ASSORTMENT OF SOLUTIONS FOR VARIABLE TASKS IN

MULTI-OBJECTIVE PROBLEMS

Gideon Avigad, Erella Eisenstadt and Uri Ben Hanan

Ort Braude College of Engineering, Karmiel, Israel

Keywords: Multiobjective, Evolution, Engineering design.

Abstract:

n the same manner that species are associated with variants in order to survive, and that human

communities, apparently in order to survive, are built up from people with different skills and professions,

we suggest in this paper to select a set of diverse solutions in order to optimally solve Multi-Objective

Problems (MOPs). As a set, the solutions may cover a wider range of capabilities within the multi-objective

space than is possible for an individual member of the set. The diversity within the set is a key issue of this

paper and hereinafter designated as an assortment. In the paper, we suggest a computational tool that

supports the selection of such an assortment. The selection is posed as an auxiliary MOP of cost versus

variability. The cost is directly related to the size of the assortment, whereas the variability is related to the

ability of the assortment to cover the objective space. A previously treated problem is adopted and utilized

in order to explain and demonstrate the approach.

1 INTRODUCTION

The use of Evolutionary Multi-objective

Optimization, (EMO) is a popular approach for

searching for solutions to MOPs (Multi Objective

Problems). Commonly when the objectives of a

MOP are contradicting the solution to the MOP is

the Pareto set. The development of Pareto-based

evolutionary algorithms has been initiated by the

procedure suggested by Goldberg, (1989). Surveys

and descriptions of EMO algorithms can be found in

several references (e.g., Deb, 2001).

Selecting a solution out of a Pareto set is

commonly based on the designers' preferences.

Choosing a set of solutions to MOPs instead of

selecting a single solution is relatively a new area of

research.

Recently a new approach to select conceptual

solutions has been investigated (e.g., Mattson, and

Messac, 2005). It involves Set-Based Concept

(SBC) representation in which a concept is

associated with the performances of multiple

solutions. When dealing with SBCs, each of the

solutions (design alternatives), of the SBC is

assumed to be associated with a point in the

objective space, representing its performances.

Therefore the concept performances can be

evaluated based on a cluster of points in the

objective space, where each of the points of the

cluster is associated with the performances of at

least one of the design alternatives (solutions) of the

SBC. According to that approach, each concept has

its related front. The global front, which is the non

dominated set over all the objective space, is the s-

Pareto (Mattson, and Messac, 2005). An approach

for choosing a concept, (a set) which has

representatives on the s-Pareto, has been suggested

in Mattson and Messac, (2005). There, it has been

assumed that, the more representatives a concept has

on the s-Pareto, the more flexible it is in

corresponding to uncertainties. Avigad and

Moshaiov (2009) have highlighted some pitfalls of

considering just the s-Pareto and suggested an

auxiliary MOP of optimality versus variability to

compare between the concepts, based on their entire

individual Pareto fronts.

Apart from selecting a set, the evolution of sets

has also been considered. For instance, there are

studies that use set domination to search for the best

approximation of the Pareto front (e.g., the

Indicator-based Evolutionary Algorithm, -IBEA of

Zitzler and Künzli, (2004). Such a search is based on

assigning a value to the degree of domination

between sets of competing approximations of the

Pareto set. For example, such an assignment is

performed using the binary additive indictor, which

269

Avigad G., Eisenstadt E. and Ben Hanan U. (2009).

ASSORTMENT OF SOLUTIONS FOR VARIABLE TASKS IN MULTI-OBJECTIVE PROBLEMS.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 269-276

DOI: 10.5220/0002277902690276

 SciTePress

was introduced by Zitzler et al. (2003). The binary

additive-indicator of two Pareto set approximations

is equal to the minimum distance among the

dimensions of the objective space by which one

Pareto set approximation needs to move or can be

moved such that the compared approximation is

weakly dominated by it.

