TELKOM
NIKA
, Vol.14, No
.1, March 2
0
1
6
, pp. 238~2
4
4
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v14i1.2740
238
Re
cei
v
ed
No
vem
ber 4, 20
15; Re
vised
Febr
uary 11,
2016; Accept
ed Feb
r
ua
ry
25, 2016
Application of A Self-adaption Dual Population Genetic
Algorithm in Multi-objective Optimization Problems
Chen
g Zhan
g
*, Hao Pen
g
Schoo
l of Information, Hu
na
n Internatio
na
l Econom
ics Univ
ersit
y
, Ch
an
gs
ha 41
02
05, Hu
nan, Ch
in
a
*Corres
p
o
ndi
n
g
author, em
ail
:
28996
25
26@
qq.com
A
b
st
r
a
ct
Multi-ob
jectiv
e evol
ution
a
ry
a
l
gorit
h
m
is
a p
o
w
e
rful tool
in
resolvi
ng
mult
i-obj
ective
opti
m
i
z
at
io
n
prob
le
ms. T
h
i
s
alg
o
rith
m
in
herits th
e a
d
v
antag
es of
pa
ralle
l ra
nd
o
m
search, stro
ng
glo
b
a
l
se
arch
in
g
capa
bil
i
ty and the ab
ility to sol
v
e hig
h
ly-co
m
p
licate
d
non-
lin
e
a
r probl
e
m
s of evol
ution
a
ry al
gorith
m
a
nd it is
usua
lly use
d
i
n
the opti
m
i
z
a
t
ion pro
b
le
ms
w
i
th mult
i
p
le
mutu
al co
nflict
s
. How
e
ver, such al
gorith
m
s
are
slow
in conv
er
genc
e an
d ea
sy to be trapp
ed in
loca
l op
tima
l sol
u
tio
n
. T
h
is pap
er p
r
opos
es a
mul
t
i
-
obj
ective d
ual
pop
ulati
on g
e
n
e
tic alg
o
rith
m (
M
ODPGA)
and
explor
es the i
m
pr
ove
m
ent strategi
es of mul
t
i-
obj
ective g
e
n
e
tic alg
o
rith
m. T
he a
d
o
p
tion
of self-ad
apt
io
n a
nd d
ual
po
pul
a
t
ion strategy c
an g
uara
n
tee t
h
a
t
the alg
o
rith
m o
f
this paper ca
n conver
ge to
Pareto sol
u
tio
n
set in a relia
bl
e and q
u
ick
mann
er an
d it can
perfor
m
more
extensiv
e sear
ch on the o
b
j
e
ctive
functio
n
space a
nd c
ond
uct mor
e
sampl
e
s on
mu
lti-
obj
ective f
uncti
ons s
o
as to
b
e
cl
oser t
o
the
ap
proxi
m
ate
o
p
timal
sol
u
tio
n
set of
gl
ob
al
opti
m
a
l
so
luti
o
n
s.
T
h
is so
lutio
n
s
e
t als
o
incl
ud
e
s
more
o
p
tima
l feas
ibl
e
poi
nt
s an
d
provi
des
rel
i
ab
le
bas
is
for the
dec
isio
n
mak
i
n
g
.
Ke
y
w
ords
: mu
lti-obj
ective o
p
timi
z
a
tio
n
, gen
e
t
ic algor
ith
m
, adaptiv
e du
al po
pul
ation
Copy
right
©
2016 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Multi-obj
ectiv
e
optimization is an
NP difficult
probl
e
m
whi
c
h wi
d
e
ly exists in reality and
the optimization of the ov
erall o
b
je
ctive is
reali
z
ed
by weig
hing
the optimi
z
a
t
ion of multip
le
obje
c
tives u
n
der
one
overall obje
c
tive, therefo
r
e, it
b
e
com
e
s very
difficult to take the
weight
s of
every su
b-obj
ective into
co
nsid
eratio
n a
nd re
qui
re hi
gh latitude
a
nd large
scal
e [1]. Worse
still,
the tradition
al optimizatio
n mean
s are even more st
ri
n
gent on the f
o
rm of the ob
jective functio
n
.
