Prof. Ga Gana Ganapati napa pati ti Pa Pand Panda, nda, a, FNAE, FNASc. Dean Academic Affairs Professor, School of Electrical Sciences IIT Bhubaneswar
Multi Mu Multiobjective ltiobj object ective ive Optimization Optim Op timiza izatio tion n
³
Mu M ulti tio obj bjeect ctiive op opttimiz izat atiion is th thee pr pro oce cess ss of simultaneously optimizing two or more conflicting objectives subject to certain constraints.´ Examples of multi-objec multi-objective tive optimization problems:
Max axiimizing pr prof ofiit and mini nim mizi zing ng the cos ostt of a pr prod oduc uctt.
Maxi Ma xim miz izin ing g pe perf rfor orma manc ncee an and d mi mini nim miz izin ing g fu fuel el co cons nsum umpt ptio ion n of a ve vehi hicl cle. e.
Minimizing weight while maximizing the strength of a particular component.
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2
Multi Mu Multiobjective ltiobj object ective ive Optimization Optim Op timiza izatio tion n
³
Mu M ulti tio obj bjeect ctiive op opttimiz izat atiion is th thee pr pro oce cess ss of simultaneously optimizing two or more conflicting objectives subject to certain constraints.´ Examples of multi-objec multi-objective tive optimization problems:
Max axiimizing pr prof ofiit and mini nim mizi zing ng the cos ostt of a pr prod oduc uctt.
Maxi Ma xim miz izin ing g pe perf rfor orma manc ncee an and d mi mini nim miz izin ing g fu fuel el co cons nsum umpt ptio ion n of a ve vehi hicl cle. e.
Minimizing weight while maximizing the strength of a particular component.
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2
Difference Single Objective Optimization
Optimize function
ltiobjective ve Optimi Optimization zation Multiobjecti
Optimize two or more than two objecti obje ctive ve funct functions ions
Single Sin gle opt optim imal al sol soluti ution on
Sett of op Se opti tim mal so solu luti tion onss
Maximum Maxim um/Mi /Minim nimum um fit fitnes nesss va value lue is se sellected as the be besst sol olut utiion on..
Com Co mpa pari riso son n of so solu luti tion onss by
only
one
objective
Minimize
Domination
Non-domination
400
100 90
f1(x) f1(x) f2(x) f2(x)
80 300
70
where
-10 < x < 20
60
) x (200 f
50
2
f
40
Optimal solution:-
30
100
20 10 0 -1 0
-5
0
5
10
15
20
x
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0 0
10
20
30
40
50
60
70
80
90
100
f 1
3
Standar d Approa Standard Approach ch :Weighted :Weighted Sum of Objective Functions T
T
T
Minimize _ f 1` ( x ), f 2 ( x ),- - , f m ( x )a Formulate as a single objective with weighted sum of all objective functions -
T
T
T
T
g ( x ) ! P1 f 1 ( x ) P2 f 2 ( x ) - - Pm f m ( x ) where P1 , P2 ,.- - , Pm are weights values & P1 and m represents the number of objective functions.
P2 -
-
Pm ! 1
Limitations: ×
R esult esult depends on weights.
×
Some solutions may be missed.
×
×
×
Multiple runs of the algorithm are required in order to get the whole range of solutions. Difficult to select proper combination of weights.
Combining objectives loses information and predetermines trade-offs between objectives.
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4
Definitions Domination
: One solution is said to dominate another if it is better in all objectives. Non-Domination [Pareto points] : A solution is said to be nondominated if it is better than other solutions in at least one objective.
