using the above two
conditions for all
objectives.
Step 2:If for any
,
is dominated by
,mark
as`dominated'.Increment
by
one and Go to Step 1.
Step 3:If all solutions (that is,when
is reached) in the set are considered,Go to
Step 4,else increment
by one and Go to Step 1.
Step 4:All solutions that are not marked`dominated'are nondominated solutions.
A population of solutions can be classi®ed into groups of different nondomination levels
(Goldberg,1989).When the above procedure is applied for the ®rst time in a population,
the resulting set is the nondominated set of ®rst (or best) level.In order to have further
classi®cations,these nondominated solutions can be temporarily omitted fromthe original
set and the above procedure can be applied again.What results is a set of nondominated
solutions of second (or nextbest) level.This new set of nondominated solutions can be
omitted and the procedure applied again to ®nd the thirdlevel nondominated solutions.
This procedure can be continued until all population members are classi®ed into a non
dominated level.It is important to realize that the number of nondomination levels in a
set of
solutions is bound to lie within
.The minimumcase of one nondomination
level occurs when no solution dominates any other solution in the set,thereby classifying all
solutions of the original population into one nondominated level.The maximum case of
nondomination levels occurs when there is a hierarchy of domination of each solution
and no two solutions are nondominated by each other.
In a set of
arbitrary solutions,the ®rstlevel nondominated solutions are candidates
for possible Paretooptimal solutions.The following de®nitions determine whether they
are local or global Paretooptimal solutions:
Local Paretooptimal Set:If for every member
in a set
there exists no solution
satisfying
,where
is a small positive number (in principle,
is obtained
by perturbing
in a small neighborhood) dominating any member in the set
,then
the solutions belonging to the set
constitute a local Paretooptimal set.
Global Paretooptimal Set:If there exists no solution in the search space that dominates
any member in the set
,then the solutions belonging to the set
constitute a global
Paretooptimal set.
We describe the concept of local Paretooptimal solutions inFigure 1,where bothobjectives
and
are minimized.By perturbing any solution in the local Paretooptimal set
(solutions marked by`x') in a small neighborhood in the parameter space,it is not possible
to obtain any solution that would dominate any member of the set.
The size and shape of Paretooptimal fronts usually depend on the number of objective
functions and interactions among the individual objective functions.If the objectives are
`con¯icting'to each other,the resulting Paretooptimal front may have a larger span than
if the objectives are more`cooperating'
.However,in most interesting multiobjective
The terms`con¯icting'and`cooperating'are used loosely here.If two objectives have similar individual
optimum solutions and similar individual function values,they are`cooperating',as opposed to a`con¯icting'
situation where both objectives have drastically different individual optimumsolutions and function values.
Evolutionary Computation Volume 7,Number 3
207
K.Deb
f
f
1
2
Space
x
x
x
x
x
Parameter
Global
Local
x
Figure 1:The illustrated concept of local and global Paretooptimal sets.
optimizationproblems,the objectives are`con¯icting'toeachother andusually the resulting
Paretooptimal front (local or global) contains many solutions.
3 Principles of Multiobjective Optimization
It is clear fromthe above discussionthat a multiobjective optimizationproblemusuallyhas a
set of Paretooptimal solutions,instead of one single optimal solution
.Thus,the objective
in a multiobjective optimization is different from that in a singleobjective optimization.
In multiobjective optimization the goal is to ®nd as many different Paretooptimal (or
near Paretooptimal) solutions as possible.Since classical optimization methods work with
a single solution in each iteration (Deb,1995),in order to ®nd multiple Paretooptimal
solutions they are required to be applied more than once,hopefully ®nding one distinct
Paretooptimal solution each time.Since GAs work with a population of solutions,a num
ber of Paretooptimal solutions can be captured in one single run of a multiobjective GA
with appropriate adjustments to its operators.This aspect of GAs makes them naturally
suited to solving multiobjective optimizationproblems for ®nding multiple Paretooptimal
solutions.Thus,it is no surprise that a number of different multiobjective GA implemen
tations exist in the literature (Fonseca and Fleming,1995;Horn et al.,1994;Srinivas and
Deb,1995;Zitzler and Thiele,1998).
Before we discuss the problemfeatures that may cause multiobjective GAs dif®culty,
let us mention a couple of matters
that are not addressed in the paper.First,we consider
all objectives to be of minimization type.It is worth mentioning that identical properties
as discussed here may also exist in problems with mixed optimization types (some are min
imization and some are maximization).The concept of nondomination among solutions
addresses only one type of problem.The meaning of`worse'or`better',discussed in Sec
tion 2,takes care of other cases.Second,although we refer to multiobjective optimization
throughout the paper,we restrict ourselves to two objectives.This is because we be
lieve that the twoobjective optimization brings out the essential features of multiobjective
optimization.
There are two tasks that a multiobjective GA should accomplish in solving multi
In multimodal function optimization,there may exist more than one optimal solution,but usually the interest
is to ®nd global optimal solutions having identical objective function value.
A number of other matters which need immediate attention are also outlined in Section 7.
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MultiObjective GAs
objective optimization problems:
1.Guide the search towards the global Paretooptimal region,and
2.Maintain population diversity (in the function space,parameter space,or both) in the
current nondominated front.
We discuss the above two tasks in the following subsections and highlight when a GAwould
have dif®culty in achieving each task.
3.1 Dif®culties in Converging to Paretooptimal Front
Convergence to the true (or global) Paretooptimal front may not occur because of various
features that may be present in a problem:
1.Multimodality,
2.Deception,
3.Isolated optimum,and
4.Collateral noise.
All the above features are known to cause dif®culty in singleobjective GAs (Deb et al.,
1993) and,when present in a multiobjective problem,may also cause dif®culty for a multi
objective GA.
In tackling a multiobjective problemhaving multiple Paretooptimal fronts,a GA,like
many other search and optimizationmethods,may converge to a local Paretooptimal front.
Later,we create a multimodal multiobjective problem and show that a multiobjective
GAcan get stuck at a local Paretooptimal front if appropriate GAparameters are not used.
Despite some criticism (Grefenstette,1993),deception,if present in a problem,has
been shown to cause GAs to be misled towards deceptive attractors (Goldberg et al.,1989).