Gene
rally, the pro
b
lem
s
u
s
ually have
more th
an on
e solutio
n
an
d the re
sult
s of every sol
u
tio
n
are i
n
comp
arable. So f
a
r,
the m
u
lti-ob
jective
o
p
timization
theo
ry not o
n
ly m
a
ke
s im
po
rta
n
t
achi
evement
s in
cludin
g
the form
ation
and imp
r
ove
m
ent of a se
t of optimizat
ion theo
ry which
parall
e
ls to t
he
single
-
o
b
j
e
ctive o
p
timization,
but it ha
s al
so
b
een a
pplie
d
more
and
m
o
re
extensively [2].
Every objecti
ve in the multi-obje
c
tive optim
ization
p
r
oble
m
is cal
l
ed a su
b-ob
jective.
Due to the
mutual influe
nce a
nd inte
ractio
n
amo
n
g
each su
b-obje
c
tive, the multi-obj
ective
optimizatio
n meets
the opt
imal conditio
n
s
of
every
su
b-obj
ective a
s
well as
the
con
s
trai
nts
of
the
intera
ctive rel
a
tionship am
ong the sub-obje
c
tives
[3]. Multi-obje
c
tive optimizati
on pro
b
lem
wa
s
first rai
s
ed
b
y
V.Pareto, a Fren
ch e
c
o
n
o
mist when
rese
archin
g th
e eco
nomi
c
balan
ce. At tha
t
time, he
had
su
mma
rize
d
num
ero
u
s o
b
jective
s
whi
c
h
are h
a
rd
to comp
are
a
s
m
u
lti-obj
ect
i
ve
optimizatio
n
probl
em from
the p
e
rspe
ct
ive of
politica
l
economy. A
fter that, Von
.
Neume
nn
an
d
J.Morgen
stern came u
p
with the multi-obje
c
tive
de
cisi
on-ma
king
proble
m
whi
c
h ha
s multi
p
le
confli
cting d
e
c
isi
on m
a
kers in
game
th
eory. T.C.
Ko
opman
s
wa
s
the first to
bri
ng forth
Pare
to
optimal sol
u
tion after putting forward m
u
lti-obje
c
ti
ve optimizatio
n probl
em
s fro
m
the analysi
s
of
prod
uctio
n
a
nd distri
butio
n. Z.
Johnse
n
systemati
c
ally propo
se
d
the rese
arch
repo
rt on multi-
obje
c
tive
de
ci
sion
-ma
k
in
g model, whi
c
h
is a
tu
rn
in
g
point
whe
n
th
e multi-obje
c
t
i
ve optimization
has b
een de
veloped g
r
eat
ly. The numerou
s evolutio
nar
y algo
rith
ms su
ch a
s
g
enetic alg
o
rit
h
m,
particl
e
swarm optimi
z
atio
n alg
o
rithm
a
nd fish
sw
arm algo
rithm
have
come
in
to bein
g
in
re
cent
years to
solv
e multi-obj
ect
i
ve optimizati
on pro
b
le
m
s
,
neverthele
ss, these algo
rit
h
ms a
r
e sl
ow in
conve
r
ge
nce
and
ea
sy t
o
be
trap
pe
d in l
o
cal o
p
timal soluti
ons an
d the
y
need
furth
e
r
improvem
ent
s [5].
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Applicatio
n of A Self-adaption Du
al Popu
lation Gen
e
tic Algorithm
in… (Ch
eng Z
h
ang)
239
This
pap
er
prese
n
ts a
mult
i-obje
c
tive g
e
netic
al
go
rith
m by co
mbini
ng self-a
dapti
on, dual
popul
ation strategy
an
d ge
netic algo
rith
m
to settle
m
u
lti-obje
c
tive optimizatio
n probl
em
s.
It
firstly
con
d
u
c
ts ma
thematical
d
e
scriptio
n of
the mu
lti-objective optimization
problems. T
hen it
elabo
rate
s th
e ba
sic p
r
in
ciples of g
e
n
e
tic alg
o
rith
m, base
d
on
whi
c
h, it de
sign
s the m
u
lti-
obje
c
tive ge
netic
algo
rith
m in a
c
co
rd
ance
wi
th self-ada
ption and dual
po
pulation
strategy.
Finally, it is about the perf
o
rma
n
ce test and an
alysi
s
of this algorit
hm.