B
f 2 n o i t c n u f e z i m i n i M
A
A dominates B (better in both f 1 and f 2 )
C
D
Minimize function
f 1
A dominates C (same in f 1 but better in f 2 ) A does not dominate D (non-dominated points) A and D are in the ³Pareto optimal front´ These non-dominated solutions are called Pareto optimal solutions. This non-dominated curve is said to be Pareto front. 2/13/2012
5
Definitions Pareto Optimal
A vector variable and
either
or, there is at least one
where
is Pareto optimal if for every
such that
is the vector of decision variables,
is the vector of objective functions, is the feasible region ,where represents the whole search space. 2/13/2012
6
Definitions Cont«. Pareto Optimal Set
For a given MOP
the Pareto optimal set is
defined as Pareto Front
For a given MOP and Pareto optimal set Pareto front is defined as
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, the
7
Desirable MOEA features Best
Solutions
Lie on true Pareto front They are uniformly
distributed on the front
f 2
Possible solutions
n o i t c n u f e z i m i n i M
True Pareto front
Minimize function
Aim:
f 1
To achieve convergence to Pareto optimal front To achieve diversity (representation of the entire Pareto
optimal front) 2/13/2012
8
Non Dominated Sorting based Genetic Algorithm II (NSGA(NSGA- II) Developed by Prof. K. Deb at Kanpur Genetic Algorithms
Laboratory (2002) Famous for F ast non-dominated search Fitness assignment - Ranking based on non-domination
sorting Diversity mechanism is based on C rowding distance Uses Elitism 2/13/2012
9
Initialize Population Minimize
x 2
f 1 x ! x , f 2 x 2
where
!
2
x 0.4678 1.7355
5 e x e 5
0.8183
Search space is of single dimension (given).
-0.414 3.2105
Objective space is of two dimension (given).
-1.272 -1.508
Let population size = 10
-1.832
Initialize population with 10 chromosomes having single dimensioned real value.
-2.161 -4.105
These values are randomly distributed in between [-5,5].
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Evaluate Fitness values Find out all objective functions values (fitness values) for all chromosomes.
x
f 1 x
f 2 x
40
-0.414
0.171
5.829
35
0.467
0.218
2.347
0.818
0.669
1.396
1.735
3.011
0.07
3.210
10.308
1.465
-1.272
1.618
10.708
-1.508
2.275
12.308
-1.832
3.355
14.682
-2.161
4.671
17.317
-4.105
16.854
37.275
30 25 ) x (
f
20
2
15 10 5 0 0
2
4
6
8
10 f 1 (x)
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12
14
16
18 11
Fast NonNon-domination Sorting
Assigning the rank to each individual of the population. R ank based on the non-domination sorting (front wise). It helps in selection and sorting. R eference chromosome
x
f 1 x
f 2 x
-0.414
0.171
5.829
x1
0.467
0.218
2.347
2
0.818
0.669
1.396
1.735
3.011
0.07
3.210
10.308
1.465
chromosomes
} n1
! {! {-
} n
x 3
! {! {-
x
-1.272
1.618
10.708
x 4
-1.508
2.275
12.308
5
-1.832
3.355
14.682
-2.161
4.671
17.317
-4.105
16.854
37.275
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Counter
Dominated
x
x 6
! { x
x 7
! { x
!
0
1 1
!
0
} n3
!
0
1
} n4
!
0
2
3
! { x
2
Rank
1 1
4
, x } n
5
1
2
3
!1
, x 2 , x 3} n 6
4
, x 2 , x 3 , x 6 } n 7
5
!1
6
!
2
12
Fast NonNon-domination Sorting 40
x
f 1 x
f 2 x
Rank
-0.414
0.171
5.829
1
35
0.467
0.218
2.347
1
30
0.818
0.669
1.396
1
1.735
3.011
0.07
1
3.210
10.308
1.465
2
-1.272
1.618
10.708
2
-1.508
2.275
12.308
3
-1.832
3.355
14.682
4
-2.161
4.671
17.317
5
-4.105
16.854
37.275
6
6
25 ) x ( f
5 2
20
4 3
15 10 5 0 0
2 1 2
4
6
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f 1 (x )
10
12
14
16
18
13
Crowding Distance Assignment To get an estimate of density of solutions surrounding a particular solution in population.
Choose individuals crowding distance.
Help for distribution.
having
obtaining
large
uniformly
1C . D . ! l C . D. ! g,
iC . D. where max m 2/13/2012
and f
¨ f [i 1]m f [i 1]m ¸ ¹¹ ! §© max min © m f m f m ª º represent
objective function value of
where i
!
2,3,-
, (l 1)
solution.
