TELKOM
NIKA
, Vol. 11, No. 6, June 20
13, pp. 2926
~ 293
2
e-ISSN: 2087
-278X
2926
Re
cei
v
ed
Jan
uary 7, 2013;
Re
vised Ma
rch 29, 2013; A
c
cepted Ap
ril 7, 2013
Orthogonal Particle Swarm Optimization Algorithm and
Its Application in Circuit Design
Xuesong Ya
n*
1
, Qinghua
Wu
2,3
, Hammin Liu
4
1
School of Co
mputer Scie
nc
e, Chin
a Univ
e
r
sit
y
of Geosci
ences, W
uha
n,
Chin
a
2
Hub
e
i Provi
n
ci
al Ke
y La
bor
ator
y
of Intel
lig
en
t R
obot, W
uha
n Institute of
T
e
chn
o
lo
g
y
, W
uhan, Ch
in
a
3
School of Co
mputer Scie
nc
e and En
gi
neer
ing, W
uha
n Ins
t
itute of
T
e
chnolo
g
y
, W
u
h
an,
Chin
a
4
W
uhan Institute of Shipb
u
i
l
di
ng T
e
chnol
og
y, W
uhan, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
y
a
n
x
s19
99@
126.com
A
b
st
r
a
ct
In this p
a
p
e
r, ai
m at t
he
disa
dvanta
ges
of st
and
ard P
a
rticl
e
Sw
arm Opti
mi
z
a
ti
on
(PSO) al
gorith
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 reve
al that
the
prop
osed
alg
o
rith
m can fin
d
better
solutio
n
s w
h
e
n
co
mp
ared to
th
e
standar
d parti
cle sw
arm op
timi
z
a
t
i
o
n
alg
o
r
ithm. W
e
us
e the prop
os
ed al
gorith
m
for digita
l circuit
opti
m
i
z
at
ion
de
sign, a
nd t
he fi
nal c
i
rcuit
is o
p
timi
z
e
d
in
ter
m
s
of co
mpl
e
xi
ty (w
ith the mi
ni
mu
m
nu
mb
er
of
gates).
Ke
y
w
ords
: par
ticle sw
arm o
p
timi
z
a
tio
n
, ortho
gon
al d
e
sig
n
, bench
m
ark func
tion, circuit des
ign
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Swarm
intelli
gen
ce i
s
an i
m
porta
nt research
topic b
a
se
d o
n
the
colle
ctive b
e
havior
of
decentrali
ze
d
and
self-o
rgani
zed
sy
stems in
co
m
putational
int
e
lligen
ce. It
co
nsi
s
ts of
a
popul
ation
which
sim
u
late
s the
ani
mals beh
avior i
n
t
he
real
wo
rld.
No
w th
ere a
r
e m
any
swarm
intelligen
ce o
p
timization al
gorithm
s, su
ch as ge
net
ic algorithms,
particle
swarm
optimization,
ant
colo
ny optimi
z
ation, b
ee colony algo
rith
m, different
ial
evolution, fish-warm
algo
ri
thm, etc. Due
to
the sim
p
le
concept, ea
sy impleme
n
tation an
d qui
ck conve
r
ge
n
c
e, PSO h
a
s
gai
ned
mu
ch
attention and
been
su
ccessfully applied in a variety
of fields mainly for optimi
z
atio
n 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
r laws of bird flo
c
ks, fish scho
ol
s and h
u
ma
n
commu
nities [1]. If we co
mpare PSO with
Geneti
c
Algo
rithms
(GAs),
we m
a
y find t
hat they
a
r
e
all man
euvered 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 [
2
-4]. 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 [5
-8]. PSO is
a
meta-heuris
tic
as
it makes
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
prove
s
t
he di
sadva
n
tage
s of
stan
dard
PSO b
e
i
ng
easily tra
ppe
d into a lo
cal
optimum a
n
d
pro
p
o
s
ed
a
new
algo
rith
m whi
c
h p
r
o
v
es to be m
o
re
simply co
ndu
cted an
d with
more efficie
n
t
global se
arching capa
bility.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
Orthog
onal P
a
rticle S
w
arm
Optim
i
zation Algorithm
and Its Applicati
on in Circuit (Xueso
ng Yan
)
2927
2. Particle Sw
arm Optimi
z
a
tion Algori
t
hm
The stand
ard
PSO
alg
o
r
ithm works by
hav
ing a
popul
ation (call
ed a swarm)
of
can
d
idate
sol
u
tions
(call
e
d
particle
s
). These parti
cle
s
are moved
aroun
d in th
e sea
r
ch spa
c
e
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 kno
w
n
positio
n in the sea
r
ch spa
c
e a
s
we
ll a
s
the entire swarm'
s b
e
st
kno
w
n po
siti
on.
