Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
7, N
o
. 3
,
Ju
n
e
201
7, p
p
. 1
643
~165
0
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v7
i
3.p
p16
43-
165
0
1
643
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Comparative Study of Meta-heuri
stics Optimization
Algorithm
using Benchmark Function
I.
Isma
il, A. Ha
nif
Ha
lim
Electrical and
Electron
i
c
Engin
e
ering
Depar
t
ment, Universiti Tekn
ologi P
ETRON
AS, 32610 Tron
oh, Perak
,
M
a
lay
s
ia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ja
n 14, 2017
Rev
i
sed
Mar
20
, 20
17
Accepte
d Apr 9, 2017
Meta-heur
i
stics
optimization is b
ecoming a popular tool for solving
numerous problems in real-world appli
cation du
e to
the
ability
to overcom
e
man
y
shortcom
ings in trad
itional
optimization. Despite of
the good
perform
ance
, th
ere
is lim
ita
tio
n in som
e
algo
rithm
s
that d
e
t
e
riorat
es
b
y
cert
a
in degr
ee o
f
problem
t
y
p
e
.
Therefor
e it is
neces
s
a
r
y
to c
o
m
p
are the
perform
ance
of
these
a
l
gorith
m
s
with cer
tain
problem
t
y
p
e
.
This p
a
pe
r
com
p
ares
7 m
e
ta-heuris
t
ics
opti
m
i
zation
with 1
1
benchmark fu
nctions that
exhibits
cer
tain
difficu
lti
es
and can
b
e
a
ssume
d a
s
a si
mul
a
t
i
o
n re
le
va
nt to
the re
al-world pr
oblem
s
.
The t
e
s
t
ed be
nchmark fu
nction has diff
er
ent ty
pe of
problem
such as
m
odalit
y,
separ
a
bilit
y,
d
i
scontinu
i
t
y
and surfa
ce
e
ffects wi
th
steep-drop global optimum, bowl- and pl
ateau-
t
yped function
.
Some of the
proposed function has the co
mbinati
on of
th
ese problems, which might
incre
a
s
e
th
e d
i
fficul
t
y
l
e
vel
o
f
s
earch
towar
d
s
global op
ti
m
u
m
.
The
performance co
mparison includ
es com
putation
time and convergence of
global optimum.
Keyword:
Meta-Heuristic Op
ti
m
i
zatio
n
Nat
u
re-
I
ns
pi
re
d Al
g
o
ri
t
h
m
Test Problem
Glob
al Op
tim
u
m
B
e
nchm
ark F
u
nct
i
o
n
Copyright ©
201
7 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
A. Ha
nif Halim
,
Depa
rtem
ent of Electrical a
n
d
El
ect
ro
ni
c E
n
gi
nee
r
i
n
g,
Uni
v
ersi
t
y
Te
k
nol
ogi
PETR
O
N
A
S
,
Ser
i
Isk
a
nd
ar
,
3
261
0 Tron
oh
, Per
a
k
,
Malaysia.
Em
a
il: h
a
li
m
_
h
a
n
i
f@ym
ail.c
o
m
1.
INTRODUCTION
Th
e Meta-h
euristic o
p
tim
iza
tio
n
is
b
eco
m
i
n
g
m
o
re
p
o
werfu
l
m
e
th
od
o
f
so
lv
i
n
g
o
p
tim
izat
io
n
pr
o
b
l
e
m
.
Such
opt
i
m
i
zati
on t
echni
que
s are cl
assi
fi
ed as st
oc
hast
i
c
o
p
t
i
m
i
zat
i
on m
e
t
hod. T
h
ei
r r
o
bu
st
ness
and
ab
ility o
f
find
in
g
g
l
ob
al so
l
u
tio
n
in
v
a
ri
o
u
s
k
i
nd
of field
is p
r
o
v
e
n
in
m
a
n
y
literatu
res. Th
e m
a
in
ch
aracteristic
of t
h
ese al
g
o
ri
t
h
m
s
i
s
t
h
e dy
nam
i
c bal
a
nce of
di
ve
rsi
f
i
c
a
t
i
on a
nd i
n
t
e
n
s
i
f
i
cat
i
on i
n
a
gra
d
i
e
nt
-f
ree
search
space [1],
[2].
Nu
m
e
rou
s
m
e
ta-h
eu
ristic alg
o
rith
m
s
in
spired
b
y
n
a
t
u
re were in
tro
d
u
ced su
ch
as Sim
u
lated
Ann
ealing
(SA), Particle Swarm
Op
ti
mizati
o
n
(PSO),
Fire
fly
Algo
rithm
(FF
A
)
,
Ha
rm
ony
Searc
h
(
H
S
)
an
d
m
a
ny
m
o
re. E
ach m
e
t
hod
ha
s i
t
s
o
w
n
back
gr
o
u
n
d
p
h
i
l
o
s
o
phy
o
f
m
i
m
i
cki
ng t
h
e
nat
u
re
and
bl
e
n
ded
w
i
t
h
t
h
e
search
st
rat
e
gy
t
o
e
xpl
ore a
n
d
ex
pl
oi
t
a
defi
n
e
d
pr
obl
em
t
o
war
d
s a
gl
obal
opt
i
m
u
m
. Such
t
y
pe o
f
al
g
o
r
i
t
h
m
i
s
al
so de
n
o
t
e
d a
s
nat
u
re-i
nspi
r
e
d al
g
o
ri
t
h
m
.