There is a difference between the dif®culties caused by multimodality and by deception.
For deception to take place,it is necessary to have at least two optima in the search space
(a true attractor and a deceptive attractor),but almost the entire search space favors the
deceptive (nonglobal) optimum.Multimodality may cause dif®culty for a GA merely
because of the sheer number of different optima where a GA can stick.We shall show how
the concept of singleobjective deceptive functions can be used to create multiobjective
deceptive problems,which may cause dif®culty for a multiobjective GA.
There may exist some problems where the optimum is surrounded by a fairly ¯at
search space.Since there is no useful information provided by most of the search space,no
optimization algorithm will perform better than an exhaustive search method to ®nd the
optimum in these problems.Multiobjective optimization methods also face dif®culty in
solving such a problem.
Collateral noise comes from the improper evaluation of loworder building blocks
(partial solutions which may lead towards the true optimum) due to the excessive noise
coming from other parts of the solution vector.These problems are usually`rugged'with
relatively large variation in the function landscape.Multiobjective problems having such
`rugged'functions may also cause dif®culties for multiobjective GAs if adequate population
size (adequate to discover signal fromthe noise) is not used.
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209
K.Deb
3.2 Dif®culties in Maintaining Diverse Paretooptimal Solutions
As it is important for a multiobjective GA to ®nd solutions near or on the true Pareto
optimal front,it is alsonecessary to®ndsolutions as diverse as possiblein the Paretooptimal
front.If most solutions found are con®ned in a small region near or on the true Pareto
optimal front,the purpose of multiobjective optimization is not served.This is because,
in such cases,many interesting solutions with large tradeoffs among the objectives and
parameter values may have been undiscovered.
Inmost multiobjective GAimplementations,a speci®c diversitymaintaining operator,
such as a niching technique (Deb and Goldberg,1989) or a clustering technique (Zitzler
and Thiele,1998) is used to ®nd diverse Paretooptimal solutions.However,the following
features might be likely to cause a multiobjective GA dif®culty in maintaining diverse
Paretooptimal solutions:
1.Convexity or nonconvexity in the Paretooptimal front,
2.Discontinuity in the Paretooptimal front,and
3.Nonuniformdistribution of solutions in the Paretooptimal front.
There exist multiobjective problems where the resulting Paretooptimal front is non
convex.Although it may not be apparent,a GA's success in maintaining diverse Pareto
optimal solutions largely depends on the ®tness assignment procedure.In some GA imple
mentations,the ®tness of a solution is assigned proportionally to the number of solutions
it dominates (Fonseca and Fleming,1993;Zitzler and Thiele,1998).Figure 2 shows how
such a ®tness assignment favors intermediate solutions,in the case of problems with convex
Paretooptimal front (the left ®gure).With respect to an individual champion
solution
f
f
1
2
f
f
1
2
(a) (b)
Figure 2:The ®tness assignment proportional to the number of dominated solutions (the
shaded area) favors intermediate solutions in convex Paretooptimal front (a),compared to
that in nonconvex Paretooptimal front (b).
(marked with a solid dot in the ®gures),the proportion of dominated region covered by an
intermediate solution is more in Figure 2(a) than in Figure 2(b).Using such a GA (with
GAs favoring solutions having more dominated solutions),there is a natural tendency to
Optimumsolution corresponding to an individual objective function.
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MultiObjective GAs
®nd more intermediate solutions than solutions near individual champions,thereby causing
an arti®cial bias towards some portion of the Paretooptimal region.
In some multiobjective optimization problems,the Paretooptimal front may not
be continuous,instead it may be a collection of discretely spaced continuous subregions
(Poloni et al.,in press;Schaffer,1984).In such problems,although solutions within each
subregion may be found,competition among these solutions may lead to extinction of
some subregions.
It is also likely that the Paretooptimal front is not uniformly represented by feasible
solutions.Some regions in the front may be represented by a higher density
of solutions
than other regions.In such cases,there may be a natural tendency for GAs to ®nd a biased
distribution in the Paretooptimal region.
3.3 Constraints
In addition to the above,the presence of`hard'constraints in a multiobjective problem
may cause further dif®culties.Constraints may hinder GAs from converging to the true
Paretooptimal region and they may also cause dif®culty in maintaining a diverse set of
Paretooptimal solutions.It is intuitive that the success of a multiobjective GA in tackling
both these problems will largely depend on the constrainthandling technique used.Tradi
tionally,a simple penaltyfunction based method has been used to penalize each objective
function (Deb and Kumar,1995;Srinivas and Deb,1995;Weile et al.,1996).Although suc
cessful applications are reported,penalty function methods demand an appropriate choice
of a penalty parameter for each constraint.Recent suggestions of penalty parameterless
techniques (Deb,in press;Koziel and Michalewicz,1998) may be worth investigating in the
context of multiobjective constrained optimization.
4 A Special TwoObjective Optimization Problem
Let us begin our discussion with a simple twoobjective optimization problem having two
variables
(
) and
:
Minimize
(1)
Minimize
(2)
where
(
) is a function of
only.Thus,the ®rst objective function
is a function
of
only
and the function
is a function of both
and
.In the function space (a
space with (
) values),the above two functions obey the following relationship:
(3)
For a ®xed value of
,a

plot becomes a hyperbola (
).There exists a
number of intuitive yet interesting properties of the above twoobjective problem:
L
EMMA
1:If for any two solutions,the second variables
(or more speci®cally
) are the same,
both solutions are not dominated by each other.
Density can be measured as the hypervolume of a subregion in the parameter space representing a unit
hypercube in the ®tness space.
With this function,it is necessary that
and
function values be strictly positive.
Evolutionary Computation Volume 7,Number 3
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K.Deb
The proof follows from
property.
L
EMMA
2:If for any two solutions,the ®rst variables
are the same,the solution corresponding
to the minimum
dominates the other solution.
P
ROOF
:Since
,the ®rst objective function values are the same.So,the solution
having smaller
(meaning better
) dominates the other solution.
L
EMMA
3:For any two arbitrary solutions
and
,where
for
,and
),there exists a solution
which dominates the solution
.
P
ROOF
:Since the solutions
and
have the same
value and since
,
dominates
,according to Lemma 2.
C
OROLLARY
1:The solutions
and
have the same
values and hence they are non
dominated to each other according to Lemma 1.