Within the context of this paper, it is important to

consider another assessment measure that allows a

comparison between two sets. This measure is the

hyper-volume measure or S metric, which has been

proposed by Zitzler and Thiele, (1998), who called it

the 'size of the space covered' or the 'size of

dominated space.' Van Veldhuizen and Lamont

(2000), described it as the Lévesque measure of the

union of hyper-cubes defined by a non-dominated

point and a reference point. When engineering

design is considered, the reference point might be

related to specified boundaries in the objective

space, namely within a 'region-of-interest'.

According to Mattson and Messac (2005), in order

to define a region-of-interest, the designer should

specify a single point in the objective space.

It is noted that choosing a set might also be

related to the notion of community of robots. In that

case, the aim is to find a set of robots which are

communicating in order to perform a task or tasks.

The idea might be related to swarms (see

http://www.swarm-robotics.org) or to a multi-agent

design (e.g., Bensaid, and Matheieu, 1998).

In contrast to previous studies, which utilize a set

of robots in order to execute an aggregated mission,

the current paper suggests choosing a set such that

its members do not always participate in the mission,

but are rather "called for" based on the mission at

hand. To elucidate the problem that will be attended

by the current paper, refer to the following

illustrative example: Suppose that robotic platforms

are operating in a multi-task environment. This

means that sometimes a fast action is needed and a

robot should quickly move from one place to the

other. In another scenario heavy loads should be

transferred by a robot from one place to the other. It

is clear that if optimality is considered, ideally the

two different tasks should be performed by different

robots. Choosing the optimal variety of robots

(solutions) is the scope of the current paper.

The paper is organized as follows. The next

section lays out the background on which we rely in

order to introduce the suggested approach. Section 3

describes the motivation for this paper, having its

origins in biology, sociology, and engineering

design. Section 4 is the methodology, where the

problem, its solutions, and the search approach are

explained and formulated. In Section 5, an example

is utilized in order to demonstrate the applicability

of the approach in choosing an optimal assortment.

2 BACKGROUND

Recently in Avigad et al. (2009), we have introduced

a new problem within the context of MOPs. The

problem treated in Avigad et al. (2009), although

defined as a common uncertain MOP, differs

inherently from that problem. This difference

influences both the search procedure as well as the

multi criteria decision making. In the following, a

brief outline of the problem and its solution as given

in Avigad et al. (2009) are described. Consider the

following MOP:

)d,x(FMinimize

)x(

(1)

where

K21

)]d,x(f),.....d,x(f),d,x(f[)d,x(F =

;

≥

Rx ⊆Ω∈

n21

]x,.....x,x[x =

R,d ⊆Γ∈

m21

]d,.....d,d[d =

)U()L()U()L(

jjjiii

ddd and xxx ≤≤≤≤

where

is the design parameters space

(parameters that are to be chosen) and

Γ is the

model's environmental parameters space (which are

not chosen but might be uncertain). x is a solution

]x,...x,..x[x =

Rx ⊂Ω∈

where

xxx

)U()L(

iii

≤≤

and

is the design space

(controlled parameter space). The interval given for

x is commonly related to an uncertainty, which is

related to the realization of an exact value of x.

Factors such as machine precision are the basis for

the interval, which is in fact a tolerance given for

each design parameter. In Avigad et al. (2009), the

origin of the interval is fundamentally different. The

interval is a span of possible values associated with

choosing a specific parameter. For example

choosing a specific motor (with no uncertainty) is

like choosing a span of output torques (as well as

weight, size, etc.). Each of the possible values for all

design parameters is performing within an

environmental situation

Rd ⊆Γ∈

mj1

]d....d,...d[d =

such that

)U()L(

jjj

ddd ≤≤

and

are the environmental space (uncontrolled

parameter space). If

Γ×

⊆

then

Ss ∈ is a

IJCCI 2009 - International Joint Conference on Computational Intelligence

270

scenario of x (which is a vector in

) . A

scenario's vector of performances in a K objective

space is

)s(Fy

where

)]s(F....),s(F),s(F[)s(F =

The corresponding set of all the scenarios'