2. The Math
e
m
atical De
sc
ription of
Mu
lti-objec
tiv
e
Optimiza
tion
Problem
At present, multi-obj
ectiv
e
optimizatio
n pro
b
lem
h
a
s b
een
app
lied mo
re a
n
d
more
widely, involving many fields. In daily life and pr
oje
c
ts, optimizatio
n is not only requi
re
d in on
e
index but it deman
ds
sev
e
ral ind
e
xes
to achiev
e th
e optimizatio
n at the sam
e
time. A great
numbe
r of proble
m
s can
be redu
ced
as the multi-obje
c
tive o
p
timization p
r
oble
m
to make
multiple objectives to
realiz
e
optimi
z
ation un
de
r ce
rtain
co
nstrai
nt con
d
itions [6].
The
mathemati
c
al
descri
p
tion
of multi-obj
ective optim
izat
ion problem i
s
mad
e
of d
e
ci
sion va
ria
b
le,
obje
c
tive fun
c
tion an
d
con
s
traint
con
d
itio
n. Becau
s
e
of the
differen
c
es
of the
ap
pl
ication
field
s
of
multi-obj
ectiv
e
optimizatio
n pro
b
lem, the co
rres
pon
ding mathe
m
atical de
scrip
t
ion is differe
nt,
inclu
d
ing
ge
neral
multi-o
b
jective o
p
timization,
dynamic multi-obje
c
tive opt
imization,
ce
rtain
multi-obj
ectiv
e
optimizatio
n and u
n
certain multi-
obj
ective optimi
z
ation. Th
e
multi-obj
ectiv
e
optimizatio
n probl
em can
be de
scribe
d as follo
ws ma
thematically:
Min(&Max)
12
(
)
[
(
),
(
)
,
...,
(
)](
1
,
2
,
...
,
)
n
yf
x
f
x
f
x
f
x
n
N
S. t.
12
()
[
(
)
,
(
)
,
.
.
.
,
(
)
]
0
k
gx
g
x
g
x
g
x
12
()
[
(
)
,
()
,
.
.
.
,
(
)
]
0
m
hx
h
x
h
x
h
x
12
_m
i
n
_
m
a
x
[
,
,
.
..,
,
.
.
.
,
]
(1
,
2
,
.
.
.
,
)
dD
dd
d
xx
x
x
x
x
xx
d
D
(1)
Her
e
:
x
is a
D-di
men
s
ion
a
l de
cisi
on v
a
riabl
e,
y
is the obj
ective
function,
N
is
the
numbe
r of total optimizati
on obje
c
tives,
()
n
f
x
is the
th
n
su
b
-
obje
c
tive function,
()
g
x
is
K
inequ
ality co
nstrai
nts,
()
hx
is
M
equality co
nstrai
nts, the
con
s
trai
nts
con
s
titute the
feasible
regio
n
an
d
_m
i
n
d
x
and
_m
a
x
d
x
are the
uppe
r an
d l
o
we
r bo
und
s of vector
se
arch. The
ab
ove
equatio
n is t
he multi-obje
c
tive optimi
z
ation p
r
oble
m
whi
c
h i
n
cl
uding th
e mi
nimizatio
n
p
r
oblem
(min), the ma
ximization p
r
oblem (m
ax)
and certai
n m
u
lti-obje
c
tive optimizatio
n probl
em [7].
The co
ncept
of the optimal solutio
n
or Pa
reto optimal sol
u
tion to multi-obje
c
tive
optimizatio
n probl
em is a
s
follows:
Definition
1: If any
d
which
meets
1,
dD
and
*
dd
x
x
and whi
c
h ha
s
0
1,
dD
and
00
*
dd
x
x
, the vector
**
*
2
*
1
[
,
,
...
,
]
D
x
xx
x
dominate
s
the vector
12
[
,
,
,
...
,
]
dD
x
xxx
x
.
If
*
()
f
x
dominate
s
()
f
x
, it must mee
t
two conditio
n
s:
*
,
(
)
(
)
1
,
2
,
...,
nn
nf
x
f
x
n
N
,
00
*
00
,1
nn
nf
x
f
x
n
N
.
The domi
nan
ce rel
a
tion of
()
f
x
is
c
o
ns
is
tent with that of
x
.
Definition 2: Pareto optim
al solution i
s
t
he solution
whi
c
h can’t be dominate
d
by any
s
o
lution in the feas
ible s
o
lution s
e
t. If
*
x
is a point in th
e sea
r
ch sp
a
c
e,
*
x
is the Pareto optimal
solutio
n
wh
e
n
and o
n
ly whe
n
no
x
(in
the feasibl
e
regio
n
of the sea
r
ch sp
ace
)
can ma
ke
0
*
()
(
)
nn
f
xf
x
whe
n
1
,
2
,
...