Based on the above discussions,we can present the following theorem:
T
HEOREM
1:The twoobjective problem described in equations (1) and (2) has local or global
Paretooptimal solutions
,where
is the locally or globally minimum solution of
,
respectively,and
can take any value.
P
ROOF
:Since solutions with a minimum
have the smallest possible
(in the
neighborhood sense,in the case of local minimum,and in the whole search space in the case
of global minimum),according to Lemma 2,all such solutions dominate any other solution
in the respective context.Since these solutions are also nondominated to each other,they
are Paretooptimal solutions,in the respective sense.
Although obvious,we shall present a ®nal lemma about the relationship between a
nondominated set of solutions and Paretooptimal solutions.
L
EMMA
4:Although some members in a nondominated set are members of the Paretooptimal
front,not all members are necessarily members of the Paretooptimal front.
P
ROOF
:Say,there are only two distinct members in a set of which
is a member of
Paretooptimal front and
is not.We shall show that both these solutions still can
be nondominated to each other.The solution
can be chosen in such a way that
.This makes
.Since
,it follows that
.Thus,
and
are nondominated solutions.
This lemma establishes a negative argument about multiobjective optimization meth
ods which work with the concept of nondomination.Since these methods seek to ®nd
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Evolutionary Computation Volume 7,Number 3
MultiObjective GAs
the Paretooptimal front by ®nding the best nondominated set of solutions,it is important
to realize that all solutions in the best nondominated set obtained by an optimizer may
not necessarily be the members of the Paretooptimal set.However,in the absence of
any better approach,a method for seeking the best set of nondominated solutions is a
reasonable approach.Postoptimal testing (by locally perturbing each member of obtained
nondominated set) may be performed to establish Paretooptimality of members in an
experimentally obtained nondominated set.
The above twoobjective problem and the associated lemmas allow us to construct
different types of multiobjective problems from singleobjective optimization problems
(de®ned by the function
).The optimality and complexity of function
is then directly
transferred into the corresponding multiobjective problem.In the following subsections,
we construct a multimodal and a deceptive multiobjective problem.
4.1 Multimodal Multiobjective Problem
According to Theorem 1,if the function
is multimodal with local (
) and global
(
) minimumsolutions,the corresponding twoobjective problemalso has local and global
Paretooptimal solutions corresponding to solutions
and
,respectively.The
Paretooptimal solutions vary in
values.
We create a bimodal,twoobjective optimizationproblemby choosing a bimodal
function:
(4)
Figure 3 shows the above function for
with
as the global minimum
and
as the local minimumsolutions.Figure 4 shows the

plot with local and
global Paretooptimal solutions corresponding to the twoobjective optimization problem.
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
g(x_2)
x_2
Figure 3:The function
has a global
and a local minimumsolution.
0
2
4
6
8
10
12
14
16
18
20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f_2
f_1
Global Paretooptimal front
Local Paretooptimal front
Random points
Figure 4:Arandomset of 50,000 solutions
shown on a

plot.
The local Paretooptimal solutions occur at
and the global Paretooptimal
solutions occur at
.The corresponding values for
functionvalues are
and
,respectively.The density of the randomsolutions marked on the plot
shows that most solutions lead towards the local Paretooptimal front and only a few
solutions lead towards the global Paretooptimal front.
Evolutionary Computation Volume 7,Number 3
213
K.Deb
To investigate how a multiobjective GA would perform in this problem,the non
dominated sorting GA (NSGA) (Srinivas and Deb,1995) is used.Variables are coded in
20bit binary strings each,in the ranges
and
.A population of
size 60 is used
.Singlepoint crossover with
is chosen.No mutation is used.The
niching parameter
is calculated based on normalized parameter values and
assumed to form about 10 niches in the Paretooptimal front (Deb and Goldberg,1989).
Figure 5 shows a run of NSGA which,even at generation 100,gets trapped at the local
Paretooptimal solutions (marked with a`+').When NSGA is tried with 100 different
0
2
4
6
8
10
12
14
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f_2
f_1
Global Paretooptimal front
Local Paretooptimal front
Initial population
Population at 100 gen
Figure 5:A NSGA run gets trapped at the local Paretooptimal solution.
initial populations,it gets trapped into the local Paretooptimal front in 59 out of 100 runs,
whereas in the other 41 runs NSGA can ®nd the global Paretooptimal front.We also
observe that in 25 runs there exists at least one solution in the global basin of function
in
the initial population and still NSGAs cannot converge to the global Paretooptimal front.
Instead,they get attractedtothe local Paretooptimal front.These results showthat a multi
objective GA can have dif®culty even with a simple bimodal problem.A more dif®cult test
problemcan be constructed by using a standard singleobjective multimodal test problem,
such as Rastrigin's function,Schwefel's function,or by using a higherdimensional,multi
modal
function.
4.2 Deceptive Multiobjective Optimization Problem
Next,we shall create a deceptive multiobjective optimization problem from a deceptive
function.This function is de®ned over binary alphabets.Let us say that the following
multiobjective function is de®ned over
bits,which is a concatenation of
substrings of
variable size
such that
:
Minimize
Minimize
(5)
This population size is determined to have,on an average,one solution in the global basin of function
in a
randominitial population.
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Evolutionary Computation Volume 7,Number 3
MultiObjective GAs
where
is the unitation
of the ®rst substring of length
.To keep matters simple,we
have used a tight encoding of bits representing each substring.The ®rst function
is a
simple onemin problem,where the optimal solution has all
s.A one is added to make all
function values strictly positive.The function
is de®ned
in the following:
if
,
if
(6)
This makes the true attractor (with all
s in the substring) have the worst neighbors with a
function value
and the deceptive attractor (with all
s in the substring) have the
good neighbors with a function value
.Since most of the substrings lead toward
the deceptive attractor,GAs may ®nd dif®culty converging to the true attractor (all
s).
The global Paretooptimal front corresponds to the solution for which the summation
of
function values is absolutely minimum.Since at each minimum,
has a value one,
the global Paretooptimal solutions have a summation of
equal to
.Since each
function has two minima (one true and another deceptive),there are a total of
local
minima,of which one is global.Corresponding to each of these local minima,there exists a
local Paretooptimal front (some of themare identical since the functions are de®ned over
unitation),to which a multiobjective GA may be attracted.
