TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 11, Novembe
r
2014, pp. 79
7
0
~ 797
8
DOI: 10.115
9
1
/telkomni
ka.
v
12i11.61
18
7970
Re
cei
v
ed Ap
ril 20, 2014; Revi
sed
Jul
y
1
5
, 2014; Acce
pted Augu
st 10, 2014
An Improved Constrained Engineering Optimization
Design Algorithm
Yuxin Sun*
1
，
Qinghua Wu
2
，
Xuesong Yan
3
1,2
Hubei Provi
n
cial Ke
y L
a
b
o
rator
y
of Intellig
ent Ro
b
e
rt, W
uhan Institute of
T
e
chnol
og
y, W
uhan, Ch
ina
1,2
School of Co
mputer Scie
nc
e and En
gi
neer
ing,W
u
h
an Inst
itute of T
e
chnolog
y, W
uha
n, Chin
a
3
School of Co
mputer Scie
nc
e, Chin
a Univ
e
r
sit
y
of Geosci
ences, W
uha
n,
Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
y
u
xi
n_s
un@
263.n
e
t
1
, 1622
375
24
9@q
q
.co
m
2
A
b
st
r
a
ct
Many e
ngi
ne
er
ing
opti
m
i
z
at
io
n pro
b
l
e
ms
ca
n be
st
ate as f
unctio
n
o
p
ti
mi
zation w
i
th c
ons
traine
d,
intell
ig
ence
op
timi
z
a
t
i
o
n
al
go
rithm c
an s
o
lv
e these
p
r
ob
lem
s
we
l
l
.
Pa
rticl
e
Swa
r
m
Opti
m
i
z
a
ti
on
(PSO)
alg
o
rith
m w
a
s deve
l
op
ed u
n
d
e
r the insp
irati
on of beh
av
i
o
r law
s
of bird flocks, fish schools a
nd hu
ma
n
communiti
es. In this pa
per, a
i
m at the
disa
dvanta
ges
of s
t
andar
d Particl
e
Sw
arm Opti
mi
z
a
t
i
o
n
alg
o
ri
th
m
like
bei
ng tra
p
ped
eas
ily i
n
to a l
o
cal
opti
m
u
m
, w
e
i
m
pr
oves the
stan
dard PSO
and
prop
oses
a n
e
w
alg
o
rith
m to s
o
lve the
overc
o
mes of the s
t
andar
d
PSO. T
he new
alg
o
rith
m kee
p
s
not only th
e fast
conver
genc
e spee
d charact
e
r
i
stic of PSO, b
u
t effectivel
y i
m
pr
oves the c
apa
bil
i
ty of glo
bal se
archi
ng
as
w
e
ll. Experi
m
e
n
t results rev
e
al that
the
pro
pose
d
al
gorit
h
m
ca
n fin
d
bet
ter soluti
ons w
hen c
o
mpar
ed
to
other he
uristic
meth
ods a
nd is
a pow
erful opti
m
i
z
at
io
n alg
o
rit
h
m for e
ngi
ne
e
r
ing o
p
ti
mi
z
a
ti
o
n
prob
le
ms.
Ke
y
w
ords
:
eng
ine
e
ri
ng o
p
timi
z
a
ti
on pr
obl
e
m
s, particl
e sw
ar
m opti
m
i
z
at
io
n, cons
traine
d opti
m
i
z
a
t
i
o
n
,
evol
ution
a
ry co
mp
utatio
n
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Can
d
idate
so
lutions to
so
me p
r
obl
em
s are
n
o
t simp
ly
deem
ed correct or
in
co
rre
ct
b
u
t
are i
n
ste
ad
rated in te
rm
s
of quality an
d
finding th
e
candid
a
te solu
tion with th
e
highe
st qu
ality is
kno
w
n a
s
op
timization. O
p
timization p
r
oble
m
s a
r
is
e in many real-wo
r
ld sce
nario
s. Ta
ke
for
example
the spreadi
ng of manu
re on a cornfield,
wh
ere
de
pendi
n
g
on
the
spe
c
ie
s of
grain,
the
soil
quality,
expecte
d a
m
ount of
rai
n
,
sun
s
hi
ne
and
so
on,
we
wish to fin
d
t
he a
m
ount
a
n
d
composition of
fertilizer
that maximizes the crop,
whi
l
e st
ill bei
ng
within the
bounds impo
sed by
environ
menta
l
law.
Several chall
enge
s ari
s
e
in optimizatio
n.
First is th
e nature of
the probl
em
to b
e
optimize
d
whi
c
h may have
several local optima t
he op
timizer
can g
e
t stuck in, the probl
em m
a
y
be discontin
uou
s, can
d
id
ate solution
s may yield
different fitness value
s
wh
en evaluated
at
different time
s, and the
r
e
may be con
s
traints a
s
to
what
candi
da
te solution
s
are fea
s
ibl
e
as
actual
solutio
n
s to
the
real
-wo
r
ld
proble
m
. Furt
h
e
rm
o
r
e, the
large
numbe
r
of ca
ndidate
soluti
ons
to an
optimization p
r
obl
em
ma
ke
s it intractabl
e to
co
nsid
er all can
d
idate sol
u
tio
n
s
i
n
tu
rn, wh
ich
is the only way to be com
p
letely su
re that the
glob
a
l
optimum ha
s bee
n foun
d
.
This difficult
y
gro
w
s mu
ch
worse with
increa
sing dimen
s
ion
a
lit
y, which i
s
f
r
equ
ently cal
l
ed the
curse of
dimen
s
ion
a
lity, a name th
at is attrib
ute
d
to Bellman,
see fo
r exa
m
ple [1]. This phen
omen
o
n
ca
n
be und
erstoo
d by first co
n
s
ide
r
ing a
n
n-dimen
s
ion
a
l binary
sea
r
ch
-sp
a
ce. He
re,
adding a
noth
e
r
dimen
s
ion to
the proble
m
means a d
oublin
g of the numbe
r of
candi
date solution
s. So the
numbe
r of candid
a
te sol
u
tions g
r
o
w
s
expone
ntially
with increa
si
ng dimen
s
io
n
a
lity. The sa
me
prin
ciple
hold
s
for
continu
ous o
r
real-v
alued
sea
r
ch
-sp
a
ces, o
n
ly it is now th
e volume of
the
sea
r
ch-sp
a
ce
that gro
w
s expone
ntiall
y with incre
a
sin
g
dime
n
s
ion
a
lity. In either
ca
se i
t
is
therefo
r
e of
great inte
re
st
to find optimizat
ion
met
hod
s which
not only pe
rform
well in f
e
w
dimen
s
ion
s
,
but do
not
req
u
ire a
n
expon
entia
l
numb
e
r of
fitness eva
l
uation
s
a
s
the
dimen
s
ion
a
lity gro
w
s.