,
nN
.
Definition 3:
For a multi
-
o
b
jective optim
ization p
r
obl
e
m
()
f
x
,
*
()
f
x
is the gl
o
bal optimal
solutio
n
wh
en
and only wh
en any
x
(in the sea
r
ch sp
a
c
e)
can m
a
ke
*
()
(
)
f
xf
x
.
Definition
4: The
set consi
s
ting of all P
a
reto o
p
timal
solutio
n
s i
s
t
he Pareto op
timal set
of the multi-o
b
jective optim
izati
on p
r
obl
e
m
and it can
also b
e
the a
c
ceptabl
e or
effective solut
i
on
s
e
t.
The optimi
z
at
ion pro
c
e
s
s o
f
multi-obje
c
tive proble
m
is indicate
d as
Figure 1.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 14, No. 1, March 2
016 : 238 – 2
4
4
240
Figure 1. Optimization p
r
o
c
ess
3. Genetic
Algorithm
The main
ch
ara
c
teri
stic of
genetic al
go
rithm
is that the strate
gy of populatio
n
sea
r
ch
and th
e info
rmation
exch
ange
bet
wee
n
the
individ
uals in th
e
popul
ation
are not
ba
sed
o
n
gradi
ent info
rmation. It ca
n be u
s
e
d
to
handl
e the
complicated a
nd no
n-lin
ea
r probl
em
s which
are difficult to be
solved
by traditional
sea
r
ch me
th
ods i
n
pa
rticular a
nd it
ca
n also be
wi
dely
applie
d in
su
ch field
s
a
s
combi
nato
r
ial
optimizat
ion,
machine l
e
a
r
ning,
self-ad
aptive co
ntro
l,
planni
ng a
n
d
desi
gn
as
well as artifici
a
l
life. As a
gl
obal o
p
timiza
tion se
arch
al
gorithm, g
e
n
e
tic
algorith
m
is o
ne of the core intelligent computati
on te
chn
o
logie
s
in
the 21st cent
ury for it is easy
and u
n
iversal
to apply, it has
stro
ng rob
u
stne
ss, it
ca
n be u
s
e
d
in
parall
e
l p
r
o
c
e
ssi
ng a
nd it h
a
s
a wide a
pplication scop
e [8].
3.1.
The Principles and Me
th
ods of Ge
netic Algorithm
(1) Chr
o
mo
so
me
Enco
ding
Encodi
ng refers to th
e tra
n
sformation
method to
tra
n
sfer th
e fea
s
ible
solutio
n
s
to on
e
probl
em to the sea
r
ch sp
a
c
e which gen
etic algo
rithm
can ha
ndle.
De
Jo
ng
on
ce propo
se
d t
w
o
pra
c
tical
coding
pri
n
ci
pl
es: o
ne i
s
to
use
the
en
co
ding
plan
whi
c
h is
rel
a
ted to the p
r
o
b
lem to be
solved and
which
ha
s lower-ord
e
r,
sho
r
t-defin
ed len
g
th
pattern
and t
he othe
r i
s
t
o
utilize
the
encodin
g
pla
n
whi
c
h
can
give natu
r
al
pre
s
e
n
tation
or
descri
p
tion to
the proble
m
or whi
c
h h
a
s
the minimum
cod
ed character set [9].
Encodi
ng me
thods in
clu
d
e
the following
: binary enco
d
ing metho
d
, gray-cod
e e
n
co
ding
method, float
ing-p
o
int nu
mber
en
codi
ng meth
od,
para
m
eter cascad
e en
co
ding m
e
thod
an
d
multi-pa
ram
e
ter cro
s
s-ove
r
encodin
g
method.
(2) Fitn
ess Computation
Bas
i
cally, there
are three
methods
to trans
form the
objec
tive func
tion value
()
f
x
of a
certai
n point
in the solutio
n
spa
c
e to th
e fitness fun
c
tion value
((
)
)
F
it
f
x
of the corre
s
po
nding
individual in the se
arch sp
ace:
(a)
Dire
ctly tran
sform the
objective fu
nction valu
e
()
f
x
to be s
o
lved to the fitn
ess
function valu
e
((
)
)
F
it
f
x
and make
()
m
a
x
.
