An Evolutionary Algorithm with Advanced Goal and Priority Specification for Multi-objective Optimization

K. C. Tan, E. F. Khor, T. H. Lee and R. Sathikannan

Volume 18, 2003

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Abstract:
This paper presents an evolutionary algorithm with a new goal-sequence domination scheme for better decision support in multi-objective optimization. The approach allows the inclusion of advanced hard/soft priority and constraint information on each objective component, and is capable of incorporating multiple specifications with overlapping or non-overlapping objective functions via logical 'OR' and 'AND' connectives to drive the search towards multiple regions of trade-off. In addition, we propose a dynamic sharing scheme that is simple and adaptively estimated according to the on-line population distribution without needing any a priori parameter setting. Each feature in the proposed algorithm is examined to show its respective contribution, and the performance of the algorithm is compared with other evolutionary optimization methods. It is shown that the proposed algorithm has performed well in the diversity of evolutionary search and uniform distribution of non-dominated individuals along the final trade-offs, without significant computational effort. The algorithm is also applied to the design optimization of a practical servo control system for hard disk drives with a single voice-coil-motor actuator. Results of the evolutionary designed servo control system show a superior closed-loop performance compared to classical PID or RPT approaches.

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19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998
Hewlett-Packard CompanyJournal of Artificial Intelligence Research 18 {2003}
183-215                     Submitted 9/01; published 2/03 2512003 AI Access
Foundation and Morgan Kaufmann Publishers. All rights reserved. An Evolutionary
Algorithm with Advanced Goal and Priority Specification for Multi-objective
Optimization Kay Chen Tan ELETANKC@NUS.EDU.SG Eik Fun Khor EIKFUN.KHOR@SEAGATE.COMTong
Heng Lee ELELEETH@NUS.EDU.SGRamasubramanian Sathikannan K.SATHI@GSK.COMNational
University of Singapore 4 Engineering Drive 3, Singapore 117576 Republic
of SingaporeAbstractThis paper presents an evolutionary algorithm with a
new goal-sequence domination scheme fo
r better decision support in multi-objective
optimization. The approach allows the inclusion of advanced hard/soft priority
and constraint information on each objective component, and
 is capable
of incorporating multiple specifications with overlapping or non-overlapping
objective functions via logical 223OR224 and 223AND224 connectives to
drive the search towards multiple regions of trade-off. In 
addition, we
propose a dynamic sharing scheme that is simple and adaptively estimated
according to the on-line population distribution without needing any a priori
paramet
er setting. Each feature in the proposed algorithm is examined to
show its respective contribution, and the pe
rformance of the algorithm
is compared with other evolutionary optimization methods. It is shown that
the proposed algorithm has performed well in the diversity of evolutionary
search and
 uniform distribution of non-dominated individuals along the
final trade-offs, without significant computational effort. The algorithm
is also applied to the design optimization of a practical servo control sy
stem for hard disk drives with a single voice-coil-motor actuator. Results
of the evolutionary designed servo control system show a superior closed-loop
performance compared to classical PID or RPT approaches.1. Introduction Many
real-world design tasks involve optimizing a vector of objective fu
nctions
on a feasible decision variable space. These objective functions are often
non-commensurable and in competition with each other, and cannot be simply
aggregated into a scal
ar function for optimization. This type of problem
is known as multi-objective {MO} optimization problem, for which the solution
is a family of points known as a Pareto-optimal set {Goldberg, 1989}, where
each objective component of any member in the set can only be improved b
y degrading at least one of its other objective components. To obtain a good
solution via conventional MO optimization techniques such as the methods
of inequalities, goal attainment or weighted sum approach, a continuous cost
function and/or a set of precise settings of weights or goals are required,
which are usually not well manageable or understood {Grace, 1992; Osyczka,
1984}. Emulating the Darwinian-Wallace principle of 223survival-of-the-fittest224
in natural selection and genetics, evolutionary algorithms {EAs} {Holland,
1975} have been found to be effective and efficient in solving complex problems
where conventional optimization tools fail to work well. 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN184
The EAs evaluate performances of candidate solutions at multiple points simultaneously,
and are capable of approaching the global optimum in a noisy, poorly understood

and/or non-differentiable search space {Goldberg, 1989}. Since Schaffer222s
work {1985}, evolutionary algorithm-based search techniques for MO optimization
have been gaining significant attention from researchers in various disciplines.
This is reflected by the high volume of publications in this topic in the
last few years as well as the first international conference on Evolutionary
Multi-criteria Optimization {EMO22201} held in March 2001 at Zurich, Switzerland.
Readers may refer to {Coello Coello, 1996;
 1999; Deb, 2001; Fonseca, 1995;
Van Veldhuizen & Lamont, 1998; Zitzler & Thiele, 1999} on detailed implementation
of various evolutionary techniques for MO optimization. Unlike most conventional
methods that linearly combine multiple attributes to form a composite scalar
objective function, the concept of Pareto's optimality or modifie
d selection
scheme is incorporated in an evolutionary MO optimization to evolve a family
of solutions at multiple points along the trade-off surface simultaneously
{Fonseca & Fleming, 1993}. 
Among various selection techniques for evolutionary
MO optimization, the Pareto-dominance scheme {Goldberg, 1989} that assigns
equal rank to all non-dominated individuals is an effective approach for
comparing the strengths among different candidate solutions in a population
{Fonseca 
& Fleming, 1993}. Starting from this principle, Fonseca and Fleming
{1993} proposed a Pareto-base
d ranking scheme to include goal and priority
information for MO optimization. The underlying reason is that certain user
knowledge may be available for an optimization problem, such as pre
ferences
and/or goals to be achieved for certain objective components. This information
could be useful and incorporated by means of goal and priority vectors, which
simplify the optimization process and allow the evolution to be directed
towards certain concentrated regions of the trade-offs. Although the ranking
scheme is a good approach, it only works for a single goal and p
riority
vector setting, which may be difficult to define accurately prior to an optimization
process for real-world optimization problems. Moreover, the scheme does not
allow advanced specifications, such as logical 223AND224 and 223OR224
operations among multiple goals and priorities. Based on the Pareto-based
domination approach, this paper reformulates the domination scheme to incorporate
advanced specifications for better decision support in MO optimization. Besides
the flexibility of incorporating goal and priority information on each objective
component, the proposed domination scheme allows the inclusion of hard/soft
priority and constraint specifications. In addition, the approach is capable
of incorporating multiple specifications with overlapping or non-overlapping
objective functions via logical 223OR224 and 223AND224 connectives to
drive the search towards multiple regions of the trade-off. The paper also
proposes a dynamic sharing scheme, which computes the sharing distance adaptively
based upon the on-line population distribution in the objective domain without
the need of any 
a priori parameter setting. The dynamic sharing approach
is essential since it eliminates the difficulty of manually finding an appropriate
sharing distance prior to an optimization process. The choice of such a distance
would be sensitive to the size and geometry of the discovered trade-offs
{Coello Coello, 1999; Fonseca & Fleming, 1993}. This paper is organized as
follows: The formulation of the proposed domination scheme for better decision
support is presented in Section 2. A dynamic sharing sch
eme that estimates
the 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c)
1998 Hewlett-Packard Company            ANEVOLUTIONARY ALGORITHM FOR MULTI-OBJECTIVE
OPTIMIZATION185 sharing distance adaptively based upon the on-line population
distribution is described in Section 3. Section 4 examines the usefulness
and contribution of each proposed feat
ure in the algorithm. The performance
comparison of the proposed algorithm with other evolutionary MO optimization
methods is also shown in the section. Practical application of the proposed
algorithm to servo control system design optimization is given in Section
5. Conclusions are drawn in Section 6. 2.  Advanced Goal and Priority Specifications
for MO OptimizationA multi-objective optimization problem seeks to optimize
a vector of non-commensurable and often competing objectives, i.e., it tends
to find a parameter set P for }{MinPFP000}000217,nR000217P, where P=
{p1,p2,205,pn} is a n-dimensional vector having n decision variables or
parameters, and 000} defines a feasible set of P.F = {f1,f2,205,fm} is
an objective vector with m objectives to be minimized. For a MO optimization
problem with simple goal or priority specification on 
each objective function,
the Pareto-based ranking scheme is sufficient {Fonseca & Fleming, 1993}.
In practice, however, it may be difficult to define an accurate goal and
priority setting ina priori to an optimization process for real-world optimization
problems. Besides goal and priority information, there could also be additional
supporting specifications that are useful or need to be satisfied in the
evolutionary search, such as optimization constraints or feasibility of a
solution. Moreover, the Pareto-based ranking scheme does not allow advanced
specifications, such as logical 223AND224 and 223OR224 operations among
multiple goals and priorities for better decision support in complex MO optimization.
In this section, a new goal-sequence Pareto-based dominationscheme is proposed
to address these issues and to provide hard/soft goal and priority specifications
for better controls in the evolutionary optimization process. 2.1  Pareto-based
Domination with Goal Information This section is about an effective two-stage
Pareto-based domination scheme for MO optimization, which is then extended
to incorporate advanced soft/hard goal and priority specifications. Consider
a minimization problem. An objective vector Fa is said to dominate another
objective vector Fbbased on the idea of Pareto dominance, denoted by Fa000SFb,iff
},...,2,1{,,miffibia000217000005000d and jbjaff,,000037 for some},...,2,1{mj000217.
Adopting this basic principle of Pareto dominance, the first stage in the
proposed domination approach ranks all individuals that satisfy the goal
setting to minimize the objective functions as much as possible. It assigns
the same smallest cost for all non-dominated individuals, while the dominated
individuals are ranked according to how many individuals in the population
dominate them. The second stage ranks the remaining individuals that do not
meet the goal setting based upon the following extended domination scheme.
Let aa000fF and ab000fF denote the component of vector aF and bF respectively,
in which aF does not meet the 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB
Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN186 goalG.
Then for both aF and bF that do not totally satisfy the goal G, the vector
aF is said to dominate vector bF {denoted by aFG000SbF}iff {aa000fF000Sab000fF}
or {abs{aF-G}000Sabs{bF-G}} {1} For this, the rank begins from one increment
of the maximum rank value o
btained in the first stage of the cost assignment.
Therefore individuals that do not meet the goal will be directed toward the
goal and the infinum in the objective domain, while those that have satisfied
the goal will only be directed further towards the infinum. Note that the
domination comparison operator is non-commutative {aFG000SbF000zbFG000SaF}.
Figure 1 shows an optimization problem with two objectivesf1 and f2to be
minimized. The arrows in Figure 1 indicate the transformation acco
rding
toGFF'000020000  of the objective function F to F'for individuals that
do not satisfy the goal, with the goal as the new reference point in the
transformed objective domain. It is obvious that the domination scheme is
simple and efficient for comparing the strengths among partially or totally
unsatisfactory individuals in a population. For comparisons among totally
satisfactory individuals, the basic Pareto-dominance is sufficient. To study
the computational efficiency in the approach, the population is divided into
two separate groups classified by the goal satisfaction, and the domination
comparison is performed separately in each group of individuals. The total
number of domination comparisons for the two-stage domination scheme is Nc
= [Gn000ff{Gn000ff-1}+Gn000f{Gn000f-1}] where Gn000ff is the number
of individuals that completely satisfy the goal G and Gn000f is the number
of individuals partially satisfy or completely not satisfy the goal G. Note
that Gn000ff + Gn000f = N for a population size of N. Hence, in any generation,
Nc is always less than or equal to the total number of domination comparisons
among all individuals in a population {each individual in the population
is compared with {N-1}individuals}, i.e., Nc000d Nnc = }1{000020NN. In
the next section, the two-stage Pareto-based domination scheme will be extended
to incorporate soft/hard priority specifications for advanced MO optimization.
19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998
Hewlett-Packard Company            ANEVOLUTIONARY ALGORITHM FOR MULTI-OBJECTIVE
OPTIMIZATION187 11562456f1f2G6522262226222Figure 1:  Advanced Pareto Domination
Scheme with Goal Information 2.2  Goal-Sequence Domination Scheme with Soft/Hard
Priority Specifications One of the advanced capabilities in evolutionary
MO optimization is to incorporate cognitive specification, such as priority
information that indicates the relative importance of the multiple tasks
to provide useful guidance in the optimization. Consider a problem with multiple
non-commensurable tasks, where each task is assigned a qualitative form of
priority indicating its relative importance. In general, there exist two
alternatives to accomplish these tasks, i.e., to consider one task at a time
in a sequence according to the task priority or to accomplish all tasks at
once before considering any individual task according to the task priority.