performances of the solution x is designated as: Y

RY ⊆Τ⊆

. This means that a solution is

represented by a cluster of points (each representing

a vector) in objective space. The cluster of possible

scenarios in the work of Avigad et al. (2009), is built

of scenarios that are related to the same solution

whereas in the uncertainty MOP case, each scenario

is a different realized solution. So if all are possible

scenarios, then a comparison between the solutions

should be based on the best. In a multi objective

space, the best might be a set of best scenarios. The

set of best scenarios of a solution x, RS

and related

front RSF

has been defined in Avigad et al. (2009)

as follows:

}RSs:)s(Fy|y{:RSF

)}s(F)s(F:Ss|Ss{:RS

xxx

x'x

∈=Γ∈=

∈∃¬∈= ≺

(2)

The set of optimal solutions P* and their

representation in objective space, the Pareto Layer,

PL, are defined as follows:

*}Px|RSF)s(F{:PL

}RSRS:RS|x{:*P

x'x'x

∈∈=

∃¬Ω⊆= ≺

(3)

The PL is associated with sets of representative sets,

each related to a solution. This front is not a clear-

cut front but rather a cloud of scenarios'

performances and therefore, it has been termed in

Avigad et al. (2009) as the Pareto layer (PL). Such a

PL possesses solution scenarios' performances that

are dominated by the performances of other

solutions’ scenario's performances. Nevertheless,

the representative sets of the optimal solutions do

not dominate each other.

In Avigad et al. (2009), an MOEA (Multi Objective

Optimization Algorithm), which applies a search for

finding the PL, has been suggested and investigated.

The current paper deals with selecting sub-sets of

the Pareto layer set, based on the motivation

explained in the following section.

3 MOTIVATION

This paper approach is motivated by the apparent

diversity within species in nature and by the

diversity of professions and expertise within human

societies. The genetic diversity carried in natural

populations is a key factor in evolution (e.g., Mayr

1982) and is one of the fundamentals of what is

termed as

modern evolutionary synthesis. The

importance of population diversity is highlighted in

many nature related studies (e.g., Booy, G. et al

2000). According to Booy, G. et al 2000, "Such

genetic variation within a population may allow

species to change over time and thereby survive

changing environmental conditions." According to

Boer et al. 1993, "… a population can only achieve

its adaptability by distribution of the variation across

its individuals". The above citation clearly implies

the importance of variability within a species. The

above biologically related differences are associated

with diversity in the genotype. A question that

needs to be investigated involves phenotypic

differences. Naturally, the dissimilarity of human

faces is one example. However, here we are more

interested in the behavior aspects. A clear

dissimilarity between individuals within a human

community is the existence of different trades, such

that different people within the community are

experts in different fields of knowledge (e.g., a

medic, a coal miner and a fisherman are trades

commonly practiced by different people).

When optimality is considered, this is somewhat

comparable to the fact that there is no world heavy

weight champion winning a 100m run against Usain

Bolt (the current world champion). These two

extremes (strong and fast) are not the only cases. A

decathlon athlete should possess characteristics that

will allow him competing both in speed and

strength. Neither the decathlon athlete, the runner,

nor the heavy lifter athletes, is superior to each other

if the bi-objective space of speed versus strength is

considered. This issue is the base for choosing the

members of an Olympic team. Instead of choosing

one solution (a single superstar that performs

reasonably in all Olympic professions), a set of

solutions (several athletes, each expert at his own

field of profession) are selected. Thus a multi-

objective problem of optimizing all objectives (i.e.,

running the fastest, lifting the heaviest, etc.) is

solved through using a set of solutions (athletes).

Motivated by the apparent importance of

diversity in species and human communities, we

suggest searching for a set of diversified engineering

solutions such that they may optimally comply with

their set related tasks.