Pref
erably
su
ch
o
p
timization
m
e
thod
s h
a
ve
a line
a
r
rel
a
tionship
betwe
en
the dime
nsio
nality of the p
r
oble
m
an
d the nu
mbe
r
of
can
d
idate
so
lutions th
ey must eval
uat
e in
orde
r to a
c
hi
eve sati
sfact
o
ry re
sult
s, that is
, optimi
z
ation
metho
d
s
sho
u
ld id
eally have lin
ea
r
time-complex
ity O(n) in the
dimensi
onalit
y n of the problem to be op
timized.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Im
proved
Con
s
trai
ned
Enginee
ring
Optim
i
zation De
sign Algo
ri
thm
(Yuxin Sun)
7971
Another chall
enge in optim
ization a
r
ises
from
how mu
ch or ho
w little is kno
w
n a
bout the
probl
em at h
and. For exa
m
ple, if
the optimization p
r
oblem is
give
n by a simpl
e
formula the
n
it
may be possi
ble to derive the inverse of that form
ula and thus find its optimum. Other families
of
probl
em
s ha
ve had
sp
ecialize
d
meth
ods
develo
p
ed to o
p
timize th
em effi
ciently. But
whe
n
nothing
is kn
own
ab
out th
e optimi
z
atio
n p
r
oble
m
at
hand, th
en th
e No F
r
ee
Lu
nch
(NFL
)
se
t of
theore
m
s by
Wolp
ert a
nd
Macrea
dy
states th
at
any
one
optimization meth
od
will be
as likely
a
s
any othe
r to
find a
satisfactory
soluti
on [2]. Thi
s
is e
s
p
e
ci
ally impo
rtant in
de
cidin
g
wh
at
perfo
rman
ce
goal
s one
sh
ould h
a
ve when d
e
sig
n
in
g ne
w optimi
z
ation m
e
tho
d
s, an
d whet
her
one shoul
d a
ttempt to devise the ultima
te optimizatio
n method
whi
c
h will a
dapt
to all probl
e
m
s
and
perf
o
rm
well. A
c
cordi
ng to
the
NF
L theo
rem
s
such
an
optimi
z
ation
meth
o
d
do
es not
e
x
ist
and the
focus of this th
esi
s
will the
r
efo
r
e
be o
n
the
op
posite: Simpl
e
optimi
z
atio
n metho
d
s th
at
perfo
rm well f
o
r a ra
nge of
probl
em
s of intere
st.
Many engi
ne
ering
optimi
z
ation de
sig
n
probl
em
s can be fo
rmu
l
ated a
s
con
s
train
ed
optimizatio
n probl
em
s. The pre
s
en
ce
of con
s
trai
nt
s may signifi
cantly affect the optimizati
o
n
perfo
rman
ce
s of any optimization al
gorit
hms for
un
co
nstrai
ned p
r
o
b
lems.
With the increa
se
of
the re
sea
r
ch
and appli
c
at
ions b
a
sed o
n
evoluti
ona
ry computatio
n techni
que
s [3], constrai
nt
handli
ng u
s
e
d
in evolutio
n
a
ry computati
on tec
hniq
u
e
s
ha
s b
een
a
hot topic in
both a
c
ad
emi
c
and en
gine
ering fields [4,
5]. A general con
s
traine
d optimizatio
n pro
b
lem m
a
y be written
as
follows
:
ma
x
(
)
f
x
(1)
Subject to:
()
,
1
,
2
,
.
.
.
,
,
()
,
1
,
2
,
.
.
.
,
.
ii
jj
g
xc
i
n
hx
d
j
m
(2)
Whe
r
e
x
is a v
e
ctor
re
sidi
ng
in a n-dimen
s
ion
a
l space,
()
f
x
is a
scalar v
a
lued
obje
c
tive
function,
()
,
1
,
2
,
.
.
.
,
ii
g
xc
i
n
and
()
,
1
,
2
,
.
.
.
,
jj
hx
d
j
m
are con
s
tra
i
nt function
s that need to be
sat
i
sf
ie
d.
Evolutionary comp
utation has
fo
und
a wide ran
ge o
f
applications in variou
s fi
elds
of
sci
en
ce an
d engin
eeri
ng. Among othe
rs, evoluti
ona
ry algorithm
s
(EA) have b
e
en proved to be
powerful glo
bal optimize
r
s. Ge
nerall
y
, evol
utionary algo
rith
ms outpe
rfo
r
m co
nventi
onal
optimizatio
n
algorith
m
s fo
r problem
s
whi
c
h a
r
e
di
scontinu
o
u
s
, non-differe
n
t
ial, multi-mo
dal,
noisy
and
no
t well
-define
d
problem
s,
such
a
s
a
r
t d
e
s
ign,
mu
sic compo
s
ition
a
nd exp
e
rim
e
ntal
desi
g
n
s
. Besi
des, evolutio
nary algo
rith
ms are al
so well suitable f
o
r multi-crite
r
i
a
probl
em
s.
Particle
Swa
r
m O
p
timizat
i
on (PSO
) a
l
gorithm
wa
s an intelli
ge
nt tech
nolog
y first
pre
s
ente
d
in
1995
by Ebe
r
ha
rt and Ke
nnedy, an
d
it wa
s develo
ped u
nde
r th
e inspiratio
n
of
behavio
ur la
ws
of bird flo
c
ks, fish
scho
ols a
nd hu
ma
n com
m
unitie
s
[6]. If we co
mpare PSO
with
Geneti
c
Algo
rithms
(GAs),
we m
a
y find t
hat they
a
r
e
all man
oeuvred o
n
the
ba
sis of po
pulat
ion
operated. Bu
t PSO doe
sn't rely
on
geneti
c
o
perat
ors li
ke
se
lection
op
era
t
ors,
crossov
e
r
operators
an
d mutatio
n
o
perato
r
s to
o
perate
i
ndivi
dual, it o
p
timize
s th
e p
opulatio
n throug
h
informatio
n e
x
chan
ge
amo
ng in
dividual
s. PSO achi
ev
es it
s o
p
timu
m solution
by sta
r
ting from
a
grou
p of ra
n
dom solution
and then
se
arching
re
p
e
a
tedly. Once PSO wa
s pre
s
ente
d
, it invited
wide
sp
rea
d
concern
s
am
o
ng schol
ars in the optim
ization fields a
nd sh
ortly afterwards it ha
d
become a
studying focus within only several ye
a
r
s.