Intuitively, the former approach provides good optimization performance for
tasks with higher priority and may result in relatively poor performance
for others. This is due to the fact that optimizing the higher priority tasks
may be at the performance expense of the lower priority tasks. This definition
of priority is denoted as 223hard224 priority in this paper. On the other
hand, the latter approach provides a distributed approach in which all tasks
aim at a compromise solution before the importance or priority of individual
task is considered. This is defined as "soft" priority. Similarly, priorities
for different objective components in MO optimization can be classified as
"hard" or "soft" priority. With hard priorities, goal settings {if applicable}
for higher priority objective components must be satisfied first before attaining
goals with lower priority. In contrast, soft priorities will first optimize
the overall performance of all objective components, as much as possible,
before attaining any goal setting of an individual objective component in
a sequence according to the priority vector. To achieve greater flexibility
in MO optimization, the two-stage Pareto-based domination scheme is further
extended to incorporate both soft and hard priority specifications with or
without goal information by means of a new goal-sequence domination. Here,
instead of having one priority vector to indicate priorities among the multiple
objective components {Fonseca & Fleming, 1998}, two kinds of priority vectors
are used to accommodate the soft/hard priority information. Consider an objective
priority vector, Pf0002170002011xm and a goal priority vector, Pg0002170002011xm,
where Pf{i} represents the priority for the ith objective component F{i}
that is to be minimized; Pg{i} denotes the priority for 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN188
theith goal component G{i} that is to be attained; m is the number of objectives
to be minimized and 000201 denotes the natural numbers. The elements of
the vector PfandPg can take any value in the natural numbers, with a lower
number representing a higher priority and zero representing a 223don222t
care224 priority assignment. Note that repeated values among the elements
in PfandPg can be used to indicate equal priority provided that Pf{i}000zPg{i}000005i000217
{1, 2, 205, m}, avoiding contradiction of the priority assignment. With
the combination of an objective priority vector Pfand a goal priority vectorPg,
soft and hard priorities can be defined provided that there is more than
one preference among the objective components as given by 000007 {{Pf :
Pf {j} > 1} 000233 {Pg : Pg {j} > 1}} for some j000217 {1, 2, 205, m}
{2} Based on this, a priority setting is regarded as 223soft224 iff000005i000217
{1, 2, 205, m}000007 {{Pf : Pf {i} = 1} 000233 {Pg : Pg {i} = 1}} {3}
else, the priority is denoted as 223hard224. For example, the settings
of Pf = [1, 1, 2, 2] and Pg = [0, 0, 0, 0] for a 4-objective optimization
problem indicate that the first and second objective components are given
top priority to be minimized, as much as possible, before considering minimization
of the third and fourth objective components. Since all elements in Pg are
zeros {don222t care}, no goal components will be considered in the minimization
in this case. On the other hand, the setting of Pf = [0, 0, 0, 0] and Pg
= [1, 1, 2, 2] imply that the first and second objective components are given
the first priority to meet their respective goal components before considering
the goal attainment for the third and fourth objective components. The above
two different priority settings are all categorized as hard priorities since
in both cases, objective components with higher priority are minimized before
considering objective components with lower priority. For soft priority as
defined in Eq. 3, the objective priority vector and goal priority vector
can be set as Pg= [1, 1, 1, 1] and Pf= [2, 2, 3, 3], respectively. This implies
that the evolution is directed towards minimizing all of the objective components
to the goal region before any attempt to minimize the higher 
priority objective
components in a sequence defined by the priority vector. To systematically
rank all individuals in a population to incorporate the soft/hard priority
specifications, a sequence of goals corresponding to the priority information
can be generated and represented by a goal-sequence matrix G222 where the
kth row in the matrix represents the goal vector for the corresponding kth
priority. The number of goal vectors to be generated depends on the last
level of priority z, where z is the maximum value of any one element of Pg
and Pf as given by z = max[Pg{i},Pf{j}]   },...,2,1{,mji000217000005
{4} For this, the goal vectors with kth priority in the goal-sequence matrix
G222k{i} for the priority index k= 1, 2,205, z is defined as ,,...,1mi000
000005G222000260000257000260000256000255000 000 000 000
000 otherwisekiifkiifiiiifgNjNjk}{}{}]{max[}]{min[}{}{,...,1,...,1PPFFG
{5} whereN denotes the population size; }]{min[,...,1iNj000 F and }]{max[,...,1iNj000
F represents the minimum and maximum value of the ith objective function
from the on-line population distribution, 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company            ANEVOLUTIONARY
ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION189 respectively. In Eq. 5, for
any ith objectivecomponent of any k priority level, the reason for assigningG222k{i}
with G{i} is to guide the individuals towards the goal regions; }]{min[,...,1iNj000
F is to minimize the corresponding objective component as much as possible;
a
nd }]{max[,...,1iNj000 F is to relax the requirements on the individuals
to give other objective com
ponents more room for improvement. According
to Eq. 5, the goal-sequence matrix G222k{i} is dynamic at each generation,
as the values of }]{min[,...,1iNj000 F and }]{max[,...,1iNj000 F are dynamically
computed depending on the on-line population distribution. After computing
the sequence of goals G222k000005k000217 {1, 2,205, z}, the individuals
are first ranked according to the computed goal G2221 for the first priority.
Then each group of individuals that has the same ranks will be further compared
and ranked according to next goalG2222 for the second priority to further
evaluate the individuals' domination
 in a population. In general, this
ranking process continues until there is no individual with the same rank
value or after ranking the goal G222z that has the lowest priority in the
goal-sequence matrix. Note that individuals with the same rank value will
not be further evaluated for those components with 223don222t care224
assignments. With the proposed goal-sequence domination scheme as given in
Eq. 5, both hard and soft priority specifications can be incorporated in
MO optimization. Without 
loss of generality, consider a two-objective optimization
problem, with f1 having a higher priority than f2, as well as a goal setting
of G = [g1,g2]. For soft priority optimization as defined in Eq. 3, the goal
priority
 vector and objective priority vector can be set as Pg= [1, 1]
and Pf= [2, 0], respectively. Let min[F{i}] and max[F{i}] denote the minimum
and maximum value of the i-objective component of F in a population, respectively.
The relevant goals in the goal-sequence matrix for each priority level as
defined in Eq. 5 are then given as G2221 = Gfor the first priority andG2222
= {min[F{1}], max[F{2}]} for the second priority. The goal-sequence domination
scheme for the two-objective minimization problem is illustrated in Figure
2. Here, the rank value of each individ
ual is denoted by r1000or2,wherer1
and r2 is the rank value after the goal-sequence ranking of the first and
second priority, respectively. The preference setting indicates that both
g1 and g2 are given the same priority to be attained in the optimization
before individuals are further ranked according to the higher priority of
f1. This is illustrated in Figure 3a, which shows the location of the desired
Pareto-front {represented by the dark region} and the expected evolution
direction {represented by the curved arrow} in the objective domain for an
example with an unfeasible goal setting G.For hard priority optimization
as defined in Eqs. 2 and 3, the goal priority vector and objective priority
vector can be set as Pg = [1, 2] and Pf = [0, 0], respectively. According
to Eq. 5, this gives a goal sequence of G2221 = [g1, max[F{2}] and G2222
= [max[F{1}], g2] for the first and second priority, respectively. It implies
that g1 is given higher priority than g2 to be attained in the optimization.
Figure 3b shows the location of the desired Pareto-front {represented b
y dark region} and the expected evolution direction {represented by curved
arrow} in the objective domain. As compared to the solutions obtained in
soft priority optimization, hard priority o
ptimization attempts to attain
the first goal component and leads to the solution with better f1 {higher
priority} but worse f2 {lower priority}. It should be mentioned that the
setting of soft/hard priority may be subjective or problem 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN190
dependent in practice. In general, the hard priority optimization may be
appropriate for problems with well-defined goals in order to avoid stagnation
with unfeasible goal settings. Soft priority optimization is more suitable
for applications where moderate performance among various objective components
is desired. Besides soft/hard priority information, there may be additional
specifications such as optimization constraints that are required to be satisfied
in the optimization. These specifications could be easily incorporated in
MO optimization by formulating the constraints as additional objective components
to be optimized {Fonseca & Fleming, 1998}. This will be discussed in the
next section.   1000o2  1000o15000o6 2000o3 4000o4   5000o57000o8f1f27000o7G'2g1g2G'1Figure
2:  Goal-sequence Domination with Goal G = {g1,g2}, Priority Pg = [1, 1]
and Pf = [2, 0] Efl+AkD:]`'eh;*V?>iZp`UT'*I!3Ne93VjD,*)S&&4=g*s'[Mq3obA"GED6l<^0'k=hG*bZINJc*`Qn"Z4K`;:S'R.93ZR"C:R0J&p,^['YA:Ufme?%]*u=n"kD+8pCJODGpA^p$1'"[VXZ/I*fQP;*D2nE`:A_D'FF%R]b;9`qf.'Bp0!O_d:VUlZ!aK1(iDD%FVm"D&IDOfWIC_l?[hT-<=WGLte9rQ5OL0]eWcD@6.DiAYT]/S,bY'=eP.KC"9G6'nK?&"+V(g.*K*=$OF$.o1J5[?i2FisQiAp-o2@`sOM'[<,h'-t3)B/K@mSKTXU@KSZqVPF!+668H45)E*9%)GP8PpJLj[l%/!uk&#]u>!2dTt+0>PD.N;-igbBkL^qS>U_GE$$YhcJXk?XM-iEk3@f'R/>E;eatD%U[_,OF1q`B$D723TA>[-7'U.i35Af.TtlHkY,WdJX:UoCQ:HWQ_N9023ng$r9sBFfT.Tf?(^`e_/9^a%#=f;nmYm]&!;STm"TqKf&SqgS*^%Op?lIH(Jlol?Z]rT-ftZ^L%#?K[R5Hegnpn9Zj%_]4&r/E=]ql@7.QGAX88pim7_kPBTK+/j7?Rh<)T5%JSTEG<]:^I$P9DsB[)?[bWBV]TIL;$E+TPFfLZ;k/1H8aIMK^fHBjZO!/;'/WB_2ip0]-SJIj*Y_m_?1?u=p13.u&jni)CYI0mXoQP0fR:N0.i-H'#bHf*TM0O_`&!nD>$P5fWVCP,g5898BoSc2T3(?;g)rn;6NtLM0RCMQ)&uhC^ki+/610#IL9SJPTWVsnI;'HV]>`m%75XhFs,UBB$1@emJ^1f5JA*K)=N+Oh:13FfeOcu#C5#o?r)(T7aVGcu:4GNa/PKERPS2@8JcaeNiB7mR`,d%h0=b!#`hLUe-(G.%L&>A_CHU$d53bUg8pbW)Xp^oL3:dl1mbV`6P04no@N`;#oaT.u*R,L7iR9jH4-J^iDE38.n0W1=KrnV3F34i%%jVF[m)P4@ZI^U_C5,gD.ZrH>]
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Q
n
Q
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510.54 192.9 m
/N182 [10.1979 0 0 10.0057 0 0] Tf
(f1f2GUnfeasibleRegionEvolutiondesired
solutiong1g2G'2G'1max[F{1}]max[F{2}]{b} Hard priority f1 higher than f2Figure
3:  Illustration of Soft and Hard Priority with Unfeasible Goal S
etting
2.3  Optimization with Soft and Hard Constraints Constraints often exist
in practical optimization problems {Luus et al.
 1995; Michalewicz & Schoenauer,
1996}. These constraints are often incorporated in the MO optimization function
as 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998
Hewlett-Packard Company            ANEVOLUTIONARY ALGORITHM FOR MULTI-OBJECTIVE
OPTIMIZATION191 one of the objective components to be optimized. It could
be in the form of "hard" constraint where the optimization is directed towards
attaining a threshold or goal, and further optimization is meaningless or
not desirable whenever the goal has been satisfied. In contrast, a "soft"
constraint requires that the value of objective component corresponding to
the cons
traint is optimized as much as possible. An easy approach to deal
with both hard and soft constraints concurrently in evolutionary MO optimization
is given here. At each generation, an updated objective function Fx#concerning
both hard and soft constraints for an individual x with its objective function
Fx can be computed in a priori to the goal-sequence domination scheme as
given by 
000>000@000>000@000257000256000255000037000 otherwiseiiiifiii}{}{&hardis}{}{}{}{#GFGFGFxxx
},...,1{mi000 000005 {6} In Eq. 6, any objective component i that corresponds
to a hard constraint is assigned to the value ofG{i} whenever the hard constraint
has been satisfied. The underlying reaso
n is that there is no ranking preference
for any particular objective component that has the s
ame value in an evolutionary
optimization process, and thus the evolution will only be directed towards
optimizing soft constraints and any unattained hard constraints, as desired.