In the experimental set up,we used
,
,
,
,such that
.
Since the functions are de®ned with unitation values,we have used genotypic niching with
Hamming distance as the distance measure between two solutions (Deb and Goldberg,
1989).Since we expect 11 different function values in
(all integers from1 to 11),we use
guidelines suggested in that study and calculate
.Figure 6 shows that when a
population size of 80 is used,an NSGA is able to ®nd the global Paretooptimal front from
the initial population shown (solutions marked with a`+').
0
2
4
6
8
10
12
1
2
3
4
5
6
7
8
9
10
11
f_2
f_1
Global Pareto front
All deceptive Pareto front
Initial Population (n=80)
n=80
n=60
n=16
Figure 6:Performance of a single run of
NSGA is shown on the deceptive multi
objective function.
0
0.2
0.4
0.6
0.8
1
10
50
100
150
200
250
300
Proportion of Successful GAs
Population size
Easy g()
Deceptive g()
Figure 7:Proportionof successful GAruns
(out of 50 runs) versus population size with
easy and deceptive multiobjective prob
lems.
When a smaller population size (
) is used,the NSGA cannot ®nd the true
substringinall three deceptive subproblems.Instead,it converges tothe deceptive substring
Unitation is the number of
s in the substring.Note that minimum and maximum values of unitation of a
substring of length
is zero and
,respectively.
It can be shown that an equivalent dual maximization function
is deceptive according to
conditions outlined elsewhere (Deb and Goldberg,1994).Thus,the above minimization problemis also deceptive.
Evolutionary Computation Volume 7,Number 3
215
K.Deb
in one subproblem and to the true substring in the two other subproblems.When a
suf®ciently small population(
) is used,the NSGAconverges tothe deceptive attractor
inall three subproblems.The correspondinglocal Paretooptimal front is showninFigure 6
with a dashed line.
In order to further investigate the dif®culties that a deceptive multiobjective function
may cause to a multiobjective GA,we construct a 30bit function with
and
for
and use
.For each population size,50 GA runs are started
fromdifferent initial populations and the proportion of successful runs is plotted in Figure 7.
A run is considered successful if all four deceptive subproblems are solved correctly.The
®gure shows that NSGAs with small population sizes could not be successful in many runs.
Moreover,the performance improves as the population size is increased.To show that this
dif®culty is due to deceptionin subproblems alone,we use a linear function for
,
instead of the deceptive function used earlier.Figure 7 shows that multiobjective GAs with
a reasonable population size worked more frequently with this easy problemthan with the
deceptive problem.
The above two problems show that by using a simple construction methodology (by
choosing a suitable
function),any problem feature that may cause singleobjective GAs
dif®culty can also be introduced in a multiobjective GA.Based on the above construction
methodology,we now present a tunable twoobjective optimization problem which may
have additional dif®culties pertaining to multiobjective optimization.
5 Tunable TwoObjective Optimization Problems
Let us consider the following
variable twoobjective problem:
Minimize
Minimize
(7)
The function
is a function of
(
) variables (
),and the function
is a function of all
variables.The function
is a function of
variables
(
) which do not appear in the function
.The function
is a
function of
and
function values directly.We avoid complications by choosing
and
functions that take only positive values (or
and
) in the search space.By
choosing appropriate functions for
,
,and
,multiobjective problems having speci®c
features can be created:
1.Convexity or discontinuity in the Paretooptimal front can be affected by choosing an
appropriate
function.
2.Convergence to the true Paretooptimal front can be affected by using a dif®cult
function (multimodal,deceptive,or others) as demonstrated in the previous section.
3.Diversity in the Paretooptimal front can be affected by choosing an appropriate (non
linear or multidimensional)
function.
We describe each of the above issues in the following subsections.
5.1 Convexity or Discontinuity in Paretooptimal Front
By choosing an appropriate
function,multiobjective optimization problems with convex,
nonconvex,or discontinuous Paretooptimal fronts can be created.Speci®cally,if the
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Evolutionary Computation Volume 7,Number 3
MultiObjective GAs
following two properties of
are satis®ed,the global Paretooptimal set will correspond to
the global minimumof the function
and to all values of the function
:
1.The function
is a monotonically nondecreasing function in
for a ®xed value of
.
2.The function
is a monotonically decreasing function of
for a ®xed value of
.
The ®rst condition ensures that the global Paretooptimal front occurs for the global
minimum value for
function.The second condition ensures that there is a continuous
`con¯icting'Paretofront.However,we realize that when we violate the second condition,
we shall no longer create problems having continuous Paretooptimal front.However,if
the ®rst condition is met alone,for every local minimum of
there will exist one local
Paretooptimal set (corresponding value of
and all possible values of
).
Although many different functions may exist,we present two such functionsÐone
leading to a convex Paretooptimal front and the other leading to a more generic problem
having a control parameter which decides the convexity or nonconvexity of the Pareto
optimal fronts.
5.1.1 Convex Paretooptimal Front
For the following function
(8)
we only allow
.The resulting Paretooptimal set is
.In Section 4,we have seen that the resulting Paretooptimal set is convex.In the
following,we present another function which can be used to create convex and nonconvex
Paretooptimal sets by simply tuning a parameter.
5.1.2 Nonconvex Paretooptimal Front
We choose the following function for
:
if
,
otherwise.
(9)
With this function,we may allow
,but
.The global Paretooptimal set
corresponds tothe global minimumof
function.The parameter
is a normalizationfactor
to adjust the range of values of functions
and
.To have a signi®cant Paretooptimal
region,
may be chosen as
,where
and
are the maximumvalue
of the function
and the minimum(or global optimal) value of the function
,respectively.
It is interesting to note that when
,the resulting Paretooptimal front is nonconvex.
In tackling these problems,the classical weightedsummethod cannot ®nd any intermediate
Paretooptimal solution by using a weight vector.The above function can also be used to
create multiobjective problems having convex Paretooptimal sets by setting
.Other
interesting functions for the function
may also be chosen with properties mentioned in
Section 5.1.
Although the condition for Paretooptimality of multiobjective problems can be established for other
functions,here,we state the suf®cient conditions for the functional relationships of
with
and
.Note that this
allows us to directly relate the optimality of
function with the Paretooptimality of the resulting multiobjective
problem.