A number
of scie
n
tific a
c
hievement
s h
a
d
emerged in t
hese fields [
7
-9]. PSO was p
r
oved to
be a so
rt o
f
high efficie
n
t optimizati
o
n
algorith
m
by nume
r
ou
s re
search
an
d experim
ents [10
]. PSO is a meta-he
u
ri
stic
as it make
s few
or n
o
a
s
sum
p
tions abo
ut
the problem
being
optim
ize
d
a
n
d
c
a
n
s
e
ar
ch
ve
r
y
la
r
g
e sp
ac
es
o
f
can
d
idate so
lutions. Ho
wever,
meta-h
euri
s
tics
such as PSO
d
o
not gu
ara
n
tee an
opti
m
al
solutio
n
is
ever foun
d. Mo
re spe
c
ifically
, PSO
does
not use the g
r
adie
n
t of the
probl
em b
e
i
ng
optimize
d
, wh
ich m
ean
s P
S
O doe
s
not
requi
re th
at
the optimi
z
ati
on p
r
obl
em b
e
differe
ntiabl
e a
s
is requi
re
d by
cla
s
sic
o
p
timization
method
s
su
ch a
s
g
r
a
d
ie
nt de
scent a
nd q
u
a
s
i-Ne
wton
method
s. PSO can the
r
ef
ore
also b
e
u
s
ed
on
optim
i
z
ation
proble
m
s that
ar
e partially irregul
a
r,
noisy,
cha
n
g
e
over time,
etc. Thi
s
p
a
p
e
r im
pr
ove
s
t
he di
sadva
n
tage
s of
stan
dard
PSO b
e
ing
easily tra
ppe
d into a lo
cal
optimum a
n
d
pro
p
o
s
ed
an imp
r
oved
PSO algorith
m
(IPSO)
wh
ich
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 79
70 – 797
8
7972
prove
s
to be
more simply
co
ndu
cted a
nd with
mo
re
efficient
glo
bal
sea
r
ching
ca
pability, then
use the n
e
w
algorith
m
for engin
eeri
ng o
p
timization p
r
oblem
s.
2. Particle Sw
arm Optimi
z
a
tion Algori
t
hm
A basic vari
a
n
t of the PSO algorith
m
works
by having a popul
ation (called a
swarm) of
can
d
idate
sol
u
tions (calle
d
pa
rticle
s).
T
hese p
a
rticl
e
s a
r
e
moved
around
in
th
e sea
r
ch
-spa
ce
according to
a few simpl
e
formulae. T
he movem
e
n
t
s of the particle
s
are g
u
i
ded by their
own
best
kn
own
positio
n in
th
e sea
r
ch-spa
ce
as
well
a
s
the
e
n
tire
swarm'
s be
st
kn
own
po
sition.
Whe
n
improved po
sition
s are bei
ng di
scovered th
e
s
e will then co
me to guide t
he moveme
nts of
the swarm. T
he pro
c
e
s
s i
s
re
peated
a
nd by doing
so it is hop
e
d
, but not guarante
ed, tha
t
a
satisfactory solution will ev
entua
lly be di
scovered. Formally, let
:
n
f
RR
be
the cost fun
c
tion
whi
c
h mu
st be minimized. The fun
c
tion
take
s a ca
ndi
date sol
u
tion
as a
r
gum
ent in the form of a
vector of real
numbers an
d prod
uces a
real
numb
e
r as output which indi
cate
s the obje
c
tive
function valu
e of the given candid
a
te so
lution. The
gradient of f is not kno
w
n. T
he goal is to find
a solution
a
for whic
h
()
(
)
f
af
b
for all
b
in th
e
sea
r
ch-sp
a
ce, wh
ich
wo
uld m
ean
a
is the
global minim
u
m. Maximization ca
n be
perfo
rmed by
con
s
ide
r
ing t
he functio
n
hf
in
stead.
PSO wa
s pre
s
ente
d
und
er the in
spi
r
atio
n of
bird
flock immig
r
ation
durin
g the
co
urse
of
finding fo
od
a
nd the
n
b
e
u
s
ed
in th
e o
p
timization
p
r
o
b
lems.
In PS
O, ea
ch
opti
m
ization
p
r
ob
lem
solutio
n
is ta
ken
as a
bird
in the se
arching
spa
c
e and it is called “particl
e”
. Eve
r
y p
a
r
t
ic
le
ha
s
a
fitness valu
e
whi
c
h is d
e
termined by targ
et functi
on
s a
nd it has al
so
a velocity wh
ich dete
r
min
e
s
its de
stinatio
n
and
di
stan
ce
. All parti
cle
s
sea
r
ch
in
the
solutio
n
spa
c
e for their be
st positio
ns an
d
the po
sition
s of the
be
st
particl
es in
the
swar
m. P
S
O is initially
a g
r
o
up
of random
pa
rticles
(ra
ndom
sol
u
tions), a
nd th
en the optim
um sol
u
tion
s are fou
nd by
repe
ated
sea
r
chi
ng. In every
iteration, a particle
will follow two best
s to r
enew itself: the best
posit
ion found for a parti
cle
calle
d p
b
e
s
t; the b
e
st
p
o
sition
foun
d
for th
e
wh
ole
swarm
called
gbe
st.
All parti
cle
s
will
determi
ne fol
l
owin
g step
s throug
h the
best expe
ri
ences of indi
viduals the
m
selve
s
and t
heir
comp
anio
n
s.
For pa
rticle id
, its velocity and its po
sition
rene
wal form
ula are a
s
foll
ows:
'
12
()(
)
()(
)
id
id
id
b
i
d
gdb
i
d
V
V
ran
d
P
X
ran
d
P
X
(3)
''
id
id
id
XX
V
(4)
In here:
is called in
ertia
weig
ht, it is a
prop
ortio
n
factor that i
s
con
c
erned
with
forme
r
veloc
i
ty,
01
,
1
and
2
are
con
s
tants and
a
r
e called a
c
cele
rating fa
ctors, no
rma
lly
1
2
2
;
()
rand
are rand
o
m
numbe
rs,
id
represents the positio
n of particle
id
;
id
V
rep
r
e
s
ent
s th
e velocity of
particl
e
id
;
idb
P
,
g
db
P
rep
r
esent
sep
a
rately the be
st positio
n pa
rti
c
le
id
has fou
nd an
d the positio
n
of the best particle
s
in the
whole
swarm
.
In formula
(3
), the first pa
rt
represents t
he
fo
rmer vel
o
city of the particle, it enables the
particl
e to p
o
s
sess exp
a
n
d
ing ten
den
cy in the sea
r
chin
g spa
c
e
and thu
s
m
a
ke
s
the alg
o
ri
thm
be mo
re
cap
able i
n
glo
bal
se
archin
g; the
se
con
d
p
a
rt is
called
cognition
pa
rt, it rep
r
e
s
e
n
ts the
pro
c
e
s
s
of ab
sorbing
indivi
dual experi
e
n
c
e kn
owl
edg
e
on
the
pa
rt
of
the p
a
rticl
e
; the thi
r
d
p
a
rt
is called
so
ci
al part, it rep
r
ese
n
ts the p
r
oce
s
s of
lea
r
ning fro
m
the
experie
nces
of other p
a
rti
c
le
s
on the pa
rt of certai
n pa
rticle, and it also
sho
w
s th
e in
formation
sha
r
ing a
nd soci
al coo
p
e
r
atio
n
among p
a
rti
c
l
e
s.
The flow of PSO can bri
e
fly descri
be a
s
follo
win
g
: First, to initialize a grou
p of particl
es,
e.g. to give
randomly
ea
ch pa
rticle
a
n
initial po
sition
i
X
and an initial velocity
i
V
, an
d then
to
cal
c
ulate
its f
i
tness valu
e f
.