Whe
n
imp
r
ov
ed po
sition
s
are
being
di
scovered,
they
are
used to
guide th
e mo
vements
of the
swarm. Th
e p
r
ocess is
rep
eated until th
at a
s
a
tis
f
ac
tory s
o
lution is dis
c
overed.
In PSO, each solution i
s
rega
rd
ed as
a bird in the
sea
r
ch sp
ace
and calle
d “particl
e”.
Each p
a
rti
c
le
has
a fitness value
whi
c
h is d
e
term
i
n
ed by a ta
rge
t
function. Th
e statu
s
of e
a
ch
particl
e in
clu
des its
po
sition a
nd vel
o
ci
ty. Its velo
city determine
s it
s flying
dire
ct
ion. All pa
rticl
e
s
(the
swarm
)
work to
gethe
r in
se
archin
g for th
e o
p
timal solution
in
the
soluti
on
spa
c
e. P
S
O
achi
eves the
sea
r
che
s
via updatin
g th
e statu
s
of
e
a
ch p
a
rti
c
le.
At the begin
n
ing, it rand
o
m
ly
initiates a gro
up of particle
s
(rando
m so
lutions) wi
th a spe
c
ific po
sition and vel
o
city for each
.
It
update
s
th
e
status
(its vel
o
city and
p
o
siti
on, referring
Formul
a
(1
) a
nd
(2) of
ea
ch pa
rticl
e
u
s
i
n
g
the
reco
rd
ed best po
sition experie
nced
f
o
r
thi
s
parti
cl
e
(calle
d
idb
P
in
Formul
a 1
)
a
nd the
be
st
positio
n of
th
e whole
swa
r
m
(called
g
db
P
in Fo
rmul
a 1
)
until
no
w. T
h
is
upd
ating
pro
c
ed
ure
contin
ue
s unt
il it reach
e
s t
he maximum
numbe
r of
iteration
s
, whi
c
h is
set up
at the begin
n
i
ng.
Thro
ugh thi
s
iteratively up
dating, PSO f
i
nds
(o
r ap
proache
s to)
o
p
timal solutio
n
s in
the
solu
tion
spa
c
e. T
h
u
s
, the statu
s
of
each p
a
rticl
e
is
u
pdate
d
n
o
t only ba
sed
on itself be
st positio
n (
idb
P
)
experie
nced, but also ba
se
d on the best
position of
th
e whol
e swarm (its com
p
a
n
ion
s
). That i
s
,
the parti
cles
insid
e
the swarm sh
are
the informati
on (
g
db
P
). Specifi
c
ally, for a particle id, its
velocity and its po
sition is u
pdated a
c
cording to the formula as follo
ws
respe
c
tively:
'
12
()
(
)
()
(
)
i
d
id
id
b
i
d
gdb
id
V
V
ran
d
P
X
ran
d
P
X
(1)
''
id
id
i
d
X
XV
(2)
whe
r
e
is call
ed the inertia
weight. It is a prop
ortion f
a
ctor
con
c
e
r
ned with former velo
city
(
01
).