It
i
s
al
so
proven that thes
e a
l
gorithm
s
are capable t
o
ove
r
com
e
m
a
ny
sh
ort
c
o
m
i
ngs
of t
r
adi
t
i
onal
al
g
o
r
i
t
h
m
ap
pl
i
cat
i
on
[4]
.
D
e
sp
ite of
go
od
p
e
rfo
r
m
an
ce, there are lim
itation that
m
a
de thes
e al
gori
t
hm
s det
e
ri
orat
es by
cert
a
i
n
degree of proble
m
types [5] especially
i
n
real
-w
orl
d
pr
o
b
l
e
m
whi
c
h ex
hi
bi
t
a l
a
rge-scal
e pr
ope
rt
y
t
h
at
gr
ow
s
ex
pon
en
tially b
y
in
creasing
n
u
m
b
e
r
o
f
v
a
ri
ab
les and
d
i
m
e
n
s
ion
s
[2
],
[8
]. Th
is
p
r
o
b
l
em
will co
n
tinu
e
t
o
gro
w
p
a
rallel with
ad
v
a
n
ce of scien
ce and
tech
no
log
y
. As a resu
lt o
f
th
e in
creasin
g
d
i
m
e
n
s
io
n
a
lity, o
t
h
e
r facto
r
s
suc
h
as interac
tion of va
riabl
e
s (also re
ferre
d
as no
n-se
parability) and se
arch sp
ace properties m
i
ght result in
d
i
fficu
lties o
f
find
ing
g
l
o
b
a
l
op
ti
m
u
m
.
Therefo
r
e an
y
d
e
v
e
lop
m
en
t, i
m
p
r
ov
em
en
t o
r
an
alysis o
f
algo
rith
m
need t
o
be v
e
ri
fi
ed
wi
t
h
b
e
nchm
ark t
e
st
fu
nct
i
ons
[7]
,
[8]
.
T
h
e so
-cal
l
e
d be
nch
m
ark fu
nct
i
o
n
i
s
a
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
7,
No
. 3,
J
u
ne 2
0
1
7
:
16
4
3
– 16
50
1
644
m
a
t
h
em
at
i
c
al
fu
nct
i
o
ns t
h
at
has a de
fi
ne
d searc
h
s
p
ac
es and e
x
hibit
certain diffic
ulty classes such as
sep
a
rab
ility, la
n
d
s
cap
e wh
ich in
clu
d
e
m
u
lti
m
o
d
a
l fu
n
c
tio
n
s
, steep-dro
p, b
a
sin
or v
a
lley-typ
e
d
and
fun
c
tio
n
with
nu
ll-space effects or p
l
ateau
sh
ap
ed. Th
e prop
er
ties o
f
su
ch
d
i
fficu
lties are in
ten
d
e
d
to
sim
u
l
a
te th
e
charact
e
r
i
s
t
i
c
of
real
-
w
orl
d
pr
o
b
l
e
m
s
. Thi
s
pa
per
p
r
ese
n
t
s
a c
o
m
p
ari
s
o
n
of
nat
u
re
-i
ns
pi
re
d al
g
o
ri
t
h
m
s
wi
t
h
defi
ned
cl
ass
of
be
nc
hm
ark
pr
o
b
l
e
m
.
A sh
ort
o
v
er
vi
ew
o
f
7
nat
u
re-i
n
s
p
i
red al
go
ri
t
h
m
s
st
art
i
n
g
fr
om
l
o
n
g
-
estab
lish
e
d algo
rith
m
u
n
til recen
t op
timizati
o
n
al
go
rith
m
i
s
p
r
esen
ted
in
Sectio
n
2
.
Sectio
n
3
d
i
scu
sses
th
e
per
f
o
r
m
a
nce com
p
ari
s
on
of
m
e
t
a
-heuri
st
i
c
m
e
t
hod wi
t
h
b
e
nchm
ark f
u
n
c
t
i
on an
d t
h
e l
a
st
sect
i
on pres
ent
s
a
concl
u
si
o
n
of
t
h
i
s
pape
r.
2.
R
E
SEARC
H M
ETHOD
The ove
r
v
i
e
w of 7
m
e
t
a
-heu
r
i
st
i
c
al
gori
t
h
m
s
are
s
u
mm
arised in Table
1. The
perform
ance
of eac
h
al
go
ri
t
h
m
i
s
com
p
ared
wi
t
h
be
nc
hm
ark f
unct
i
o
n
as
pr
o
v
i
d
e
d
i
n
Ta
bl
e 2
.