In every ite
r
ation,
evalu
a
ted a
pa
rticl
e
'
s
fitne
s
s valu
e by a
nalyzi
n
g
the velocity a
nd po
sition
s
of ren
e
wed p
a
rticle
s
i
n
formula (3)
and
(4).
Wh
en a
parti
cle find
s a
better p
o
sitio
n
than p
r
evio
usly, it will m
a
rk this
co
ordinate into ve
ctor P1, th
e
vector
differe
nce
betwe
en P1 and the present positio
n of the particl
e will ran
d
o
m
ly be adde
d to next vel
o
city
vector, so that the followi
ng rene
wed particl
es will search
around
th
is poi
nt, it's also called in
formula
(3
)
cognition
co
m
pone
nt. The
weig
ht differe
nce
of the
prese
n
t po
sitio
n
of the
pa
rticle
swarm
an
d th
e be
st p
o
sitio
n
of the
swa
r
m
g
db
P
will
also be added to veloci
ty vector f
o
r adjusting
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Im
proved
Con
s
trai
ned
Enginee
ring
Optim
i
zation De
sign Algo
ri
thm
(Yuxin Sun)
7973
the next population veloci
ty. This is also called in formul
a (3)
social comp
on
ent. These t
w
o
adjustments will
enable
particles to search
around two best
s
.
The m
o
st
ob
vious
advant
age
of PSO i
s
that
th
e co
nverge
nce speed
of
the
swarm
is
very high, schol
ars like
Clerc
[11] h
a
s p
r
esented
proof on it
s co
nverg
e
n
c
e. He
re a f
a
tal
w
e
ak
ne
ss
may r
e
su
lt fr
o
m
th
is
c
h
ar
ac
te
r
i
s
t
ic
.
W
i
th
con
s
tant in
crea
se of iter
atio
ns, the velocity
of
particl
es
will grad
ually dim
i
nish a
nd re
a
c
h zero
in the
end. At this time, the whol
e swarm
will be
conve
r
ge
d at one point in the solution
space,
if gbest particle
s
hav
en't found gb
est, the whol
e
swarm
will b
e
trap
ped i
n
to a lo
cal
opt
imum; and
the capa
city
of swarm
ju
mp out of
a
local
optimum is ra
ther we
ak.
3. Impro
v
ed
PSO Algorithm
In the
stan
d
a
rd
PSO al
g
o
rithm, the
converg
e
n
c
e
spe
ed
of pa
rticles is fa
st, but th
e
adju
s
tments
of cognitio
n
compon
ent an
d so
cial
com
pone
nt make
particl
es
sea
r
ch
aro
und
g
db
P
and
idb
P
. Accord
ing to vel
o
cit
y
and
po
sition re
ne
wal fo
rmula, on
ce
the b
e
st i
ndiv
i
dual in
the
swarm
is tra
pped
into
a l
o
cal
optimu
m
, the info
rmat
ion
sha
r
ing
mech
ani
sm i
n
PSO
will at
tract
other
parti
cle
s
to a
pproa
ch
this lo
cal
opti
m
um g
r
ad
uall
y
, and in the
end the
whol
e swa
r
m
will
be
conve
r
ge
d at this positio
n. But acco
rdin
g to ve
locity and po
sition
rene
wal form
u
l
a (3), on
ce t
he
whol
e
swarm
is tra
ppe
d int
o
a l
o
cal opti
m
um, its
co
g
n
ition
comp
o
nent a
nd
so
ci
al compo
nent
will
become
zero in the end; still, because
01
and
with th
e
numbe
r
of ite
r
ation i
n
cre
a
se, the
velocity of pa
rticle
s
will be
come
zero
in
the end,
th
us
the wh
ole
swarm i
s
h
a
rd
to
jump o
u
t of t
h
e
local o
p
timu
m and h
a
s n
o
way to achie
v
e the global
optimum. Here a fatal we
a
k
ne
ss may re
sult
from thi
s
characteri
stic.
With
constant increase
of iterations
, t
he velocity of parti
cles
will
grad
ually di
m
i
nish
an
d
rea
c
h
ze
ro
in
the
end.
At this time, the
wh
ol
e swa
r
m
will
be
conve
r
g
e
d
at
one poi
nt in the sol
u
tion space, if gbest
particl
es h
a
ven't found g
b
e
st, the wh
ol
e swarm
will
be
trappe
d into
a local optim
um; and the
cap
a
city of swarm jump
o
u
t of a local
optimum i
s
rather
wea
k
. In o
r
d
e
r to g
e
t through thi
s
di
sadvantag
e, in this p
ape
r
we p
r
e
s
ent
s
a ne
w alg
o
rit
h
m
based on PS
O.
3.1. Information Sharing Mecha
n
ism
In order to
a
v
oid bei
ng t
r
appe
d into
a
local
optim
u
m
, the n
e
w a
l
gorithm
ad
o
p
ts a
n
e
w
informatio
n sharin
g me
ch
anism.
We al
l kno
w
that
when a pa
rticl
e
is searchi
n
g in the sol
u
tion
spa
c
e, it doe
sn't kn
ow the
exact positio
n of t
he optimum solutio
n
. But we can not only record
the best p
o
si
tions an i
ndiv
i
dual pa
rticle
and t
he
whol
e swarm
hav
e experi
e
n
c
e
d
, we can al
so
record the
worst
po
sition
s an in
dividual
parti
cle a
nd
the wh
ole
swarm h
a
ve ex
perie
nced, th
us
we m
a
y ma
ke individu
al
particl
es mo
ve in the
direction
of ev
ading
the
worst
po
sition
s a
n
individual pa
rticle and the
whole flock have exper
i
enced, this
will su
rely e
n
larg
e the gl
obal
sea
r
ching
sp
ace
of pa
rticl
e
s
and
ena
bl
e them to
av
oid bei
ng t
r
a
pped
into a
l
o
cal
optimum
too
early, in the
same time, it will improve the possib
ility of finding gbest in the searching space. In
the new
strat
egy, the particle velo
city and
po
sition re
newal formul
a are a
s
follo
ws:
'
12
()(
)
(
)
(
)
i
d
id
id
id
w
i
d
g
dw
V
V
ra
nd
X
P
ra
nd
X
P
(5)
''
id
id
id
XX
V
(
6
)
In here:
id
w
P
,
g
dw
P
re
pre
s
ent
the
worst
po
sitio
n
particl
e
id
ha
s found an
d
the worst
positio
ns of the wh
ole swa
r
m ha
s found.
3.2. Elite Selection Str
a
te
g
y
In stand
ard P
S
O algo
rithm
,
the next flying direct
io
n o
f
each p
a
rti
c
l
e
is ne
arly de
finite, it
can
fly to th
e be
st in
dividual
and th
e
be
st individ
uals for th
e
whol
e
swa
r
m
.
From t
he a
bove
con
c
lu
sio
n
we may easily to kno
w
it will be the dang
er for bein
g
trappe
d into a local o
p
timum
.