1
and
2
are
con
s
tant a
c
cele
rating facto
r
s, norm
a
lly
12
==
2
. The rando
m fu
nctio
n
()
rand
is to gene
rate
rando
m num
bers.
id
X
rep
r
e
s
e
n
ts the po
sition of parti
cle
id
.
id
V
r
e
p
r
es
en
ts
the velocity o
f
particl
e
id
.
id
b
an
d
g
db
re
pre
s
e
n
t the be
st po
siti
on of the
particle
id
found a
n
d
the best po
sit
i
on of the wh
ole swarm
found re
sp
ectiv
e
ly until this moment.
In the formul
a (1)
above, the first pa
rt
repre
s
e
n
ts the
impact of the
former velo
ci
ty of the
particl
e. It en
able
s
the
pa
rt
icle to
po
sse
s
s exp
andi
ng t
ende
ncy i
n
th
e sea
r
chi
n
g
space, an
d thu
s
make
s
the
algorithm
be m
o
re
cap
able i
n
glob
al s
earchin
g. The
se
con
d
pa
rt is
called a
co
gnit
i
on
part. It repre
s
ents the process of abso
r
b
i
ng indivi
dual
experie
nce knowl
edge of
the parti
cle. The
third pa
rt is
calle
d a soci
al part. It rep
r
esents
th
e p
r
ocess of le
a
r
ning f
r
om th
e experi
e
n
c
e
of
other pa
rticl
e
s. It also sho
w
s the info
rm
ati
on sh
arin
g and soci
al co
operation am
ong pa
rticle
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,
scholars like
Cl
erc [9] h
a
s p
r
ese
n
ted
pro
o
f on it
s
co
nvergen
ce.
Here
a fatal
wea
k
n
e
ss
may res
u
lt from this
c
h
arac
teris
t
ic
. With c
o
ns
tant
in
crease of iterati
ons, t
he velo
city of particl
es
will gra
dually
diminish an
d rea
c
h zero
in the end. At
this time, the whole swarm
will be
conve
r
ge
d at
one poi
nt in the solutio
n
spa
c
e, if g
best
particle
s
h
a
ven't found
g
best
, the whole
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 i
s
rather
we
ak.
The p
r
o
babili
ty of the
o
c
curren
ce i
s
e
s
peci
a
lly high
so fa
r fo
r mu
lti-
pea
ks fun
c
tio
n
s, we h
a
ve test the algorithm for
the multi-pe
aks b
enchma
r
k fu
nction
s to verify
these. We select
th
ree
multi-pe
aks
f
unctio
n
s
fo
r
our
experi
m
e
n
t, the functi
ons de
scrib
e
d
a
s
followin
g
and
Table 1 is the
experime
n
t result
s.
Table 1. Experime
n
t Re
su
lts
Function
Best Value
Mean Value
Worst Value
min
f
F1
1495.71
4224.775
7032.89
0
F2
72.5069
101.410452
123.954
0
F3
-5038.62
-4005.02
-3233.13
-12569.5
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No. 6, June 20
13 : 2926 – 2
932
2928
2
mi
n
1
1
:
(
)
,
(
10
0
,
10
0
)
,
0
n
ii
i
Ff
x
x
x
f
Figure 1. Be
nchm
ark Fun
c
tion F1
2
mi
n
1
1
10
0
1
2
:
(
)
(
1
0
0
)
c
o
s
(
)
1
,
(
3
00
,
3
00)
,
0
40
00
n
n
i
ii
i
i
x
Ff
x
x
x
f
i
Figure 2. Benchma
r
k Fun
c
t
i
on F2
mi
n
1
3
:
(
)
si
n
(
),
(
5
00
,
5
0
0
)
,
12
56
9.5
n
ii
i
i
Ff
x
x
x
x
f
Figure 3. Be
nchm
ark Fun
c
tion F3
3. Orthog
on
al Particle Sw
a
r
m Optimi
zatio
n
Algori
t
hm
3.1. Orthogonal Initializ
ation
The tra
d
ition
a
l metho
d
of
parti
cle
swa
r
m optim
i
z
ati
on algorithm
is randomly initialized
popul
ation, th
at is, ge
ne
rat
e
a
se
ries of
rand
om
n
u
m
bers in
the
solution
spa
c
e
of the q
u
e
s
tion.