Evolutionary Computation Volume 7,Number 3
217
K.Deb
Test problems having local and global Paretooptimal fronts being of mixed type (some
are convex and some are nonconvex shape) can also be created by making the parameter
a
functionof
.These problems may cause dif®culty toalgorithms that work by exploiting the
shape of the Paretooptimal front simply because the search algorithmneeds to adapt while
moving froma local to global Paretooptimal front.Here,we illustrate one such problem,
where the local Paretooptimal front is nonconvex,and the global Paretooptimal front is
convex.Consider the following functions (
) along with function
de®ned in
Equation 9:
if
if
(10)
(11)
(12)
where
and
are the local and the global optimal function value of
,respectively.
Equation 12 is set to have a nonconvex local Paretooptimal front at
and a convex
global Paretooptimal front at
.The function
is given in Equation 9 with
.
A random set of 40,000 solutions (
) is generated and the corresponding
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
f_2
f_1
Global Paretooptimal Front
Local Paretooptimal Front
Random points
Figure 8:A twoobjective function with a nonconvex local Paretooptimal front and a
convex global Paretooptimal front.40,000 randomsolutions are shown.
solutions in the

space are shown in Figure 8.The ®gure clearly shows the nature
of the convex global and nonconvex local Paretooptimal fronts (solid and dashed lines,
respectively).Notice that only a small portion of the search space leads to the global Pareto
optimal front.An apparent front at the top of the ®gure is due to the discontinuity in the
function at
.
Another simple way to create a nonconvex Paretooptimal front is to use Equation 8
but maximize both functions
and
.The Paretooptimal front corresponds to the
maximumvalue of
function and the resulting Paretooptimal front is nonconvex.
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MultiObjective GAs
5.1.3 Discontinuous Paretooptimal Front
As mentioned earlier,we have to relax the condition for
being a monotonically decreasing
function of
to construct multiobjective problems with a discontinuous Paretooptimal
front.In the following,we show one such construction where the function
is a periodic
function of
:
(13)
The parameter
is the number of discontinuous regions in a unit interval of
.By choosing
the following functions:
and allowing variables
and
to lie in the interval [0,1],we have a twoobjective opti
mization problemwhich has a discontinuous Paretooptimal front.Since the
(and hence
) function is periodic to
(and hence to
),we generate discontinuous Paretooptimal
regions.
Figure 9 shows the 50,000 random solutions in

space.Here,we use
and
.When NSGAs (population size of 200,
of 0.1,crossover probability of
1,and no mutation) are applied to this problem,the resulting population at generation
300 is shown in Figure 10.The plot shows that if reasonable GA parameter values are
2
0
2
4
6
8
10
12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f_2
f_1
Paretooptimal front
Random solutions
Figure 9:50,000 random solutions are
shown on a

plot of a multiobjective
problem having discrete Paretooptimal
front.
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
f_2
f_1
Paretooptimal front
NSGA
Figure 10:The population at genera
tion 300 for a NSGA run is shown to have
found solutions in all four discontinuous
Paretooptimal regions.
chosen,NSGAs can ®nd solutions in all four discontinuous Paretooptimal regions.In
general,discontinuity in the Paretooptimal front may cause dif®culty to multiobjective
GAs which do not have an ef®cient way of implementing diversity among discontinuous
regions.Functionspace niching may have dif®culty in these problems because of the
discontinuities in the Paretooptimal front.
5.2 Hindrance to Reach True Paretooptimal Front
It is shown earlier that by choosing a dif®cult function for
alone,a dif®cult multiobjective
optimization problemcan be created.Some instances of multimodal and deceptive multi
objective optimization have been created earlier.Test problems with standard multimodal
Evolutionary Computation Volume 7,Number 3
219
K.Deb
functions used in singleobjective GA studies,such as Rastrigin's functions,NKlandscapes,
and others can all be chosen for the
function.
5.2.1 Biased Search Space
The function
plays a major role in introducing dif®culty to a multiobjective problem.
Even though the function
is not chosen to be a multimodal function nor to be a deceptive
function,with a simple monotonic
function the search space can have adverse density of
solutions toward the Paretooptimal region.Consider the following function for
:
(14)
where
and
are the minimum and maximum function values that the function
can take.The values
and
are minimumand maximumvalues of the variable
.
It is important to note that the Paretooptimal region occurs when
takes the value
.
The parameter
controls the bias in the search space.If
,the density of solutions
away fromthe Paretooptimal front is large.We showthis on a simple problemwith
,
,and with the following functions:
We also use
and
.Figures 11 and 12 show 50,000 random solutions
each with
equal to 1.0 and 0.25,respectively.It is clear that for
,no solution
0
0.5
1
1.5
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f_2
f_1
Paretooptimal front
Random solutions
Figure 11:50,000 random solutions are
shown for
.
0
0.5
1
1.5
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f_2
f_1
Paretooptimal front
Random solutions
Figure 12:50,000 random solutions are
shown for
.
is found in the Paretooptimal front,whereas for
,many Paretooptimal solutions
exist in the set of 50,000 randomsolutions.Randomlike search methods are likely to face
dif®culty in ®nding the Paretooptimal front in the case with
close to zero,mainly due to
the low density of solutions towards the Paretooptimal region.
5.2.2 Parameter Interactions
The dif®culty in converging to the true Paretooptimal front may also arise because of
parameter interactions.It was discussed before that the Paretooptimal set in the two
objective optimization problem described in Equation 7 corresponds to all solutions of
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Evolutionary Computation Volume 7,Number 3
MultiObjective GAs
different
values.Since the purpose of a multiobjective GA is to ®nd as many Pareto
optimal solutions as possible and,since in Equation 7 the variables de®ning
are different
fromvariables de®ning
,a GA may work in two stages.In one stage,all variables
may
be found and in the other,optimal
may be found.This rather simple mode of a GA
working in two stages can face dif®culty if the above variables are mapped to another set of
variables.If
is a randomorthonormal matrix of size
,the true variables
can ®rst
be mapped to derived variables
using
(15)
Thereafter,objective functions de®ned in Equation 7 can be computed using the variable
vector
.Since the components of
can now be negative,care must be taken in de®ning
and
functions so as to satisfy restrictions suggested on themin previous subsections.A
translation of these functions by adding a suitable large positive value may have to be used
to force these functions to take nonnegative values.Since the GA will be operating on
the variable vector
,and the function values depend on the interaction among variables of
,any change in one variable must be accompanied by related changes in other variables
in order to remain on the Paretooptimal front.This makes this mapped version of the
problem dif®cult to solve.We discuss more about mapped functions near the end of the
following section.