T
h
ese
f
u
nct
i
o
n
s
have
di
ffe
re
nt
ch
aracteristic based
on
th
e d
i
fficu
lty class that
can
be sim
u
lated as
a r
eal-w
or
ld pr
ob
lem.
Tabl
e
1. T
h
e
o
v
er
vi
ew
o
f
7 m
e
t
a
-he
u
ri
st
i
c
al
go
ri
t
h
m
Num
Algorithm
(
y
ea
r)
Main fe
atur
es
1
Genetic Algor
ith
m, GA (1960s)
Inspired from ev
ol
ution’s theor
y
. Governed
b
y
3
operators: 1)
selection, 2) mutation
,
3)
crossover [2]
2
Differential Evo
l
ution, DE (1996)
Improvement of
GA with same o
p
erators. Adv
a
n
t
age: no
coding
needed
. De
cis
i
o
n
fac
t
or b
y
diff
er
enti
al we
ight
,
and
crossover proba
bilit
y
[2]
.
3
Simulated Annealing
,
SA (1983)
Trajector
y
-
b
a
sed
algor
ithm inspir
ed from metal
co
oling pro
cess. [2
]
4
Particl
e
Swarm
Optim
izatio
n,
PSO
(1995)
Swarm-based algorithm inspired
from swarming of creatur
e
s. Solution
is
at
trac
ted
to
lo
cal
and
globa
l b
e
s
t
in
ea
ch i
t
er
ati
on [2]
,
[3]
.
5
Firefly
Algor
ith
m, FFA (2008)
Inspired from flashing be
hav
i
our
of firef
l
ies.
Each
solution
is
attr
acted
to potential solu
tion based
on f
itn
ess [2]
.
6
Cuckoo Sear
ch,
CS (2009)
Inspired from
C
u
ckoo bird
par
a
sitism
m
e
thod. Solution m
oved
randomly
with
L
ѐ
v
y
flights. Some solution will b
e
removed
b
y
probability
,
[2]
,
[13]
7
Tree
Ph
y
s
io
log
y
Optim
iza
tion
,
T
P
O
(2013)
Inspired from plant growth
s
y
stem
with shoots an
d roots var
i
ab
les
.
Potential solu
tio
n (shoots) sear
ch
fo
r optimum driven
b
y
amplification
of root: root-sho
ot corr
elation
search str
a
teg
y
[6]
.
Each
d
e
fin
e
d
meta-h
eu
ristic
alg
o
rith
m
is co
m
p
ared
with
11
test fun
c
tio
ns as su
mmarized
in
Tab
l
e 2
.
The cha
r
acteri
s
tic of each test function incl
ude
s m
oda
lity,
separa
bility a
nd c
o
nti
nuity. W
i
t
h
hi
ghe
r modality,
th
e algo
rith
m
mig
h
t
trap in
l
o
cal m
i
n
i
m
a
,
wh
ich
resu
lts
a ne
gative im
pact on the
sea
r
ch
process a
w
ay from
tru
e
so
lu
tion
[7
]. Th
e sep
a
rab
ility is a
m
e
a
s
u
r
e of
fun
c
tion
d
i
fficu
lty, non
-sep
arab
le
fun
c
tio
n is
h
a
rd
t
o
so
lve
due t
o
c
o
m
poun
de
d effect
b
e
t
w
een eac
h v
a
ri
abl
e
s. Di
sc
o
n
t
i
n
u
o
u
s f
unct
i
on ha
s st
ep p
r
o
p
ert
i
e
s,
whi
c
h ha
s
certain flat and steep
surface
due
to the
fl
oo
r
effect of the function. T
h
i
s
might lead t
o
a slow c
o
nverge
n
ce
and
l
o
cal
t
r
a
p
ped
o
p
t
i
m
u
m
. Ot
he
r
pr
ope
rt
i
e
s o
f
t
e
st
fu
nc
t
i
on i
n
cl
u
d
e
b
o
wl
-sha
pe
d,
v
a
l
l
e
y
-
shape
d
o
r
st
eep
drops a
n
d flat
surface. Flat surface
problem
will lead a poor
algorithm to
be t
r
appe
d in l
o
cal opti
m
u
m
as
fl
at
ness
of
t
h
e
fu
nct
i
o
n
di
d
n
o
t
gi
ve a
n
y
i
n
f
o
r
m
at
i
on t
o
w
a
r
d
s gl
obal
o
p
t
i
m
u
m
.
Tabl
e
2. B
e
nc
h
m
ark f
u
nct
i
o
n
charact
e
r
i
s
t
i
c
Bench
m
ar
k function
M
U
S
NS
D
B
SD
P
F1 Ackley
✓
✓
F2 Dam
a
vandi
✓
✓
✓
✓
F3 Easo
m
✓
✓
✓
✓
F4 Griewank
✓
✓
✓
F5 Matyas
✓
✓
✓
F6 Michalewic
z
✓
✓
✓
F7 Rosenbr
ock
✓
✓
✓
F8 Shekel.F.
✓
✓
✓
✓
F9 Step
✓
✓
✓
✓
F10
WW
avy
✓
✓
F11
X.