De
sign th
e n
e
w
algo
rithm
,
we u
s
in
g t
he o
r
thog
on
al initializatio
n [10-12]
in
the initialization
pha
se. Fo
r the gen
eral condition, bef
ore
see
k
in
g
out the optim
al solutio
n
th
e location of
the
global
optim
al solution
is impo
ssible
to kn
ow, fo
r so
me
high
-dimen
sion
al
and
multi-mo
de
function
s to
o
p
timize, the f
unctio
n
itself
has
a lo
t of
p
o
les, a
nd th
e
global
optimu
m
location of
the
function is u
n
kn
own. If
the initial popul
ation of
chro
moso
me
s ca
n be evenly distrib
u
ted in
the
feasibl
e
solut
i
on spa
c
e, th
e algorithm
can evenl
y se
arch in the solution sp
ace
for the global
optimum. O
r
thogo
nal initi
a
lizatio
n is to u
s
e
the
o
r
thogo
nal ta
ble ha
s th
e
dispersio
n
and
uniformity
comparable; the individual
will be in
itiali
zed uniforml
y
disp
ersed into
the search
spa
c
e, so
th
e
o
r
thog
onal
desi
gn meth
od can
b
e
u
s
ed
to
gen
erate unifo
rmly
dist
ribute
d
i
n
itial
popul
ation.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
Orthog
onal P
a
rticle S
w
arm
Optim
i
zation Algorithm
and Its Applicati
on in Circuit (Xueso
ng Yan
)
2929
3.2. Intergen
eration
a
l Elite Mecha
n
is
m
In stand
ard
PSO algo
rith
m, the next flying direct
io
n
of ea
ch
parti
cle i
s
n
early
definite, 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
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:
Step1: Usin
g cro
s
sove
r an
d mutation o
peratio
ns fo
r popul
ation P1 whi
c
h si
ze i
s
N then
gene
rating th
e next genera
t
ion of sub-po
pulation
s
P2;
Step2: The current pop
ula
t
ion P1 and t
he nex
t gen
e
r
ation of sub
-
popul
ations P
2
mixed
together fo
rm
a tempora
r
y popul
ation;
Step3: Temp
ora
r
y populati
on acco
rdin
g to fitne
ss values
in desce
nding o
r
de
r, to retain
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.
We al
so u
s
e the three
multi-pe
aks functi
o
n
s to test our n
e
w algorithm a
nd the
experim
ent result sho
w
ed
in Table 2. From
the e
x
perime
n
t re
sults, we ca
n say the n
e
w
algorith
m
ha
s got the better solutio
n
.
Table 2. Experime
n
t results
Function
Algorithm
Best Value
Mean Value
Worst Value
min
f
F1
PSO
1495.71
4224.775
7032.89
0
Ne
w 8.13E-29
10.46E-26
5.08E-24
0
F2
PSO
72.5069
101.410452
123.954
0
Ne
w 12.18E-12
0.070377
18.63E-25
0
F3
PSO
-5038.62
-4005.02
-3233.13
-12569.5
Ne
w -8535.19
-7741.27
-5203.56
-12569.5
4. Cicuit De
s
i
gn Experiment
Evolutionary
Electro
n
ics a
pplie
s the co
nce
p
ts
of ge
netic alg
o
rith
ms to the ev
olution of
electroni
c
circuits.