Before going on to present the methodology, we

would like to note the following two remarks.

ASSORTMENT OF SOLUTIONS FOR VARIABLE TASKS IN MULTI-OBJECTIVE PROBLEMS

271

1. Choosing a name for the set of solutions within

the context of this paper was not an easy task.

Community, group, team, unit, and other names

were considered. The main drawback of all of these

notions is the inherent interaction between their

members. The definition for an assortment seems to

best fit the idea. According to

en.wiktionary.org/wiki/assortment an assortment is

"a collection of varying but related items

No interaction is reminded in that definition, which

is fine, with the relation possibly being interpreted

here as the relation to the same objective space.

2. The current paper is bound to the ideas

presented in Avigad et al. (2000). Here a solution

may have a span of possible performance vectors

rather than a single performance vector. Consider a

100 meters runner that runs slower than his/her best,

or would run slower if s/he carries a load (or if s/he

runs uphill). This is comparable to a design of a cart

to move as fast as possible and to carry the highest

loads. Carrying heavier loads means moving slower

and vice versa. We could think of choosing a set of

cars that could carry as high as possible loads and

move as fast as possible or on a set of telescopic

arms that should carry high as possible loads to most

distant horizontal locations. In these cases there is a

fundamental contradiction in the objective space.

However, more importantly, each solution has a

span of possible performances. This is why the

methodology and the example are built upon the

Pareto Layer notion.

4 METHODOLOGY

4.1 An Assortment of Solutions

An assortment A

is a sub-set of all possible

solutions

Ω⊆

A ,

∪

where n

A .

Notes:

1. An assortment might possess a single

member.

2. A solution might be a member of more than

one assortment.

3. An assortment might possess identical

members.

The performance of an assortment, Y

is represented

in the problem objective space by the union of the

representative sets of the assortment's members,

∪

RSFY

= where

RSF is the representative-set

related front (see Section 2) of the i-th member of

the assortment.

In the current paper we shall follow some

assumptions, which are given and explained here.

1: Following the motivation for optimality, we

only consider solutions that belong to the Pareto set

as candidates for members in an assortment.

2: The boundaries of the performances within the

objective (task) space are known beforehand. In

other words, the task WOI is given

a priori to the

design process. For example, it is assumed that the

maximal carried load is known.

3: The maximal cost involved with the assortment

is known. This might be based on costs of

manufacturing and transportation, among others.

Based on the above assumptions, the assortment

set and related performances are:

PA ⊆ and PLY

⊆

In the current paper, we assume that the Pareto set

and related Pareto Layer are given.

4.2 The Competency of an Assortment

It is suggested here that comparing and selecting an

assortment out of all possible assortments is carried

out by considering their performances. The

performances of an assortment is termed here as the

competency of the assortment. There might be

several measures used to assess this competency. In

the current paper, we consider just two: The first is a

straight forward one, the cost of the assortment. As

the number of members within an assortment

increase, so does its cost. The cost of an assortment

is the sum of the individual members' costs:

∑

)x(Cost)A(Cost (4)

The second measure is the variability of the

assortment, V

. It is a measure of the capability of

the assortment to cover the objective space. The

issue of variability has been extensively treated in

Avigad and Moshaiov, (2009). Here it is the hyper-

volume rendered by the community related scenarios

as formulized for an assortment A

as follows: Let

F be the union of all representative sets of an

assortment such that:

∪

RSF

= . Thus the

variability measure may be defined as:

∏

−

WOI

))HV)(HV(HV(

xxs

∩∪

(5)

IJCCI 2009 - International Joint Conference on Computational Intelligence

272

where

HV is the hyper-volume measure of the i-th

scenario belonging to the set F

. To elucidate the

measure, refer to Figure 1. The figure depicts the

representative sets of three solutions within a WOI

(designated by dashed lines). The size of the grey

area in the figure is the variability measure. As it

grows, the assortment may comply with more tasks

within the WOI.