S.
Yang
4
✓
✓
✓
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Co
mp
ara
tive
Stu
d
y
o
f
Met
a
-
Heu
r
istics Op
timiza
tio
n Algo
rith
m u
s
i
n
g Ben
c
h
m
a
r
k
Fun
c
tion
(I. Isma
il)
1
645
B
e
nchm
ark
f
unct
i
o
n i
s
s
e
l
ect
ed fr
om
[7]
wi
t
h
differen
t
ch
aracteristics; M
=
Mu
lti
m
o
d
a
l,
U= Un
im
o
d
a
l, S= Sep
a
rab
ility, NS= No
n-sep
a
rab
l
e, D=
Discon
tin
uou
s, B
=
Bo
wl-typ
e,
SD= Steep-d
rop
,
and
P= Plateau-sha
ped.
The pa
ram
e
ter
s
of each algorithm is designed
with
the best setting that it
can converge towards
gl
o
b
al
o
p
t
i
m
u
m
.
Di
ffere
nt
a
l
go
ri
t
h
m
m
a
y
have
di
f
f
ere
n
t
set
t
i
ng de
pen
d
i
n
g o
n
t
h
ei
r
nat
u
re o
f
co
di
ng a
n
d
search. T
h
e
para
m
e
ter settings
are ta
bulated in Ta
ble
3.
Tabl
e
3.Si
m
u
l
a
t
i
on
param
e
t
e
r fo
r eac
h al
g
o
r
i
t
h
m
Alg
o
.
Para
m
e
ters
Alg
o
.
Para
m
e
ters
GA
I
t
er
ation = 30
Population = 50
Mutation = Gauss.
Cr
ossover
= Scatter
e
d
Selection = stoch.
un.
SA
Init. te
m
p
. =
10
Final = 1e-
1
0
alpha=0.
9
5
DE
I
t
er
ation = 30
Diff.
weight = 0.
7
Cr
ossover
p.
= 0.
9
PSO
I
t
er
ation = 30
Pop.
= 100
α
=
0.6
;
β
=0
.6
FFA
Fir
e
flies = 100
Gen.
= 100
CS
Nests = 25
Gen.
= 100
pa r
a
te= 0.
25
T
P
O
I
t
er
ation = 30
Pop.
= 30
leaves = 30
α
= 0.
3
β
= 50
θ
=
0.9
3.
R
E
SU
LTS AN
D ANA
LY
SIS
The e
v
al
uat
i
o
n
i
s
base
d
on
c
o
m
put
er
pr
oce
ssor
o
f
2.
6G
H
z
. Eac
h
al
g
o
ri
t
h
m
i
s
sim
u
l
a
t
e
d
hu
n
d
re
d
ti
m
e
s fo
r ev
ery test fun
c
tion and
th
e
o
p
t
i
m
ized
p
a
ra
m
e
ters a
r
e c
o
m
p
ared
. T
h
e
para
m
e
ters fo
r c
o
m
p
arison
i
n
cl
ude
com
p
u
t
at
i
on t
i
m
e and co
nve
rgence
towa
rds global optim
u
m
.
3.
1.
C
o
mp
ut
at
io
n t
i
me
The com
putation tim
e
of each algorithm
is
carried
out onl
y
with a sim
p
le unim
odal function, which
m
a
ke i
t
sui
t
a
bl
e f
o
r
be
nc
hm
arki
ng
t
h
e c
o
nv
erge
nce s
p
ee
d
of
m
e
t
a
-heuri
s
t
i
c
al
gori
t
h
m
[
1
]
.
I
n
t
h
i
s
pap
e
r t
h
e
com
put
at
i
on t
i
m
e i
s
com
p
ared wi
t
h
u
n
i
m
odal
–t
y
p
ed
f
unc
t
i
on F
3
a
nd
F
5
, as
sh
o
w
n i
n
Fi
gu
re
8.
U
n
i
m
odal
t
e
st
fu
nct
i
o
n c
a
n
be
use
d
as
a b
e
nc
hm
ark f
o
r
n
o
t
onl
y
con
v
e
r
ge
nce
s
p
eed
,
but
al
so
ex
pl
oi
t
a
t
i
o
n
of
t
h
e
al
go
ri
t
h
m
[1]
.
The com
put
at
i
on t
i
m
e
i
s
dep
e
nde
nt
m
a
i
n
l
y
on
num
ber o
f
i
t
e
rat
i
on an
d
po
pul
at
i
o
n si
ze. I
n
t
h
i
s
stu
d
y
, each
algo
rith
m
is ex
ecu
ted
u
n
til ex
ceed
ing
th
e
n
u
m
b
e
r
o
f
iteration o
r
u
n
til no
imp
r
ov
em
en
t is a
c
h
i
ev
ed
(g
l
o
b
a
l op
ti
m
u
m
so
lu
tio
n
)
.
Fi
gu
re
8.