In
orde
r to de
crease the po
ssibility of bei
ng trap
ped i
n
to the local optimum, the
improved P
S
O
introdu
ce
s eli
t
e sele
ction st
rategy. Traditi
onal gen
et
ic
algorith
m
is u
s
ually comple
te the sele
ction
operation
ba
sed
on
the i
ndividual'
s
fit
ness val
ue,
i
n
the
me
cha
n
ism
of elite
sel
e
ctio
n, the
popul
ation of
the front g
eneration mi
xed with
the
new p
opul
a
t
ion whi
c
h g
enerate thro
ugh
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 79
70 – 797
8
7974
geneti
c
op
era
t
ions, in the
mixed pop
ula
t
ion sel
e
ct
th
e optimum i
n
dividual
s accordin
g to a
ce
rtain
probability. The specifi
c
pr
ocedure is as follows:
Step 1: Using
cro
s
sover an
d mutation op
eration
s
for p
opulatio
n P1 whi
c
h si
ze is
N then
gene
rating th
e next genera
t
ion of sub-po
pulation
s
P2;
Step 2: The
current po
pula
t
ion P1 and t
he nex
t ge
ne
ration of
sub
-
popul
ations P
2
mixed
together fo
rm
a tempora
r
y popul
ation;
Step 3: Te
mp
ora
r
y po
pulati
on a
c
cording
to fit
ness val
ues
in de
sce
nding
order,
to retai
n
the best N in
dividual
s to form ne
w pop
ul
ations P1.
The cha
r
a
c
te
ristic
of this
strategy i
s
m
a
inly
in
th
e
fo
llo
w
i
ng
as
pe
c
t
s
.
F
i
r
s
t is r
o
b
u
s
t
,
becau
se of u
s
ing thi
s
sele
ction
strate
gy, even
wh
en
the gen
etic o
peratio
ns to
prod
uce mo
re
inferio
r
indivi
dual
s, as th
e
results of th
e
majority
of in
dividual resi
d
ues
of the o
r
i
g
inal p
opulati
o
n
,
doe
s not ca
u
s
e lo
wer the
fitness valu
e of the individual. The seco
nd is in
geneti
c
diversity
maintainin
g, the op
eratio
n
of large
pop
u
l
ations,
you
can better
mai
n
tain the g
e
n
e
tic diversity of
the po
pulatio
n evolutio
n
pro
c
e
ss.
Thi
r
d is in
th
e
sortin
g m
e
th
od, it is go
o
d
to ove
r
co
me
prop
ortio
nal t
o
ad
apt to th
e calculation
of scale.
Thi
s
pro
c
e
s
s of
this
strategy i
n
imp
r
ove P
S
O
like this: To set particle nu
mber in the swarm as
m, father po
pulati
on and son p
opulatio
n add
up
to 2m. To sel
e
ct ra
ndo
mly q pairs from
m; as to e
a
ch
individual p
a
r
ticle i, if the f
i
tness value
of
i
is smalle
r tha
n
its o
ppo
ne
nts, we
will
win o
u
t and
then a
dd o
n
e
to its ma
rk,
and finally
se
lect
those
pa
rticle
s
whi
c
h
have
the m
a
ximu
m ma
rk valu
e into th
e n
e
x
t gene
ration
. The
experi
m
ent
result shows that this strategy greatly reduce
s the possi
bility of being
trapped into a local
optimum whe
n
solving
cert
ain functio
n
s.
4. Cons
train
e
d Engineeri
ng Optimiza
tion Problems
In this
section, we will
carry
out num
eri
c
al simul
a
tion
b
a
sed on some
we
ll-kn
own
con
s
trai
ned engin
eeri
ng optimizatio
n desi
gn
p
r
obl
em
s to inve
stigate the pe
rforma
nces
o
f
th
e
prop
osed IP
SO. The
sel
e
cted
proble
m
s h
a
ve
b
e
en well
studi
ed befo
r
e
as ben
chm
a
rks by
variou
s app
roache
s, whi
c
h is useful to sho
w
the
validity and effectivene
ss of the prop
ose
d
algorith
m
. For each testing
proble
m
, the param
eters
of the IPSO a
r
e set a
s
follo
ws: the nu
mb
er
of particle i
s
100, c1
=c2
=
2
.
0 and the nu
mber of iterati
on is 50
0.
4.1. Tension/
Compre
ssio
n
String Problem
This
problem
is d
e
scribe
d
by Arora [12],
Co
ello a
nd
Montes [13]
and Bel
egu
n
du [14]. It
con
s
i
s
ts of m
i
nimizin
g
the
weig
ht (
()
f
x
) of a
tensio
n/com
p
re
ssi
on
strin
g
su
bje
c
t to con
s
trai
nt
s
on she
a
r st
re
ss, surg
e fre
quen
cy and
minimum def
l
e
ction a
s
sh
own in Figu
re 1. The design
variable
s
are the
mea
n
coil
diamete
r
1
()
Dx
; the wi
re
diam
eter
2
()
dx
and
the
nu
mber of a
c
tive
coil
s
3
()
N
x
. The problem can be
stated as:
Minimize:
2
32
1
()
(
2
)
f
xx
x
x
(7)
Subject to:
3
23
1
4
1
2
21
2
2
34
2
21
1
1
1
3
2
23
12
4
()
1
0
,
717
85
4
1
()
1
0
,
12
566(
)
510
8
140.
45
()
1
0
,
()
1
0
.
1.
5
xx
gx
x
xx
x
gx
xx
x
x
x
gx
xx
xx
gx
(
8
)
This p
r
obl
e
m
has be
en
solved by Belegun
du u
s
ing ei
ght di
fferent math
ematical
optimizatio
n t
e
ch
niqu
es [1
4], Arora al
so solv
ed thi
s
problem
u
s
i
ng a
nu
meri
cal
optimi
z
ation
techni
que
cal
l
ed
con
s
trai
n
t
co
rre
ction
at co
nsta
nt
co
st [12], Ad
ditionally, Co
ello
solved
this
probl
em u
s
in
g GA-b
ased
method [15] a
nd a fea
s
ib
ilit
y-based tou
r
n
a
ment sele
ction sch
e
me [1
3],
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Im
proved
Con
s
trai
ned
Enginee
ring
Optim
i
zation De
sign Algo
ri
thm
(Yuxin Sun)
7975
He
solved thi
s
p
r
oble
m
u
s
ing co-evol
u
tionary p
a
rti
c
l
e
swarm opti
m
ization
met
hod [3]. In th
is
pape
r, the I
PSO is
run
50 time
s ind
epen
dently. T
able
1 p
r
e
s
ents the
be
st solution
of
this
probl
em obta
i
ned u
s
ing t
he IPSO algorithm an
d compa
r
e
s
the
IPSO result
s with soluti
ons
repo
rted
by other research
ers. It
is o
b
vious from the
Table 1 th
at the re
sult o
b
ta
ined u
s
ing IP
SO
algorith
m
is b
e
tter than tho
s
e re
po
rt
ed p
r
eviou
s
ly in the literature.
Figure 1
.
