TELKOM
NIKA
, Vol. 11, No. 10, Octobe
r 2013, pp. 5
579 ~ 5
587
ISSN: 2302-4
046
5579
Re
cei
v
ed Ap
ril 23, 2013; Revi
sed
Jun
e
22, 2013; Accepted July 1
0
,
2013
A Comparison of Improved Artificial Bee Colony
Algorithms Based on Differential Evolution
Jianfe
ng Qiu
1
, Ji
w
e
n Wang
2
, Dan Yang
3
, Juanxie*
4
1, 2,
3
School of Comp
uter Scie
nce an
d T
e
chnolo
g
y
, An
hu
i U
n
iversit
y
(AHU)
, Hefei 230
03
9, Anhui, Ch
in
a
4
Department o
f
Mathematics & Ph
y
s
ics, An
h
u
i Univ
ersit
y
of
Architecture (
A
IAI),
Hefei 23002
2, Anhu
i,
Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: qiuji
anf@
a
h
u
.
edu.cn
1
, w
j
w@ah
u.edu.c
n
2
, y
a
ngd
an
198
9
102
5@
ya
ho
o.cn
3
,
qou
jia
nf@so
h
u
.
com*
4
A
b
st
ra
ct
T
he Artificial
Bee Co
lo
ny (ABC) alg
o
rith
m is a
n
active
field of opti
m
i
z
at
io
n bas
ed
on sw
ar
m
intell
ig
ence
in
recent ye
ars.
Inspire
d
by t
he
muta
ti
on s
t
rategies
use
d
in D
i
fferenti
a
l
Evoluti
on (
D
E)
a
l
go
ri
thm
,
th
i
s
p
a
p
e
r
i
n
trod
u
c
e
d
th
ree
typ
e
s
stra
te
g
i
e
s
(“rand”
, “be
s
t”
, an
d
“cu
rre
n
t
-to
-
be
st”
)
a
n
d
on
e
o
r
two
nu
mb
ers of dis
t
urbanc
e vecto
r
s to ABC alg
o
r
i
thm. Alth
ou
gh
indiv
i
d
ual
muta
tion strateg
i
es i
n
DE hav
e b
e
e
n
used i
n
ABC alg
o
rith
m by s
o
me rese
arch
ers in di
ffere
nt occasio
n
s, there hav
e not
a compre
hens
i
v
e
app
licati
on a
n
d
comparis
on
of the
mutati
on strategi
es used i
n
ABC alg
o
rith
m. In this pa
per, the
s
e
improve
d
ABC
alg
o
rith
ms c
a
n be
an
aly
z
e
d
by a s
e
t of testing fu
nctio
n
s
incl
udi
ng th
e rap
i
dity
of th
e
conver
genc
e. T
he results sh
ow
that
those i
m
pr
ove
m
ents base
d
on D
E
a
c
hiev
e better p
e
rformanc
e in t
h
e
w
hole tha
n
bas
ic ABC alg
o
rith
m.
Ke
y
w
ords
:
Artificial B
ee Co
lo
ny, Differentia
l Evoluti
on,
Sear
ch Strategy, Best, Rand, Curr
ent-to-Best.
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Optimizatio
n
is on
e of
attractive fiel
ds in
not o
n
ly aca
demi
c
re
se
arch
but also
engin
eeri
ng
pra
c
tice. M
o
st problem
s in re
al world ca
n be
redu
ced to
solve a
class of
optimizatio
n probl
em
s. Usually, a
class
of unco
n
st
rai
ned optimi
z
at
ion task ca
n
be formul
ated
as
follows
:
12
m
i
nf(
)
,
[
,
,
..
.,
]
D
x
xx
x
x
(1)
whe
r
e
x
is o
p
timized
varia
b
l
e
and
D
i
s
t
he nu
mbe
r
o
f
param
eters to be o
p
timized. A
s
the
compl
e
xity of the optimizat
ion pr
oblem i
n
crea
sing, tra
d
itional opt
imi
z
ation m
e
tho
d
s
cann
ot sol
v
e
su
ch pr
oble
m
s well.
Re
cently, biol
ogical-in
spi
r
e
d
optimi
z
atio
n al
go
rithm
s
have be
en
propo
sed to
sol
v
e su
ch
as hig
h
-dime
n
sio
n
, nonlin
ear o
p
timizati
on problem
i
n
real
wo
rld.
The ant col
ony optimization
(ACO
) is in
sp
ired by assig
n
ment
and
co
operation am
ong differe
nt colo
nie
s
to solve optimizat
ion
probl
em
s [1]. The pa
rticle
swarm
optimi
z
ation
(PSO) is a meta
-he
u
risti
c
sea
r
ch
method b
a
sed
on so
cial b
e
h
a
vior of bird
s
and ha
s be
en
widely
used
to solve vario
u
s optimi
z
ati
on pro
b
lem
s
[2]
[3].
The Differenti
a
l Evolution (DE) algo
rith
m wh
ich is si
mulating biol
ogy evolution
process
has bee
n o
n
e
of comp
etitive form evol
ution alg
o
rith
m [4] [5]. It has
been
successful in
solvi
ng
high-dimen
s
i
on, non-li
nea
r, large
-
scal
e
,
mult
imodal optimize
d
problem
s usi
n
g
DE algorith
m
s
and th
eir vari
ants. T
he
performan
ce
of t
he
DE al
gorit
hms is relyin
g on
three
st
age
s: mutatio
n
,
cro
s
sove
r a
n
d
sele
ction.