C
o
m
put
at
i
o
n t
i
m
e for
F3
(
r
i
g
ht
) a
n
d F
5
(l
eft
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
7,
No
. 3,
J
u
ne 2
0
1
7
:
16
4
3
– 16
50
1
646
Based on the c
o
m
p
arison, TPO and DE out
p
erform
ed
ot
h
e
r al
go
ri
t
h
m
i
n
t
h
e spee
d o
f
c
o
n
v
e
r
ge
nce
and
f
o
l
l
o
we
d
b
y
PSO.
Am
on
g t
h
e sl
owe
r
c
o
n
v
e
r
ge
d a
r
e
C
S
, FF
A, a
n
d
GA
. D
E
i
s
p
r
o
v
en t
o
c
o
n
v
er
g
e
bet
t
e
r
th
an
GA
wh
ich is supp
orted in [2
].
3.
2.
Co
ver
g
ence t
o
w
a
rds gl
ob
al
opti
m
u
m
The 1
1
benc
h
m
ark f
unct
i
o
ns
have
di
f
f
ere
n
t
di
ffi
c
u
l
t
y
l
e
vel
of p
r
o
b
l
e
m
-
t
ype as desc
ri
be
d i
n
p
r
e
v
i
o
u
s
section. This
will show t
h
e
capa
b
ility
of
each algorithm
whether ea
ch
of
t
h
em
ca
n sea
r
ch efficiently in
vari
ous
p
r
obl
e
m
t
y
pes.
B
a
sed
o
n
Fi
gu
re
9 a
n
d Ta
bl
e
4,
t
h
e
l
o
west
vari
a
n
ce
o
f
so
lu
tio
n is
foun
d in
TPO in all cases. C
S
sho
w
t
h
e sec
o
nd
g
o
od
pe
rf
or
m
e
r as t
h
is algo
rith
m
co
nv
erg
e
s co
nsisten
tly with
lower
variatio
n
ex
cep
t
for
F8
and F
1
1. G
A
al
so sh
ows co
ns
i
s
t
e
nt
l
y
good c
o
n
v
e
r
ge
nce ex
cept
fo
r F4, F
8
, F1
0 and F
1
1,
t
h
ese t
e
st
func
t
i
ons
have
m
a
ny
l
o
cal
opt
i
m
u
m
s wi
t
h
hi
g
h
e
r
di
ffi
cul
t
y
of
su
rfa
c
e
. T
h
e
per
f
o
r
m
a
nce
of
G
A
ca
n
be i
m
pro
v
ed
fu
rt
he
r
b
y
add
i
ng
d
i
fferen
t strateg
y
su
ch
as m
u
lti-p
a
ren
t
cro
s
so
v
e
r
[16
]
,
d
y
n
a
m
i
c ad
ap
tatio
n of cro
s
sover and
m
u
t
a
t
i
on [1
7]
,
fi
ne-t
u
n
i
n
g cr
oss
ove
r [
18]
and m
a
ny
m
o
re. DE has som
e
di
ffi
cul
t
y
t
o
track gl
ob
al
op
t
i
m
u
m
co
nsisten
tly in
m
u
lti
m
o
d
a
l with
steep
-d
rop
,
an
d p
l
ateau-sh
a
p
e
d
.
Fu
rtherm
o
r
e DE trapp
e
d
m
o
stly in
lo
cal
opt
i
m
u
m
for
F
3
.
SA
i
s
t
r
a
p
pe
d m
o
st
l
y
i
n
l
o
c
a
l
opt
i
m
u
m
for
F4
, F
8
a
n
d
F
11
. T
h
ese
f
unc
t
i
ons
ha
ve
feat
ure
o
f
m
u
lt
im
odal
and n
o
n
-se
p
a
r
abl
e
wi
t
h
st
eep
-d
ro
p. F
F
A
has
a go
o
d
co
n
v
er
gence i
n
pl
at
e
a
u-s
h
a
p
ed
f
u
n
c
t
i
o
n
ex
cep
t
with d
i
sco
n
tinuo
us
p
r
ob
lem
.
Th
is m
i
g
h
t be th
e reas
on
o
f
bro
a
d
e
r search ab
ility o
f
fireflies si
n
ce i
t
h
a
s
uni
que
fu
nct
i
o
n o
f
com
p
ari
s
o
n
wi
t
h
di
ffe
re
n
t
fi
refl
y
com
p
ani
o
n [
9
]
,
[
10]
.
Ho
we
ver F
F
A
al
so t
r
ap
pe
d i
n
l
o
cal
opt
i
m
u
m
as i
n
F8,
F9 a
n
d F
1
0. P
S
O al
s
o
s
h
ow
g
o
o
d
c
o
n
v
e
rge
n
ce e
x
cept
i
n
F2
, F
5
a
nd
F8.
Thi
s
m
i
ght
due t
o
the fast
be
havi
our
of pa
rticles res
u
lte
d in immature convergence
for flat
s
u
rface
problem. Based on all t
e
sted
benc
hm
ark fu
n
c
t
i
on,
F
8
has
t
h
e bi
g
g
est
vari
at
i
on of
s
o
l
u
t
i
o
n fo
r
al
l
al
go
ri
t
h
m
s
.