T
e
n
s
ion/compression Str
ing
Problem
T
able 1. Comparison of
th
e Best
Solu
ti
on for
T
ensio
n/compression String
Pro
b
lem
Desi
gn
vari
abl
es
IPS
O
Bele
gu
nd
u
(1
98
2)
Aro
r
a
(1
98
9)
Co
ello
(2
00
0)
Coello
(2
00
2)
H
e
(2
00
7)
1
()
x
d
0
.
05
1
1
54 0
.
05
00
00
0
.
05
33
96
0
.
05
14
80
0
.
05
19
89
0
.
05
17
28
2
()
x
D
0
.
34
98
71 0
.
31
59
00
0
.
39
91
80
0
.
35
16
61
0
.
36
39
65
0
.
35
76
44
3
()
x
N
12.
07
64
32
14.
25
00
00
9.1
8
5
4
0
0
1
1
.
6
3
2
2
0
1
10.
89
05
22
1
1
.
2
4
4
5
4
3
1
()
g
x
0.0
0
0
0
0
0
-0
.0
00
014
0.0
0
0
0
1
9
-0
.0
02
080
-0
.0
00
013
-0
.0
00
845
2
()
g
x
-
0
.
00
00
07
-0
.0
03
782
-0
.0
00
018
-0
.0
00
1
1
0
-0
.0
00
021
-1
.2
60
0e
-0
5
3
()
g
x
-
4
.
02
78
40
-3
.9
38
302
-4
.1
23
832
-4
.0
26
318
-4
.0
61
338
-4
.0
51
300
4
()
g
x
-
0
.
73
65
72
-0
.7
56
067
-0
.6
98
283
-4
.0
26
318
-0
.7
22
698
-0
.7
27
090
()
f
x
0
.
01
26
70
6 0
.
01
28
33
4
0
.
01
27
30
3
0
.
01
27
04
8
0
.
01
26
81
0
0
.
01
26
74
7
4.2. Pressur
e
Vessel Pro
b
lem
A cylindrical vessel is ca
p
ped at both e
nds
by hemi
s
pheri
c
al h
ead
s as sho
w
n i
n
Figure
2. The obj
ecti
ve is to mini
mize the total
co
st, includi
n
g
the co
st of
material, form
ing and
wel
d
i
n
g
.
There a
r
e fo
u
r
de
sig
n
va
ria
b
les:
s
T
(thickn
e
s
s of the
shel
l,
1
x
),
h
T
(thickne
ss of th
e h
ead
,
2
x
),
R
(inne
r ra
diu
s
,
3
x
) and
L
(lengt
h of cylindri
c
al se
ction of
the vessel, no
t includin
g
the head,
4
x
).
s
T
and
h
T
are integer m
u
ltiple
s of 0.062
5 inch, wit
c
h
a
r
e the availabl
e thickne
ss
o
f
rolled ste
e
l
plates, an
d
R
a
nd
L
are
contin
uou
s.
Usi
ng the sa
me notation g
i
ven by Coell
o
[
16], the problem can be
stated as foll
ows:
Minimize:
22
1
3
42
31
4
1
3
(
)
0.6224
1.7781
3.1661
19.
84
f
x
x
x
x
x
x
xx
xx
(9)
Subject to:
11
3
22
3
23
33
4
3
44
(
)
0.0193
0
,
(
)
0.00954
0
,
4
(
)
1
,
2
9
6,
0
0
0
0,
3
()
2
4
0
0
.
gx
x
x
gx
x
x
gx
x
x
x
gx
x
(10)
This p
r
obl
em
has b
een
sol
v
ed before by
Sandgren u
s
ing a bran
ch
and bo
und te
chni
que
[17], by Kannan and K
r
am
er u
s
ing
an a
ugmente
d
La
gran
gian M
u
ltiplier a
pproa
ch [18], by De
b
and G
ene
u
s
ing Ge
netic
Adaptive Sea
r
ch
[19], by
Coell
o
u
s
ing
GA-ba
s
e
d
co
-evolution
mo
del
[15] and a fe
asibility-b
a
se
d tourn
a
ment
sele
ction
sh
eme [13], an
d by He u
s
in
g co
-evolutio
nary
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 79
70 – 797
8
7976
particl
e
swa
r
m optimizatio
n metho
d
[3]. In this
p
ape
r, the IPSO is run
50 time
s indep
end
ently.
The
comp
ari
s
on
s of
re
su
lts are sho
w
n in Ta
ble
2. The resul
t
s obtain
ed
usin
g the IP
SO
algorith
m
, we
re better o
p
timized tha
n
a
n
y other ea
rli
e
r sol
u
tion
s reporte
d in the
literature.
Figure 2. Pre
s
sure Ve
ssel
Problem
T
able 2. Comparison of
th
e Best
So
lu
tion for
Pressure V
e
ssel
Problem
Desi
gn
vari
abl
es
IPS
O
S
a
nd
gr
en
(1
98
8)
K
a
nn
an
(1
99
4)
D
eb
(1
99
7)
Coello
(2
00
0)
Coello
(2
00
2)
He
(2
00
7)
1
()
s
x
T
0
.
81
25
00 1
.
12
50
00
1
.
12
50
00
0
.
93
75
00
0
.
81
25
00
0
.
81
25
00
0
.
81
25
00
2
()
h
x
T
0
.
43
75
00 0
.
62
50
00
0
.
62
50
00
0
.
50
00
00
0
.
43
75
00
0
.
43
75
00
0
.
43
75
00
3
()
x
R
38.
86
01
00
47.
70
00
00
58.
29
10
0
48.
32
90
00
40.
32
39
00
42.
09
73
98
42.
09
12
66
4
()
x
L
221
.3
65
00
0
1
1
7.7
0
1
0
0
0
43.
69
00
00
1
1
2.6
7
9
0
0
0
200
.0
00
00
0
176
.6
54
05
0
176
.7
46
50
0
1
()
g
x
-0
.0
00
000
-0
.2
04
390
0.0
0
0
0
1
6
-0
.0
04
750
-0
.0
34
324
-0
.0
00
020
-0
.0
00
139
2
()
g
x
-0
.0
04
300
-0
.1
69
942
-0
.0
68
904
-0
.0
38
941
-0
.0
52
847
-0
.0
35
891
-0
.0
35
949
3
()
g
x
-0
.0
00
000
54.
22
60
12
-2
1.
22
010
4
-
365
2.
87
68
38
-2
7.
10
584
5
-2
7.
88
607
5
-
1
1
6
.
38
27
00
4
()
g
x
-1
8.
63
500
-
122
.2
99
00
0
-
196
.3
10
00
0
-1
27
.3
210
00
-4
0.
00
000
0
-6
3.
34
595
3
-6
3.
25
350
0
()
f
x
585
0.
38
00
812
9.
10
36
719
8.
04
28
641
0.
38
1
1
628
8.
74
45
605
9.
94
63
606
1.