The
m
a
in id
ea
behi
nd thi
s
re
sea
r
ch
field
is that ea
ch
po
ssible
ele
c
tro
n
i
c
circuit
can
be
rep
r
e
s
ente
d
as a
n
individ
ual or
a
chro
moso
me of a
n
evolution
a
ry pro
c
e
ss,
which
perfo
rms sta
ndard g
eneti
c
o
peration
s
over the
ci
rcuits. Due to t
he b
r
oa
d
sco
pe of the
are
a
,
resea
r
chers
h
a
ve bee
n focusin
g
on
different p
r
obl
em
s, such a
s
pl
a
c
eme
n
t, Field
Prog
ramm
ab
le
Gate Array (FPGA) map
p
ing, optimization of
com
b
ination
a
l an
d seq
uential
digital circui
ts,
synthe
sis of
digital circu
i
ts,
synthesi
s
of passive and active a
nalog ci
rcuit
s
, synthe
sis o
f
operational
a
m
plifiers, an
d tran
sisto
r
size opt
imi
z
a
t
ion. Of gre
a
t relevan
c
e
are th
e wo
rks
focu
sing o
n
“intrinsi
c
” hardwa
r
e evoluti
on in whi
c
h f
i
tness evalua
tion is
pe
rformed in sili
co
n,
allowin
g
a hig
her de
gree of
exploration o
f
the phy
sical
prope
rtie
s of the m
edium.
This pa
rticul
ar
area i
s
freq
ue
ntly called Evolvable Hard
ware [13-15].
In the sequ
e
n
ce of this
work, Coello,
Chri
stian
s
e
n
and Agui
rre
pre
s
ente
d
a comp
uter
prog
ram
that
automati
c
all
y
gene
rate
s
high-quality
circuit
de
sign
s [16]. Miller,
Thomp
s
o
n
a
n
d
Foga
rty appli
ed evolution
a
ry algo
rithm
s
for the
de
sign of arithm
etic circuit
s
[17]. Kalgano
va
,
Evaluation Warning : The document was created with Spire.PDF for Python.
TEL
K
2930
Mille
r
orde
r
evol
u
Bas
e
Parti
c
Evol
u
cir
c
u
i
4.1.
O
effic
i
e
wa
s
i
the f
u
cir
c
u
i
(with
this
d
4.2.
T
ca
se
,
siz
e
o
cir
c
u
i
(with
desi
g
K
OM
NIKA
V
r
an
d Lip
n
its
r
to s
o
lv
e
c
u
tion. T
h
e id
e
e
d
on
th
e M
c
le S
w
arm
O
u
tionary Alg
o
i
ts
.
O
ne-bit Add
Evolving
e
nt ci
rcui
t.
T
i
ntimatel
y re
u
lly function
a
The o
r
ig
i
i
t de
sig
n
ed
b
three gates
d
e
s
ign i
s
an
o
F
T
wo
-
b
i
t
A
d
d
A two-bit
,
our al
g
o
rit
h
o
f 3
×
3.
The
i
t d
e
si
gn
ed
b
six gate
s
).
F
g
n is a
n
opti
m
V
ol. 11, No.
6
kaya propo
s
c
ompl
ex sy
s
e
a i
s
to
e
v
ol
v
iller’s meth
o
O
ptimization
A
o
rithm [25]
e
r
the one-bit
a
T
hat is ma
ny
lated to t
he
s
a
l solutions.
i
na
l c
i
rc
u
i
t i
s
b
y Miller’s
al
g
).
From t
he
f
o
ptimum sol
u
A
B
C
1
F
igure 4. On
Figure 5
Figure
6
d
e
r
full adde
r
c
h
m use sm
a
original circ
u
b
y Mill
er’
s
al
F
ro
m the fig
u
m
um solutio
n
6
, June 20
1
3
s
e
d
anoth
e
r
t
s
tems
, Torr
e
v
e a
syste
m
o
d, Yan
ap
p
A
lgorithm
s (
P
and evoluti
o
a
dd
er w
a
s e
geneti
c
al
g
o
s
i
z
e of the
g
s
s
how
e
d
i
n
g
orithm (wit
h
f
ig
ur
es
w
e
k
u
tion.
e-bit Full Ad
d
. One-bit Fu
l
6
. One-bit F
u
c
ircuit, which
a
ll geomet
ry
u
it is
showe
d
go
rithm (wi
t
h
u
re
s w
e
kn
o
w
n
.