5.3 Nonuniformly Represented Paretooptimal Front
In all the test functions constructed above (except the deceptive problem),we have used
a linear,singlevariable function for
.This helped us create a problem with a uniform
distribution of solutions in
.Unless the underlying problemhas discretely spaced Pareto
optimal regions (as in Section 5.1.3),there is no bias for the Paretooptimal solutions to be
spread over the entire range of
values.However,a bias for some portions of range of
values for
may also be created by choosing any of the following
functions:
1.The function
is nonlinear,or
2.The function
is a function of more than one variable.
It is clear that if a nonlinear
function (whether single or multivariable) is chosen,the
resulting Paretooptimal region (or,for that matter,the entire search region) will have
bias towards some values of
.The nonuniformity in distribution of the Paretooptimal
region can also be created by simply choosing a multivariable function (whether linear
or nonlinear).Multiobjective optimization algorithms,which are poor at maintaining
diversity among solutions (or function values),will produce a biased Paretooptimal front
in such problems.Thus,the nonlinearity in function
or dimension of
measures how
well an algorithmis able to maintain distributed nondominated solutions in a population.
Consider the singlevariable,multimodal function
:
(16)
The above function has ®ve minima for different values of
as shown in Figure 13.The
®gure also shows the corresponding nonconvex Paretooptimal front in a

plot with
function de®ned in Equation 9 having
and
(since
,the Paretooptimal
front is nonconvex).The right ®gure is generated from 500 uniformlyspaced solutions
in
.The value of
is ®xed so that the minimumvalue of
is equal to 1.The ®gure
shows that the Paretooptimal region is biased for solutions for which
is near one.
Evolutionary Computation Volume 7,Number 3
221
K.Deb
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.2
0.4
0.6
0.8
1
f_1
x_1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f_2
f_1
Figure 13:A multimodal
function and corresponding nonuniformly distributed non
convex Paretooptimal region.In the right plot,Paretooptimal solutions derived from500
uniformlyspaced
solutions are shown.
5.3.1 FunctionSpace and ParameterSpace Niching
The working of a multiobjective GA on the above function provides interesting insights
about functionspace niching (Fonseca and Fleming,1993) and parameterspace niching
(Srinivas andDeb,1995).It is clear that when functionspace niching is performed,diversity
in the context of objective function values is anticipated,whereas when parameter space
niching is performed,diversity in the phenotype (or genotype) of solutions is expected.
We illustrate the difference by comparing the performance of NSGAs with both niching
methods on the above problem.NSGAs with a reasonable parameter setting (population
size of 100,15bit coding for each variable,
of 0.2236 (assuming 5 niches),crossover
probability of 1,and no mutation) are run for 500 generations.A typical run for both
niching methods are shown in Figure 14.Although it seems that both niching methods are
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f_1
Parameterspace niching
Paretooptimal front
f_2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x_1
f_1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
x_1
f_1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f_1
Functionspace niching
Paretooptimal front
f_2
Figure 14:The left plot is with parameterspace niching and the right is with function
space niching.The ®gures showthat bothmethods ®nd solutions with diversity in the

space.
able to maintain diversity in function space (with a better distribution in

space with
functionspace niching),the left plot (inside ®gure) shows that the NSGA with parameter
space niching has truly found diverse solutions,whereas the NSGA with functionspace
niching (right plot) converges to about 50% of the entire region of the Paretooptimal
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MultiObjective GAs
solutions.Sincethe®rst minimumandits basinof attractionspans thecompletespacefor the
function
,the functionspace niching does not have the motivationto ®ndother important
solutions.Thus,in problems like this,functionspace niching may hide information about
important Paretooptimal solutions in the search space.
It is important to understand that the choice between parameterspace or function
space niching depends entirely on what is desired in a set of Paretooptimal solutions in
the underlying problem.In some problems,it may be important to have solutions with
tradeoff in objective function values without concern for the similarity or diversity of the
actual solutions (
vectors or strings).In such cases,functionspace niching will,in general,
provide solutions withbetter tradeoff inobjective functionvalues.Since there is noinduced
pressure for the solutions to differ from each other,the Paretooptimal solutions may not
be very different,unless the underlying objective functions demand themto be so.On the
other hand,in some problems the emphasis could be on ®nding more diverse solutions and
with a tradeoff among objective functions.Parameterspace niching would be better in
such cases.This is because,in some sense,categorizing a population using nondomination
helps to preserve some diversity among objective functions and an explicit parameterspace
niching helps to maintain diversity in the solution vector.
To show the effect of parameter interactions (Section 5.2.2),we map the solution
vector
into another vector
(obtained by rotation and translation).Now the distinction
between parameterspace and functionspace niching is even more clear (see Figure 15).
GA parameter values identical to those in the unmapped case above are used here.Clearly,
parameterspace niching is able to ®nd more diverse solutions than functionspace niching.
However,an usual

plot would reveal that the functionspace niching is also able to
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
x_2
x_1
Parameterspace niching
0
0.2
0.4
0.6
0.8
1
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
x_2
x_1
Functionspace niching
Figure 15:Solutions for a mapped problem are shown.The plots are made with all 100
solutions at generation 500.
®nd diverse solutions.A plot as in Figure 15 truly reveals the diversity achieved in the
solutions.
6 Summary of Test Problems
The twoobjective optimization problem discussed above requires three functionsÐ
,
,and
Ðwhich can be set to various complexity levels to create complex twoobjective
optimization test problems.In the following,we summarize the properties of a two
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K.Deb
objective optimization problemdue to each of above functions:
1.The function
tests a multiobjective GA's ability to ®nd diverse Paretooptimal
solutions.Thus,this function tests an algorithm's ability to handle dif®culties along
the Paretooptimal front.