T
is the maximum value of function f m
in the Pareto front. 14
Crowding Distance Assignment 40 35
Crowning distance can be calculated for all chromosomes of same Pareto front.
6
30 25
5
) x
( f
2
20
4 3
15
x -0.414
f 1 x f 2 x Rank 0.171
5.829
1
10
C . D.
5
g
0 0
0.467
0.218
2.347
1
0.945
0.818
0.669
1.396
1
1.378
2 1 2
4
6
8
10
12
14
16
18
f (x) 1
6
1
5
1.735
3.011
0.07
1
3.210
10.308
1.465
2
-1.272
1.618
10.708
2
-1.508
2.275
12.308
3
-1.832
3.355
14.682
4
-2.161
4.671
17.317
5
g
-4.105 16.854 37.275
6
g
g g
g g g
4 ) x ( f
3
2
2
2
i- 1
3 i
1
i+ 1
0 4
0 .5
1
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1 .5 f 1 (x )
2
2 .5
3
15
Selection S election
is the stage of a genetic algorithm in which individual are chosen from a
population for later breeding (recombination or crossover). Crowding
operator based sorting
The crowding operator e n guides the selection process at the various stages of the algorithm toward a uniformly spread-out Pareto optimal front.
i e n j
i
rank
jrank
or
i
rank !
jrank and iC . D. " jC . D.
where irank shows non-domination rank & iC . D . is crowding distance of i th individual. 2/13/2012
16
Tournament Selection Runs
a µtournament¶ among a few individuals chosen at random from the population and selects the winner (the one with the best fitness) for crossover . In tournament selection, a number T our size of individuals is chosen randomly from the population and the best individual from this group is selected as parent. (Based on the crowding operator)
x
f 1 x f 2 x
Rank
C . D.
1.378
0.818
0.669
1.396
1
-1.508
2.275
12.30
3
g
1rank 2rank 0.818
0.669
x
f 1 x f 2 x
1
1.378
C . D.
0.467
0.218
2.347
1
0.945
0.818
0.669
1.396
1
1.378
1rank ! 2rank 1.396
Rank
0.818
0.669
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1C . D. 2C . D. 1.396
1
1.378 17
Crossover C rossover is
a genetic operator that combines (mates) two individuals (parents) to produce two new individuals ( C hilds). The idea behind crossover is that the new chromosome may be better than both of the parents if it takes the best characteristics from each of the parents. Simulated Binary Crossover ¨ 1 ¸ ® ©© ¹¹ 1 2 * r if r e 0.5 ª º ± ± ¨ 1 ¸ ©© ¹¹ b !¯ 1 ª º ±¨©© 1 ¸¹¹ if r " 0.5 ± °ª 2 * 1 r º
where
Q
Q
child 1 ( j ) ! child 2 ( j ) !
1 2 1 2
r
is a random number {0,1} Q is a crossover operator j represent dimension of individual.
1 b* parent ( j ) 1 b* parent ( j )
1
2
1 b * parent ( j ) 1 b* parent ( j ) 1
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2
18
Mutation Mutation is a genetic operator that alters one ore more gene values in a chromosome from its initial state. Mutation is an important part of the genetic search as helps to prevent the population from stagnating at any local optima. Polynomial Mutation ¨ 1 ¸ ® ± 2 * r ©©ª 1 º¹¹ 1 if r e 0.5 d ! ¯ ¨ 1 ¸ ©© ¹ ± 1 º¹ 1 2 * 1 r if r " 0.5 ª ° L
L
child ( j ) ! parent ( j ) d
where
r
is a random number {0,1} L is a mutation operator j represent dimension of individual.