2.4  Logical Connectives among Goal and Priority Specifications For MO optimization
problems with a single goal or priority specification, the decision maker
often needs to 223guess224 an appropriate initial goal or priority vector
and then manually observe the optimization progress. If any of the goal components
is too stringent or
 too generous, the goal setting will have to be adjusted
accordingly until a satisfactory solution can be obtained. This approach
obviously requires extensive human observation and intervention, which can
be tedious or inefficient in practice. Marcu {1997} proposed a method of
adapting the goal values based upon the on-line population distribution at
every generation. However, the adaptation of goal values is formulated in
such a way that the search is always uniformly directed towards the middle
region of the trade-offs. This restriction may be undesirable for many application
s, where the trade-off surface is unknown or the search needs to be directed
in any direction other than the middle region of the trade-off surface. To
reduce human interaction and to allow multiple sets of goal and priority
specifications that direct the evolutionary search towards a different portion
of the trade-off surface in a single run, the goal-sequence domination scheme
is extended in this section to enable logical statements such as 223OR224
{000211} and 223AND224 {000210} operations among multiple goal and
priority specifications. These logical operations can be built on top of
the goal-sequence domination procedure for each specification. By doing this,
the unified rank value for each individual can be determined and taken into
effect immediately in the evolution towards the regions concerned. Consider
ranking an objective vector Fx by comparing it to the rest of the individuals
in a population with ref
erence to two different specification settings
of Si and Sj, where Si and Sj are the specifications concerning any set of
objective functions with or without goals and priorities. Let these ranks
be denoted by rank{Fx,Si} and rank{Fx,Sj}, respectively. The 223OR224 and
223AND224 operations for the two goal settings are then defined as, 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN192
000^000`},{},,{min},{jxixjixrankrankrankSSSSFFF000 000211 {7a} 000^000`},{},,{max},{jxixjixrankrankrankSSSSFFF000
000210 {7b} According to Eq. 7, the rank value of vector Fx for an 223OR224
operation between any two specifications Si and Sj takes the minimum rank
value with respect to the two specification settings. This is in order to
evolve the population towards one of the specificati
ons in which the objective
vector is less strongly violated. In contrast, an 223AND224 operation takes
the maximum rank value in order to direct the evolutionary search towards
minimizing the amount of
 violation from both of the specifications concurrently.
Clearly, the 223AND224 and 223OR224 operations in Eq. 7 can be easily
extended to include general logical specifications with more complex connectives,
such as 223{Si OR Sj} AND {Sk OR Sl}224, if desired. 3.  Dynamic Sharing
Scheme and MOEA Program Flowchart 3.1  Dynamic Sharing Scheme Fitness sharing
was proposed by Goldberg and Richardson {1987} to evolve an equally distributed
population along the Pareto-optimal front or to distribute the populatio
n at multiple optima in the search space. The method creates sub-divisions
in the objective domain by degrading an individual fitness upon the existence
of other individuals in its neighborhood defined by a sharing distance. The
niche count, 000ff000f000246000 Njjiidshm,, is calculated by summing
a sharing function over all members of the population, where the distance
di,j represents the distance between individual i and j. The sharing function
is defined as 000ff000f000260000257000260000256000255000037000270000270000271000267000250000250000251000247000020000
otherwisedifddshsharejisharei,jji01,,000V000V000D {8} with the parameter
000D being commonly set to 1. The sharing function in Eq. 8 requires a good
setting of sharing distanc
e 000Vshare to be estimated upon the trade-off
surface, which is usually unknown in many optimization problems {Coello Coello,
1999}. Moreover, the size of objective space usually cannot be predefined,
as the exact bounds of the objective space are often undetermined. Fonseca
and Fleming {1993} proposed the method of Kernel density estimation to determine
an appropriate sharing distance for MO optimization. However, the sharing
process is performed in the 221sphere222 space which may not reflect the
actual objective space for which the population is expected to be uniformly
distributed. Miller and Shaw {1996} proposed a dynamic sharing method for
which the peaks in the parameter domain are 221dynamically222 detected
and recalculated at every generation with
 the sharing distance remains
predefined. However, the approach is made on the assumption that the number
of niche peaks can be estimated and the peaks are all at the minimum distance
of 2000Vshare from each other. 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB
Copyright (c) 1998 Hewlett-Packard Company            ANEVOLUTIONARY ALGORITHM
FOR MULTI-OBJECTIVE OPTIMIZATION193 Moreover, their formulation is defined
in the parameter space to handle multi-modal function optimization, which
may not be appropriate for distributing the populati
on uniformly along
the Pareto-optimal front in the objective domain. In contrast to existing
approaches, we propose a dynamic sharing method that adaptively computes
the sharing distance 000Vshare to uniformly distribute all individuals along
the Pareto-optimal front at each generation. This requires no prior knowledge
of the trade-off surface. Intuitively, the trade-offs for an m-objective
optimization problem are in the form of an {m-1} dimensional hyper-volume
{Tan et al. 1999}, which can be approximated by the hyper-volume Vpop{n}
of a hyper-sphere as given by, 1}{2/}1{}{2!21000020000020000273000274000272000253000254000252000u000270000271000267000250000251000247000020000
mnmnpopdmV000S {9} where}{nd is the diameter of the hyper-sphere at generation
n. Note that computation of the diameter }{nd depends on the curvature of
the trade-off curve formed by the non-dominated individuals in the objective
space. For a two-objective optimization problem, the diameter }{nd is equal
to the interpolated distance of the trade-off curve covered by the non-dominated
individuals as shown in Figure 4. Although computation of }{nd that accurately
represents the interpolated curvature of the non-dominated individuals distribution
is complex, it c
an be estimated by the average distance between the shortest
and the longest possible diameter given by dmin{n} and dmax{n}respectively
{Tan et al. 1999}. Let Fx and Fy denote the objective function of the two
furthest individuals in a population. Then dmin{n} is equal to the minimum
length between Fx and Fy, and dmax{n} can be estimated by d1{n} + d2{n} as
shown in Figure 4. The same procedure can also be extended to any multi-dimensional
objective space. To achieve a uniformly distributed population along the
trade-off set, the sharing distance 000Vshare{n} could be computed as half
of the distance between each individual in the {m-1}-dimensional hyper-volume
Vpop{n} covered by the population size N at generation n,000ff000f}{1}{2/}1{!21npopmnsharemVmN000
000u000270000271000267000250000251000247000020000u000020000020000V000S
{10} Substituting Eq. 9 into Eq. 10 gives the sharing distance 000Vshare{n}
at generation n in term of the diameter }{nd and the population size N as
given by 2}{}1/{1}{nmnsharedN000u000 000020000V {11} Clearly, Eq. 11
provides a simple computation of 000Vshare that is capable of distributing
the population evenly along the Pareto front, without the need for any prior
knowledge of the usually 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB
Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN194 unknown
fitness landscape. Moreover, adopting the computation of sharing
 distance
that is dynamically based upon the population distribution at each generation
is also more appropriate and effective than the method of off-line estimation
with pre-assumed trade-off surface as employed in most existing sharing methods,
since the trade-off surface may be changed any time along the evolution whenever
the goal setting is altered. dmin{n}d{n}d1{n}d2{n}Discoveredtrade-off curvef2f1FyFxFigure
4:  The Diameter }{nd of a Trade-off Curve 3.2  MOEA Program Flowchart The
overall program flowchart of this paper222s multi-objective evolutionary
algorithm {MOEA} is illustrated in Figure 5. At the beginning of the evolution,
a population of candidate solutions is initialized and evaluated according
to a vector of objective functions. Based upon the user-defined specifications,
such as goals, constraints, priorities and logical operations, the evaluated
individuals are ranked according to the goal-sequence domination scheme {described
in Section 2} in order to evolve the search towards the global trade-off
surface. The resulted rank values are then further refined by the dynamic
sharing scheme {described in Section 3.1} in order to distribute the non-dominated
individuals uniformly along the discovered Pareto-optimal front. If the stopping
criterion is not met, the individuals will undergo a series of genetic operations
which are detailed within the 223genetic operations224 in Figure 6. Here,
simple genetic operations consisting of tournament selection {Tan et al.
1999}, simple crossover with mating restriction that selects individuals
within the sharing distance for mating {Fonseca & Fleming, 1998} as well
as simple mutation are performed to reproduce offspring for the next generation.
After the genetic operations, the newly evolved population is evaluated and
combined with the non-dominated individuals preserved from the previous generation.
The combined population is then subjected to the domination comparison scheme
and pruned to the desired population size according to the Switching Preserved
Strategy {SPS} {Tan et al. 1999}. This maintains a set of stable and well-distributed
non-dominated individuals along the Pareto-optimal front. In SPS, if the
number of non-dominated individuals in the combined population is less than
or equal to the desired population size, extra individuals are removed according
to their rank values in order to promote stability in the evolutionary search
towards the final trade-offs. Otherwise, the 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company            ANEVOLUTIONARY
ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION195 non-dominated individuals with
high niched count value will be discarded in order to distribute the individuals
uniformly along the discovered Pareto-optimal front. After the process, the
remaining individuals are allowed to survive in the next generation and this
evolutionary cycle is repeated until the stopping criterion is met. Population
initializationFunction evaluationDomination comparisonDynamic sharingIsstopping
criterionmet?Genetic operationsFunction evaluationDomination comparisonSize{nondom}>
popsize?Dynamic sharingFiltering - based onPareto ranked costFiltering -
based onshared costsFinalpopulationYesNoYesNonon-dominated individualsnew
populationevolvedpopulation0006combinedpopulationFigure 5:  Program Architecture
of the MOEA Genetic Operations for MOEA: Let,  pop{n} = population in current
generation nStep 1} Perform tournament selection to select individuals from
pop{n}. The selected population is calledselpop{n}.Step 2} Perform simple
crossover and mating restriction for selpop{n} using the dynamic sharing
distance in Step 1. The resulted population is called crosspop{n}.Step 3}
Perform simple mutation for crosspop{n}. The resulted population is called
evolpop{n}.Figure 6:  Detailed procedure within the box of 223genetic operations224
in Figure 5 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright
(c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN196 4.  Validation
Results on Benchmark Problems This section validates the proposed algorithm
in two ways. The first kind of validation {presented in Section 4.1} illustrates
how each of the proposed features, including goal-sequence domination scheme,
hard/soft goal and priority specifications, logical operations among multiple
goals and dynamic sharing, enhances the performance of MOEA in MO optimization.
As shown in Section 4.2, the second type of validation compares performance
of the proposed MOEA with various evolutionary algorithms based upon a benchmark
problem. Various performance measures are used in the comparison and the
results are then discussed. 4.1  Validation of the Proposed Features in MOEA
In this section, various proposed features in MOEA are examined for their
usefulness in MO optimization. This study adopts the simple two-objective
minimization problem {Fonseca & Fleming, 1993} that allows easy visual observation
of the optimization 
performance. The function has a large and non-linear
trade-off curve, which challenges the algorithm222s ability to find and
maintain the entire Pareto-optimal front uniformly. The two-objective functions,
f1 and f2, to be minimized are given as 000270000270000271000267000250000250000251000247000246000270000270000271000267000250000250000251000247000ffi000020000020000
000270000270000271000267000250000250000251000247000246000270000270000271000267000250000250000251000247000020000020000020000
000 000 281812281811811},...,{811},...,{iiiixexpxxfxexpxxf {12} where,8,...,2,1,22000
000005000037000d000020ixi. The trade-off line is shown by the curve
in Figure 7, where the shaded region represents the unfeasible area in the
objective domain.  f1f2>7(?1%KHJ>!!!Q1!tYH:&HDeC!'q4t>8.''(][b[!!",A!$D8I,61po!*^cM>:'>U.0'W+!!=hR>;ZCb49,@G!!#7a!C-Wh63'A@!!>Uh>=JSm7fZac!!#[m!)NXq;?0p'!!$:)>@@M9@fQL>!!$X3!,_d@Du^)]!*sjM!.4c#NrT0E!*bfj!1He*3ZiC*9!*d2_>mfY!*dME!6kLJci='u!*e4Y>N,Wej8]2H!!)?c"7-!pli:>Z!!)Kg!:g*ioDem`!*f3u!<)s"rW")@!'pSc-NF03!]
cs 
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357.84 250.74 m
/N169 [13.4255 0 0 13.5471 0 0] Tf
(Trade-offCurveUnfeasibleRegionFigure
7:  Pareto-optimal Front in the Objective Domain19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company            ANEVOLUTIONARY
ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION197 The simulations are run for
70 generations with a population size of 100. Standard mutation with a probability
of 0.01 and standard two-point crossover with a probability of 0.7 are used.