2.The function
tests a multiobjective GA's ability to converge to the true (or global)
Paretooptimal front.Thus,this function tests an algorithm's ability to handle dif®
culties lateral to the Paretooptimal front.
3.The function
tests a multiobjective GA's ability to tackle multiobjective problems
having convex,nonconvex,or discontinuous Paretooptimal fronts.Thus,this func
tion tests an algorithm's ability to handle different shapes of the Paretooptimal front.
In the light of the above discussion,we summarize and suggest in Tables 1,2,and 3 a
few test functions for the above three functionals,which may be used in combination with
each other.Unless speci®ed,all variables
mentioned in the tables take real values in the
range [0,1].The functions mentioned in the third column in each table are representative
Table 1:Effect of function
on the test problem.
Function
(
)
Controls search space along the Paretooptimal front
Type
Example and Effect
F1I
Single
variable
(
) and
linear
Example:
Effect:Uniformrepresentation of solutions in the Paretooptimal
front.Most of the Paretooptimal region is likely to be found.
F1II
Multi
variable
(
) and
linear
Example:
Effect:Nonuniform representation of Paretooptimal front.
Some Paretooptimal regions are not likely to be found.
F1III
Nonlinear
(any
)
Example:Eqn (16) for
or,
where
Effect:Same as above.
F1IV
Multimodal
Example:Eqn (4) with
replaced by
or other standard
multimodal test problems (suchas Rastrigin's function,see Table 2)
Effect:Same as above.Solutions at global optimum of
and
corresponding function values are dif®cult to ®nd.
F1V
Deceptive
Example:
,where
is same as
de®nedinEqn (6)
Effect:Same as above.Solutions at true optimumof
are dif®cult
to ®nd.
functions which will produce the desired effect mentioned in the respective fourth column.
While testing an algorithmfor its ability to overcome a particular feature of a test problem,
we suggest varying the complexity of the corresponding function (
,
,or
) and ®xing
the other two functions at their easiest complexity level.For example,while testing an
algorithm for its ability to ®nd the global Paretooptimal front in a multimodal,multi
objective problem,we suggest choosing a multimodal
function (GIII) and ®xing
as in F1I and
as in HI.Similarly,using
function as GI,
function as HI,and by
®rst choosing
function as F1I,test a multiobjective optimizer's capability to distribute
solutions along the Paretooptimal front.By only changing the
function to F1III (even
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MultiObjective GAs
Table 2:Effect of function
on the test problem.
Function
(
),say
Controls search space lateral to the Paretooptimal front
Type
Example and Effect
GI
Unimodal,
single
variable (
),and linear
Example:
(
),or Eqn (14) with
Effect:No bias for any region in the search space.
GII
Unimodal
and
nonlinear
Example:Eqn (14) with
Effect:With
,bias towards the Paretooptimal region and
with
,bias against the Paretooptimal region.
GIII
Multimodal
Rastrigin:
Example:
Effect:Many (
) local and one global Paretooptimal fronts
Schwefel:
Example:
Effect:Many (
) local and one global Paretooptimal fronts
Griewangk:
Example:
Effect:Many (
) local and one global Paretooptimal fronts
GIV
Deceptive
Example:Eqn(6)
Effect:Many (
) deceptive attractors and one global attractor
GV
Multi
modal,
deceptive
Example:
if
,
if
.
where
Effect:Many (
) deceptive attractors and
global attractors
Table 3:Effect of function
on the test problem.
Function
(
)
Controls shape of the Paretooptimal front
Type
Example and Effect
HI
Monotonically non
decreasing in
and convex
on
Example:Eqn (8) or Eqn (9) with
Effect:Convex Paretooptimal front
HII
Monotonically non
decreasing in
and non
convex on
Example:Eqn (9) with
Effect:Nonconvex Paretooptimal front
HIII
Convexity in
as a func
tion of
Example:Eqn (9) along with Eqn (12)
Effect:Mixed convex and nonconvex shapes for local
and global Paretooptimal fronts
HIV
Nonmonotonic periodic
in
Example:Eqn(13)
Effect:Discontinuous Paretooptimal front
Evolutionary Computation Volume 7,Number 3
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K.Deb
with
),the same optimizer can be tested for its ability to ®nd distributed solutions in
the Paretooptimal front.
Along with any such combination of three functionals,parameter interactions can be
introducedto create even more dif®cult problems.Using a transformationof the coordinate
system as suggested in section 5.2.2,all the abovementioned properties can be tested in
a space where simultaneous adjustment of all parameter values is desired for ®nding an
improved solution.
7 Future Directions for Research
This study suggests a number of immediate areas of research for developing better multi
objective GAs.A list of themare outlined and discussed in the following:
1.Compare existing multiobjective GA implementations
2.Understand dynamics of GA populations with generations
3.Investigate scalability issue of multiobjective GAs with number of objectives
4.Develop constrained test problems for multiobjective optimization
5.Study convergence properties to the true Paretooptimal front
6.Introduce elitismin multiobjective GAs
7.Develop metrics for comparing two populations
8.Apply multiobjective GAs to more complex realworld problems
9.Develop multiobjective GAs for scheduling and other kinds of optimization problems
As mentioned earlier,there exists a number of different multiobjective GAimplementations
primarily varying in the way nondominated solutions are emphasized and in the way the
diversity in solutions are maintained.Although some studies have compared different GA
implementations (Zitzler and Thiele,1998),they all have been done on a speci®c problem
without much knowledge about the complexity of the test problems.With the ability
to construct test functions having controlled complexity,as illustrated in this paper,an
immediate task would be to compare the existing multiobjective GAs and to establish
the power of each algorithm in tackling different types of multiobjective optimization
problems.
The test functions suggested here provide various degrees of complexity.The con
struction of all these test problems has been done without much knowledge of how multi
objective GAs work.If we knowmore about howsuchGAs work basedona nondomination
principle,problems can be created to test more speci®c aspects of multiobjective GAs.In
this regard,an interesting study would be to investigate howan initial randompopulationof
solutions moves fromone generation to the next.An initial randompopulation is expected
to have solutions belonging to many nondomination levels.One hypothesis about the
working of a multiobjective GA would be that most population members soon collapse
to a single nondominated front and each generation thereafter proceeds by improving
this large nondominated front.On the other hand,it may also be conjectured that GAs
work by maintaining a number of nondomination levels at each generation.Both these
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Evolutionary Computation Volume 7,Number 3
MultiObjective GAs
modes of working should provide enough diversity for the GAs to ®nd new and improved
solutions and are likely candidates,although the actual mode of working may depend on the
problem at hand.Thus,it will be worthwhile to investigate how existing multiobjective
GA implementations work in the context of different test problems.