((
)
)
()
m
i
n
f
x
T
h
e
o
b
j
e
c
tiv
e
f
un
c
tio
n
i
s
i
m
i
ze
d
Fi
t
f
x
f
x
T
h
e
o
b
j
e
c
tiv
e
f
un
c
tio
n
i
s
i
m
i
ze
(b) As fo
r the mini
mization
p
r
oblem, p
e
rf
orm th
e fo
llowing
tra
n
s
form
ation
max
m
ax
()
()
((
)
)
0
Cf
x
f
x
C
Fit
f
x
oth
e
rs
. Here,
ma
x
C
is the maximum input value of
()
f
x
.
(c) If the objective function
is the minimi
zation pro
b
lem
,
1
((
)
)
,
0
,
(
)
0
1(
)
Fit
f
x
c
c
f
x
cf
x
(2)
If the objective function i
s
the maximization pro
b
lem,
1
((
)
)
,
0
,
(
)
0
1(
)
Fi
t
f
x
c
c
f
x
cf
x
(3)
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TELKOM
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Applicatio
n of A Self-adaption Du
al Popu
lation Gen
e
tic Algorithm
in… (Ch
eng Z
h
ang)
241
3.2. The Process o
f
Gen
e
tic Algori
t
h
m
The g
eneti
c
operation
s
of
gen
etic al
go
rithm
in the
en
tire evolutio
n
pro
c
e
s
s a
r
e
random,
but the cha
r
a
c
teri
stic it pre
s
ent
s is full search.
It can effectively use the previou
s
informatio
n to
predi
ct the o
p
timization
p
o
int set with i
m
prov
e
d
exp
e
cted p
e
rfo
r
mance in the
next generat
ion.
After the co
ntinuou
s evo
l
ution fr
om g
eneration to gene
ration, it
is finally co
nverge
d to the
individual
whi
c
h
can a
dapt
to the enviro
n
ment at
mo
st and the
op
timal solutio
n
to the probl
e
m
can
be
obtai
ned.
Ge
netic algo
rithm i
n
volves five
ele
m
ents: para
m
eter co
ding
,
setting
of
in
itial
popul
ation, d
e
sig
n
of fitness functio
n
, des
ig
n of geneti
c
ope
ration and
setting of co
ntrol
para
m
eters [10]. The ope
rations of ge
n
e
tic algo
rithm
are as follo
ws:
(1) Selection
Selection op
eration com
b
ines
elite sel
e
ction
a
nd roulette wh
eel
sele
ction. At first, it
dire
ctly copi
e
s
several
elite individual
s
to t
he popul
a
t
ion in the ne
xt generatio
n
and
sele
ct the
rest in
dividua
ls with
roulett
e
whe
e
l met
hod. In
this
way, it can n
o
t only pre
s
e
r
ve the excell
ent
individual
s in the popul
atio
n, but also protect the
dive
rsity of the individual
s in the popul
ation.
(2)
Cro
s
sove
r an
d Mutation
In the geneti
c
ope
ratio
n
s,
perform cro
s
sover
and m
u
tation ope
ra
tions at the
crossove
r
probability
c
P
and mutation
prob
ability
m
P
. After c
r
oss
o
v
e
r and mutation operations
,
c
o
nduc
t
validation te
st on the
ne
wly-gene
rate
d
individual
s to
ch
eck
wh
ether t
he
soluti
ons of the
n
e
w
individual
s m
eet the seq
u
ence co
nstra
i
nts. If so
, it proves that
these ne
w
individual
s are
effective; if not, they are i
n
valid and
adj
ustment
s n
e
e
d
s to
be m
a
d
e
on th
em. Redistri
bute
so
me
operation
s
to make the
m
effective gene
s [11, 12].
The ba
sic flo
w
chart of gen
etic algo
rithm
is indicated a
s
Figu
re 2.