Acco
rdi
ng to
different
mu
tation strateg
i
es
and
ado
pted n
u
mbe
r
of
differen
c
e
ve
ctor, th
e fre
quently u
s
e
d
mutation
st
rategie
s
in li
terature in
cl
ude
DE/ra
n
d
/
1
,
DE/ran
d/2, DE/best/1, DE/best/2
, DE/
c
urrent-to
-
be
st
/1, and DE/
c
urrent-to
-
be
st
/2.The detail
e
d
analysi
s
and
comp
ari
s
o
n
s
can
be
de
scri
bed i
n
[6]. Ea
ch
DE al
go
rithm vari
ants
may be
effect
ive
over
som
e
p
r
oblem
s a
n
d
p
oor over othe
r p
r
obl
em
s.
It is not
po
ssi
b
l
e to m
a
ke o
n
e DE
alg
o
rith
m
alway
s
availa
ble over all p
r
oblem
s [7].
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 557
9 –
5587
5580
The Artificial
Bee Colo
ny (ABC) algo
rith
m is a meta-heuri
s
tic al
go
rithm introd
u
c
ed by
Karab
oga in
2005 [8]. A result abo
ut the perform
an
ce
about ABC, PSO, and DE sho
w
that ABC
is better tha
n
or simil
a
r to
those a
bove
listed in [9]. The sta
nda
rd
ABC algorit
hm divided i
n
to
three
stage
s,
su
ch a
s
em
ployed be
es
stage, o
n
loo
k
er be
es
stag
e, and sco
u
t bee
s stag
e, by
imitating fora
ging beh
avior of bee colon
y
.
T
he Bee Swarm Intelli
gen
ce ha
s b
een sh
owed
b
y
division
of l
abor an
d lo
cal inte
ra
ctin
g amo
ng
be
e colony. To
achieve m
o
re n
e
cta
r
, e
a
ch
employed
be
e in be
e colo
ny adju
s
ts its sea
r
ching
di
rection
acco
rd
ing to its visu
al inform
ation
in
the neigh
borh
ood of the on
e in its memo
ry.
The
stand
ard
ABC alg
o
rith
m is
ea
sy to i
m
pl
eme
n
t an
d fewer
para
m
eters. However, the
sea
r
ching
strategie
s
used
in stand
ard
ABC is mo
re
getting trap
ped into lo
ca
l optimizatio
n
in
solving
multi
m
odal
proble
m
s o
r
slo
w
e
r
co
nverg
e
n
c
e spee
d in
solving uni
mo
dal p
r
obl
ems. In
orde
r to improve the perf
o
rma
n
ce of ABC, so
me
mutation stra
tegies in
DE will be u
s
ed
and
some a
nalysi
s
and
comp
arison
will also
be made
sim
u
ltaneo
usly in
this pape
r.
The rest
of this p
ape
r i
s
orga
nized a
s
follows. Section 2 introd
uce
s
the
ba
sic ABC
algorith
m
an
d som
e
vari
ants. A com
p
reh
e
n
s
ive i
m
prove
d
ABC alg
o
rithm
s
based o
n
the
mutation
strat
egie
s
u
s
e
d
in
DE
algo
rith
m are el
abo
rated in
Se
ction 3. Exp
e
ri
mental
settin
g
an
d
results a
r
e prese
n
ted in Se
ction 4. Finall
y
, conclu
sio
n
s are
sum
m
arized in Se
ctio
n 5.
2. Artificial Bee Colony
Optimizer
In this
se
ctio
n, we
outline
the p
r
o
c
ed
ure of
b
a
si
c A
B
C alg
o
rithm
and
so
me v
a
riant
s of
ABC algorith
m
. Meanwhil
e
, some qu
estions
existin
g
in the above
have bee
n propo
sed.
2.1. Basic Ar
tificial Bee
Colon
y
Optimizer
ABC algo
rith
m imitates th
e foragi
ng be
havior
of ho
n
e
y bee. The i
ndividual of b
ee col
ony
are cl
asse
d into one of three types a
c
cording the
d
i
fferent divisi
on of labor, that is, emplo
y
ed
bee
s, onloo
ker bee
s, and
scout bee
s.
Before
se
arching i
n
sea
r
ch spa
c
e, the
first thi
ng
we shoul
d d
o
is initiali
zatio
n
. The
initialization
i
n
ABC alg
o
ri
thm is
ran
d
o
m
ly pr
od
uci
n
g food
so
urces to
cove
r t
he whole
se
arch
spa
c
e a
s
mu
ch a
s
it possi
bly can. The
positio
n
of a food sou
r
ce repre
s
e
n
ts a
possibl
e sol
u
tion
in the D-di
me
nsio
n se
arch
spa
c
e. The p
o
sition
i
x
is pro
duced a
s
follows [11]:
mi
n
m
a
x
mi
n
(0
,
1
)
(
)
ij
j
j
j
x
x
rand
x
x
(2)
whe
r
e
12
,
(,
,
,
,
)
ii
i
i
j
i
D
x
xx
x
x
is the D-dim
e
n
s
ion
position ve
ctor of the ith food source;
ij
x
is the jth co
mpone
nt of the ith vector;
(0
,1
)
rand
is a rand
om
numbe
r in the ran
ge [0, 1].
After initializa
t
ion, the search
process is condu
cted b
y
the
employed bee
s, onl
ooker be
es,
and
scout b
e
e
s
. In ABC, em
pl
oyed be
es p
r
odu
ce
modif
i
cation
to the
cu
rrent food
so
urce i
n
th
e
neigh
bor of the food source a
c
cordi
ng
to its memo
ry
(that is simil
a
r to the mutation in DE).