07
77
4.3. Welded
Beam Proble
m
The
weld
ed
beam
structu
r
e, sho
w
n i
n
Figure 3, i
s
a
pra
c
tical d
e
sign p
r
obl
em t
hat ha
s
been
often u
s
ed
as
a b
e
n
c
hma
r
k for te
sting diffe
rent
optimization
method
s. Th
e obje
c
tive is to
find the mini
mum fabri
c
ati
ng cost of the
weld
ed
be
a
m
subj
ect to
con
s
trai
nts o
n
sh
ear
stress
()
,
bendi
ng stre
ss
()
, buckli
ng l
oad
()
c
P
, end de
flection
()
,
and side co
nstrai
nt.
There are
four
desi
gn varia
b
l
es:
1
()
hx
;
2
()
lx
;
3
()
tx
and
4
()
bx
.
Figure 3. Wel
ded Beam Problem
The m
a
them
atical fo
rmul
a
t
ion of the
obj
ective fun
c
tio
n
()
f
x
, whic
h is the total fabricating
co
st mainly compri
se
d of the set-up, we
lding
labo
r, a
nd materi
al costs, is a
s
foll
ows:
Minimize:
2
12
3
4
2
(
)
1.10
471
0
.
04
811
(
1
4
.
0
)
f
xx
xx
x
x
(11)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Im
proved
Con
s
trai
ned
Enginee
ring
Optim
i
zation De
sign Algo
ri
thm
(Yuxin Sun)
7977
Subject to:
1
2
31
4
2
41
3
4
2
51
6
7
(
)
(
)
13
000
0
,
(
)
(
)
3
000
0
0
,
()
0
,
(
)
0.
10
471
0.
04
811
(
1
4
.
0
)
5.
0
0
,
(
)
0.
12
5
0
,
()
()
0
.
2
5
0
,
(
)
60
00
(
)
0
,
c
gx
x
gx
x
gx
x
x
gx
x
x
x
x
gx
x
gx
x
gx
P
x
(12)
Whe
r
e:
'2
'
'
'
'
'
2
2
'
12
''
2
2
2
13
2
2
2
13
2
12
2
43
4
34
3
33
4
()
(
)
2
(
)
,
2
6
000
,
2
,
60
00(
14
)
,
2
()
,
42
22
(
)
,
12
2
504
000
()
,
2.
19
5
2
()
,
(
)
6474
6.
022(
1
0
.
0
2823
46
)
.
c
x
x
R
xx
MR
J
x
M
xx
x
R
xx
x
Jx
x
x
xx
x
xx
P
xx
x
x
(13)
The a
pproa
ches appli
ed t
o
this
proble
m
incl
ude
ge
ometri
c p
r
og
ramming [2
0], geneti
c
algorith
m
wit
h
bina
ry rep
r
esentation
a
nd tra
d
itional
penalty fun
c
tion [21], a
GA-ba
s
e
d
co-
evolution mo
del [15] and
a feasibility-b
a
se
d tour
n
a
m
ent sele
ctio
n scheme in
spired by the
multi-
obje
c
tive opti
m
ization
tech
nique
s [13],
and
co
-evolut
i
onary
parti
cl
e swa
r
m
optimization
met
hod
[3]. In
this pa
per, the IPSO is run 50 tim
e
s ind
epen
de
ntly. The com
pari
s
on
s of re
sults a
r
e sho
w
n
in Table
3. T
he re
sult
s obt
ained
usin
g the IPSO
algo
rithm, we
re b
e
tter optimi
z
e
d
than a
n
y other
earlie
r solutio
n
s re
po
rted in
the literature.
T
able 3. Comparison of
th
e Best
Solu
ti
on for
W
e
lde
d
Beam
Prob
lem
Desi
gn
vari
abl
es
IPS
O
Ragsdell
(1
97
6)
De
b (1
99
1)
Coello
(2
00
0)
Co
ello
(2
00
2)
H
e
(2
00
7)
1
()
x
h
0
.
20
57
30
0
.
24
55
00
0
.
24
89
00
0
.
20
88
00 0
.
20
59
86 0
.
20
23
69
2
()
x
l
3
.
47
04
90
6
.
19
60
00 6
.
17
30
00 3
.
42
05
00 3
.
47
13
28 3
.
54
42
14
3
()
x
t
9
.
03
66
20
8
.
27
30
00 8
.
17
89
00 8
.
99
75
00 9
.
02
02
24 9
.
04
82
10
4
()
x
b
0
.
20
57
30
0
.
24
55
00 0
.
25
33
00 0
.
21
00
00 0
.
20
64
80 0
.
20
57
23
()
f
x
1
.
72
48
00 2
.
38
59
37
2
.
43
31
16
-
1
.
7
4
8
3
0
9
1
.
72
82
26
1
.
72
80
24
5. Conclusio
n
This
pap
er in
trodu
ce
a ne
w alg
o
rithm
based o
n
the
stand
ard PSO algo
rithm,
for th
e
stand
ard PS
O algo
rithm t
he ne
w alg
o
rithm has
don
e two imp
r
ov
ements: 1
)
B
y
introdu
cing
a
new info
rmat
ion sh
arin
g mech
ani
sm, make p
a
rti
c
l
e
s moved o
n
the contra
ry
directio
n of the
worst individ
ual po
sition
s and the worst whole swa
r
m position
s
, thus e
n
large
global
sea
r
ch
ing
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 79
70 – 797
8
7978
space and reduce the possibility of particl
es
to
be trapped i
n
to a local
optimum; 2) By
introducing elite selection stra
tegy, decreased the
possibility
of being trapped into a local
optimum. Co
mpared with
the standa
rd PSO al
gorithm, the new algo
rithm
enlarg
e
s th
e
sea
r
ching
sp
ace
and
the
compl
e
xity is
not high.
E
x
p
e
rime
nt
re
sult
s ba
se
d o
n
s
o
me
well-
kn
o
w
n
con
s
trai
ned engin
eeri
ng optimizatio
n probl
em
s
an
d com
pari
s
o
n
s with
prev
iously repo
rted
results de
mo
nstrate th
e effectivene
ss,
efficien
cy and robu
stne
ss of
the IPSO.
Referen
ces
[1]
R Bellma
n
. D
y
namic Pro
g
ram
m
ing. Princ
e
to
n: Princeton U
n
iversit
y
Press.
1957.
[2]
DH W
o
lp
ert, W
G
Macread
y. No fr
ee lu
n
c
h theor
ems
for optimiz
atio
n.
IEEE Transactions
on
Evoluti
onary C
o
mputati
o
n
. 19
97; 1(1): 67-8
2
.
[3]
Qie He,
Lin
g
W
ang. An
e
ffective co-ev
o
luti
onar
y
part
i
cle s
w
a
rm o
p
timizati
on for
constrai
ne
d
eng
ine
e
ri
ng de
sign pr
obl
ems.
Engi
neer
in
g Applic
atio
ns of Artificial Intel
lig
e
n
ce
. 200
7; 20: 89-9
9
.
[4]
Coel
lo
CAC,
Montes EM.
C
onstrai
nt-ha
ndl
ing
in
ge
netic
alg
o
rithms thr
o
ugh
the
use
of
domi
n
a
n
ce-
base
d
tourn
a
m
ent selecti
on.