The reas
on
of suc
h
di
f
f
i
cul
t
y
i
s
com
b
i
n
at
i
on of m
u
l
t
i
m
odal
,
a st
eep gl
o
b
al
opt
i
m
u
m
sur
r
o
u
n
d
e
d
wi
t
h
bo
wl
-s
ha
pe
d wi
t
h
se
veral
l
o
cal
optim
u
m
and a
l
so a
plateau
shape that
covers
50% of
the se
arch
space
.
Table 4.
Mea
n
of
100
runs of each
algorithm
Algo
Count
F1
F2
F3
F4 F5
F6
F7
F8
F9
F10
F11
GA
100
0.
05
2.
2
-
0
.
75
0.
01
0
-
1
.
8
0.
38
-
5
.
03 0.
19
0.
03
-
0
.
48
DE
100
0
2.
18
-
23
0.
02
0
-
1
.
83 0.
24
-
4
.
65
0
0.
08
-
0
.
09
SA
100
0.
01
2.
11
-
0
.
24
0.
02
0
-
1
.
3
0
-
3
.
35 0 0.
01
-
0
.
01
PSO
100
0.
12
101.
48
-
0
.
98
0
0
-
1
.
8
0.
01
-
4
.
81 0 0.
01
-
0
.
97
FFA
100
0
0
0
0
0
-
1
.
76 0 -
6
.
09
11.
64
0.
1
-
0
.
25
CS
100
0.
01
1.
66
-
0
.
86
0
0
-
1
.
8
0.
01
-
6
.
36 0 0.
01
-
0
.
7
T
P
O
100
0
0
-
1
0
0
-
1
.
8
0
-
6
.
45 0
0
-
1
Th
e co
nv
erg
e
n
ce co
m
p
arison
with
ano
v
a
test fro
m
Fig
u
re 9
is tabu
lated
in
Tab
l
e 5 th
at shows
significa
nt
difference
of al
gorith
m
by each t
e
st function
(with p-val
u
e <
0.05) at
95%
c
o
nfi
d
ence
level. The
m
e
t
hod
bei
n
g
used t
o
di
sc
ri
m
i
nat
e
am
ong t
h
e m
eans i
s
Fi
sher'
s
l
east
si
gni
fi
ca
n
t
di
ffe
rence
(
L
SD
)
pr
oce
d
u
r
e
[1
5]
.
W
i
t
h
t
h
i
s
m
e
t
h
o
d
, t
h
ere
i
s
a 5.
0%
ri
s
k
of
cal
l
i
ng eac
h
p
a
i
r
o
f
m
eans s
i
gni
fi
ca
nt
l
y
di
f
f
ere
n
t
whe
n
t
h
e actua
l diffe
re
nce e
q
uals
0.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Co
mp
ara
tive
Stu
d
y
o
f
Met
a
-
Heu
r
istics Op
timiza
tio
n Algo
rith
m u
s
i
n
g Ben
c
h
m
a
r
k
Fun
c
tion
(I. Isma
il)
1
647
F1: F2:
F3:
F4: F5:
F6:
F7: F8:
F9:
F10: F11:
Fi
gu
re
9.
Di
st
ri
but
i
o
n
o
f
c
o
n
v
e
rge
n
ce
of
eac
h al
g
o
r
i
t
h
m
by
benc
hm
ark f
u
n
c
t
i
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E
V
o
l
.
7,
No
. 3,
J
u
ne 2
0
1
7
:
16
4
3
– 16
50
1
648
Tab
l
e
5
.
Sign
ifican
tly d
i
f
f
e
r
e
nt alg
o
r
ith
m
acc
o
r
d
i
ng
t
o
A
nova test
CS
DE
FFA
GA
PSO
SA TPO
F1
GA, P
S
O
GA, P
S
O
GA, P
S
O
SA, PS
O,
TPO
SA,
TPO
SA
F2
DE, FFA,
GA, P
S
O,
SA, TPO
FFA, P
S
O,
TPO
GA, P
S
O,
SA
PSO,
TPO
SA,
TPO
TPO
CS, DE
,
G
A
,
SO
,
SA
F3
DE
ALL
DE
DE
DE
DE
DE
F4
DE, GA,
SA
FFA, GA,
PSO, TPO
GA, S
A
PSO, S
A
,
TPO
SA
CS, FFA
,
GA, P
S
O,
TPO
DE, TPO
F5
PSO
PSO
PSO
PSO
ALL
PSO
PSO
F6 SA
FFA,
S
A
DE,
SA
SA
SA
ALL
SA
F7
DE, GA
FFA, GA,
PSO, S
A
,
TPO
PSO, S
A
,
TPO
F8 DE,
FFA,
GA, P
S
O,
SA
FFA, GA,
SA, TPO
GA, P
S
O,
SA, TPO
SA, TPO
SA, TPO
TPO
DE, FFA,
GA, P
S
O,
SA
F9 FFA
FFA
GA,
P
S
O,
SA, TPO
F10
DE,
FFA,
GA
ALL
GA, P
S
O,
SA, TPO
SA, TPO
DE, FFA
DE, FFA,
GA
DE, FFA,
GA
F11
ALL
FFA, GA,
PSO, TPO
GA, P
S
O,
SA, TPO
PSO, S
A
,
TPO
SA
TPO
CS, DE
,
FFA,GA
,
S
A
The co
n
v
er
ge
n
ce dy
nam
i
c
i
n
a si
ngl
e r
un i
s
com
p
ared wi
t
h
t
w
o be
nc
hm
ark f
unct
i
o
ns:
F
6
and F
7
as
depi
ct
ed
i
n
Fi
gu
re
1
0
.