To study the merit of the dynamic sharing scheme in MOEA as proposed in Section
3.1, 4 different types of simulations have been performed. The first type
is without fitness sharing. The second and third employ a fixed sharing distance
of 0.01 and 0.1, respectively. The fourt
h uses the dynamic sharing scheme
which does not require any predefined sharing distance setting. Figure 8
illustrates the respective population distribution in the objective domain
at the end of
 the evolution. It can be observed that all of the four simulations
are able to discover the final trade-off, but with some performance difference
in terms of the closeness and uniformity of the population distribution along
the trade-off curve. For the MOEA without fitness sharing as shown in Figure
8a, the population tends to converge to an arbitrary part of the trade-off
curve. This agrees with the findings of Fonseca and Fleming, {1993}. For
the MOEA with fitness sharing, as shown in Figures 8b and 8c, the population
can be distributed along the trade-off curve rather well, although the sharing

distance of 0.01 provides a more uniform distribution than that of 0.1.
This indicates that although fitness sharing contributes to population diversity
and distribution along the trade-off curve, the sharing distance has to be
chosen carefully in order to ensure the uniformity of the population distribution.
This often involves tedious trial-and-error procedures in order to 221guess222
an appropriate sharing distance, since it is problem dependent and based
upon the size of the discovered trade-offs as well as the number of non-dominated
individuals. These difficulties can be solved with the proposed dynamic sharing
scheme, which has the ability to automatically adapt the sharing distance
along the evolution without the need of any predefined parameter, as shown
in Figu
re 8d. .lMoh@!g-I+7F<21S$5/A/rVI nVaqDRp(:'3#U8L7":Nr`ZKVEEOTsIgo"4r%qC4`0pbJABqC4`0p$b5]AQlkKfCkV5]XmSt[^H/7_?JZ%R5rGY^q8sT=265e"/lUl:9F/4FQ0]
)p+3Jrgb;Tah?gu2s&`d^[^+AF0nB)`jq;Mh&n)2L2B)Ve]'oLRbXKV&KOI0@?$KM4KXf-Y`J_KBn&+*cG_uo+AH:3g3AG3b;R*s/cHX4NI3N@1@Smn}>GI9BU)?a8[oUiLRML3DZ")4QNY1H,s9B5TTcS[]JjB&rk-gSjVcgqSJ'ELoBM)IG67?Y=:NMR2amMR'Ie.jdLs2-X;s6sWa&h4-Eq,g;:!+I'a+dQMI?c!P/]7R[b3"WZ.3FIT:]JUcniu^%Vt/onB0)1id:H3+oRXTg#r-A*W;6^p/f#7c%eSm&U_[9I2-=%<`WAm5PX3&DuYWCq&[HRZO0C]Vu#L6@Dd8j<`8N*4,$nqQDn*,=5:p9ehDmGn0*`JYc(Uf*W11Pcr(N@Oss7TO/q1M8bA`Id?&1in#s(gX0`T`!7ASigpo"6,#6fhfH"!n7c7ddm`o[$hQ[L'balUtYfB_Unf3XIXbSAP`?t,323HkTW(*"CVi@p/:%LC420>>b$Os17!Fc;@,nTDD7(4d2s;SMs+LPJJ^oh.LKsV+!9`bIL/;ca./^r2T	h0q'8@;]]$c*-QiGLlJSF{a}
No sharing (MkVqq7&=I'6425C.-2kI8?43c<=e_i<;K9BPA^Q5-+#b-t,!KEOP:he`&KgiP#Sj!u'^5ZQ^;;>N@0BjY5_48hp0Ae1&d/ITOk,'d;lp*($HoN;li6hWuq$A0A'pM`r@7=`BPZCXi7=Y@;s8W-!s8W*=Rt':-2g&q?eL_D%*IJDQ6H1inpfaaVsn8Mr2]jP;7MZ3=`s7I2a<$nju:TtAiG2D9@P2Uf;gJEpqtT4MUAuEeC:]3p4_*^n]US^@s:=]e[hQ=/!S,QGTH2uiKm/R)ja4^Y&X<37Y)EF2aA8.p5_uG5EmJW82puVOeIg:c&bPr'E_uC5XiLSXDpgjpgmJJ`3nSH88g40/&a8@`Cm63qkikbZ0O=,p8L1SgJ&_KKJ+8#$*W0qBpqMP^roQ!9_ZYlQ%--TiKYNMAT?d7LSeWus0BE>nbYS&23/=kjg_rdkH1j&I=TAD4?fF9s_O5o;?iU%Z?;3s=]>A#q?W+rs-$h:r@Uds>gkces2m59B7.FIjOe0rbnp>R$eb9&KR-(mQXM6Y7-KjQ,b,g?o'BA"ubZ=/rk"OEK,ca$uN,*AL.$(ka.;-$N@<`d[L-li6sNs8G77s8S/[s%EAYs8V`+s8W+qk839=c7f4ts#?2/1gt5<8Bg;Vjs64Jf[.)(!!$D=M~>
Q
Q
0 0 0 sc
354.12 180.72 m
({b} Sharing distance = 0.01 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN198
.lMoh@!g-I+7F<21S$5/A/rVI nVaqDRp(:'3#U8L7":Nr`ZKVEEOTsIgo"4r%qC4`0pbJABqC4`0p$b5]AQlkKfCkV5]XmSt[^H/7_?JZ%R5rGY^q8sT=265e"/lrn@qY>B+&(Xtp+,H1X7"$fcTMTRIuXJ3"Y$7dO+U9][a/-#YLs8W-!s8VlTgb"$RO>"g'+5*do)2J,-H25jKb'W'i0R,=XVj2RA+2]NscC`2/?$^u9E.?i23JAQnM'ZFjL*>#%M4oY__R?@gk8=iI?n=ABk>$F[c!!$*5#ipk[mn5^tOGn)9NjnbuV.B1S,]Ddq#:QtT??0ZjDCJ>Y#Mb-79NSo5-P!@qOmKP.s4N'rlUl"YPj_0rKSU7BA&o3:d]1c@H%4APi7B_^H#;pgB6].6!k)uD.RJ*a87p/V-uVu%uRJ,PY;05&qhL/u+gQ*;q@b1VheEV;3rlW!rq#CBTD3M/tXJtc,2uiLloFW*fs6tmfr@Uds>WL]6r_^14k[$utI=M4q"L.pBQ%[D6j])9&2H9lXQi3/Xq8N/ZZj7=LHpiJOR/YfhJ,f3Rs#L%C&-;410FRNXK7Sl9+`[BXV/QEd47Ve'#_p_s8W#(WQL5Q.PB;3<2=L`8P"L4,6oSId,qb!pfC%.P$jq]JT0,-$O+L5N:NMR'Ie.%++:'kJ%.b+08YEBn[^#;@a8c27-3*OM5"EB]^L.$;?4/]KL]DLoY:FrrkJL2J,f@PqA923V>pSqr@^.F1gY0Os8Vo5QYE3&BYm0#+@{c}
Sharing distance = 0.1 ) show
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Y79N_mQ(RjKs7--MhYpc.o"2dZe33Q6J@e,G[3UBqRZ`(+4MrKJo6s0@?.q^>DCs()N6-qjTGLH$*Zs8U.>q);'_at`qeIuu'kYtq*qYN4J*s.G26s5WnSp/+'M1OA+"s-.C0V18OFfh+/]0DL@iTUs:2s4Kj@s8VGZ*m,As8W-!s71%c^<4qGZZqCu]7'.*?9$o.WXV/Rp8,rVgb!5q+rlV/=s8L05ncjW@IR!e0j].EiBS[TkabG5-%%RqVI*+p,P76sqN;F_kq]`%tZA+7B^5B5#X!jB:pa,t2Bg,&%p:ju7qHaT)I4sb7/`KZQ_LRWSV0L?s7F)+nc$<'r."bC?htT=s/#bqs8=T1^GK8:J,fQE`0Y25$Y[CFOs!#YBYrI+${d}
Dynamic sharing Figure 8:  Performance Validation of Dynamic Sharing Scheme
in MOEA To validate the contribution of the switching preserved strategy
{SPS} in MOEA, the above simulation was repeated with different scenarios
and settings. Figure 9a depicts the simulation result without the implementation
of SPS, in which the evolution faces difficulty converging to the trade-off
curve. The solid dots represent the non-dominated individuals while the empty
circles represent the dominated individuals. As can be seen, the final population
is crowded and the non-dominated individuals are distributed with some distance
away from the trade-off curve. Figure 9b shows the simulation result for
the MOEA with SPS and filtering solely based upon the Pareto domination.
The final population has now managed to converge to the Pareto-optimal front.
However, the non-dominated individuals are not equally distributed and t
he diversity of the population is poor: they only concentrate on a portion
of the entire trade-off curve 
{c.f. Figures 8d, 9b}. These results clearly
show that SPS in MOEA is necessary in order to achieve good stability and
diversity of the population in converging towards the complete set of trade-offs.
