TELKOM
NIKA
, Vol.12, No
.4, Dece
mbe
r
2014, pp. 97
7~9
8
4
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v12i4.531
977
Re
cei
v
ed Au
gust 28, 20
14
; Revi
sed O
c
t
ober 1
9
, 201
4; Acce
pted
No
vem
ber 5,
2014
Multi-objective Optimization Based on Improved
Differential Evolution Algorithm
Shuqiang Wang*
1
, Jianli Ma
2
1
School of Info
rmation a
nd el
ectrical e
ngi
ne
erin
g,
Hebe
i U
n
iversit
y
of En
gin
eeri
ng, Ha
n
dan 05
60
38,
Heb
e
i, Chi
n
a
2
Suburba
n
w
a
t
e
r and p
o
w
e
r supp
l
y
m
ana
ge
ment office of Han
dan C
i
t
y
, H
and
an 0
560
01,
Hebe
i, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: 1780
38
139
@
qq.com
A
b
st
r
a
ct
On the
bas
is
of the
fund
a
m
ental
differ
enti
a
l
ev
ol
utio
n (
D
E), this
pa
p
e
r p
u
ts forw
ar
d sev
e
ral
improve
d
DE
alg
o
rith
ms to
f
i
nd
a
ba
la
nce
betw
e
e
n
g
l
o
b
a
l
and
l
o
cal
s
earch
a
nd
get
opti
m
al s
o
l
u
ti
ons
throug
h
r
api
d conver
genc
e. Meanw
hi
le, a
r
and
o
m
mut
a
tio
n
mec
h
a
n
is
m i
s
ad
opte
d
to
p
r
ocess ind
i
vid
u
a
ls
that show
sta
g
natio
n
beh
avi
o
ur. After that, a
seri
es
of
frequ
ently-us
ed
be
n
c
hmark test fu
nctions
are
us
e
d
to test the performanc
e of the funda
menta
l
and i
m
prove
d
DE alg
o
rith
ms.
After a comp
a
r
ative an
alysis
of
severa
l a
l
gor
ithms, th
e
pap
er
real
i
z
e
s
its d
e
s
ired
effect
s b
y
ap
plyi
ng th
e
m
to
the c
a
lc
ul
ation
of si
ngl
e
and
mu
ltipl
e
ob
jecti
v
e functions.
Ke
y
w
ords
: differenti
a
l evo
l
uti
on, effect
of parameters, nu
merical
exper
i
m
e
n
t, multi-o
b
j
e
cti
v
e opti
m
i
z
at
ion
1. Introduc
tion
In the scie
ntific re
se
arch
and
engin
e
e
ring
de
sign,
many spe
c
i
f
ic problem
s can
be
summ
ari
z
ed
as the problem
s of para
m
eter o
p
timization.
Ho
wever, in
practi
ce, these
optimizatio
n
probl
em
s u
s
u
a
lly have m
u
l
t
iple de
sign
o
b
ject
iv
e
s
,
w
h
i
c
h
co
nt
ra
ct
w
i
t
h
and
re
st
ri
ct
each oth
e
r [1
]. The pe
rformance o
p
timi
zation
of
on
e p
r
ob
le
m u
s
ua
lly le
a
d
s
to
th
e
pe
r
f
or
ma
nc
e
degradatio
n of at least one of the ot
her problem
s,
which indi
ca
tes t
hat it is difficult to make
many o
b
je
ctives to
rea
c
h
o
p
timization
si
multaneo
us
ly
. The
r
efore, t
he
re
sea
r
ch
of multi-o
b
je
ctive
optimizatio
n
algorith
m
h
a
s be
com
e
a
re
sea
r
ch
hospo
t in current
science a
nd
en
ginee
ring
de
sign
[2]. Evolutionary algo
rithm
is the
gene
ral term of
he
uristi
c research a
nd o
p
timi
zation
algo
rithms
inspi
r
ed a
n
d
develope
d
from natural biology
a
nd system
and to solv
e multi-obj
e
c
tive
optimizatio
n probl
em
s wit
h
evolutiona
ry algorithm h
a
s be
en wi
del
y used [3].