The
modificatio
n
can be de
scrib
ed as follo
ws:
()
ij
ij
i
j
i
j
k
j
vx
x
x
(3)
After produ
ci
ng
i
v
, the fitness value is
cal
c
ulate
d
for every
i
v
:
1/
(
1
)
ii
f
itn
e
s
s
f
if
0
i
f
1(
)
i
ab
s
f
if
0
i
f
(4)
whe
r
e
i
f
is function value of
the candi
dat
e solutio
n
i
v
.
For onl
oo
ker bees, they alway
s
appe
ar in the foo
d
sou
r
ce wh
ere ab
und
ant
nectar
amount is. So
, the probabili
ty value
i
P
for e
v
ery food sou
r
ce i
s
de
scrib
ed as follo
ws:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Com
pariso
n
of Im
proved
Artificial Bee Colo
ny Algo
ri
thm
s
Based o
n
… (Juan
xie
)
5581
1
i
i
NP
i
i
f
it
ne
ss
P
f
it
ne
s
s
(5)
In onlo
o
ker
b
ees’
sta
ge, a
rand
om
num
ber in th
e ran
ge
(0, 1
)
i
s
p
r
odu
ced
for ev
ery foo
d
source. If probability value
i
P
is g
r
eate
r
th
an the
ran
d
o
m
numb
e
r, t
he onl
oo
ker
bee b
e
come
s
employed b
e
e
. After that,
the onloo
ke
r bee
s’ se
arch
strategy for e
v
ery food sou
r
ce i
s
the sa
me
as the empl
o
y
ed bee
s in (3).
Whe
never a f
ood
so
urce i
s
depl
eted
by
empl
oyed
be
es, the
empl
o
y
ed be
es a
s
sociate
d
with it will a
band
on the
food source,
and b
e
co
m
e
scout. Th
e
scout be
es perfo
rm gl
o
bal
exploratio
n in
sea
r
ch sp
ace and the se
arch pr
ocess can be d
e
fined a
s
(2
). The flowcha
r
t of
basi
c
ABC al
gorithm i
s
given in Figu
re 1
.
Initial
Population
iter=1
Employed bees:
1. searching in th
e neighbor of the curre
nt food source
2. evaluate
the candidate solution
P
r
obability calculation
fo
r every food source
Onlooker bees:
1. choose a food
s
o
urce which is abundance
in
nectar amo
unt according to the probab
ility value
2. search stra
tegy the same as employed
bees
using
3. evaluate th
e quality of the candidate
solution
Scout bees:
1. abandon some f
ood source, if the quality of a
food source a
nd its neighbors are not enhanc
ed for
several tria
ls
2. regenera
te some food source for e
xploration
iter<iter_AMX
iter=iter+1
End of ABC
Figure 1. Flowchart of the
basi
c
ABC.
2.2. Some Variational ABC
Algorith
m
s
Since it
wa
s fi
rst int
r
od
uced
in 20
05
by D.
Karab
oga [
8
], it has
attracted many
attentions
in recent years
.
In this
s
e
c
t
ion, s
o
me of
t
he variation
s
of ABC are briefly reviewed
.
A modified A
B
C alg
o
rithm
whi
c
h th
e fre
quen
cy an
d t
he ma
gnitud
e
have p
e
rtu
r
b
a
tion is
introdu
ce
d by
D. Kara
bog
a
[11]. In basi
c
ABC, in orde
r to produ
ce
a ne
w solutio
n
, there i
s
o
n
ly
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 557
9 –
5587
5582
one pa
ram
e
ter in
i
x
to be chang
ed. In [11], a modification
rate is introdu
ced t
o
improve th
e
conve
r
ge
nce
sp
eed.
In
spi
r
ed
by PSO,
an i
m
prove
d
ABC
algo
ri
thm, GABC,
is p
r
o
posed
b
y
inco
rpo
r
ating
the inform
ation of glo
bal be
st sol
u
tion to the
cu
rre
nt se
arch
spa
c
e.
The
improvem
ent of
the search strat
egi
es enhances the exploitation abilit
y which i
s
poor
at the basic
ABC algo
rith
m [12].
Litera
ture [13
-
16] i
n
trodu
ce
d so
me varia
n
ts o
f
ABC based
“be
s
t” a
nd “ra
nd”
strategi
es wh
ich
are
u
s
ed
in DE.
The
s
e im
provem
ents whi
c
h a
r
e
b
a
sed on
DE
a
r
e not
a
comp
re
hen
si
ve versi
o
n
s
a
nd la
ck of
cro
s
swi
s
e
co
m
p
arison. So, i
n
this
pap
er,
we will
ma
ke
u
s
e
of
six
m
u
tatio
n
st
rategi
es i
n
DE to improve
basi
c
ABC algo
rithm. Mean
while, a
comprehe
nsi
v
e
comp
ari
s
o
n
a
m
ong the six
improve
d
ABC algo
rithm will be explaine
d later.
3. Impro
v
ed
ABC Algo
rithms based o
n
DE
There are
six mutation strategie
s
whi
c
h a
r
e used in DE
algorithm. Th
e formula
s
ca
n be
expres
sed as follows
[6]:
“DE/ra
nd/1:”
12
3
,
,,
,
()
ii
i
iG
rG
r
G
rG
VX
F
X
X
(6)
“DE/bes
t/1:”
12
,,
,,
()
ii
iG
b
e
s
t
G
rG
rG
VX
F
X
X
(7)
“DE/current-t
o-be
st/1:”
12
,,
,
,
,,
()
(
)
ii
iG
iG
b
e
s
t
G
i
G
rG
rG
VX
F
X
X
F
X
X
(8)
“DE/ra
nd/2:”
12
3
4
5
,
,,
,
,
,
()
()
ii
i
i
i
iG
r
G
r
G
rG
rG
rG
VX
F
X
X
F
X
X
(9)
“DE/bes
t/2:”
12
3
4
,,
,,
,,
()
()
ii
i
i
iG
b
e
s
t
G
rG
rG
rG
r
G
VX
F
X
X
F
X
X
(10
)
“DE/current-t
o-be
st/2:”
12
,,
,
,
,,
()
(
)
ii
iG
iG
b
e
s
t
G
i
G
rG
rG
VX
F
X
X
F
X
X
34
,,
()
ii
rG
r
G
FX
X
(11
)
In these mut
a
tion
strategi
es, the
r
e
are
two
asp
e
ct
s differentiate one DE
fro
m
anoth
e
r.