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Selection for next generation Non-dominated sorting (Rank)
C rowding
distance sorting
F1
Pt
F2 F3
Pt+1 Qt Rejected
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R t ={Pt , Qt }
Elitist Replacement 20
Initial State
) x (
After 20 generation
40
4
35
3. 5
30
3
25
2. 5 ) x
20
(
2
f
f
2
2
15
1. 5
10
1
5
0. 5
0 0
2
4
6
8
10
12
14
16
0 0
18
0 .5
1
1 .5
2
2 .5
3
3 .5
4
f (x)
f (x)
1
1
After 10 generation
After 4 0 generation 4
3 .5
3. 5
3
3
2 .5 2. 5
2
) x
( f
) x ( f
2
1 .5
2
2 1. 5
1
1
0 .5
0. 5
0 0
0 .5
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1
1 .5
2
2 .5
3
3 .5
0 0
0 .5
f (x)
1
1 .5
2
2 .5
3
3 .5
4
f (x) 1
1
21
Flowchart of NSGANSGA-II Begin
: Initialize Population (N)
Evaluate objective f unctions Non-dominated Sorting Tournament Selection No Crossover
& Mutation
Evaluate objective f unctions
Stopping criteria met ?
Yes
R eport Final Population and Stop
Combine
parent and child populations , Non-dominating Sorting
Select N individuals
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22
Performance Measures There are two main goals in a multi-objective optimization:
1 ) C onvergence to the Pareto-optimal set 2) Maintenance of diversity in solutions of the Paretooptimal set .
D I S T A NC E M E A S U R E B E T W E E N P A R E T O F R O N T S
12
m in
distance
10
TRUE
P AR E T O F R O N T
P AR E T O F R O N T 8
Convergence metric 2 n o i t c n u f
6
m in
distance
4
2
0 0.1
0.2
0 .3
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0.4
0.5 0.6 function - 1
0 .7
0 .8
0.9
23
Cont«. DIVERSITY PLOT
4
Non uniformity in the distribution,
3.5 3 2.5 2 n o i t c n u f
2
solution
d 1 d 2
d 3
d 4
1.5
If distance between the solutions is equal to average distance gives uniformly distribution.
, that
Pareto front
Extreme
Extreme
1
solution 0.5
d n1 0 0
0.5
1
1.5
2 2.5 function -1
3
3.5
4
4.5
The parameters and are the Euclidean distances between the extreme solutions of true Pareto front and the boundary solutions of the obtained non-dominated set. The parameter is the average of all distances , , assuming that there are solutions on the best non- dominated front. 2/13/2012
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Simulation and R esults NSG A
II parameters Population ( N ) = 100 Crossover Probability ( P c)= 0.9 Mutation Probability ( P m) = 0.1 Distribution index for crossover (µ)= 20 Distribution index for mutation () = 20 Tour size (selection) = 2
MOP SO parameters Population = 100 particles R epository (Archive) size = 100 particles Mutation rate = 0.5 Divisions for Archive Grid = 30
Implementation use real numbers representation. * These parameters were kept in all test functions optimization. * Only changed the total number of fitness function evaluations. 2/13/2012
25
Test Problem : SCH
N ondo m inated
N ondo m inated
solutions with M O P S O on S C H
solutions w ith N S G A-II on S C H
5
5 Pareto-optim al Front MOPS O
Pareto-opti m al Front N S G A-II
4
4
3
3
2
2
F
F
2
2
1
1
0
0
0.5
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1
1.5
2 F1
2.5
3
3.5
4
0 0
0.5
1
1.5
2 F1
2.5
3
3.5
4 26
Comparison for SCH Test Function Convergence Metric
NSGA-II
MOPSO
Best
0.0148
0.0093
Worst
0.9578
0.1569
Mean
0.2096
0.0259
Tab. 1: R esults of the SCH Test Function
Convergence Metric
for the
Diversity Metric
NSGA-II
MOPSO
B est
0.5104
0.6947
Worst
0.7904
1.3575
Mean
0.6425
0.8582
Tab. 2: R esults of the SCH Test Function
Diversity Metric
for the
2/13/2012
27
Test Problem : DEBEB-1
N ondo m inated
solutions w ith M O P S O o n D E B -1
N ondo m inated
1. 4
solutions w ith
N S G A-II
on D E B -1
1.4
Pareto-optim al Front MO P SO
1. 2
Pareto-optim al Front N S G A-II
1.2
1
1
0. 8
0.8
2 F
2 F
0. 6
0.6
0. 4
0.4
0. 2
0.2
0 0
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0.2
0.4
0.6
0.8
1
0
0
0.2
F1
0.4
0.6 F1
0.8
1
28