As a
n
imp
o
rt
ant pa
rt of
e
v
olutionary
al
gorit
hm,
differential
evoluti
on h
a
s be
en
wid
e
ly
use
d
in solvin
g optimizatio
n probl
em
s b
e
ca
use it
has simple theo
ry, simple ope
ration an
d strong
robu
stne
ss [4
]. The ba
sic p
r
inci
ple of
differential
ev
olu
t
ion is to
dist
urb
a certain
i
ndividual i
n
the
grou
p and
search the
se
arch spac
e; however, it is too ra
ndo
m in cho
o
sin
g
the individ
uals
gene
rating dif
f
eren
ce
s an
d it is easy to cause algo
rith
m prem
aturity or long
-time
optimizatio
n so
as to ma
ke i
t
unable to
o
b
tain glo
bal
optimizatio
n
solutio
n
[5]. Beside
s, whe
n
settling m
u
lti-
obje
c
tive opti
m
ization
p
r
ob
lems, diffe
ren
t
ial evol
ution
is affe
cted
by its o
w
n
limitations, m
a
ki
n
g
the sele
ction
of mutatio
n
strategy
an
d
the
setti
ng
of
pa
ramete
r
value
s
se
rio
u
sly re
strict
the
perfo
rman
ce
of the algorith
m
[6].
In ord
e
r to
solve th
e ab
ove-me
ntione
d
problem
s,
this
pape
r
has inve
stig
ated the
sele
ction an
d
param
eter value
s
of mutation stra
te
gy when u
s
ing
differential ev
olution in mul
t
i-
obje
c
tive opti
m
ization.Fi
rst
,
the pape
r
make
s a
n
nu
meri
cal exp
e
r
iment o
n
ste
p
length
F a
nd
cro
s
sove
r o
p
e
rato
r
CR of
the fu
ndam
ental
DE al
g
o
rithm. An
d t
hen it
ma
ke
s a
com
p
a
r
ati
v
e
analysi
s
to g
e
t the ra
nge
of the optim
al value of
t
he two. T
o
a
v
oid the sho
r
tcoming
s
of
DE
algorith
m
in
handli
ng gl
ob
al optimi
z
atio
n, we
ma
ke
some
imp
r
ov
ements to
ke
ep the
variet
y of
grou
p an
d a
c
celerate po
pulation
co
n
v
ergen
ce. S
e
co
ndly, the
pape
r ma
kes a
n
nu
me
rical
experim
ent o
n
the pe
rform
ance of
the i
m
prove
d
alg
o
rithms
and
m
a
ke
s a
co
mp
arative a
nalysis.
Finally, the
pape
r u
s
e
s
t
he imp
r
oved
DE alg
o
ri
th
ms to
solve
the optimi
z
ation of mult
iple
obje
c
tive functions.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 4, Dece
mb
er 201
4: 977
– 984
978
2. DE Algorithm
2.1 Basic Id
eas of
DE Al
gorithm
DE al
gorith
m
is an
evoluti
onary
algo
rit
h
m
b
a
sed
on
re
al-n
umbe
r en
codi
ng to
optimize
the minimum
value of function
s. The
con
c
ept
wa
s put forward on the ba
sis of po
pul
ation
differen
c
e
s
when sch
o
lars tried to solve the C
heby
shev polyno
m
ials. Its overall st
ru
cture is
analo
gou
s to
that of genetic algo
rith
ms. The
two
both have the sa
me op
eration
s
su
ch as
mutation, crossover, and
selection [7]. But ther
e
are still some
differenc
es. Here are the
basic
idea
s of DE algorith
m
: the mutation betwee
n
parent
individual
s gives ri
se to mu
tant individua
ls;
cro
s
sove
r op
eration b
e
twe
en parent ind
i
viduals a
nd
mutant individual
s is ap
pli
ed acco
rdin
g
to
certai
n p
r
ob
a
b
ility to gene
rate test in
dividual
s;
greedy
sele
ction
bet
wee
n
pa
rent i
ndividual
s a
n
d
test individual
s is carried o
u
t in accord
a
n
ce
wi
th the degree of fitness;
the better one
s are ke
pt
to realize the evolution of the pop
ulation
[8].