3
: 2926 – 2
9
t
echni
que fo
e
se
n pro
p
o
s
gradu
ally a
s
p
lied Gen
e
E
P
SO) [7], C
u
o
nary alg
o
ri
t
as
ier
to
d
o
o
o
rithm wa
s
a
g
eomet
ry, b
u
n
Figu
re 4 (
w
h
three
gate
s
k
no
w it is
a
g
d
er Circ
uit
D
l
l Adde
r Circ
u
ll Adder Cir
c
with a trut
h
to find the
f
d
in
F
i
g
u
re
7
h
six gate
s
)
w
it is a grat
9
32
r desi
gning
m
s
ed the me
t
a
kind
of di
v
E
xpr
e
ss
ion
P
u
ltural
A
lgori
t
t
hm [26, 27
]
o
n a la
rg
e
r
g
a
ble to di
sc
o
t our al
gorit
h
w
ith five ga
t
) [28] and
Fi
g
ratifying re
s
D
esi
gned wit
h
uit Desi
gne
d
c
uit Desi
gne
d
h
table with
5
f
ully functio
n
7
(with ten
g
[28] and
Fi
g
ifying result
t
e
-
m
ultiple-valu
t
h
od of incr
v
ide-an
d-co
n
P
rogrammi
n
g
t
hms
(C
A)
[
2
]
for the de
g
eo
me
tr
y b
u
t
o
ver 10
0% f
u
h
m use sma
t
es
)
,
F
i
gu
r
e
gu
r
e
6
is
ou
r
s
ult to obtai
n
C
0
S
h
out Optimu
m
d
by Miller
d
by Ou
r
5
inputs a
n
d
n
al solution
s
,
g
ates) Fig
u
r
e
g
ur
e
9 is o
u
r
t
o obtain as
-
ISSN: 2087
u
ed
circuits [
1
ea
sed com
p
n
quer meth
o
d
g
(GEP)
[2
0
2
2-24], Orth
o
sig
n
of ele
c
t
resulted in
u
n
c
tional sol
ll
geomet
r
y
t
5 i
s
the
re
s
r
algorithm’
s
n
as it
is cl
e
a
m
3 output
s
.
I
,
the matrix
e
8 is the
re
s
r
al
gorithm’
s
it is clear th
a
-278X
1
8]. In
p
le
xi
t
y
d
[19].
0
, 21],
o
g
o
nal
c
tronic
a l
e
ss
utio
ns
t
o find
s
ulting
result
a
r that
I
n this
ha
s a
s
ultin
g
re
s
u
lt
a
t this
Evaluation Warning : The document was created with Spire.PDF for Python.
TEL
K
Ort
h
Fi
g
5. C
o
st
an
d
st
ag
e
poss
i
st
rat
e
st
an
d
not
h
repo
r
We
u
opti
m
Ack
n
No.6
1
Sc
ie
n
Nati
o
Ref
e
[1]
J
N
[2]
C
C
[3]
C
I
[4]
J
c
[5]
E
E
[6]
X
3
K
OM
NIKA
h
og
onal P
a
r
t
g
ure 7. Two
-
w
o
nclusio
n
This pap
d
ard
PSO al
g
e
,
we u
s
e or
t
i
bility of
par
t
e
gy, d
e
crea
s
d
ard PS
O al
g
h
igh. Experi
m
r
t
ed re
su
lt
s
d
u
se the
pro
p
m
ized in term
n
ow
l
e
d
g
m
e
n
This pap
1
203
307
),
N
n
ce Fo
u
ndat
o
nal Univ
ersi
t
ren
ces
J
Ke
nn
edy
,
R
N
etw
o
rk
s
.
19
9
C
lare M, Ken
n
C
ompl
e
x
Spa
c
C
oel
lo CA
C,
M
I
EEE Procee
d
J
Ke
nn
edy
.
T
h
c
omputati
on.