Comparison for DEBEB-1 Test Function Convergence Metric
NSGA-II
MOPSO
Best
0.0066
0.0070
Worst
0.5140
0.1664
Mean
0.0078
0.0079
Tab. 1: R esults of the DEB-1 Test Function
for the
Diversity Metric
NSGA-II
MOPSO
B est
0.3467
0.5112
Worst
0.5140
0.7168
Mean
0.4243
0.5938
Tab. 2: R esults of the DEB-1 Test Function
Convergence Metric
Diversity Metric
for the
2/13/2012
29
EB-2 Test Problem : DEB-
N ondo m inated
solutions w ith M O P S O using D E B -2
N ondo m inated
8 Pareto-optim al Front MO P S O
7
N S G A-II
on D E B -2
Pareto-optim al Front N A G A-II
7
6
6
5
5 4
4
2 F
2 F
3
3
2
2
1
1
0 0.1
solutions w ith
8
0.2
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0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
0 0.1
0.2
0.3
0.4
0.5
0.6 F1
F1
0.7
0.8
0.9
1
30
Comparison for DEBEB-2 Test Function Convergence Metric
NSGA-II
MOPSO
Best
0.0449
0.0515
Worst
0.0559
0.0725
Mean
0.0516
0.0608
Tab. 1: R esults of the Convergence Metric for the DEB-2 Test Function
Diversity Metric
NSGA-II
MOPSO
B est
0.7248
0.6800
Worst
0.7939
0.7582
Mean
0.7597
0.7193
Tab. 2: R esults of the Diversity Metric for the DEB-2 Test Function
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31
Test Problem : K UR
N ondo m inated
solutions w ith M O P S O on K U R
2 Pareto-optim al Front MO P S O
0 -2 -4 2 F
-6 -8 -10 -12 -20
-19
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-18
-17 F1
-16
-15
-14
32
Comparison for K UR Test Function Convergence Metric
NSGA-II
MOPSO
B est
0.0021
0.0021
Worst
0.0041
0.0034
Mean
0.0028
0.0026
Tab.1: R esults of the KUR Test Function
for the
Diversity Metric
NSGA-II
MOPSO
Best
0.3344
0.4803
Worst
0.7825
0.6413
Mean
0.4399
0.5602
Tab.2: R esults of the KUR Test Function
Convergence Metric
Diversity Metric
for the
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33
Layout Optimization for a Wireless Sensor Network using NSGA - II
a) Coverage b) Lifetime
2/13/2012
34
Wireless Sensor Network (WSN) Data Processing Unit (DPU) Node 1
High Ener gy Communication Node (HECN)
Node 9 Node 6 Node 2
Node 7 Node 8
Node 3
Node 4
Node 5
Example of a WSN where sensor nodes are communicating with 2/13/2012the DPU through HECN
35
Optimization of Coverage Coverage is defined as the ratio of the union of areas covered by each node and the area of the entire R OI.
C !
7i !1,..., N Ai
A
- Area covered by the ith node N - Total number of nodes A - Area of the R OI Ai
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36
Optimization of Lifetime The lifetime of the whole network is the time until one
of the participating nodes run out of energy. In every sensing cycle, the data from every node is routed to HECN through a route of minimum weight Lifetime !
Tf ailure Tmax
= =
Tfailure Tmax
maximum number of sensing cycles before failure of any node maximum number of possible sensing cycles
Competing Objectives Lifetime
Coverage
HECN
try to arrange the nodes as close as possible to the HECN for maximizing lifetime
try to spread out the nodes for maximizing coverage
2/13/2012
38
Simulation Parameters Parameters of NSGA-II Number
of chromosomes
100
Number
of generations
50
Crossover Probability
0.9
Mutation Probability
0.5
Distribution index for crossover
20
Distribution index for mutation
20
Tour size
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2
39
NSGA--II R esults NSGA Pareto optimal front 1
0. 9
0. 8 0. 7
e 0. 6 m i t 0. 5 e f i L0. 4 0. 3
0. 2
0. 1
0 0
0. 1
0 .2
0 .3
0. 4
0. 5
0 .6
0. 7
0. 8
0. 9
1
Coverage
Pareto
Front obtained f or a WSN with 10 sensors, 100 chromosomes and 50 generations
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40
NSGA--II R esults (Cont¶d) NSGA Initial Digsco0.63709 nnect Network Covera
6
Best Lif ei tiime 0.999 0.333
e =
Coverage = 6
4
4
2
2
HECN
0
0
-2
-2
-4
-4
-6
-6
-6
-4
-2
0
2
4
5 L f et me =
HECN -6
6
-4
-2
0
2
4
6
Best Covera ge = 0 5353 ifetime = 0 249
Coverage
L
.