One i
s
the va
rian
ce type (“rand
”,” b
e
st
”),
the other
i
s
the num
bers o
f
the
disturba
nce ve
ctor
(o
ne
or two). T
he
DE/ran
d a
c
hi
eved g
ood
re
sults in
solvi
ng
u
n
imod
al and se
parabl
e
fun
c
tion
s. The
differen
c
e
be
tween
DE/ra
nd/1 a
nd
DE/rand/2
is
the
numb
e
r
of d
i
sturb
a
n
c
e ve
ctor.
DE/ran
d
/
2
gets b
e
tter
o
p
timization
a
b
ility in multimodal
and
n
on-sep
a
rable
pro
b
lem
s
th
an DE/rand/1
.
The
gree
dy varia
n
ts (“be
st” or “cu
rre
nt-to-b
e
st”) introd
uce the best
food so
urce (solutio
n) to the
curre
n
t po
pu
lation. The
fast
conve
r
ge
nce
speed
can b
e
a
c
hie
v
ed in
solvi
ng o
p
timizati
on
esp
e
ci
ally multimodal p
r
o
b
lems. But, o
n
the ot
he
r h
and, the in
creasi
ng of co
nverge
nce sp
eed
may lead to some p
r
obl
e
m
s su
ch a
s
prem
ature
co
nverge
nce in
solvi
ng multimodal proble
m
s.
DE/current-to-best
which
utilize
not only the current
sol
u
ti
on
(food source), bu
t also the
best
solutio
n
red
u
c
e the ch
an
ce prem
ature
relative
to the DE/best. F
r
om the previous
studie
s
, it
can
not be ex
pecte
d that an algorith
m
can find opt
imi
z
ed
solutio
n
for any type p
r
oble
m
s. Eve
r
y
mutation strategie
s
in DE a
dapt to one cl
ass pro
b
lem.
In ba
sic AB
C, em
ployed
bee
s find
n
e
w fo
od
so
u
r
ce
whi
c
h i
s
simil
a
r to
mutation
strategi
es i
n
DE. Inspired
by this, we will impor
t the
six mutation
strategi
es to
the basi
c
ABC
algorith
m
an
d make an
al
ysis and
com
pari
s
on
s
sy
stemati-cally and com
p
letel
y
. The improved
ABC algorith
m
s ba
sed o
n
DE are d
e
scri
bed a
s
follows:
“ABC/ra
nd/1:”
1,
3
2
,,
,
()
rm
im
r
m
r
m
VX
X
X
(12
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Com
pariso
n
of Im
proved
Artificial Bee Colo
ny Algo
ri
thm
s
Based o
n
… (Juan
xie
)
5583
“ABC/bes
t/1:”
12
,,
,
,
()
im
b
e
s
t
m
r
m
r
m
VX
F
X
X
(13
)
“ABC/current-to-be
st/1:”
12
,,
,
,
,
,
()
(
)
im
im
b
e
s
t
m
i
m
r
m
r
m
VX
X
X
X
X
(14
)
“ABC/ra
nd/2:”
12
3
4
5
,,
,
,
,
,
()
(
)
i
m
r
m
rm
r
m
rm
r
m
VX
X
X
X
X
(15
)
“ABC/bes
t/2:”
12
3
4
,,
,
,
,
,
()
(
)
i
m
b
e
s
t
m
r
mr
m
r
mr
m
VX
X
X
X
X
(16
)
“ABC/current-to-be
st/2:”
12
,,
,
,
,
,
()
(
)
im
im
b
e
s
t
m
i
m
r
m
r
m
VX
X
X
X
X
34
,,
()
rm
r
m
XX
(17
)
In orde
r to test and verify the ability (12
)
– (17) in
solv
ing unimo
dal
and multimo
d
a
l
probl
em
s, we will make m
u
lti-
group experiment
s for dif
f
erent
type of test functions.
4. Experiment Arr
a
ngem
e
nt and
Res
u
lts
4.1. Test Fun
ctions
In this se
ctio
n, a set of b
a
si
c test fun
c
tions (unimo
d
a
l and multi
m
odal
) will b
e
use
d
to
test six impro
v
ed ABC algo
rithms
(12
)
-(1
7
).
The testin
g function
s are listed in tabl
e1.
Table 1. Basi
c Fun
c
tion
Function
Global min
Search range
Formula
Shpere(
UM)
0
[-100,100]
D
2
1
()
n
i
i
f
xx
Rosenbrock(UM
)
0
[-2.048,2.0
48]
D
1
22
2
1
1
(
)
[
100(
)
(
1
)
]
n
ii
i
i
fx
x
x
x
Ackley
(MM
)
0
[-32.768,3
2
.768]
D
2
1
1
1
(
)
20
ex
p(
0.2
)
1
e
xp(
c
o
s(
2
)
)
2
0
n
i
i
n
i
i
fx
x
n
x
e
n
Grie
w
ank
(MM)
0
[-600,600]
D
2
1
1
1
()
c
o
s
(
)
1
4000
n
n
i
ii
i
x
fx
x
i
Weierstrass(MM)
0
[-0.5,0.5]
D
ma
x
10
ma
x
0
()
(
[
c
o
s
(
2
(
0
.