Figure 2. Basic flowcha
r
t of genetic al
gorithm
4. De
sign
of Mul
t
i-obj
ectiv
e Gene
tic Alg
o
rith
m Bas
e
d o
n
Self-adap
t
ion an
d
Dual
Population S
t
rateg
y
In the ge
neti
c
evolutio
n, the differen
c
e
s
am
ong
the
fitness of th
e individu
als in the
popul
ation vary from the differences
of the evolution. At the early ev
olution, the differen
c
e is
big,
but it becom
es sm
all in the late evolu
t
ion. In
order to guarante
e
that the individual
s can
be
sele
cted in e
a
rly evolution
to preserve
the di
versity
of the individual
s in the popul
ation a
nd
highlight the
excelle
nt indi
viduals in th
e late
evoluti
on to improve t
he co
mpet
itiveness of the
individual
s, this pa
per h
a
s
com
e
up with a Mult
i-ob
jective Du
al Population G
enetic Algo
rit
h
m
(MO
D
PGA).
The ste
p
s of
MODPGA a
r
e as follo
ws [
13, 14]:
(1) When
the
variabl
e p
a
rt
of the
indivi
dual
(1
)
(
1
,
...,
)
l
in
j
ll
Xj
n
in the
p
opulatio
n I
(1
,
.
.
.
,
)
ui
u
Pi
n
mutates, ran
domly sel
e
ct
individual
s to
gene
rate th
e
variable
pa
rt of the mutati
on
vector fro
m
the variable p
o
pulation
(the size is
u
n
).
(2)
Whe
n
th
e v
a
riable
p
a
rt
(1
)
l
in
j
l
X
of the individual
(1
)
(
1
,
...,
)
l
in
j
ll
Xj
n
in the
popul
ation
(1
,
.
.
.
,
)
ui
u
Pi
n
mutates, ra
n
domly sel
e
ct
individual
s to
gene
rate the
variable p
a
rt
of
i
u
X
u
P
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93-6
930
TELKOM
NIKA
Vol. 14, No. 1, March 2
016 : 238 – 2
4
4
242
the mutatio
n
vector from
th
e vari
able
po
pulation
II
l
P
(t
h
e
si
ze
i
s
l
n
)
without b
e
ing
li
mited to th
e
popul
ation I
ui
P
.
(3) In o
r
de
r t
o
protect th
e
evolution
stru
cture of
po
pulation
I, MODPGA
upd
ates th
e
popul
ation I
ui
P
in a dynamic mann
er so as to r
eali
z
e the dyna
mic upd
ate of the entire
popul
ation I.
(4) Pre
s
e
r
ve t
he no
n-domin
ant elite in
dividual
s with
the
level of
and
by
usin
g externa
l
archival
stra
tegy.
Figure 3. Flowchart of MO
DPGA
The p
r
o
c
e
ss
for MO
DPGA
to con
d
u
c
t
mutation an
d
crossove
r o
peratio
ns t
o
gene
rate
new exp
e
rim
ental individu
als in
clud
es t
w
o sta
g
e
s
.
Stage I: Ran
domly
sele
ct t
h
ree
different
indivi
du
als from the
existi
ng o
b
je
ctive i
ndividual
in
u
n
different
variabl
e p
opul
ations
to pe
rf
orm va
ria
b
le
mutation
and
cro
s
sove
r o
p
e
ration
s of
popul
ation I and gen
erate t
he variabl
e p
a
rt of the experime
n
tal indi
viduals
(1
)
(
1
,
...,
)
l
in
j
ll
Uj
n
.
Stage II: As for the vari
able part
(1
)
l
in
j
l
X
of all the indi
viduals in
p
opulatio
n
ui
P
,
rand
omly sel
e
ct individu
al
s to pe
rform
muta
tion an
d crossove
r
operat
ion
s
from the enti
r
e
popul
ation
(the si
ze i
s
u
n
) and g
ene
rate
the variabl
e
part of the
experim
ental
individual
(1
)
l
in
j
U
. The flowcha
r
t of MODPG
A
is indicate
d
as Figu
re 3.
5. Performan
ce Tes
t
and
Analy
s
is of the Algorith
m
in This Pa
per
Based
on t
he the theo
retical
re
sea
r
ch
and exp
e
rime
ntal verification, thi
s
pap
e
r
prop
oses
a
new im
proved Multi-o
b
jective Dual
Po
pulation
Gen
e
tic Algo
rith
m (MO
D
PGA
)
. In
1
u
ND
1
l
ND
u
P
P
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TELKOM
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ISSN:
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930
Applicatio
n of A Self-adaption Du
al Popu
lation Gen
e
tic Algorithm
in… (Ch
eng Z
h
ang)
243
orde
r to test the perfo
rma
n
c
e of this alg
o
rithm, this p
aper int
r
odu
ces two g
r
ou
p
s
of rand
om test
function
s to
see
k
the ma
ximum and
minimum. Th
ese fun
c
tion
s sel
e
ct two
objective
s, two
deci
s
io
n vari
a
b
les and
on
e
con
s
trai
nt an
d when th
ese
two
obje
c
tives ta
ke th
eir
o
w
n
extremum
s,
the po
sition
s
of two p
a
rticl
e
s
cont
radi
ct
with ea
ch
oth
e
r
so a
s
to
b
e
tter ob
se
rve
the ap
proxim
ate
Pareto
optim
al solution.