.
6
4
2
0
-2
-4
-6
-6
-4
-2
0
2
4
2/13/2012
HEC N 6
41
R adial Basis Function Network 1
x1
w0
x2
w1 w2 w3
x3
x4
Ö d i
w4
w5
Input Layer
Hidden Layer
Output Layer
Accuracy Complexity of the model 42
Multiobjective Problem Formulation Structure determination of R BF network can be considered as the multiobjective optimization problem concerning with accuracy and complexity of the model. f 1
f 2
!
!
!
M
mse 1
n
§ d d n 2
Ö
i
i
i !1
¨ ¨ M 1 © d i © § w jJ xi c j ! § © j 1 n i 1 ©ª ª n
!
!
¸ ¸ w0 ¹¹ ¹¹ º º
2
here M is total number of basis functions (centers) in R BF network, d i: Desired output d i : Estimated output during the training of R BF network. w : Weight vector of the R BF network c j : Center vector of the R BF network J . : Gaussian Function Ö
¸ ¹ ª º is the spread of the Gaussian function .
J x c j
where
W
! exp ¨© 2W1
2
x c j
2
43
Structure selection of R BF network Selected centers n o i n t a u i b s s i r u t s a i G D
Input Data Points Chromosome
Input Data Points
1
0
1
1
0
0
1
+1
+1
-1
+1
0
0
0
0
+1
+1
-1
+1
0
0
0
0
+1
+1
-1
+1
0
0
0
0
+1
+1
-1
+1
In the chromosome, the position of gene value ³1´ indicate the center position Desired Output 0 0 +1 0 0 of the basis f unction0 (selected center) and number of 0³1´ genes in chromosome indicates the number of basis f unctions (number of centers). 44
Pareto Fronts
This Pareto Front shows that for the different number of centers, MSE changes.
The performance of an R BF network critically depends upon the chosen centers.
10
Pareto Front for 13-elem ent B arker C od e
0
)
e l a c s
10
-10
g o l ( r o r r
10
-20
E
e r a u q
10
-30
E
n a e M
10
-40
2
4
6
8
10 12 14 16 18 N um bers of C enters
20
22
24 25 45
R eferences 1.
K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan,´ A fast and elitist multi-objective genetic algorithm: N SGA-II ´, I EEE T ransaction on Evolutionary C omputation , 6 (2), 181-197,2002.
2.
K. Deb and R . B. Agrawal, ³ S imulated binary crossover for continuous search space,´ in C omplex S yst., vol. 9, pp. 115± 148. , Apr. 1995.
3.
N. Srinivas and K. Deb, ³ Multiobjective function optimization using nondominated sorting genetic algorithms ,´ Evol. C omput. , vol. 2, no. 3, pp. 221 ± 248, Fall 1995.
4.
J. Horn, N. Nafploitis, and D. E. Goldberg, ³ A niched Pareto genetic algorithm for multiobjective optimization,´ in Proceedings of the F irst I EEE C onference on Evolutionary C omputation , Z . Michalewicz , Ed. Piscataway, NJ: IEEE Press, pp. 82 ± 87 , 1994.
5.
J. D. Knowles and D.W. Corne, ³ A pproximating the nondominated front using the Pareto archived evolution strategy ,´ Evol. C omput. , vol. 8, pp. 149 ± 1 72, 2000.
6.
Carlos A. Coello Coello, Member, IEEE, Gregorio Toscano Pulido, and Maximino Salazar Lechuga, ³ H andling multiple objectives with particle swarm optimization ´, Evol. Comput., vol. 8, pp. 256 ± 279, No. 3, June 2004
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