Figure 1: Assortment related variability.

The two measures explained above are mapping an

assortment, which is represented by its related

representative sets' fronts in the problem objective

space, into an auxiliary space, which represents the

competency of the assortment. The competency of

an assortment is a point in the auxiliary space

representing the cost associated with the assortment

and its variability. Therefore, the problem of

comparing and selecting between communities is

transformed from the original objective space where

each assortment is represented by a sub-Pareto layer

into the auxiliary objective space where each

assortment is represented by a single competency

point of variability and cost.

4.3 Problem Definition

The problem of finding a sub-set of a Pareto set (an

assortment) is defined as follows:

Find

Ω⊆

A , In order to

))A((max

Ψ ,

)

Cost

,V()A(

sss

=Ψ

(6)

Following assumption 1 in Section 4.1, the problem

of Equation 6 may be restated as a search for a sub-

set of the Pareto set:

Find

PA ⊆ , (7)

In order to

)

Cost

,V(max

s.t:

)s(Fy|RSsAx

WOIx

≺∈∃∈∀

Observing Figure 1, it may be understood that as

the size of the assortment grows, its variability also

grows. This means that the MOP that is defined in

Equation 7 involves contradicting objectives.

Therefore the solution, (which is defined in Section

4.4) may involve a Pareto front within the auxiliary

MOP of variability versus cost.

4.4 Problem Solution

The solution to the problem, which has been defined

by Equation 7, is an assortment A

* and related

competency Pareto front in the auxiliary MOP, FC*:

*}AA:)A(Z|ZZ{:FC

)}A()'A(:'A|*PA{:*A

sss

***

sssss

≡Ψ=∈=

ΨΨ∃¬⊆= ≺

(8)

To elucidate the notions of Equation 8, refer to

Figure 2. In order to simplify the example, suppose

that the cost is the number of members in an

assortment.

Figure 2: The PL and the optimal assortments.

4.5 Auxiliary MOP's Boundary

Solutions

The boundary solutions of the problem of Equation

8 are; a single solution with the maximal hyper

volume on one side and a set of all the solutions,

which have the scenario/s' performances on global

Pareto front on the other side. This might be seen by

inspecting Figure 3, in which a PL in a bi-objective

space is depicted.

ASSORTMENT OF SOLUTIONS FOR VARIABLE TASKS IN MULTI-OBJECTIVE PROBLEMS

273

Figure 3: Four PL's solutions RSFs.

The PL is associated with four RSF's of the

solutions. It can be seen that the black related RSF

possesses the highest variability (biggest hyper

volume) when just one member for the assortment is

sorted. Nevertheless, the highest possible variability

would be if all three grey RSFs are combined,

unfortunately at the expanse of cost (three

members). The question is how to find the solutions

which are not the boundary solutions. Here we

suggest using EMO for that search.

4.6 The Evolutionary Search

In this paper, it is assumed that the Pareto set and the

related PL, are given. Therefore, in the current

paper, the focus is on the search for sub-sets of the

optimal solutions' set in order to comply with

Equation 8. The evolutionary search involves a

single chromosome integer value code for each

individual. The length of the individual is predefined

and is usually constrained by transportation volume

or maximal cost boundary. Decoding an individual

results in both the size of the assortment as well as

which of the solutions (found by using the procedure

of (Avigad et al., 2009)) are to be used for the

assortment. For example, depict the coded individual

of Figure 4.

Figure 4: An Individual.

Decoding the individual of Figure 4, results in a four

member assortment with solutions 3, 7 (twice) and 9

as its members. Any MOEA may be used for the

evolutionary search. Here we have used the NSGA-

II (Deb et al. 2002), which is given in the following

with some added details that relate the algorithm to

the current methodology:

Store the RSFs of all P* solutions (see equations…

and (

Avigad et al. 2000) for details how to evolve

them).