E
O
scan, C
K
E
ngi
neer
in
g
S
X
S Y
an, Q
i
n
g
3
rd Internatio
n
t
icle S
w
arm
bit full adde
r
w
ithout optim
Figure
er in
trodu
c
e
g
orithm t
he
t
ho
gon
al de
s
t
icl
e
s to be
s
ed
the p
o
s
s
g
o
r
ithm, the
m
en
t r
e
su
lts
d
em
on
stra
te
p
osed al
g
o
ri
s of
com
p
le
x
n
ts
er i
s
su
ppo
r
N
atio
nal Ci
vil
ion of Hu
bei
t
y, C
h
ina U
n
R
C Eb
erhart.
9
5: 1942-
19
4
8
n
ed
y
J. T
he P
c
e.
IEEE Tra
n
M
S Lechu
ga,
d
i
ngs W
o
rld C
o
h
e p
a
rticle
s
w
1
997: 30
03-
3
0
K
Moha
n. A
n
S
ystem
s Thro
u
g
Hua
W
u
.
A
N
n
al S
y
m
pos
iu
m
e-I
Optimiz
a
tio
n
r
circuit de
si
g
um
9. Two
-
bit f
u
e
a ne
w alg
o
n
e
w algo
rit
h
s
i
gn meth
o
d
trap
ped
int
o
s
ibility of be
i
ne
w alg
o
ri
t
h
based on
b
the effectiv
e
thm for
digi
t
x
ity (with the
r
ted b
y
Nat
u
Ae
r
o
s
p
ac
e
(No. 2012
F
n
iversity of G
P
a
rticle S
w
a
r
8
.
article S
w
arm
n
s. on
Evol
uti
o
Mopso.
A
p
r
o
ngress on C
o
w
ar
m: soc
i
a
l
a
d
0
08
.
n
aly
s
i
s
o
f
A
u
gh Artificial
N
N
ew Op
ti
m
i
z
a
m
on
Intellige
n
SSN: 2087
-
2
n
Algorithm
a
g
n
ed
F
i
u
ll adde
r cir
c
o
rithm base
d
h
m has don
e
, thus enl
ar
g
o
a lo
cal
op
t
i
ng trapp
ed
h
m enla
r
ge
s
en
chma
rk f
u
e
ne
ss,
ef
f
i
ci
e
t
al circuit o
p
minimum n
u
u
ral Sci
ence
Pre-re
se
ar
c
F
FB0410
1) a
eo
sci
en
ce
s
(
r
m Optimizati
E
x
plosi
on, S
t
o
nar
y Comput
a
r
opo
sa
l
fo
r
m
u
o
mp
utation
a
l
d
aptation of k
n
Sim
p
le P
a
rti
c
N
eu
ral
N
e
two
r
k
it
on
Alg
o
rith
m
n
ce Computa
t
2
78
X
a
nd Its Appli
c
gure 8. Two
c
uit de
sign
ed
d
on
th
e s
t
a
e
two impro
v
g
e glob
al se
a
t
i
m
um; 2. B
y
into a local
the se
archi
n
u
n
c
tion
s
an
d
e
ncy a
nd
ro
b
p
timiz
a
tion
d
u
mbe
r
of gat
e
Fo
und
a
t
ion
c
h Proj
ect o
f
nd the Fund
(
Wuhan
).
on
.
IEEE Int
e
t
ab
ilit
y, an
d C
a
tion
.
2002; 6
u
lti
p
l
e
obj
ecti
v
Intel
l
i
genc
e.
2
n
ow
le
dge
. Pr
o
c
le
S
w
a
r
m
O
ks
.
1998: 253
-
m
for
F
unction
t
ion &
Applic
a
t
c
ati
on in Cir
c
-bit full add
e
by Miller
by Our
a
nd
ard PSO
v
ements: 1.
I
a
rchi
ng spa
c
y
i
n
trodu
cin
g
optimum.
C
n
g s
p
ac
e
a
n
d
d
compa
r
i
s
o
n
b
us
tn
es
s o
f
t
d
esig
n, and
e
s).
of China
.
(
N
f
Chin
a, the
a
m
ental Re
s
e
rnati
o
nal
Co
onv
erg
ence i
n
(1): 58-73.
v
e p
a
r
t
icle s
w
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