The follo
win
g
is
the co
mpari
s
o
n
of
MO
DPGA and ba
sic
g
enetic
algorith
m
.
(1) F
u
n
c
tion 1: Seek the
maximums of
two obje
c
tive function
s.
2
11
2
2
1
2
(2
;
(
/
3
1
)
M
A
Xf
x
x
M
A
Xf
x
x
;
Con
s
trai
nts:
12
25
,
2
5
xx
.
Figure 4 is th
e com
p
a
r
iso
n
of Pareto fro
n
t-end
s of F
u
nction
1 by b
a
si
c ge
netic a
l
gorithm
and MO
DPG
A
resp
ectivel
y
.
(a) Ge
netic a
l
gorithm
(b) M
O
DPGA
Figure 4. Co
mpari
s
o
n
of pareto fro
n
t-en
ds by two alg
o
rithm
s
(fun
ct
ion 1)
(2) F
u
n
c
tion 2: Seek the
minimum
s
of two obje
c
tive function
s.
2
11
2
2
1
2
1
((
3
)
2
)
;
(
3
)
2
MIN
f
x
x
MIN
f
x
x
;
Con
s
trai
nts:
12
0,
2
0
xx
.
In orde
r to prese
r
ve the un
iversality, this paper
al
so
choo
se
s the fitness fun
c
tion
to seek
the minimum
function a
nd the Pareto fro
n
t-end
s of
two algorith
m
s
are indi
cate
d in Figure 5.
(a) G
eneti
c
al
gorithm
(b) M
O
DPGA
Figure 5. Co
mpari
s
o
n
of pareto fro
n
t-en
ds by two alg
o
rithm
s
(fun
ct
ion 2)
It can be see
n
clea
rly from
Figure 4 an
d
Figure 5 that
after combini
ng self-ada
ption and
dual p
opul
ation st
rategy,
MODPGA
ca
n qui
ckly p
u
s
h the
pop
ul
ation to conv
erge
to the
real
Pareto fro
n
t-end an
d to be unifo
rmly distrib
u
ted
along Pa
reto
front-e
nd a
nd it can
h
a
ve
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 14, No. 1, March 2
016 : 238 – 2
4
4
244
approximate Pareto soluti
on
with bette
r
diversity.
In one
word, the optimizatio
n perfo
rma
n
ce of
MODPGA h
a
s
bee
n gre
a
tly improved compa
r
ed
with
the traditiona
l algorithm
s.
6. Conclusio
n
This
pap
er e
x
plore
s
the
i
m
provem
ent
strat
egy
of m
u
lti-obje
c
tive
geneti
c
al
gori
t
hm and
desi
g
n
s
a h
i
ghly-effective
adaptive d
ual pop
ul
atio
n gen
etic al
gorithm
whi
c
h ca
n qui
ckl
y
conve
r
ge
to t
he
real
Pareto fro
n
t-en
d
shape.
The
M
O
DPGA
met
hod
propo
se
d in
this pa
pe
r h
a
s
lowe
r t
i
me
c
o
mplex
i
t
y
,
make
s t
h
e
sel
e
ct
ion
st
rate
gy have
hig
her se
l
e
ctio
n
pressu
re
a
nd
guarantee th
at the populat
ion ca
n co
nverge to the re
al Pareto fron
t-end pa
rt.
Ackn
o
w
l
e
dg
ements
This work
wa
s su
ppo
rted by High
er Sc
ie
ntific Research
Project Of
Huna
n
Province(1
5C0779,1
4
B104
), Plann
ed S
c
ien
c
e
an
d
T
e
ch
nolo
g
y Project
of Hun
an Province
of
Chin
a(No.20
14FJ304
0).
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