5
)
)
]
)
[c
o
s
(c
o
s
(
2
0
.
5
)
)
]
,
0
.
5
,
3
,
m
a
x
2
0
Dk
kk
i
ik
k
kk
k
k
fx
a
b
x
Da
a
b
a
b
k
Rastrigin(MM)
0
[-5.12,5.12
]
D
2
1
(
)
[
1
0
c
os
(2
)
+
10]
n
ii
i
fx
x
x
Schw
efel(M
M)
0
[-500,500]
D
1
(
)
*
4
1
8
.9
82
88
7
(
sin(
)
)
n
ii
i
f
xn
x
x
4.2. Parameter Setting
s a
nd Arra
nge
ment
The exp
e
rim
ents
will be
d
i
vided into th
ree pa
rt
s. Fi
rst, aimed at th
ese te
sting fu
nction
s
in table1, th
e
popul
ation
size wa
s
10, a
n
d
the m
a
ximu
m functio
n
ev
aluation
s
wa
s 30,000
for
1
0
-
dimen
s
ion. Al
l experi
m
ent
s we
re
re
peat
ed 3
0
time
s.
In orde
r to
co
mpare o
u
r
al
gorithm
s to
ot
her
existed al
go
ri
thms b
a
sed
swarm intellig
e
n
ce,
what
th
e
paramete
r
setting we
u
s
e
d
wa
s th
e
sa
me
as [11]. The e
x
perime
n
tal result
s are li
sted in Table
2
. Secon
d
, the compa
r
ison
s for co
nvergen
ce
ability among the
six improved ABC were
shown in Figure
2
– Figure
13.
Fi
nally, in order to
investigate th
e adapta
b
ility for every improved ABC
algorith
m
s, th
e ability for optimizing eve
r
y
testing function by all six improved ABC algori
thm
s
will be showed in Figure 14 – Figure 21.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 557
9 –
5587
5584
4.3. Experimental Results
In table 2, we compare the
optimization ability of th
e six improved ABC algorit
hms for
unimod
a
l and
multimodal p
r
oble
m
s. Fo
r unimod
a
l pro
b
lem, ABC/…/2 outperfo
rm
s better me
a
n
and
stand
ard
deviation th
a
n
ABC/…/1.
Und
e
r th
e
sa
me
di
sturban
ce ve
ctor, th
e intro
d
u
c
tion
of
best info
rmati
on can g
e
t be
tter co
nverg
e
n
ce
re
sults
th
an “ran
d”
stra
tegy. We al
so
find that there
exist ce
rtain
risks getting i
n
to prematu
r
e for the
“b
e
s
t” strategy. The
exp
e
rim
ental
result
s also
sho
w
that ABC/cu
rre
nt-to
-
be
st perfo
rm
s better
tha
n
other ABC variant
s on al
most multimo
dal
function.
For Unimod
al function, th
e grap
h of co
nverge
nce ca
n be dra
w
n i
n
Fig 1-Fig
7
. Und
e
r
the
same st
rategy (“best
”
or
“rand”),
the two di
sturbance ve
ctor will
be achieve faster
conve
r
ge
nce
spee
d than
one distu
r
b
ance vector
.
The “curre
nt-to-b
e
st
” st
rategy whi
c
h
is
inco
rpo
r
ating
the current
solutio
n
acqu
ires
stabl
e
re
sult in multimodal expe
ri
ments. From
the
view of the final out
come i
n
solving
mul
t
imodal,
the “curre
nt-to-be
st” strategy
d
oes better
th
an
the other
stra
tegies in
mo
st case
s. Th
e “best”
strategy
is ea
sy trap
p
ed into lo
cal
minimum val
u
e
while thin
gs g
e
t better if we
incre
a
se the numbe
r of disturban
ce ve
ctor.
In Fig 15.-Fi
g
21, we ad
opt the different va
riants
of ABC to optimize differe
nt testing
function
s. Fo
r unim
odal
fu
nction, the
in
trodu
cti
on
of the be
st solu
tion ma
ke
s the di
re
ction
of
popul
ation m
o
ve faster to
ward to the o
p
timized val
u
e. For Weie
rstra
s
s and
Rastrigi
n fun
c
tion,
the “ABC/rand/2”,
”
ABC/
best/2”,
”ABC/
c
urre
nt-to-
be
st/1”,an
d “ABC/cu
rre
nt-t
o-be
st/2”
h
a
ve
rea
c
he
d the minimal value
.