2.1.1 Mutati
on Opera
t
io
n
For ea
ch indi
vidual
t
i
x
, generate mutant individual
12
(,
,
,
)
tt
t
t
T
ii
i
i
D
vv
v
v
in accordan
ce
with the follo
wing exp
r
e
ssi
on:
12
3
()
tt
t
t
ij
r
j
r
j
r
j
vx
F
x
x
1,
2
,
3
,
j
D
(1)
In the expre
ssi
on,
11
1
1
12
(,
,
,
)
tt
t
t
T
rr
r
r
D
xx
x
x
,
22
2
2
12
(,
,
,
)
tt
t
t
T
rr
r
r
D
xx
x
x
, and
33
3
3
12
(,
,
,
)
tt
t
t
T
rr
r
r
D
xx
x
x
are three indi
viduals rand
o
m
ly
selecte
d
from the pop
ulation;
1
t
rj
x
,
2
t
rj
x
,
and
3
t
rj
x
are
the comp
one
nts in
the
jth di
mensi
on of
1
r
,
2
r
, and
3
r
, respec
tively;
F
is the
mutation ope
rator that lies
betwe
en [0.5,1]. So we can
obtain mutan
t
individual
t
i
v
.
2.1.2 Cros
so
v
e
r
Operatio
n
We o
b
tain te
st individual
12
(,
,
,
)
tt
t
t
T
ii
i
i
D
uu
u
u
from mutant
individual
t
i
v
and pa
ren
t
individual
t
i
x
in line with the followin
g
pri
n
ciple:
0,
1
_
0,
1
_
t
ij
t
ij
t
ij
v
i
f
r
and
C
R
o
r
j
j
ra
nd
u
x
if
rand
C
R
or
j
j
rand
(2)
In the expre
ssi
on,
[0
,1
]
ra
nd
is a random n
u
mb
er between [
0
,1];
CR
, the crossover
operator, i
s
a
con
s
tant
at [0
,1]; the bigg
er
CR
is,
the m
o
re
likely
cro
s
sov
e
r
occu
rs;
_
jr
a
n
d
is an
integer
ran
d
o
m
ly cho
s
en
b
e
twee
n [1,D], whi
c
h gu
ara
n
tees th
at for test individu
al
t
i
u
, at least
one ele
m
ent
must be o
b
tained from
mutant indi
vidual
t
i
v
. The mutation and cros
sover
operation
s
are both call
ed
rep
r
od
uctio
n
[9].
2.1.3 Selecti
on
DE algo
rithm
adopts the
“gree
d
y” sele
ction
st
rategy
, which m
e
a
n
s sele
cting
one that
has th
e be
st fitness val
ue
from pa
rent i
ndividual
t
i
x
an
d test individ
ual
t
i
u
as the in
dividual of
the next gene
ration. The
se
lection i
s
de
scrib
ed a
s
:
1
()
()
tt
t
t
ii
i
i
t
i
x
if
fitn
e
s
s
x
fitn
e
s
s
u
x
uo
t
h
e
r
w
i
s
e
(3)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Mutli Objecti
v
e Optim
i
zation Based o
n
Im
prove
d
Differential E
v
olu
t
ion …. (Shuq
iang Wang
)
979
Whe
r
e fitne
s
s, the o
b
je
ctive function
to
be
optimized, is
reg
a
rd
ed to b
e
the
fitness
function.
Unl
e
ss state
d
, the fitness fun
c
tion in t
he pa
per i
s
an
obj
ective fun
c
tio
n
with a mi
ni
mal
value[10].