In this paper,we have not considered more than two objectives,although extensions of
the concept toproblems having more thantwoobjectives canalsobe done.It is intuitive that
as the number of objectives increases,the Paretooptimal region is represented by multi
dimensional surfaces.With more objectives,multiobjective GAs must have to maintain
more diverse solutions in the nondominated front in each iteration.Whether GAs are able
to ®nd and maintain diverse solutions (as demanded by the search space of the problem) with
many objectives would be an interesting study.Whether population size alone can solve
this scalability issue or a major structural change (implementing a better niching method)
is required would be the outcome of such a study.
We also have not considered constraints in this paper.Constraints can introduce
additional complexity in the search space by inducing infeasible regions in the search
space,thereby obstructing the progress of an algorithmtowards the global Paretooptimal
front.Thus,creation of constrained test problems is an interesting area which should be
emphasized in the future.With the development of such complex test problems,there is
also a need to develop ef®cient constraint handling techniques that would be able to help
GAs to overcome hurdles caused by constraints.
Most multiobjective GAs that exist to date work with the nondomination ranking of
population members.Ironically,we have shown in Section 4 that all solutions in a non
dominated set need not be members of the true Paretooptimal front,although some of
themcould be.In this regard,it would be interesting to introduce special features (such as
elitism,mutation,or other diversitypreserving operators),the presence of which may help
us to prove convergence of a GA population to the global Paretooptimal front.Several
attempts have been made to achieve such proofs for singleobjective GAs (Suzuki,1993;
Rudolph,1994) and similar attempts may also be made for multiobjective GAs.
Elitismis a useful andpopular mechanismusedinsingleobjective GAs.Elitismensures
that the best solutions in each generation will not be lost.What is more important is that
these good solutions get a chance to participate in recombination with other solutions in
the hope of creating better solutions.In multiobjective optimization,all nondominated
solutions of the ®rst level are the best solutions in the population.Copying all such
solutions to subsequent generations may make GAs stagnate.Thus,strategies for copying
only a subset of nondominated solutions must be developed.
Comparison of two populations in the context of multiobjective GAs also raises
some interesting questions.As mentioned earlier,there are two goals in a multiobjective
optimizationÐconvergence to the true Paretooptimal front and maintenance of diversity
among Paretooptimal solutions.Amultiobjective GAmay have found a populationwhich
has many Paretooptimal solutions but with less diversity among them.How would such
a population be compared with respect to another which has a fewer number of Pareto
optimal solutions but with wider diversity?Although there exists a suggestion of using a
statistical metric (Fonseca and Fleming,1996),most researchers use visual means of com
parison which causes dif®culty in problems having many objectives.The practitioners of
multiobjective GAs must address this issue before they would be able to compare different
GA implementations in a reasonable manner.
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227
K.Deb
Test functions test an algorithm's ability to overcome a speci®c aspect of a realworld
problem.In this respect,an algorithm which can overcome more aspects of problem
dif®culty is naturally a better algorithm.This is precisely the reason why so much effort is
spent on doing research in test function development.As it is important to develop better
algorithms by applying them on test problems with known complexity,it is also equally
important that the algorithms are tested in realworld problems with unknown complexity.
As mentioned earlier,the advantages of using a multiobjective GA in realworld problems
are many and there is a need for interesting application case studies which would clearly
show the advantages and ¯exibilities in using a multiobjective GA,as opposed to a single
objective GA.
With the advent of ef®cient multiobjective GAs for function optimization,the con
cept of multiobjective optimization can also be applied to other search and optimization
problems such as multiobjective scheduling and other multiobjective combinatorial opti
mization problems.Since in tackling these problems using permutation GAs,the main dif
ferences frombinary GAs are inthe way the solutions are representedandinthe construction
of GAoperators,anidentical nondominationprinciple along witha similar niching concept
can still be used in solving such problems having multiple objectives.In this context,similar
concepts can also be implemented in developing other populationbased,multiobjective
EAs.Multiobjective evolution strategies,multiobjective genetic programming,or multi
objective evolutionary programming may better solve speci®c multiobjective problems
which are ideally suited for the respective evolutionary method.
8 Conclusions
For the past few years,there has been a growing interest in the studies of multiobjective
optimization using genetic algorithms (GAs).Although there exists a number of multi
objective GA implementations and applications to interesting multiobjective optimization
problems,there is no systematic study to speculate what problemfeatures may cause a multi
objective GA to face dif®culties.In this paper,a number of such features are identi®ed
and a simple methodology is suggested to construct test problems from singleobjective
optimization problems.The construction method requires the choice of three functions,
each of which controls a particular aspect of dif®culty for a multiobjective GA.One
function,(
),tests an algorithm's ability to handle dif®culties along the Paretooptimal
region;function (
) tests an algorithm's ability to handle dif®culties lateral to the Pareto
optimal region;and function (
) tests an algorithm's ability to handle dif®culties arising
because of different shapes of the Paretooptimal region.This allows a multiobjective GA
to be tested in a controlled manner on various aspects of problem dif®culties.Since test
problems are constructed from singleobjective optimization problems,most theoretical
or experimental studies on problem dif®culties or on test function development in single
objective GAs are of direct importance to multiobjective optimization.
This paper has made a modest attempt to reveal and test some interesting aspects of
multiobjective optimization.A number of other salient and related studies are suggested
for future research.We believe that more studies are needed to better understand the
working principles of a multiobjective GA.An obvious outcome of such studies would be
the development of new and improved multiobjective GAs.
228
Evolutionary Computation Volume 7,Number 3
MultiObjective GAs
Acknowledgments
The author acknowledges support from the Alexander von Humboldt Foundation,Ger
many,and The Department of Science and Technology,India,during the course of this
study.The comments of G
È
unter Rudolph,Eckert Zitzler,and anonymous reviewers have
improved the readability of the paper.
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