1. Initialize a population

with n individuals.

create Q

2. Create a combined population

ttt

QPR ∪=

3. Decode R

and compute the competency of all

assortments using equations 4 and 5.

4. Perform a non-dominated sorting for

*Z and

find fronts,

, i=1,…,n

where n

is the

number of fronts in a generation.

5. Initialize a new parent population

∅

+1t

P . Set

a non-dominance level counter i=1.

While

nFrP

i1t

≤+

, include the i-th front in

the new parent population:

i1t1t

FrPP +=

and

set i=i+1.

6. Perform the Crowding Sort procedure (see

(Deb et al., 2002)), and complete the filling of

with the most widely spread

−

solutions using the Crowding Distance measure

of (Deb et al., 2002).

7. Create offspring population

from

Tournament Selection.

8. Perform crossover to obtain

from

9. Perform mutation to obtain

from

10. If last generation Go to 12

11. Go to 2

12. Introduce the FC* (Equation 8) to decision

makers.

5 EXAMPLE

The example described in this section proceeds from

Avigad et al. (2000). A cart of mass m(m') driven by

a motor and gear with a mass of m' is to be designed.

m(m'), meaning that as the chosen motor gets

heavier, the carrying cart should be bigger and

heavier in order to support the motor. The cart is

carrying a load M as depicted in Figure 5.

Figure 5: The cart.

Considering a movement of the cart along the x axis,

and that the overall mass is m*=m(m')+m'+M, the

following relaxed equation has been shortly

developed in (Avigad et al., 2009)

IJCCI 2009 - International Joint Conference on Computational Intelligence

274

0gm)x(f

)'m(T

=−



, (9)

where T, is the driving moment beyond the

transmission gear, which depends on the motor

size/mass (i.e., the bigger the stronger) and may

change such that:

max

TT0

≤

≤ . R=0.05m is the

wheel diameter, and f

is the rolling resistance force

that may be computed for inflation wheel-pressure

of 30 psi,

5.2

)x(002.001.0f



. Let

m(m')=mo+2*m' where mo=4kg. The following bi-

objective problem, which maximizes the speed and

carried load of the cart, i.e.,

)M,xmax(



has been

considered. T and m' are motor dependent and are

taken from Pittman

motor data. The resulting PL

is depicted in Figure 6, which is borrowed from

Avigad et al. (2000).

Figure 6: The Pareto Layer of the problem.

The algorithm, which was given in Section 4.6, has

been utilized in order to search for the optimal

assortments. A population of 100 individuals with

50%, 3% crossover and mutation rates respectively

were used. The Pareto front of the auxiliary MOP is

found and is depicted in Figure 7.

Figure 7: The most variable yet expensive assortment.

It is observed that there are three possible different

assortments consisting of one, two or three

members. The solution to equation 8 is depicted in

Figure 8. In the figure, each square represents an

optimal assortment. The leftmost square represents

the three member assortment (all solutions of Figure

7). Although its cost is high compared to other

assortments, its variability is the largest. This means

that if all there carts are available more performance

demands may be complied with. The middle square

represents the two cart assortment with its medium

competency. The rightmost square represents the

single cart assortment, which has the least variability

but with the least cost. Choosing one of the

assortments is up to the DMs, who should consider

their available resources, versus the gain of

variability.

Figure 8: The auxiliary problem Pareto front.

The algorithm was run 50 times for the current

problem. The statistical data is depicted in Figure 9.

Figure 9: Statistical results for the cart problem.

The Figure depicts the spread of the resulting

variability for the three assortments as related to

their variability.

ASSORTMENT OF SOLUTIONS FOR VARIABLE TASKS IN MULTI-OBJECTIVE PROBLEMS

275

6 CONCLUSIONS

In the paper, we introduce the notion of an

assortment, suggest an auxiliary MOP whose

solution may aid decision makers in choosing an

optimal assortment. Furthermore, an EMO to solve

the auxiliary MOP is suggested. The paper

contributions are: a. A new kind of a set (the

assortment) is represented and motivated, b. A need

to choose a set based on a set of sets has been

encountered here for the first time by using EC, c.