Table 2. NP
=10, D=10, Ma
x.Eval =30, 000,
runtime
=
30, limit=200,
UM: Unimo
d
a
l; MM:
Multimodal
UM
UM
MM
MM
MM
MM
MM
Shpere
Rosenbrock
Ackley
Grie
w
ank
Weierstras
s
Rastrigin Schw
efel
Basic ABC
Mean
7.09e-017
4.11e-017
2.08e+000
2.44e+000
4.58e-016
1.76e-016
1.57e-002
9.06e-003
9.01e-006
4.61e-005
1.61e-016
5.20e-016
7.91e+000
2.95e+000
Std
MABC[11]
Mean
7.04e-017
4.11e-017
4.42e-001
8.67e-001
3.32e-016
1.84e-016
1.52e-002
1.28e-002
1.18e-016
6.38e-016
1.14e-007
6.16e-007
3.96e+000
2.13e+000
Std
ABC/best/1
Mean
1.46e-002
4.17e-002
9.82e+000
1.57e+001
4.08e-001
6.72e-001
1.59e-001
2.03e-001
5.44e-002
6.19e-002
1.31e+000
1.40e+000
1.10e+002
1.32e+002
Std
ABC/rand/1
Mean
4.28e-002
1.93e-001
5.25 e+000
9.02 e+000
3.33e-001
5.74e-001
1.95e-001
4.01e-001
4.76e-002
8.45e-002
1.52e+000
1.22e+000
1.04e+002
1.17e+002
Std
ABC/current
-
to-best/1
Mean
5.39e-124
2.69e-123
7.87e-001
1.57 e+000
8.5857e-0
1
5
1.8853e-0
1
5
9.31e-003
6.72e-003
0
0
0
0
1.25e-004
4.45e-004
Std
ABC/best/2
Mean
4.02e-156
2.20e-155
2.24 e+000
2.26 e+000
6.2172e-0
1
5
1.8067e-0
1
5
2.42e-002
2.28e-002
0
0
3.32e-002
1.81e-001
1.27e-004
2.30e-003
Std
ABC/rand/2
Mean
1.38e-148
7.60e-148
2.66e-001
3.88e-001
7.7568e-0
1
5
9.0135e-0
1
6
9.82e-003
7.51e-003
0
0
0
0
2.43e+001
5.71e+001
Std
ABC/current
-
to-best/2
Mean
2.84e-112
1.51e-111
1.0e-001
8.23e-002
7.8752e-0
1
5
1.4703e-0
1
5
7.23e-003
9.01e-003
0
0
0
0
2.20e-001
1.20e+000
Std
Figure 2 – Figure 7. Conv
erge
nce grap
h fo
r Unim
od
al function
s u
s
ing (12
)
-(17
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Com
pariso
n
of Im
proved
Artificial Bee Colo
ny Algo
ri
thm
s
Based o
n
… (Juan
xie
)
5585
0
50
0
1000
1500
2000
25
00
3000
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
C
ycl
e
Er
r
o
r
C
o
nv
er
genc
e gr
aphs
of
U
n
i
m
odal
F
unc
t
i
ons
sp
h
e
r
e
ro
s
e
n
b
ro
c
k
Figure 2. ABC/be
s
t/1
0
50
0
1000
1500
2000
25
00
3000
10
-250
10
-200
10
-150
10
-100
10
-50
10
0
10
50
C
ycl
e
Er
r
o
r
C
onv
er
genc
e gr
aphs
of
U
n
i
m
odal
F
u
nc
t
i
ons
s
p
her
e
r
o
s
enbr
oc
k
Figure 3. ABC/be
s
t/2
0
500
1000
1500
2000
2500
3000
10
-4
10
-2
10
0
10
2
10
4
10
6
Cy
c
l
e
E
rro
r
C
onv
ergenc
e grap
hs
of
U
n
i
t
i
m
odal
F
unc
t
i
ons
sp
h
e
r
e
r
o
s
enbroc
k
Figure 4. ABC/ran
d/1
0
50
0
1000
1500
2000
25
00
3000
10
-200
10
-150
10
-100
10
-50
10
0
10
50
C
ycl
e
E
rro
r
C
o
nv
er
genc
e gr
aphs
of
U
n
i
t
i
m
odal
F
unc
t
i
ons
sp
h
e
r
e
ro
s
e
n
b
ro
c
k
Figure 5. ABC/ran
d/2
0
500
10
00
15
00
20
00
25
00
30
00
10
-
120
10
-
100
10
-8
0
10
-6
0
10
-4
0
10
-2
0
10
0
10
20
Cy
c
l
e
Er
r
o
r
C
o
nv
er
ge
nc
e
g
r
ap
hs
of
U
n
it
i
m
od
al
F
u
n
c
t
i
on
s
sp
h
e
r
e
ros
e
n
b
r
o
c
k
Figure 6. ABC/cu
r
rent-to
-
best/1
0
500
100
0
1
500
2
000
2
500
3000
10
-
120
10
-
100
10
-8
0
10
-6
0
10
-4
0
10
-2
0
10
0
10
20
Cy
c
l
e
E
rro
r
C
o
nv
er
g
enc
e graphs
of
Uni
t
i
m
odal
F
unc
t
i
o
n
s
s
p
here
r
o
se
n
b
r
o
ck
Figure 7. ABC/cu
r
rent-to
-
best/2
Figure 8 – Figure 1
3
. Con
v
ergen
ce g
r
a
ph fo
r Multim
odal fun
c
tion
s usi
ng (1
2)-(17)
0
50
0
10
00
15
00
20
00
25
00
30
00
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
Cy
c
l
e
Er
r
o
r
Co
nv
e
r
ge
nc
e g
r
ap
hs
of
M
u
l
t
i
m
o
d
a
l
F
u
n
c
t
i
on
s
ac
k
l
ey
gri
e
wa
nk
wei
e
rs
t
r
a
s
s
ra
s
t
r
i
g
i
n
s
c
h
w
ef
el
Figure 8. ABC/be
s
t/1
0
500
10
00
15
00
20
00
25
00
30
00
10
-1
5
10
-1
0
10
-5
10
0
10
5
Cy
c
l
e
Er
r
o
r
Con
v
erg
e
n
c
e
gra
p
h
s
o
f
M
u
l
t
i
m
od
al
F
u
nc
t
i
on
s
ac
k
l
ey
gr
i
e
w
a
nk
we
i
e
r
s
t
r
as
s
ras
t
ri
g
i
n
s
c
h
w
ef
el
Figure 9. ABC/be