_PMR&sF;#a+P:SbgVbXN0b(R$S!F@Xe4MkWCB2dh1`>o6q2o5V:W,N]S`1?cYb&cLhJQ`0g:66.llr!D&4P"M#T'oKA^U]i+@(K*G?!Q3m]!7XLcN&^]1M&Z23pqXODiOrs9Wd+RDs1s3bp1PQ+^]2Yj`gb)oMg>?n7=UcoMg>?lem>Cb-ca!YO+*IH$/%rD/91MKF>2Zr"$fcTMTRIuXJ3"Y$7dO+U9]YKEh'.eo!JGsr@O:hs8W-!oE`gkLkr9%P"`&d5`@$h0-@"4DJHrN17QLh?SK0Y$gcUnmBV-4T[=MIMp8S,!$sEP%@/YB2CL;!_F^FQ&FUc&%,*?Y<4Hm>tF86*@+-H2Kk>!Q5N:GU!l[IKMl1US/[?!Sk5.aW!L9-(g{Jeq"Z0"OJA"-GD-72;Qm3"#^1)X19m"MsnH>7lF_)#-D`n.HU,ms519l<;U:%:s+BDF/BW1rD:gt3k_&,iikq@cf0Lci#H;r$C.Mpc+V<%a'6YcB9*iu*7H0Ui$%
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With SPS solely based on Pareto ranked costFigure 9:  Performance Validation
of SPS in MOEA 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright
(c) 1998 Hewlett-Packard Company            ANEVOLUTIONARY ALGORITHM FOR
MULTI-OBJECTIVE OPTIMIZATION199 The proposed goal-sequence domination scheme
was also validated for problems with different goal settings, including a
feasible but extreme goal setting of {0.98, 0.2} and an unfeasible goal setting
of {0.7, 0.4} as shown in Figures 10 and 11, respectively. As desired, the
population is seen to concentrate on the preferred region of the trade-off
curve at the end of the evolution, regardless of the unattainable or extreme
goal settings. As shown in Figures 10 and 11, MOEA is capable of uniformly
distributing the non-dominated individuals along any trade-offs size resulting
from different goal settings, with the help of the dynamic sharing scheme
that automatically computes a suitable sharing distance for optimal population
distribution at each ge
neration. 0!k`;TH(IWYbDSZBJhLlDk2UH78'E1JA^Q=N,QC#[!ji4WSKLB`YOU^jRE^jM?h:P!s#?1e`BU^8'3JjB'DMaB'MSbG2WB/mD-@j2^dP/h+*33jnp"NS>l;`1$F6sJ;ucbO2imtCA_%3m9n5M-i,=8_K#f@lB>lPpKA3V)aiL+^TKns^STRUoE6;H^b`+Bf](fhh/bo>rir7)*X31'AB3NU.sVEYcF?*V3oW@4l_jY:Y-5_lFWSi<57FC3q)9^/s8Vloj`,FaRljhYWH;.*7C1)aWW1qB-,9YM[nnnq:=s7pT*nK4(grYTLp3@!iAP$$'p/I9*cQI^l_ZC:gnrrfSKi6&oK_.>0Iau37aV>k*Yrc*BdQY6pW[SU=eO8ibpI`bdd;(qr&q3JTfgVP5fgUDNp/7O_ALVm`-t!#Sq)9<(s]kt/^7?'f-UG$UGGflgYfA_D64QY"in?'f-UG$UGGflgYfA_D64QY"in?'f-UG$UGGflgYfA_D64QXH1mfsY_5&*ZFj@Q=GF-qr/k#Dtua"rVtdP+2@ibnCdL3s8R$:5JOu5[UF[bhuES7_d%dXkc63ZXV/QEd47Vi'!t9Y.n$WtOoPI]ZA0oZs8Vurs8Vin+D;*L4Fd6*aWi!'d1A3aP77=8)*/m6q36j8-5kelWRX=D@>
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(f1f2) show
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316.05 558.39 200.34 -155.4 re
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408.3 412.02 22.56 -11.4 re
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(f1f2Figure 10:  Feasible but Extreme Goal SettingFigure
11:  Unfeasible Goal
 Setting Figure 12 shows the trace of sharing distance
during the evolution. The thin and thick lines represent the average sharing
distance without any goal setting {see Figure 8d for the corresponding Pareto-front}
and with the goal setting of {0.7, 0.4} {see Figure 11 for the corresponding
Pareto-front}, respectively. Generally, MO optimization without a goal setting
has an initially small size of discovered Pareto-front, which subsequently
grows along with the evolution to approach and cover the entire trade-off
region at the end of evolution. This behavior is explained in Figure 12 where
the sharing distance increases asymptotically along the evolution until a
steady value of 0.0138 is reached. It should be noted that this value is
close to the fixed sharing distance of 0.01 in Figure 8b, which was carefully
chosen after trial-and-error procedures. For the case of MOEA with a goal
setting of {0.7, 0.4}, the sharing distance increases initially and subsequently
decreases to 0.0025 along the evolution, which is lower than the value of
0.0138 {without goal setting}. The reason is that the concentrated trade-off
region within the goal setting is smaller than the entire trade-off region
{without goal setting}, and hence results in a smaller distance for uniform
sharing of non-dominated individuals. These experiments show that the proposed
dynamic sharing scheme can effectively auto-adapt the sharing distance to
arrive at an appropriate value for uniform population distribution along
the discovered trade-off region at differe
nt sizes, without the need for
any a priori parameter setting. 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB
Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN200 )
[5.04012 4.98 5.04012 2.53512 ] pdfxs
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203.19 701.79 205.62 -164.34 re
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( )9"[4B^?^93[1e6YM;YaQLr2qRej`)[=V']_j+MNI(?VBX30;0)_$0=Z'Tq$Op`PAZ*^J_0boV*pLc<[*#m)fG'c?bkq]e)XR_hu?>B#%M,EPmN]`91b@%`kqUC'3ZXG64ClNR#$Rn1E-f"5g4$JklWZF5=cZOJpn-7@@iV8Ms11m"#,HH5oT0C$:@;0rs!$&K_eLE_4+N,ql9T;Afq[,c8:Q?iV/
( L0lZtj-T7J4nK#pTLJKtHON_9MM&]:R=,,hrfHO32kDjqr%QR6G-u,bhtV!+E85sCs6p!W5`#DOS"K`Jf@&=M^X-?&s24lOhuDSAMLYIP6GAcQUu'g/KF>/S4Foe0s7F)Qs8W,@,pI5k?iF*NAWYGls8W(bAO9rps0Qc'l#@:Ws7F))0E;$edCjgaDIrDIRI6Z>hu,5.r]OKc?^-WOoUFiGb5Hip0.q-PV_DE@fqK_eL@e"XHL+Tc7/^TZT!hrl`Ns5BWcoY)LJ=7hsf/-#M5&PrEq#QHc'7qfj%DSrKpkK&7#hHl'92k-aVHcp;lIn1]nlJ<(Y=8tf6^-?<7(M.SSpPX2Y^b(,0Dfu[G#CQ#&6q_080GGQpP7$4Ir!-&V$].$e7Jp!kLsPf`1WZBdN0?2>?J@qJV2eIp;pPgO_Fs1d!sn,Ei0FI"B6+kg=bL#f8ic,RD7m/*Sh"17^2),!,Gts/#bqs8$[6qHp`3r@U]eACuM6qHT]PA6Hair-XlkLW"Tfhe)5R8CY:ZuIqSmI:c#nDDd2mJh+shtI)SriA?N5N.Co$m4uVqM$*]@K-f:%K/X)o]?QpgFcDFD__>:Figure
12:  Trace of the Dynamic Sharing Distance Along the Evolution Figures 13
and 14 show the MOEA simulation results for the case of an infeasible goal
setting with soft and hard priorities, respectively. In the figures, diamonds
represent goals, small circles represent non-dominated individuals and solid
dots represent dominated i
ndividuals. For the soft priority setting in
Figure 13, goals are treated as first priority followed by the objective
component off1 as second priority, i.e., Pg = [1, 1] and Pf = [2, 0]. As
can be seen, it provides a distributive optimization approach for all goals
by pushing the population towards the objective component of f1 that has
a higher priority, after taking the goal vector into consideration {c.f.
Figures 3a, 13b}. In contrast, Figure 14 shows the minimization results with
hard priority setting where priority of f1 is higher than f2, i.e., Pg =
[1, 2] and Pf = [0, 0]. Unlike the soft priority optimization, hard priority
minimizes the objective of f1 until the relevant goal component of g1 = 0.5
is satisfied before attaining the objective component of f2 with the second
goal component of g2 = 0.5, as shown in Figure 14 {c.f. Figures 3b, 14b}.
As can be seen, objective values with hard priority settings are better with
higher priority but are worse with lower priority, as compared to the solutions
obtained in soft priority optimization {c.f. Figures 13b, 14b}. (MkVqq7&=I'6425C.-2kI8?43c<=e_i<;K9BPA^Q5-+#b-t,!KEOP:he`&KgiP#Sj!u'^5ZQ^;;>N@0BjY5_48hp0Ae1&d/ITOk,'d;lp*($HoN;li6hWuq$A0_TIRo4r=,7CT'ReN0*Mi?8up(%&W-A/Wh^$OfZ,L"Cj&63L1]GC&sX?J5D)%_,m?p_RWoshG72IN50YSWS2nuhH/06HX5hT.BI$9+92/mFtnL>_3il-&,sEjWBH1ubfX7NfmJ'L-@M`^7K5Ff?i8nV3u%5P!X.'lNMhSm%'sa@ZJXW2f[n#!`JRnJK/U[J#%#+'K)+-XU%'5.16fe_'dZ)b1/%UW1pm6m._k?9Jku=.cb)%J"j$G.a#kq:T+6^bdt-5&-)D'U9ln%KG.uS6tZ579N_f5c5@n+9#6):+$X,7pl0MY,VHrAmCO5&14Vo>UUC#*&s	!/RfsknlYfFUeCS77%PXoDGU(5d3nWeP^Z1L]/ClJ&uX,8,7'cIMl$!p3@!j>s	!$3&)/QgF8]r%b&Qil[?!ZD;!`;G$5S'Cr_fb*'F$cbF$_7Z+LUX"-7oJcCdcqHb1^5&p/]VS=^,,(hRqC?^D4>.j[V_LGfsYaeB%aV0RqC?^D4>.j[V_/Qj2tW;:(g'.#jesKfNMYRMt,J)/D:45Xb'-0ZFiSKR9LCgGd=b)r`;pa>8$u9m&c3Fb1O1].q668fVc$`-i_?S'.[O!Zg;Jsb1Ig&Xat`sASfDm[Zg`Ve053]$4TW*L-L@9-EYj8"kGs8Vj?M#RZ['/g4IA,]?2XJITI.<"g9b3Bes":;8C/RftX?t6i[2dPB#4SO8`3W^R5h>FD'L:M_nG!KiqRYj5N<'s-%h;dZZV1Ktg5Xl3`V%76dXKoY$JdZsb.]rZX>-L&_#u>Q9u<0Dd/!s8+3Ca8bj0T7eZF'Z2aq
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At generation 5 {b} At generation 70 Figure 13:  MOEA Optimization with Unfeasible
Goal Setting: f1has Soft Priority Higher than f219980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company            ANEVOLUTIONARY
ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION201 IVakM+Yq-KGCIe;9D`qO]GFPWZUIWUuQ2q1um4U@MnL/r9@8d<%n%k3NX)bG,I95&N:r7!uj,H@,R=r3,HV"QVcj3/'pi&VfDkOB<^5p`?X>rj&ju]NPtrM%s7e8IrlkQC(]X=]4Ke_arecd:rX>T^5A!OrHo/0fiG,r_-JH:$^^N2r<+&KVrkH(G[?LG;W)Knr?bDAO8X0g3L:H6NZo[2..`%_@cK]q&_?kr'q_q34.lks888ZTLU#NQ&%pP:Z`Wc:%(,6-VmFeA[BbE?]D"P+"IPob1+iKUl?q2qO@0B&-+r,MO>(Ubg?<#huDA&O8;aiQ6&V)ns<6:N5&!oW?qT2nlsg%]WRq4<0jd$p3A3<]VF'jEpo='UcL-ou_//<3#uhqJ4l^]0@.&15/JO8QI%?Yk>UdlY-Xeg9J[^YV*Xa;R650DU"Q..-]aa[l0fh[m_-:&_^f,EZF0qsrKX]1b1]Yj=&>;h-i!`Anc(YUFnA!5QAm
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At generation 5 {b} At generation 70 Figure 14:  MOEA Optimization with Unfeasible
Goal Setting: f1has Hard Priority Higher than f2Figure 15 shows the MOEA
minimization result with f1 being a hard constraint. The population continuously
evolves towards minimizing f2 only after the hard constraint of f1 has been
satisfied. In general, objective components with hard constraints may be
assigned as hard priorities in order to meet the hard constraints before
minimizing any other objective components. )
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15:  MOEA Minimization with Hard Constraint on f1Figures 16 and 17 show the
MO optimization results that include multiple goal settings specified by
logical 223OR224 {000211} and 223AND224 {000210} connectives, respectively.