New motivation to correlate nature and sociology to

engineering design has been suggested d. A new

motivation for variability within engineering design

has been highlighted, e. Yet another use of MOEA's

has been explored.

Future work should consider searching for

assortments based on the auxiliary MOP directly

from the beginning without relying on an

a priori

search of the PL. Furthermore, some more examples

and test cases should be explored. Among the

investigated cases, problems with more objectives,

both in the original and the auxiliary MOPs should

be interesting. Finally, robustness, while choosing an

assortment, should be an important issue. Gaining

more robustness may call for a need for overlapping

of RSFs, which may reduce variability.

REFERENCES

Avigad G., Eisenstadt, E., and Goldvard A., Report no

Br.11709, at http://mech.braude.ac.il/gideonavigad/

Avigad, G., Moshaiov, A., Set-based Concept Selection in

Multi-objective Problems: Optimality versus

Variability Approach, Journal of Engineering Design,

Vol. 20(3) pp: 217 – 242, 2009

Bensaid, N. and Matheieu, P. An Autonomous Agent

System to Simulatea Set of Robots Exploring a

Labyrinth, In the proceedings of 11th International

FLAIRS Conference, pp: 384-388, Sanibel, Florida

May, 17-20, 1998.

Boer, P., Szyszko, K., and Vermeulen, R. Spreading the

risk of extinction by genetic diversity in populations of

the carabid beetle, Netherlands Journal of Zoology, 43,

242-259, 1993.

Booy, G. et al., Genetic Diversity and the Survival of

Populations. J. Planet Biology 2, 379-395, 2000.

Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. A

Fast and elitist multiobjective genetic algorithm:

NSGA–II, IEEE Transactions on Evolutionary

Computation, 6(2):182–197, April 2002.

Deb K., Multi-objective optimization using evolutionary

algorithms. J. Wiley & Sons, Ltd, 2001.

Goldberg, D.E., Genetic algorithms in search,

optimization and machine learning, Addison-Wesley,

1989.

Marler, R.T., and Arora, J.S. Survey of multi-objective

optimization methods for engineering, Structural

Multidisciplinary Optimization, 26: 369–395, 2004.

Mattson, C. A., and Messac, A. Pareto frontier Based

concept selection under uncertainty, with

visualization, Optimization and Engineering, 6: 85–

115, 2005.

Mayr E. 1982. The growth of biological thought: diversity,

evolution and inheritance. Harvard, Cambs. p567 et

seq.

Pareto, V. 1906: Manuale di Economica Politica, Societa

Editrice Libraria. Milan; translated into English by

A.S. Schwier as Manual of Political Economy, edited

by A.S. Schwier and A.N. Page, 1971. New York:

A.M. Kelley

Van Veldhuizen, D.A., and Lamont, G.B. Multiobjective

evolutionary algorithms: analyzing the State-of-the-

Art, Evolutionary Computation, 8(2): 125-147, 2000.

Zitzler, E., Thiele, L.: Multiobjective Optimization Using

Evolutionary Algorithms—A Comparative Study. In

Eiben, A.E., ed.: Parallel Problem Solving from

Nature V, Amsterdam, Springer-Verlag 292–301,

1998.

Zitzler, E., L. Thiele, M. Laumanns, C. M. Fonseca, and

V. G. da Fonseca (2003). Performance assessment of

multiobjective optimizers: An analysis and review.

IEEE Transactions on Evolutionary Computation 7(2),

117–132, 2003.

Zitzler, E., and K¨unzli, S. Indicator-Based Selection in

Multiobjective Search, Proceedings of the 8th

International Conference on Parallel Problem Solving

from Nature (PPSN VIII) September 2004,

Birmingham, UK

IJCCI 2009 - International Joint Conference on Computational Intelligence

276