Advanc
ed En
gi
neer
ing Infor
m
atics
. 2002; 1
6
: 193-2
03.
[5]
Michal
e
w
icz Z
.
Evolutio
nar
y A
l
gorit
hms in En
gin
eeri
ng Ap
pli
c
ations. Berl
in:
Spring
er. 199
5: 497-5
14.
[6]
J Kennedy
,
RC Eberhart. Pa
rticle S
w
arm
Optimization.
IEEE International Conf
erenc
e
on Neur
al
Netw
orks
. 199
5: 1942-
19
48.
[7]
Clare M, Kennedy
J. T
h
e Particle S
w
arm
- E
x
pl
osio
n, Stabil
i
ty, an
d C
onv
erge
nce i
n
a
Multidim
ens
ion
a
l Com
p
le
x Sp
ace.
IEEE Trans. on Evoluti
on2ary Computation
. 200
2; 6(1): 58-7
3
.
[8]
Coel
lo C
A
C, MS Lech
uga,
Mopso. A pr
op
osal fo
r mu
ltipl
e
ob
jective
par
ticle s
w
arm
op
timizatio
n
.
In
IEEE Proceedi
ngs World C
o
n
g
ress on C
o
mp
utation
a
l Intel
l
i
genc
e
. 200
2; 1051-
105
6.
[9]
J.Kenne
d
y
. T
he particl
e s
w
ar
m: social ad
ap
tation of kno
w
l
edg
e.
Proc.IEEE int.Conf.on evolutionar
y
computati
o
n
. 1
997; 30
03-
300
8.
[10]
E Oscan, CK Mohan. An
al
ysis of A Si
mple Partic
le
S
w
a
rm Optimi
zation S
y
stem
.
Intellige
n
ce
Engi
neer
in
g Systems T
h
ro
ugh
Artificial Ne
ura
l
Netw
orks
. 1998; 253-
25
8.
[11]
Clare M, Kennedy
J. T
h
e Particle S
w
arm
- E
x
pl
osio
n, Stabil
i
ty, an
d C
onv
erge
nce i
n
a
Multidim
ens
ion
a
l Com
p
le
x Sp
ace.
IEEE Trans. on Evoluti
on2ary Computation.
200
2; 6(1): 58-7
3
.
[12]
Arora JS. Introductio
n
to Optimum Desig
n
. Ne
w
York: McGra
w
-
Hil
l. 198
9.
[13]
Coel
lo
CAC,
Montes EM.
C
ons
trai
nt-ha
ndl
ing
in
ge
netic
alg
o
rithms thr
o
ugh
the
use
of
domi
n
a
n
ce-
base
d
tourn
a
m
ent selecti
on.
Advanc
ed En
gi
neer
ing Infor
m
atics.
2002; 1
6
: 193–
20
3.
[14]
Bele
gun
du AD.
A stud
y
of ma
thematic
al pro
g
r
amming meth
ods
for structural optimiz
atio
n.
Depart
m
e
n
t
of Civil a
nd En
viron
m
e
n
tal En
gin
eeri
ng U
n
iv
ersity of Iow
a
, Iow
a
City.
Io
w
a
.
1982.
[15]
Coel
lo, C.A.C.
Use
of a s
e
lf-ada
ptive pe
nalt
y
appr
oac
h
for
e
n
g
i
ne
e
r
ing optimiz
ati
on prob
lems.
Co
mp
uters in Industry
. 200
0: 41; 113
–1
27.
[16]
Coel
lo CA
C. T
heoretical a
n
d
numer
ical c
onstrai
nt han
d
ling tec
hni
qu
e
s
used
w
i
th
evol
ution
a
r
y
alg
o
rithms: a s
u
rve
y
of the st
ate of the art.
Co
mp
uter Met
hods
in Ap
pli
e
d Mech
anics
a
nd En
gin
eer
ing
.
200
2: 191 (1
1/12), 124
5–
128
7.
[17]
Sand
gre
n
E. Nonli
n
e
a
r inte
ge
r and d
i
screte
progr
ammin
g
i
n
mech
anic
a
l d
e
sig
n
.
In Proce
edi
ngs of t
h
e
ASME Desig
n
T
e
chno
logy C
o
nferenc
e
. Kissi
mine, F
L
. 198
8
:
95–10
5.
[18] Kann
an BK, Kr
amer SN. An
a
ugme
n
ted
Lagr
ang
e multi
p
li
er
base
d
meth
od
for mixed
inte
ger d
i
scret
e
contin
uo
us opti
m
izatio
n an
d it
s appl
icati
ons t
o
mech
anic
a
l d
e
sig
n
.
T
r
ansac
tions of the AS
ME, Journa
l
of Mechan
ical
Desig
n
. 19
94: 116; 31
8–
32
0.
[19]
Deb
K. Gen
e
A
S
: a ro
bust
op
timal d
e
si
gn t
e
chni
que
for m
e
cha
n
ica
l
c
o
mpon
ent
desi
gn.
Evol
ution
a
ry
Algorit
h
m
s in E
ngi
neer
in
g App
licatio
ns.
19
97:
497–
51
4.
[20]
Rags
del
l KM,
Phi
lli
ps DT
. Optimal
des
i
gn
of
a
clas
s of
w
e
ld
ed
structures us
i
ng
geom
etric
progr
ammin
g
.
ASME Journa
l of Engin
eer
ing
for Industries
. 197
6: 98(3); 10
21–
10
25.
[21]
Deb
K. Optima
l d
e
sig
n
of
a
weld
ed
be
am vi
a g
enetic
a
l
gor
ithms.
AIAA Journal.
19
91: 2
9
(11); 201
3–
201
5.
[22]
Xu
e So
ng Y
a
n
,
Qing Hu
a W
u
, Che
ng Y
u
Hu, Qing Z
h
on
g Li
ang.
Circu
i
t
Desig
n
Bas
e
d on P
a
rticl
e
S
w
a
rm Optimiz
a
tion Al
gorit
hms.
Key Engin
e
e
r
ing Mater
i
als
.
201
1: 474-
476:
1093-
10
98.
[23]
X.S.Yan, Qin
g
Hua W
u
. A N
e
w
Optim
i
za
it
on
Algor
ithm for
F
unction Opti
mizatio
n
.
Proc
eed
ings of
the
3rd Internati
o
n
a
l Sympos
iu
m
on Inte
ll
ig
ence
Co
mp
utation &
Applic
ations
. 2
009: 14
4-1
50.
[24]
Xu
eso
ng Y
an,
W
e
i L
i
, W
e
i
Che
n
, W
enj
in
g Lu
o,
C
an Z
han
g, Ha
nmin
Liu.
Cultur
al
Algorit
hm fo
r
Engi
neer
in
g D
e
sig
n
Probl
em
s.
Internation
a
l
Journa
l of
Co
mp
uter Scie
nc
e Issues
. 201
2: 9(6): 53-61.
Evaluation Warning : The document was created with Spire.PDF for Python.