It
ca
n
be
o
b
se
rve
d
t
h
at
G
A
,
FF
A,
PSO
an
d
TP
O
co
nve
r
g
e t
o
w
a
rds
gl
ob
al
o
p
t
im
u
m
faster c
o
m
p
are
d
to ot
hers
.
Fi
gu
re
1
0
. C
o
n
v
er
ge
nce
of
be
st
sel
ect
ed al
g
o
r
i
t
h
m
i
n
one
r
u
n
fo
r F
6
(ri
ght
)
an
d F
7
(l
eft
)
4.
CO
NCL
USI
O
N
Meta-h
euristic op
ti
m
i
zatio
n
alg
o
rith
m
is
ab
le to so
l
v
e wid
e
ran
g
e o
f
non
lin
ear op
ti
m
i
zatio
n
pr
o
b
l
e
m
s
opt
i
m
al
ly
[1
9]
. T
h
e reas
on
f
o
r
t
h
ese ad
va
nt
ages
i
s
fr
om
i
t
s
uni
que
cha
r
act
eri
s
t
i
c
of
di
ve
rsi
f
i
cat
i
on
and e
x
ploitation capa
b
ility. Howe
ve
r each al
gorithm
has di
ffe
rent bac
k
ground philosophy that blende
d wit
h
dynam
i
c search strategy. T
h
is leads to the diffe
re
n
ce i
n
c
o
n
v
e
r
ge
nce an
d com
put
at
i
o
n
t
i
m
e
, whi
c
h m
i
ght
reveal
t
h
e abi
l
i
t
y
of rea
c
hi
ng
gl
o
b
al
o
p
t
i
m
u
m
by
di
ffere
nt
t
y
pe of
di
f
f
i
c
ul
t
i
e
s. In t
h
i
s
pa
per
,
7
m
e
t
a
-heuri
st
i
c
s
are
com
p
are
d
wi
t
h
11 be
nch
m
ark
f
u
nct
i
o
n
s
.
B
a
sed o
n
t
h
e
statistical
co
mparis
on,
TP
O per
f
o
r
m
s
si
gni
fi
cant
l
y
bet
t
e
r com
p
are
d
t
o
ot
he
r al
g
o
r
i
t
h
m
s
i
n
m
o
st
benc
hm
ark fu
n
c
tio
n. Th
is is d
u
e
to
th
e p
a
rallel search
o
f
lo
ca
l
opt
i
m
u
m
(i
ndi
vi
d
u
al
l
eaf) an
d gl
o
b
al
o
p
t
i
m
u
m
(branc
hes)
.
W
i
t
h
t
h
e am
pl
i
f
i
cat
i
on searc
h
fr
om
root
sy
st
em
,
th
e search
p
r
ocess b
eco
m
e
b
r
o
a
d
e
r, th
u
s
the p
r
ob
ab
ility
o
f
find
ing
a tru
e
so
lu
tio
n
will
b
e
g
r
eater [11
]
. CS is
al
so abl
e
t
o
search
gl
o
b
al
opt
im
u
m
for vari
o
u
s t
y
pe of
pr
ob
l
e
m
except
for
benc
hm
ark fu
n
c
t
i
on F1
1 (
f
u
n
c
t
i
o
n
with m
a
ny local optim
a and single
stee
p
global optim
u
m
). This m
a
y due
to t
h
e ra
ndom
search vi
a L
ѐ
vy
f
lig
h
t
s [2
].
The searc
h
usi
ng C
S
al
go
ri
t
h
m
can be im
pr
ove
d f
u
rt
her i
f
t
h
e n
u
m
b
er of
nest
s are m
o
re t
h
an
num
ber
o
f
lo
cal op
tima [12
]
. Th
is id
ea is also
sup
ported with
TPO feat
ure
of wide
r searc
h
. PSO
has suc
cessful
so
lu
tion
in
some b
e
n
c
h
m
ark fun
c
tio
ns, bu
t
its well-k
n
o
w
n
stab
ility p
r
ob
lem
restricts
t
h
e su
ccess rate o
f
th
is
algorithm
.
FFA shows s
u
cce
ssful sea
r
c
h
in
m
o
st plat
eau
-typ
e prob
lem
.