s
t/2
0
500
1000
1
500
2000
2500
3000
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
Cy
c
l
e
Er
r
o
r
C
onv
er
genc
e gr
aphs
o
f
M
u
l
t
i
m
odal
F
unc
t
i
ons
a
ckl
e
y
g
r
i
e
w
ank
we
i
e
r
s
t
r
a
s
s
ra
s
t
ri
g
i
n
s
c
hw
ef
el
Figure 10. ABC/ran
d/1
0
50
0
1000
1500
2000
25
00
3000
10
-15
10
-10
10
-5
10
0
10
5
C
ycl
e
E
rro
r
C
o
nv
er
genc
e gr
aphs
of
M
u
l
t
i
m
odal
F
u
nc
t
i
ons
ac
k
l
ey
gr
i
e
w
a
nk
we
i
e
r
s
t
r
a
s
s
ra
s
t
ri
g
i
n
s
c
hw
ef
el
Figure 11. ABC/ran
d/2
0
500
100
0
1
500
2
000
2
500
3000
10
-1
5
10
-1
0
10
-5
10
0
10
5
Cy
c
l
e
Er
r
o
r
C
o
n
v
e
r
ge
nc
e graphs
of
M
u
l
t
i
m
odal
F
unc
t
i
ons
a
ckl
e
y
gri
e
wank
we
i
e
r
s
t
r
a
s
s
ra
s
t
r
i
g
i
n
sch
w
e
f
e
l
Figure 12. ABC/cu
r
rent-to
-
best/1
0
500
100
0
1
500
2
000
2
500
3000
10
-1
5
10
-1
0
10
-5
10
0
10
5
Cy
c
l
e
E
rro
r
C
o
n
v
e
r
ge
nc
e graphs
of
M
u
l
t
i
m
odal
F
unc
t
i
ons
ac
k
l
ey
gri
e
wank
we
i
e
r
s
t
r
a
s
s
ras
t
r
i
gi
n
s
c
hwef
el
Figure 13. ABC/cu
r
rent-to
-
best/2
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ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 557
9 –
5587
5586
Figure 1
5
–
F
i
gure
21
Con
v
ergen
ce
abil
i
ty for every t
e
sting
fun
c
tio
n
by all
six va
riants of
ABC
Figure 14. Le
gend u
s
e
d
in Figure 15 – F
i
gure 2
1
0
500
1000
1
500
2000
2500
3000
10
-200
10
-150
10
-100
10
-50
10
0
10
50
Cy
c
l
e
E
rro
r
Sp
h
e
r
e
Figure 15. Sphere
(UM
)
0
50
0
100
0
15
00
2000
250
0
3
000
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
Cy
c
l
e
Er
r
o
r
R
o
s
enb
r
o
c
k
Figure 16. Ro
sen
b
ro
ck (UM)
0
50
0
1000
1500
2000
2500
3000
10
-15
10
-10
10
-5
10
0
10
5
Cy
c
l
e
Er
r
o
r
Ac
k
l
e
y
Figure 17. Ackley(MM
)
0
500
1000
1500
2000
25
00
3000
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
C
ycl
e
E
rro
r
Gr
i
e
w
a
n
k
Figure 18. Gri
e
wa
nk(MM
)
0
500
1000
1500
2000
25
00
3000
10
-15
10
-10
10
-5
10
0
10
5
C
ycl
e
E
rro
r
We
i
e
r
s
t
r
a
s
s
Figure 19. Weierstra
s
s (M
M)
0
500
1000
15
0
0
2000
2500
3000
10
-15
10
-10
10
-5
10
0
10
5
Cy
c
l
e
E
rro
r
Ra
s
t
ri
g
i
n
Figure 20. Ra
strigin
(MM)
0
500
1000
1500
2000
2500
3000
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
C
ycl
e
E
rro
r
S
c
hw
ef
el
Figure 21. Schwefel (MM)
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TELKOM
NIKA
ISSN:
2302-4
046
A Com
pariso
n
of Im
proved
Artificial Bee Colo
ny Algo
ri
thm
s
Based o
n
… (Juan
xie
)
5587
5. Conclusio
n
In this wo
rk, we inve
stigat
ed ro
undly th
e perfo
rma
n
ce of variant
s
of ABC base
d
on the
mutation stra
tegies u
s
e
d
in DE. Besid
e
s compa
r
in
g with that already exist
s
, we espe
cial
ly
analyze the
ability of co
n
v
ergen
ce
u
s
i
ng the
di
fferent improved
ABC alg
o
rit
h
ms. F
r
om t
h
e
results, we can co
ncl
ude
that the vari
ants of
ABC algorithm
s
can enh
an
ce the
optimization
abilities for dif
f
erent type problem
s.
Ackn
o
w
l
e
dg
ments
This
work i
s
sup
porte
d by
National
Na
ture
Scie
nce
Found
ation
of China
(G
rant No.
6107
5049
), the Excelle
nt Young T
a
l
ents F
oun
da
tion Proje
c
t
of Anhui
Province (Grant
No.20
11SQ
R
L018
), the
Y
outh F
oun
da
tion of An
hui
Unive
r
sity (Grant
No. K
J
Q
N
10
15),
the
University Na
tural Scie
nce
Re
sea
r
ch Project of
Anh
u
i
Province
(G
rant No. KJ2
012B03
8), th
e
Provinci
al Natural S
c
ien
c
e Foun
dation
of the An
h
u
i Hig
her E
d
ucatio
n Instit
utions
of Chi
na
(Grant No. KJ2013A0
0
9
)
.
Referen
ces
[1]
Dorig
o
M, Man
i
ezzo V, Co
lor
n
i A. “Ant s
y
st
em
: optimizati
o
n b
y
a co
lon
y
of
coop
eratin
g age
nts”.