For the 223OR224 operation as shown in Figure 16, the population is automatically
distributed and equally spread over the different concentrated trade-off
regions to satisfy the goal settings separately, regardless of the overlapping
or feasibility of the goals. With the proposed dynamic sharing scheme, the
sub-population size for each goal is in general based upon the relative size
of the concentrated trade-off surface of that goal, 19980209064900RelativeColorimetricsRGB
IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN202
and thus individuals are capable of equally distributing themselves along
the different concentrated trade-off regions. For the 223AND224 operation
as illustrated in Figure 17, the whole population evolves towards minimizing
all the goals G1,G2 and G3 simultaneously. As a result, the individuals are
equally distributed over the common concentrated trade-off surface formed
by the three goals, as desired.    )
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For a comprehensive comparison, various performance measures are used and the comparison results are discussed in the section. 4.2.1 The Test Problem 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company ANEVOLUTIONARY ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION203 The test problem used for the performance comparisons is a two-objective minimization problem {Deb, 1999}. The problem is chosen because it has a discontinuous Pareto-front which challenges the evolutionary algorithm222s ability to find and maintain the Pareto-optimal solutions that are discontinuously spread in the search space. The problem involves minimizing the objective functionsf1 and f2 as given below, 000ff000f111xxf000 {13a} 000ff000f,110101,...,102102000020000246000ffi000 000 iixxxg {13b} 000ff000f000ff000f000ff000f000ff000f1125.01110sin1,fgfgfgfh000S000020000020000 {13c} 000ff000f000ff000f000ff000fgfhxxgxf,,...,110212000 {13d} All variables are varied in [0, 1] and the true Pareto-optimal solutions are constituted with xi = 0 000005i = 2, 205, 10 and the discontinuous values of x1 in the range of [0, 1] {Deb, 1999}. Figure 18 depicts the discontinuous Pareto-optimal front {in bold}. The shaded region represents the unfeasible region in the search space. ) [5.49917 5.49917 3.65978 4.88223 4.88223 4.27655 3.0541 5.49917 3.0541 4.88223 2.76 3.65978 4.92188 5.49917 3.0541 5.49917 5.49917 2.76 3.0541 5.49917 2.76 3.0541 5.49917 4.88223 2.76 4.27655 4.88223 4.88223 3.65978 4.88223 5.49917 2.76 4.27655 5.49917 4.88223 4.88223 4.88223 2.74572 2.76 ] pdfxs q 208.68 0 0 165 201.66 267.12 cm q n 0 0 1 1 re q W [/Indexed /N168 /ColorSpace findRes 15 <~!!!"K!!!"K!.TIu!!%Lu!.OuuJ%rb5It)tI!!!#u!<2rt!!*#t!<)utrr2or~>] cs << /ImageType 1 /Width 504 /Height 400 /ImageMatrix [504 0 0 -400 0 400] /BitsPerComponent 4 /Decode [0 15 ] /_Filters [currentfile /ASCII85Decode filter dup /LZWDecode filter dup ] >> pdf_image J3Vsg3$]7K#D>EP:q1$o*=mro@So+<5,H7Uo<*jE<[.O@Wn[3@'nb-^757;Rp>H>q_R=AlC^cenm@9:1mM8F2A1_!A4KP ( DMiCJPuWO4d_WhBNo9c#jc%76`Mg1]'66B^i2<424$hOZ72XDK/n( ( bV`e3h92Ek*Q%S5kB:SlQ9:*e ( o=.06j'1_ds*Dq#Qt5mT:f#JG)7GmZnK4<"^HN/T*_m$Iq&47$CK+^O+1;=Ml0`5&Y`C`g2$:Z_>B')G'lu*'E-QU_^sIlZ+35!%EtcX!"HJ$]dS'B^TC'jmq_o.&33:9NS8e0["/I!?3@Y3[V4&Oc7TLWD2*3(rm:Hd$[XI`$FohZZPW&cRpaSohVp$i.%ZL`=X1h8d,d1:+Akg+&QEmEBgIeoaaTEo6fPn;Tj_75j^r6VLE81Rgl_oaITmd1KVb8CS>IAO<1kH^_%VeT.KjI<9=#k2dSG,U%T/Xl4Z?pVlpt?e>1eJDa2:$%f`j40m`Hj!?bUcechi&Q6&S'H8WkE"_LhWLIP=pVPun%Xi"M`80>hFD"u.t>KOqiZc%!smC7Ej6>"`P[BDaMWah:jsra-na90=i6+BD!diRLe;-h5U2A"(8"^[r/IZnNT?fh7TacfP/9qROZ`fckd))HX0Es*p,O-od+)?J>9]3bNh98+,,=40Ofdn=O@UFbZom`&`ek_knb3sJLAbbTi.[1X8JCZNK[1o:SA];qd3nMVVd&=F,Brp>_l-`?1^//gF7L-QHK!.#G<+MG3S=NMt'iVDB!P`cjF31OUfrf0C7ek5rEtV&T"EX,,@Ot[;4pd<^X?u$7oY+CEYi@?d3?!-%bt2kIfCsQBHQo=j2dLDXP3d,BY?V4*t"na?=IL1qr=nM^9jS%i1c3p/$/of37@Var9?g-hH#cW%m?!OqjXp8)9Or911LA3Tud4icNf]5.d<:q2$IAHVAi3A4UmtScDYVZAXr#P+/j$n$% "'@eVR90PakA=7/j3RC/=""/"=m_l@ND^EmQ8@t!g]+!Og$_J_F!<2A8KDDWjY[1drP:rD#L'+FI&6*4',kM*6kp7emFGaIo,G^`Gg!$mihSK>j5K*BQn'+Zt^/*ABE&:p'Et=LWGASKYA/<]]%uQSJ!>H+c!u"lS'XE1bl>`&AI0i-6%jmBS=Ipmo/W-_nn*9b2Nq6&5$uHiFd'Epp0XQ"&22b?#.i#; ( ,!cAmmKaKoJk7rj=sXNE/,9M&J^.V+#/(D,begHXWB2#'lBWb$p>di.iYbcnkMCoO.?oCo-=o4+OA^Q1.pLL-R#c8D3>&FE[;IJfZ#[T,3=.h1/T4c!odk430A ( E02@3H96.j=8%-15d!4>oSrQ[Gt4s11qmG;^,LZB;P,AAVs!*&ZHVEFsBCn]d,bU@Y^T/+PnbPD;0>@lmVtn7G0OPJ8A`j!H0%4Hmqo&f!67WSr3F],WF0fP%a-L`dXf1o31sTk?a;!jff2!oGH>E7*Sa*W#N&XEcc?nPeJtPI*bdYI.Pu@ ( L;9i.(;(*N/P_W]1j0,!>0L6Z3@16`E^=d'G$[K8VOq*,Wa^I@Hs$op1%R:S9B%#*.qPdI!I@(d2/P%5sneNfED16*B6;O4$Q^&pj"FGg9244-erN/q.W[/,hAGW0[XVDV+RapC4aP@9mpTPk:t8YC,LESDu4?EE%fdN?.ACRD!>`a?#p_=1j9A,(D175-X6<_D$0U7*dA/u^%YR@=*4/qG88SO2g0:%HIDP:VO*D>.tM!74*e9tH1U!$oXs]QNER."o[DM`gdb]=8Qg@t(QEPd0>j^7tS&b,D`ddC;?ER.&h5FI4LkEN'#[QLR3(FkL8eNJPQFd?n!l=EkS^?uuW=NJObbrRE.iJ2Z-OLXmIB[MlC7uhiOOE;OR/:bTJfqLPe[VeBfNXHijGb#bL%q4!1g;F(UuPRU7:sq`1:<+oZ]?O?LR<@LGsOiD7?%#>LB_gZNBrS-23jQdOdUWaH#VO'6Z'@3Pd6TUq(F_[ac6GKMl/(`Gi3V@#`c;T0N$3J!IWEnQ;C@rC'q;eS+crR2s#Y]rbD0>FEmhDa-$U>`Q.]B%Nd+JH=PQ,@6-EGdpK/t?DHqIIBn$r.I]j<3qP1SIq`IC5[#L'Tn@&KU)aqT)4Y^)gf/#R$TL8]DKF/42nU!dC3?)]_""C *Psq1u8K6X%IPDYbV0dTgpPPROZ_h&g4NFc,?N0n.Z:oL0]dEjID+C6?ee3A25YZ+LGUqo2XETV31ASd2+(aj4s>I2&9uTIHTHXs4kS(dAp3"<%m+P?rqt^*(*r?.66_S>-*No:1B/@^^.IHk,2j5"8?j5+6EhJ;0nG:uHXk8mLbKCS-U0BldZh]@e<*KB93Cl^b'l.uC&?C0pc"hM`mBKi@5)j,[_-Z3HVkOKt_gts5/]oo4_TC[Omo=mH[s$?9A7J)($oV7eVhb5JNKC`GJ-3%-@U0NBO@!:F7p1?&Q`67lWg%N1&,p'boUT@P]^A/hmU(c22Jr6hOlr@$._h2CU;u?0TI_CIq4FN4L/7MMXsF%TqdN4k+2.S:`Vb5-r1Z_Figure 18: Pareto-optimal Front in the Objective Domain 4.2.2 Current Evolutionary MO Optimization Methods Besides MOEA, five well-known multi-objective evolutionary optimization methods are used in the comparison. These approaches differ from each other in their working principles and mechanisms and have been widely cited or applied to real-world applications. The algorithms are summarized below and readers may refer to their respective references for detailed information. {i}Fonseca and Fleming222s Genetic Algorithm {FFGA}: For MO optimization, Fonseca and Fleming {1993} proposed a multi-objective genetic algorithm {MOGA} with Pareto-based ranking scheme, in which the rank of an individual is based on the number of other individuals in the current 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN204 population that dominate it. Their algorithm was further incorporated with fitness sharing and mating restriction to distribute the population uniformly along the Pareto-optimal front. {ii} Niched Pareto Genetic Algorithm {NPGA}: The method of NPGA {Horn & Nafpliotis, 1993} works on a Pareto-dominance-based tournament selection scheme to handle multiple objectives simultaneously. To reduce the computational effort, a pre-specified number of individuals are picked as a comparison set to help determine the dominance. When both competitors end in a tie, the winner is decided through fitness sharing {Goldberg and Richardson, 1987}. {iii} Strength Pareto Evolutionary Algorithm {SPEA}: The main features of SPEA {Zitzler & Thiele, 1999} are the usage of two populations {P and P222} and clustering. In general, any non-dominated individual is archived in P222 and any dominated individual that is dominated by other members in P222 is removed. When the number of individuals in P222 exceeds a maximum value, clustering is adopted to remove the extra individuals in P222. Tournament selection is then applied to reproduce individuals from P + P222 before the evolution proceeds to the next generation. {iv} Non-Generational Genetic Algorithm {NGGA}: In NGGA {Borges & Barbosa, 2000}, a cost function of an individual is a non-linear function of domination measure and density measure on that individual. Instead of evolving the whole population at each iteration, a pair of parents is selected to reproduce two offsprings. An offspring will replace the worst individual in a population if the offspring has lower cost function than the worst individual. {v}Murata and Ishibuchi222s Genetic Algorithm {MIGA}: Unlike the above evolutionary optimization methods, MIGA {Murata & Ishibuchi, 1996} applies the method of weighted-sum to construct the fitness of each individual in a population. To keep the diversity of the population along the Pareto-optimal front, the weights are randomly specified when a pair of parent solutions is selected from a current population for generating the offspring. 4.2.3 Performance Measures This section considers three different performance measures which are complementary to each other: Size of space covered {SSC}, uniform distribution {UD} index of non-dominated individuals and number of function evaluation {Neval}.{i} Size of Space Covered {SSC}: This measure was proposed by Zitzler and Thiele {1999} as a measure to quantify the overall size of phenotype space covered {SSC} by a population. In general, the higher the value of SSC, the larger the space covered by the population and hence the better th e optimization result. {ii} Uniform Distribution {UD} of Non-dominated Population: Besides the size of space covered by a population, it is also essential to examine the ability of an evolutionary optimization to distribute their non-dominated individuals as uniformly as possible along the discovered 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company ANEVOLUTIONARY ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION205 Pareto-optimal front, unless prohibited by the geometry of the Pareto front. This is to achieve a smooth transition from one Pareto-optimal solution to its neighbors, thus facilitating the decision-maker in choosing his/her final solution. Mathematically, UD{X'} for a given set of non-dominated individuals X' in a population X, where X'000216X, is defined as {Tan et al. 2001a}, ncSUD000ffi000 11}'{X {14} whereSnc is the standard deviation of niche count of the overall set of non-dominated individuals X'.It can be seen that larger value of UD{X222} indicates a more uniform distribution and vice versa. {iii} Number of Function Evaluation {Neval}: The computational effort required to solve an optimization problem is often an important issue, especially when only limited computing resources are available. In the case that a fixed period of CPU time is allocated and the CPU time for each function evaluation is assumed to be equal, then more function evaluations being performed by an optimization indirectly indicates less additional computational effort is required by the algorithm. 