Howev
e
r it h
a
s li
m
i
tatio
n
with
m
a
n
y
lo
cal o
p
tim
a a
n
d
d
i
sco
n
tinuou
s pro
b
l
em
. D
i
scon
tin
uou
s fun
c
tio
n, F9
h
a
s
sev
e
r
a
l
p
l
ateaus th
at mig
h
t
r
e
su
lt in
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Co
mp
ara
tive
Stu
d
y
o
f
Met
a
-
Heu
r
istics Op
timiza
tio
n Algo
rith
m u
s
i
n
g Ben
c
h
m
a
r
k
Fun
c
tion
(I. Isma
il)
1
649
po
o
r
con
v
e
r
ge
nce. G
A
has
be
t
t
e
r perf
orm
a
nce com
p
ared t
o
DE as
m
o
st
of t
h
e sol
u
t
i
o
n i
n
10
0 ru
ns c
o
n
v
e
rge
d
t
o
wa
rds
gl
o
b
al
opt
i
m
u
m
. SA sho
w
s sl
i
g
ht
l
y
po
o
r
co
n
v
er
ge
nce i
n
s
p
eci
fi
c
m
u
lt
im
odal
t
yped
wi
t
h
st
ee
p
gl
o
b
al
opt
i
m
u
m
funct
i
on
(F
4,
F
6
, F
8
an
d F
1
1 a
r
e m
u
l
t
i
m
odal
wi
t
h
m
o
re l
o
cal
o
p
t
i
m
u
m
).
C
o
m
put
at
i
on t
i
m
e
i
s
co
m
p
ared usi
n
g u
n
i
m
odal
-
t
y
ped f
u
n
c
t
i
on. DE a
nd
TPO com
put
es
t
h
e pro
b
l
e
m
fast
er com
p
are
d
t
o
ot
her al
g
o
r
i
t
h
m
.
A
m
ong
t
h
e sl
owe
r
co
m
put
at
i
on t
i
m
e
i
s
FFA an
d GA
.
The fi
ndi
ng
fr
om
this paper suggests a necessity of
m
o
re
comparison st
udies in di
fferent field especially
in real worl
d proble
m
as i
t
wi
l
l
gro
w
paral
l
e
l
wi
t
h
adva
nce o
f
sci
e
nce an
d t
ech
no
l
ogy
. S
o
m
e
prop
ose
d
exam
pl
e of areas t
h
at
m
i
ght
be c
o
n
s
i
d
ere
d
f
o
r
real
wo
rl
d
a
r
e m
a
nufact
uri
n
g
i
m
provem
e
nt
, c
ont
rol
e
n
gi
neeri
n
g
an
d
ro
ut
i
n
g
sy
st
em
ACKNOWLE
DGE
M
ENTS
Thi
s
researc
h
was s
u
pp
ort
e
d
by
U
n
i
v
e
r
si
t
y
Tek
nol
ogi
PE
TR
ON
AS
. T
h
e
aut
h
o
r
s
wo
ul
d
l
i
k
e t
o
t
h
a
n
k
t
h
e uni
versi
t
y
or
ga
ni
zat
i
on f
o
r t
h
e su
pp
o
r
t
.
The aut
h
o
r
s w
oul
d al
so l
i
k
e t
o
t
h
an
k t
h
e re
v
i
ewers f
o
r t
h
o
u
ght
ful
l
comments that
adde
d t
o
the
efficacy of this
paper.
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I
S
SN
:
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08
IJEC
E
V
o
l
.
7,
No
. 3,
J
u
ne 2
0
1
7
:
16
4
3
– 16
50
1
650
BIOGRAP
HI
ES OF
AUTH
ORS
Idr
i
s Ismail
receiv
e
d Bach
elor
Degree in El
e
c
tri
cal Eng
i
ne
er
ing from
W
i
chita S
t
at
e Uni.,
Kansas, USA in 1986. He receiv
e
d Master Degree in
Control S
y
s
t
em Engineering f
r
om Sheffield
Uni,, UK in 2000 and PhD in
Electrical & Electronic Eng
i
neering from The University
of
Manchester
, UK in 2009. He is
an Assoc Prof at
Universiti
Tek
nologi Petronas
and regist
ered
Professional Eng
i
neer
(with
practice)
with
Board
of Engin
eers, Malay
s
ia
Abd
u
l Hanif A
bdul Halim
was born in Sarawak, Mala
y
s
ia in 1
982. He receiv
e
d Bachelor an
d
Master degr
ee
i
n
Ele
c
tr
ica
l
Au
tom
a
tion Eng
i
n
eering
from
Universit
y
of
Applied Sc
ien
ces
Cologne, German
y
in 2009
. He
is currently
purs
u
i
ng the Ph.D. d
e
gree in
Electr
i
cal
Engineering
from Universiti
Teknologi PETRONAS, Malay
s
ia. He
is also currently
emplo
y
ed
as a
pro
cess
R&D engin
eer f
r
om
a m
a
nufac
t
u
ring com
p
an
y
i
n
M
a
la
ys
i
a
.
The
res
ear
ch
areas
o
f
P
h
.D. deg
r
ee
includ
e con
t
rol sy
stem
and optimization
algorith
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.