2.2 Calcula
t
ion Proces
s of DE
Algorithm
From the pri
n
ciple
s
of fund
amental DE
algorith
m
me
ntioned ab
ove, we can u
n
derstand
the cal
c
ulatio
n pro
c
e
ss of
DE algo
rithm as follo
ws:
(1) Parameter
initialization:
NP
: popul
ation
si
ze;
F
: s
c
a
le fac
t
or;
D
: spati
a
l dimen
s
io
n
of
mutation ope
rator; evolutio
n gene
ration
0
t
.
(2)
Ran
dom in
itialization
of the initial popula
t
ion
12
()
{
,
,
,
}
tt
t
NP
Xt
x
x
x
, where
12
(,
,
,
)
tt
t
t
T
ii
i
i
D
xx
x
x
.
(3)
Individual eva
l
uation: cal
c
ul
ate t
he fitness value of ea
ch individ
ual.
(4)
Mutation o
p
e
r
ation: m
u
tation op
eration
is a
pplie
d t
o
ea
ch i
ndivi
dual in
a
c
co
rdan
ce
with
Expressio
n
(1
) to work out
mutant individual
t
i
v
.
(5)
Cro
s
sove
r op
eration:
cro
s
sover o
p
e
r
ati
on is ap
plied
to each indi
vidual in accorda
n
ce with
Expressio
n
(2
) to work out test individu
al
t
i
u
.
(6)
Selection:
select o
n
e
from
pare
n
t individ
ual
t
i
x
and
te
st i
ndividual
t
i
u
as
the individ
ual
of th
e
next generation in acco
rda
n
ce
with Expression (3).
(7)
Test te
rmin
ation: the
next gen
eratio
n
of pop
ulation
gen
erate
d
from the
ab
ove p
r
o
c
e
s
s
i
s
11
1
12
(1
)
{
,
,
,
}
tt
t
NP
Xt
x
x
x
; suppo
se the
optimal individual in
(1
)
Xt
is
1
t
bes
t
x
; if it reache
s
the maximum
evolution ge
neratio
n or m
eets
the criteria of erro
rs,
merg
e and o
u
tput
1
t
bes
t
x
as
the optimal solution; otherwise, make
1
tt
and retu
rn to step (3).
2.3 Parameter Selection
of DE
Algorithm
2.3.1 Selecti
on of Popula
t
ion Size NP
In view of
computation
complexity, the la
rger the
popul
ation
si
ze i
s
, the g
r
eater th
e
likeliho
od of
global
optima
l
solutio
n
be
comes. Bu
t
it also nee
ds more cal
c
ul
ation
amo
unt
a
n
d
time. Noneth
e
less, the q
uality of the optimal
sol
u
tion doe
s n
o
t simply ge
ts better a
s
the
popul
ation si
ze expan
ds.
Sometimes, it’s the other
way rou
nd. The accu
ra
cy of solution
s e
v
en
decli
ne
s after popul
ation
size NP in
creases to
a certain nu
mbe
r
. This is b
e
c
au
se a la
rg
er
popul
ation si
ze re
du
ce
s the rate of co
nverge
nce,
thoug
h it can kee
p
the vari
ety of population.
Variety and the rate of
converge
nce m
u
st be kept in balance.
He
nce, the accuracy
will decrease
if the populat
ion si
ze g
e
ts large
r
but th
e ma
ximum evolution
ge
n
e
ration rem
a
i
n
s
u
n
chang
e
d
.
The larg
er th
e populatio
n size is, the greater the va
ri
ety is. Therefore, a larg
er
popul
ation si
ze is
need
ed to expand vari
ety and prevent p
r
emat
u
r
e con
v
ergen
ce of a
populatio
n [11].
Acco
rdi
ng to
our p
r
eviou
s
resea
r
ch re
sults,
the app
ropriate
pop
ul
ation si
ze fo
r simple
low-dime
nsio
nal problem
s sho
u
ld lie
betwe
en
15
and 3
5
in the case of
given maxim
u
m
evolution
gen
eration.