4.2.4 Simulation Settings and Comparison Results The decimal coding scheme {Tan et al. 1999} is applied to all the evolutionary methods studied in this comparison, where each parameter is coded in 3-digit decimals and all parameters are concatenated together to form a chromosome. In all cases, two-point crossover with a probability of 0.07 and standard mutation with a probability of 0.01 are used. A reproduction scheme is applied according to the method used in the original literature of each algorithm under comparison. The population size of 100 is used in FFGA, NPGA, NGGA and MOEA, which only require a single population in the evolution. SPEA and MIGA are assigned a population size of 30 and 70 for their external/archive and evolving population size, respectively, which form an overall population size of 100. All approaches under comparison were implemented with the same common sub-functions using the same programming language in Matlab {The Math Works, 1998} on an Intel Pentium II 450 MHz computer. Each simulation is terminated automatically when a fixed simulation period of 180 seconds is reached. The simulation period is determined, after a few preliminary runs, in such a way that different performance among the algorithms could be observed. To avoid random effects, 30 independent simulation runs, with randomly initialized population, have been performed on each algorithm and the performance distributions are visualized in the box plot format {Chambers et al. 1983; Zitzler & Thiele, 1999}. Figure 19 displays the performance of SSC {size of space covered} for each algorithm. In general, SPEA and MOEA produce a relatively high value of SSC indicating their ability to have a more distributed discovered Pareto-optimal front and/or to produce more non-dominated solutions that are nearer to the global trade-offs. It can also be observed that, compared to the others, FFGA, SPEA and MOEA are more consistent in the performance of SSC. The performance of UD{uniform distribution} for all algorithms is summarized in Figure 20. In general, the UDdistributions are mostly overlapping with each other and thus there is too little evidence to draw any 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN206 strong conclusion. However, as the average performance is concerned {see bold horizontal line in the box plots}, SPEA, MIGA and MOEA outperform others slightly and are more consistent in terms of the measure of UD. Figure 21 shows the distribution of Neval {number of function evaluation} performed by each algorithm in a specified time. More function evaluations in a fixed CPU time indirectly indicates that less CPU time is required by the algorithm. Intuitively, this means less computational efforts are required by the algorithm to find the trade-offs. As shown in Figure 21, MIGA requires the least algorithm effort while the performances of FFGA, NPGA and MOEA are moderate in terms of Neval. It can also be observed that SPEA and NGGA are suitable for problems with time-consuming function evaluations: the effects in algorithm effort become less significant in these problems. In summary, the results show that MOEA requires moderate computational effort and exhibits a relatively good performance in terms of SSC and UD on the test problem, as compared to other MO evolutionary optimization methods in this study. WPm'J"W5r2n2Fq-3ED5QCc`s8W+Z9u>N^s8W-!s8W-!s5geQ5EGj?2RWSbq#CAXCL@,@7bRKCn^1B<'kKCK?cj]"+MacdO.KBGC[H-&hhlm(*bsD9L;.KCsOK^QhUE;61%`c_-hINRKrBE.D,'V2D15PA7q:V]+s.,H#ZV^'Tq?cuU9(FGb/jV_s8S3Ms6K^^#`b_1&:KR:o]Ku8a-EsVSUiHos#?2Ag&q_`Ih<_d-J#s'ZGlf-s5W1of:@Hp8+MUgs87935k[pTB1E,rn$=@8=3cZs8W,sAU6!k/78GWM#[LGQ6Nnos8W,Y5PZ72fn8osQ%`@9<@;@6Y+^@Fs8$+%e=V6_U`FOgI>`UQ8,jg;g&q_Z]L59is8W#6pci257Z6t:-i_jg]p*t?Xag3tS05*uiVRs+0-$*Y[F]dJJ$9MAe=!_bXqF=miIR?TLIJs15gK/(RX[UN%*O+5Rgr;nUVoU/.]Pm`Qp[RIJ.giSgL;3YhQEgJi"no9oMq00f3+LF Ao*ni"f+F8Zhb;+d[gs!r[F@=FAHs2k$88l_.0f.,oL8,2YA"Qi%odf9>,^3Ur2u@#s8O2@o"&$ ZDbPr_C&jB3Q-;dcpo8blZAI@*fr8UA5#A$F#dU`j-+-61Z2;sg.j[]?g=DRjg_dDYX.R3W.;-6Y[Ku&.QW!/-,$ic:fq#C@2rlWRUZFYq1J;V[C8?=5:m&mY*MSb'!HRl%r-jnPJjdlC.6'VH)aQ;%q^XmFH=+97/?JjMb0TO^X_E/N6p_Ci2M,U'"G.%*K[=-4[*VMO&9pM5r9EEi^hldmZVL:d/Mco[+dp2IZr?Jn"a8bp"M#[)AI"21roY$#1+91STiP#Sr&-f;qK>%oN@Zt5_qb;rB2BqBBkTrM?fVU6rdqXF_DlODi1.i#(pb1kgiZl5A!'gk~> Q Q 0 0 0 sc 242.941 330.72 m (Figure 19: Box Plot of SSC) [5.46032 5.51968 7.29608 ] pdfxs n 204.69 310.65 211.44 -162.06 re 0.996002 0.996002 0.996002 sc f q 211.5 0 0 162.12 204.66 148.56 cm 0 0 0 sc q n 0 0 1 1 re << /ImageType 1 /Width 487 /Height 401 /ImageMatrix [487 0 0 -401 0 401] /BitsPerComponent 1 /Decode [0 1 ] /_Filters [currentfile /ASCII85Decode filter dup <> /CCITTFaxDecode filter dup ] >> pdf_imagemask ,_0P8;M>J8Q+nhj.t1OsThk^gs8W,u9u>NDFp,>s7oqPJ,fL4s5e@1O@B$/AW+o'FoFQ9=WJ%?r1:k.QK`&Y,GQh,j,a6#O5p9>=%cp#s,[2NMM(_1/V!g"G$'P"s73aFGb0U+LFLD/KqZ#n+3g3i]A/'JTXd0Z77MD&cFs!;?G6sMUas7G?qbQ$$@Ma?D'r2tPLqu?$nOCbjE+o7tBUE/tN.:l6@]%7s8W#!/eA!Bs-,K-s!oW"3.Lu.Lbr0Nd1a,j8]/Z_>jQ)7Nqg0XXlc*>Q.Z)pbIPQs8+3CjB+:So?!4Eo`$$):hqCBrh'*1VuI#!?<3CBa5gA:#ct6"#.D:-a!K+e^H7nas2Y/[s7--hs!tTDs-8g5s..Das8W-!q#C=Xs,ja7^ mY;1I=J,eV$0QoMU0Dp>"s8=B-s5lU;r1a57E^f5Bs8W)GL%33>N^XQ;UFMW1s7hZ8s5[#tqYr=MJ,fP`hllW$QiI*O5p9$;'#N519#d!)e{-MR'Ie.j@4Qj8].An,"f7`erZlcPjVrpRs#@c,&15%XJ,fQKs8W-!s8W-!s8W-!s8W-!s2`qB?Ac%daJ(cgGNeI.n'U$rs8W*b&_I'_s8W+qa;P,BYPC+[%K-1U9n?K&c(5d,DpK'dH(e;L/lW&aII&qp.%W8'8Yk47Kko2<,Bgb3]P4Enfk?+VV)t26k2#4ZXTM2ECnS'T] G$_dU`d=;>fHGPQtbipsN_TN^"WTGb2H_9.i=L9T;+?Z>Dr`"C6Y+i.o.gpt(fh`%np)!`Ns7--g1&[B0hdppFK`Mp30@4u#p=ATa7Q-2.L=B.Wk&8_QsHgIj4V> /CCITTFaxDecode filter dup ] >> pdf_imagemask j9ZksfnNqc;-$NT=[SHc)lt0OmJ4s8W-!s8W,udnP>lLkq'c_Unhfs8W-!s8W,\r^]RI6E;7bNN)R"'2ZA0pP*31SSs8W-!rkJL"+M[q'DGqYWjVrpRs7.Fus8Qf1Rdgt3s8W-!s8W-!s8W,i-Hn7S_t6rn3EAu<=[SHcBE/#3s8Qm7r=rajr0bg]I]#hYgiXC)k+^k*[?N @kQ/t]`Zs,;4;>?RC*m2"Z`a3OX3&_:+@LSdR[IJnl`X!i"iTusZr.a$*;j8]/I(]XO,BKer$s8O#D0E76>P@^MAYQ*]#jOhR?(r-+N`#4;Zs8NKH9f1Js845uG+MYc$r#ic$?grmOr<@$c0DL2"R[5,W;1IX*5QBeAJ,fL4s,f)ibVKbRMj9!!6^;3ms7oqPJ,fLL.Lg;le,TG`b3<^>MPG"*0`P*PPW;c@j]P_1M!raKXRFQoXTV.$s8W'_s8W-!s8W-!r.1Li)ZTe2ecUF%s8Ve]C*Fnc@g+*7EKSZ1%',+_d9"DZCao'J8!+I@k`/+94Gd$Rc*BAiaQW1=Z'_EA*?ITY1QA_MSm68,dNbFDQ!fOrnq`FO's69LXT&4AcgS0ABm`<:64X>[JnR1JJd`fUcs[0dcqO]UGX%R?Ll&jBs?U0P5&Uqpe,3]/6IZo9LPAPcqrnC/SgHq*QPLc#fSh6f=s7ibN8Upbtj71#/s8W,X![I`cs7ej8s7?#5J,fQ@]n*[s65$l^'LYJs8W,k^]4;L'`Q Q Q 0 0 0 sc 239.28 510.72 m /N169 10.9931 Tf (Figure 21: Box Plot of NevalFigure 22 shows the distribution of non-dominated individuals in the obj ective domain, where the range of each axis is identical to the range shown in Figure 18. For each algorithm, the distribution is the best selected, among the 30 independent runs, with respect to the measure of SSC.It can be seen from Figure 22 that MOEA benefits from evolving more non-dominated individuals than the other methods. MOEA222s individuals are also better distributed within the trade-off region. ) show n 100.41 400.59 122.82 -102 re 0.996002 0.996002 0.996002 sc f q 122.88 0 0 102.06 100.38 298.56 cm 0 0 0 sc q n 0 0 1 1 re << /ImageType 1 /Width 271 /Height 209 /ImageMatrix [271 0 0 -209 0 209] /BitsPerComponent 1 /Decode [0 1 ] /_Filters [currentfile /ASCII85Decode filter dup <> /CCITTFaxDecode filter dup ] >> pdf_imagemask 0FM=:f#gT6Co+FfW'C:]r'X$c_12Lbs8W-!n6iIcs8W-!S,G*,nULB+p]&;6s8W-!s87ZKJ,eRGo`+s3s8W-!s8VjLQ%Hh!&-FFGAO8o7[s8W-!n6iIcs.,U-TE"]ds8W-!s8W-!R8s8W-!s8U<+^JFn-ItIoV#QO ( s8W-!s8W-!s89#kqLAP*s8W-!s8V2'T@=BtUZrYfYs8NZC(]]'l$3~> Q Q 0 0 0 sc 290.94 285.721 m (NPGAb?4_s8V2!o`+s3s8W-!s8W,smJfu^r."aloCW(_a3N*/J,eR/s8W!6/.b&Gs8V7oJ#E>.rtPM5a8c2>s8W-!r5iaHs8U"5J,eR/s8W,]7UN(f_;MOm Q Q 0 0 0 sc 436.441 285.72 m (SPEA]XO8&J>'Cs8W-!s8V#n@"J@Xs8W-!s8W-!s77lA5QC9Rs8W-!s8W*LXo=smq#C?u$Nq"?s8W-!s8W-!s8W-!s-:gC:]L?Bs8W-!s8W-!K*?i3s8W-!s8W-!s8W-!s8W-!s8W-!s8W-!s+:K,#QO ( s8W-!s8W-!s8U6q.7gS8[_MkBpkXaA"9:[o~> Q Q 0 0 0 sc 145.98 165.721 m (NGGA ) [7.91823 7.91823 7.9776 7.91823 2.72966 ] pdfxs n 242.19 278.79 127.26 -98.4 re 0.996002 0.996002 0.996002 sc f q 127.32 0 0 98.46 242.16 180.36 cm 0 0 0 sc q n 0 0 1 1 re << /ImageType 1 /Width 271 /Height 209 /ImageMatrix [271 0 0 -209 0 209] /BitsPerComponent 1 /Decode [0 1 ] /_Filters [currentfile /ASCII85Decode filter dup <> /CCITTFaxDecode filter dup ] >> pdf_imagemask 0FM=N/X2Jtfh5lWJ,fQ=&Kh&Qs8W-!s8EEB#Ts*Hs8W-!s8W J,eR/s1s<'"97OuJ#E>G5'CN6J,eHaVuQ`[A8D$krV1=2rP/C&s8W-!s"[dV5QCHWs2Y&Z/+ii@pY`ll= Q Q 0 0 0 sc 291.24 165.721 m (MIGAMOEAFigure 22: Best Selected Distribution of Non-dominated Individuals from Each Algorithm with Respect to the Measure of SSC19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN208 5. Application to Practical Servo Control System Design 5.1 The Hard Disk Drive Servo System A typical plant model of hard disk drive {HDD} servo system includes a driver {power amplifier}, a VCM {Voice Coil Motor} and a rotary actuator that is driven by the VCM. Figure 23 {Goh et al. 2001} shows a basic schematic diagram of a head disk assembly {HDA}, where several rotating disks are stacked on the spindle motor shaft. 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i[n_Ref1@JkQ[:dR/:0)MbUR[j&$S1+:8V4C-!S1'q^+;$^XRNTeicDpDIuJeWc6).3t'B8kC:k;NaTsPbaI6,q1+&8P2Q"P#@H7,a>*f8L5f2P"A8;,a_k8URP$8q*d3I&`?