IEEE
T
r
ans on Syste
m
s, Man, an
d Cy
ber
netics, Part B: Cybernet
ics.
1996; 2
6
(1
): 29-41.
[2
]
R
C
Ebe
r
ha
rt an
d
J Ke
nn
edy
.
“A n
e
w
opti
m
i
z
e
r
usi
n
g
parti
cle sw
ar
m th
e
o
ry
”
.
in Proc. 6th Int. Sy
mp.
Micromach
i
n
e
Huma
n Sci, Na
go
ya, Jap
an. 1
995: 39-
43.
[3
]
J Ke
n
n
edy
a
nd R
C
Eb
e
r
ha
rt.
“Particle sw
ar
m o
p
timi
z
a
ti
on
”.
in Proc. IEEE Int. Conf. Neural Net
w
orks.
199
5: 194
2–
19
48.
[4]
Storn R, Price K.
“Differenti
a
l evo
l
utio
n–
a simple a
nd effi
cient he
uristic
for glob
al opti
m
izatio
n ove
r
contin
uo
us spa
c
es”.
Journa
l o
f
globa
l opti
m
i
z
ation
. 19
97; 11
(4): 341-3
59.
[5]
Mezura-Mo
nte
s
E, Veláz
q
u
e
z-Re
ye
s J, Coel
lo
Coe
l
l
o
CA. “
A co
mparativ
e study
of differe
ntial
evol
ution
var
i
a
n
ts for g
l
o
bal
o
p
timi
z
a
ti
on
”.
Pr
ocee
din
g
s of
the 8th ann
ua
l confere
n
ce
on
Genetic an
d
evol
ution
a
r
y
c
o
mputat
io
n. 200
6: 485-4
92.
[6]
S Das
and
PN
Suga
ntha
n.
“Di
fferential evo
l
u
t
ion—A
surv
ey
of the stat
e-
of-the-art
”
.
IEEE Trans.
Evol.
Co
mp
ut
. 2011;
15(1): 4-31.
[7] E
Mezura-Montes,
J Velázquez-Rey
e
s, CA Coello. “
A c
o
mpar
ative stu
d
y of
differe
nti
a
l
evol
utio
n
variants for glo
bal optim
i
z
a
tion
”. in Proc. GECCO. 2006: 4
8
5
-49
2
.
[8]
D Kara
bo
ga. “
An i
dea
bas
ed
on
hon
eyb
ee s
w
arm for
nu
me
rical
opti
m
i
z
a
t
i
o
n
”.
Technical Report
TR06
,
Erci
yes Univ
er
sit
y
, Eng
i
ne
eri
ng F
a
cult
y,
Co
mputer Eng
i
n
e
e
rin
g
Dep
a
rtment; 200
5.
[9]
D Karab
o
g
a
, B Basturk. “A comparativ
e st
ud
y of artifici
al b
e
e
colo
n
y
al
gorit
hm”,
Appli
ed M
a
the
m
atic
s
and C
o
mput
ati
o
n
. 200
9; 214:
108-
132.
[10]
E Bon
abe
au,
M Dori
go, G T
hera
u
laz, “
Sw
a
r
m Inte
lli
ge
nce
:
F
r
om N
a
tural to Artificial System
”. Ne
w
York, NY: Oxford Univ
ersit
y
P
r
ess; 1999.
[11]
B Aka
y
, D K
a
rabo
ga. “A mo
difie
d
artifici
al
bee
c
o
lo
n
y
al
gorithm for re
al-p
arameter
o
p
timizati
on”.
Information Sci
ences
. 20
12; 1
92: 120-
14
2.
[12]
GP Z
hu, S K
w
ong,
“Gbest-g
u
id
ed
artificia
l
bee
co
l
o
n
y
al
g
o
rithm for
num
erical
functi
on
optimiz
ation”
.
Appl
ied Mat
h
e
m
atics a
nd C
o
mp
utatio
n
. 201
0; 217(7): 3
166
-317
3.
[1
3
]
We
i
f
e
n
g
Ga
o
,
Sa
ny
ang
Li
u. “Imp
ro
ve
d arti
fi
ci
al
be
e c
o
lo
n
y
a
l
g
o
rith
m for gl
ob
al
optimiz
ation”
.
Information Pr
ocessi
ng L
e
tters
. 2011; 11
1: 871-8
82.
[14]
W
e
ifeng
Ga
o, San
y
a
ng Liu, Lin
g
li
ng hua
ng
,
“A
glob
al
bes
t artificial
be
e
colo
n
y
al
gor
ith
m
for glo
b
a
l
optimiz
ation”.
J
ourn
a
l of Co
mputatio
na
l and
Appl
ied Mat
h
e
m
atics
. 2
012; 2
36: 274
1-2
753.
[15] Parvan
eh
Man
s
ouri.
T
he
M
o
d
i
f
y
Versi
on of Artifici
a
l
Be
e Co
l
ony
Alg
o
r
i
t
hm to
sol
v
e R
e
a
l
Op
ti
mi
za
ti
on
prob
lems.
Internatio
nal J
ourn
a
l of Electrica
l
and C
o
mput
er Engi
neer
in
g
. 2012; 2(4): 4
73-
480.
[16]
Vimal N
a
yak,
Hares
h
A Suth
ar, Jagrut Gadi
t.
Implementati
on of Artifici
a
l
Bee Co
lo
n
y
A
l
gorithm.
IAES
Internatio
na
l Journ
a
l of Artificial Intel
lig
enc
e
. 2012; 1(3): 1
1
2
-12
0
.
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