%I]>teWffu'W_%T teWffu'W_%T teWffu'W_%T teWffu'W_%T teWffu'W_%T teWffu'W_%T teWffu'W_%T Q n Q Q q 0.977005 0 0 -1.02361 0 0 cm 1 J 1 j 0.485992 w n 353.628 -419.76 m 335.449 -450.241 l S Q q 0.977005 0 0 -1.02361 0 0 cm 1 J 1 j 0.485992 w n 333.914 -448.658 m 333.546 -453.582 l 337.66 -450.886 l 333.914 -448.658 l h q eofill Q S Q q 0.977005 0 0 -1.02361 0 0 cm 1 J 1 j 0.485992 w n 353.505 -419.76 m 359.217 -419.76 l S Q n 351.18 436.74 58.2 -13.56 re 1 1 1 sc f 0 0 0 sc 352.62 426.66 m /N132 [7.89729 0 0 8.3161 0 0] Tf (DATA TRACKSUSPENSION ANDRECORDING HEADARMVOICE COIL MOTORACTUATORDISKFigure 23: A HDD with a Single VCM Actuator Servo System The dynamics of an ideal VCM actuator is often formulated as a second-order state-space model {Weerasooriya, 1996}, uKvyKvyvy000270000270000271000267000250000250000251000247000ffi000270000270000271000267000250000250000251000247000270000270000271000267000250000250000251000247000 0002700002700002710002670002500002500002510002470000000 000 {15} whereu is the actuator input {in volts}, y and v are the position {in tracks} and the velocity of the R/W head, Kv is the acceleration constant and Ky the position measurement gain, wheremKKty000 with Kt being the current-force conversion coefficient and m being the mass of the VCM actuator. The discrete-time HDD plant model used for the evolutionary servo controller design in this study is given as {Tan et al. 2000}, ukxkx000270000270000271000267000250000250000251000247000270000270000271000267000250000250000251000247000ffi000 000ffi664.1384.1}{10664.11}1{ {16} 5.2 Evolutionary HDD Controller Design and ImplementationA two-degree-of-freedom {2DOF} control structure is adopted for the read/write head servo system as shown in Figure 24. For simplicity and easy implementation, a simple first-order discrete-time 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company ANEVOLUTIONARY ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION209 controller with a sampling frequency of 4 kHz is used for the feedforward and feedback controllers, which is in the form of 000270000270000271000267000250000250000251000247000ffi000ffi000 21ffzffzKKfp000270000270000271000267000250000250000251000247000ffi000ffi000 21fbzfbzKKbs {17} respectively. The control objective during the tracking in HDD is to follow the destination track with a minimum tracking error. Note that only time domain performance specifications are considered in this paper, and the design task is to search for a set of optimal controller parameters {Kf,Kb,ff1,ff2,fb1,fb2} such that the HDD servo system meets all design requirements. These requirements are that overshoots and undershoots of the step response should be kept less than 5% since the head can only read or write within 000r5% of the target; the 5% settling time in the step response should be less than 2 milliseconds and settle to the steady-state as quickly as possible {Goh et al. 2001}. Besides these performance specifications, the system is also subject to the hard constraint of actuator saturation, i.e., the control input should not ex ceed 000r2 volts due to the physical constraint on the VCM actuator. Feedback controller+FeedforwardcontrolleryurPlantVCMKsKp-Figure 24: The Two Degree-of-freedom Servo Control System The multi-objective evolutionary algorithm {MOEA} proposed in this paper has been embedded into a powerful GUI-based MOEA toolbox {Tan et al. 2001b} for ease-of-use and for straightforward application to practical problems. The toolbox is developed under the Matlab {The Math Works, 1998} programming environment, which allows users to make use of the versatile Matlab functions and other useful toolboxes such as Simulink {The Math Works, 1999}. It allows any trade-off scenario for MO design optimization to be examined effectively, aiding decision-making for a global solution that best meets all design specifications. In addition, the toolbox is equipped with a powerful graphical user interface {GUI} and is ready for immediate use without much knowledge of evolutionary computing or programming in Matlab. A file handling capability for saving all simulation results and model files in a Mat-file format for Matlab or text-file format for software packages like Microsoft Excel is also available in the toolbox. Through the GUI window of MOEA toolbox, the time domain design specifications can be conveniently set as depicted in Figure 25, where Tr,OS,Ts,SSE,u and ue represents the rise time, overshoot, settling time, steady-state error, control input and change in control input, respectively. 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN210 >1KIkEp600aq>5(5.LIP40?5":#T7XdNjIQg8N01Kk57rLRdIQg8?5#-#:N&EmDa$p65#6&`ME?U% ( :1%?m?=#j'?;>*>:1%=&AmS8F?VY6?:M4!/Aku07:JYk7:2Wj'Dbj/H:JYk':N'!7Dbj/P:L83?:NfK>G>DOh:/?$t:NoQ0G<]D_:0r-/?@G+GInj="5@8nEIm.4o5@0(.?^jQL-At"5&Q->?^:PN]pg105ch7DN?GXO"Xk$/oHt.DN?DgQ7m-A/oR(0DjN(oQS*3J/oI!tDOqqgT.Y&R+)dbnI.^#VB9`K*e't.Dl+S 7&J$0oCY7HquNjfOK[gnG4&!"@6J%cDQ^CGb.%ZO=NkYRT`X$K!0>/1NlM-c45oz5h>dPY!5>%3?bQLmm]J52mQTJ[!3Q;%!!'194ok#R!:F_)/hV$+5Ai,-09.Xb!!#2H!'Y=>Dba&&cH"5Fa1o@&^9>#L^:1i6k222SNdcG+zzzzzzzzzzzzzzzzzzzzzzzz~>] cs << /ImageType 1 /Width 664 /Height 383 /ImageMatrix [664 0 0 -383 0 383] /BitsPerComponent 8 /Decode [0 255 ] /_Filters [currentfile /ASCII85Decode filter dup /LZWDecode filter dup ] >> pdf_image J,g]g3$]7K#D>EP:q1$o*=mro@So+<5,H7Uo<*jE<[.O@Wn[3=lBh-^757;Rp>H>q_R=AlC^cenm@9:1mM9jS"!dTMT<$3[GQ$8"tNa1.l+9SPQ1`C/mHcW_6^o4^F&0mXZ"Jj5/Ja7uU@Dp7g[>sfINnl[ct5'1rdQ#pR+(mML4Q:?!'iiKDZ^3;^iCD"6,7[L64eMi*XACsI0rns+Lf":?+iGQ=8J2;&;VNdJi$LP#kJp]V5EH!S"@%u9*Rf/e=)HAP&7cpk64hbNmDG"5gVN`2Hg5mEmX3?!O1j+Qi6:TNd^4/K!h:=0gkC_$0S('PX"=;JQQ7b+p6DdZq]L#IdEqJMA1)#f9EcLnm>0IU_On/M.B"Mld*F<5E)D*t0>Z<m84=bcgj#STBH.P6^"_b:^pRC`R{%%&o[O/A0=J1I;9j4#]$OWp7J>&;I*2L5P3LfVfdnV;636*0'SCm[[%XVA?k@j3&I1@D*/"3Xf,ELN+pDrl--X!#6L9F.)0uQtAg-DboqioO*Ym$,&&^Rr'7tBO6t4);=F1OV7C@r.kb?&Kb8XqV0oe!pYfPK0N+DTUQ*R/5bLpm*Ah;B4ZY"8[bsOR*bkA<^!')#.gIQ%$XPo[-T$u`[;if17Xu1j-f1rASC[I+ef@.mqgLYq!%Cru,oBBZZ1i/giP%7Zr8:g)JoICA`&MrRj".u0&ZBip=k=dQgrl4/c?[bt4%tW24RG+!S0

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&nFgrf:^R@`LW0k9mFbO9iTZ1Xj)H#U1;_?cIp*<`U04q9jgN"T*SFg;lVClgtUOdMcgYKr[?90CV92Jn>&r0IN4iB%Z<'Fr9C-?6SZc7u<7f+%>KqLlHJcb[^IMOO>Q)Aj*;`r?Y?>_>YUI'oUaiQ0$"dJ>`H="29K./0hgEGgs+#FmEJo'b?I]ch$f34`Qqg?m5Kg>tZ)A-.-"6eI8:f?&dNCS_@AOgKhMJ?-V22h;2HFigure 25: MOEA GUI Window for Settings of Design Specifications The simulation adopts a generation and population size of 200, and all the design specifications listed in Figure 25 have been successfully satisfied at the end of the evolution. The design trade-off graph is shown in Figure 26, where each line representing a solution fou nd. The x-axis shows the design specifications and the y-axis shows the normalized cost for each objective. Clearly, trade-offs between adjacent specifications result in the crossing of the lines between them {e.g., steady-state error {SSE} and control effort {u}), whereas concurrent lines that do not cross each other indicating the specifications do not compete with one another {e.g., overshoots {OS} and settling time {Ts}).) 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+TY%D>qM`*%G4sohUn>=<;_$]Q&8l^VcA00"CuB0JRZ5@aWr7G/]qFD&u='!$%#tm"oq,hC;IU67MKWFQ2`,);is-hs@_1"p<`I!T6-_)MT>i2Ba1i+5i^UhMpq@$':Ag#ia+!A<[EfllA-X4un#!E5U$=OeNindV%=@"n(/:2K((NX/*Ziml7]9P7d?A4N5&AmL&O'n]HIH.Y#HjGfB08M+D]S_dt-QRKui[H:r$JXS>BO-$!":5"[_/*Hm:BD-n?Yfg]DK&sLoB1![d+?,CNd5RQF0_na!RNH:_4:M&Bi=WhMsY=kEGT=Yl2u:s`lDnma1^8.7>B%Qf$3R2_sFPPm'b8tKOsNo&EW"Km2"4=#1^G:R-eM;m8hm@gi?o'L.mmcs< Q n Q Q 148.381 105.72 m (Figure 26: Trade-off Graph of the HDD Servo Control System Design 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard Company ANEVOLUTIONARY ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION211 The closed-loop step response of the overall system for an arbitrary selected set of MOEA designed 2DOF controller parameters given as {Kf,Kb,ff1,ff2,fb1,fb2} = {0.029695, -0.58127, 0.90279, -0.3946, -0.70592, 0.83152} is shown in Figure 27. With a sampl ing frequency of 4 kHz, the time domain closed-loop performance of the evolutionary designed controller has been compared with the manually designed discrete-time PID controller as given in Eq. 18 {Goh et al. 2001} as well as the Robust and Perfect Tracking {RPT} controller {Goh et al. 2001} as given in Eq. 19,}{25.025.11.023.013.022yrzzzzu000020000ffi000020000ffi000020000 {18} }{18.0}{04.0}{1043.3}{}{453681}{15179}{04.0}1{7kykrkxkukykrkxkx000020000ffi000u000020000 000020000ffi000020000 000ffi000020 {19} It can be seen in Figure 27 that the evolutionary designed 2DOF controller has outperformed both the PID and RPT controllers, with the fastest rise time, smallest overshoots and shortest settling time in the closed-loop response. Its control performance is excellent and the destination track crossover occurs at approximately 1.8 milliseconds. 00.0010.0020.0030.0040.0050.0060.0070.0080.0090.0100.20.40.60.811.21.41.61.8: MOEA Based 2DOF Controller: PID ControllerTime in SecondsHead Position {Tracks}: RPT Controller123132Figure 27: Closed-loop Servo System Responses with Evolutionary 2DOF, RPT and PID ControllersThe performance of the evolutionary 2DOF servo control system was further verified and tested on the physical 3.5-inch HDD with a TMS320 digital signal processor {DSP} and a sampling rate of 4 kHz. The R/W head position was measured using a laser doppler vibrometer {LDV} and the resolution used was 1 000Pm/volt. Real-time implementation result of the evolutionary HDD servo control system is given in Figure 28, which is consistent with the simulated step response in Figure 27, and shows an excellent closed-loop performance. 19980209064900RelativeColorimetricsRGB IEC61966-2.1RGB Copyright (c) 1998 Hewlett-Packard CompanyTAN,KHOR,LEE,&SATHIKANNAN212 00.0010.0020.0030.0040.0050.0060.0070.0080.0090.0100.20.40.60.811.21.4Time in SecondsTracks { Actuator Output }Output ResponseFigure 28: Real-time Implementation Response of the Evolutionary 2DOF Servo System 6. ConclusionsThis paper has presented a multi-objective evolutionary algorithm {MOEA} with a new goal-sequence domination scheme to allow advanced specifications such as hard/soft priorities and constraints to be incorporated for better decision support in multi-objective optimization. In addition, a dynamic fitness sharing scheme that is simple in computation and adaptively based upon the on-line population distribution at each generation has been proposed. Such a dynamic sharing approach avoids the need for a priori parameter settings or user knowledge of the usually unknown trade-off surface often required in existing methods. The effectiveness of the proposed features in MOEA has been demonstrated by showing that each of the features contains its specific merits and usage that benefit the performance of MOEA. In comparison with other existing evolutionary approaches, simulation results show that MOEA has performed well in the diversity of evolutionary search and uniform distribution of non-dominated individual s along the final trade-offs, without significant computational effort. The MOEA has been applied to the practical engineering design problem of a HDD servo control system. 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