In th
e same
ci
rcu
m
stan
ce
, th
e
popul
ation
si
ze th
at mai
n
tains bet
wee
n
15
and 50 h
e
lp
s kee
p
a goo
d balan
ce b
e
tween vari
ety a
nd the rate of
convergen
ce
[12].
2.3.2 Selecti
on of Scale
Factor F
Let us te
st the perfo
rma
n
ce of scale factor
F
. Set the populatio
n si
ze at 15. Make su
re
the cro
s
sove
r operato
r
an
d
the maximum
evolution g
eneration sta
y
uncha
nge
d.
Based
on the
test on
scale
factor
F
of the bana
na fun
c
t
i
on, we
kn
ow
that in the ca
se of
the sam
e
initial popul
ation,
the results of
every
30 times of ru
nnin
g
vary greatly from ea
ch
oth
e
r
whe
n
0.7
F
, and
we
can
get b
e
tter lo
cal o
p
t
imization
an
d faste
r
rate
of conve
r
g
e
n
c
e at th
e
expen
se of lower
su
ccess
rate of opt
imi
z
ation a
nd lo
nger
run
n
ing
time; when
0.7
F
, there a
r
e
no signifi
cant
differe
nces betwe
en
th
e results
of
ev
ery 3
0
time
s of runni
ng,
and
we
can
get
better glob
al optimizatio
n, sho
r
ter
runni
ng
time, and faster
rate of converg
e
n
c
e.
To sum up,
F
, to a
ce
rtain
deg
ree,
can
re
gulate
the
local a
nd
gl
obal
se
arch
of an
algorith
m
. A bigge
r
F
helps keep th
e va
riety of popul
ation and i
n
crea
se the
global sea
r
ch
ability, while a
smaller
F
help
s
in
cre
a
se
the l
o
ca
l se
arch
abil
i
ty as
well a
s
the
rate
o
f
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ISSN: 16
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930
TELKOM
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Vol. 12, No. 4, Dece
mb
er 201
4: 977
– 984
980
conve
r
ge
nce. Hen
c
e, the
value of
F
sh
ould b
e
neit
her too
big
nor
small
e
r t
han a
sp
ecifi
c
value[13]. This explain
s
wh
y the algorith
m
has go
od e
ffects wh
en
[0
.
7
,
1
]
F
.
2.3.3 Selecti
on of Cro
s
s
o
v
e
r Operator CR
To test the effect of cro
s
so
ver operator
on algo
rithm perfo
rman
ce,
we make th
e scale
fac
t
or
0.9
F
and set population
size at 20. Make the
cro
ssover ope
rato
r lie between
0 and 1.
Set the interval at 0.1. Let the maximu
m
evolution ge
n
e
ration
remai
n
the same.
The te
st sho
w
s the
bana
na fun
c
tion
can
cha
nge
CR
. Thus, i
n
the
ca
se
of the
same
initial po
pulat
ion, the
re
sul
t
s of eve
r
y 3
0
times
of
ru
nning
vary g
r
eatly from
ea
ch
other whe
n
0.3
CR
, and we ca
n get better local optimizatio
n at
the expense of slo
w
e
r
rate of conve
r
gen
ce,
lowe
r
su
cce
s
s
rate
of o
p
timization
a
n
d
lon
g
e
r
run
n
ing tim
e
; when
0.3
CR
,
there are no
signifi
cant differen
c
e
s
bet
ween
th
e re
sul
t
s
of ev
ery
3
0
time
s of
ru
nning,
and
we can
get
bet
ter
global o
p
timi
zation; b
u
t when
0.3
0
.6
CR
, we get
slo
w
er
rate
of conve
r
ge
n
c
e a
nd lon
g
e
r
runni
ng time; whe
n
0.6
CR
, we ge
t faster rate o
f
conver
g
e
n
c
e and shorte
r runnin
g
time[14].
3. Simulation Testing o
f
Fiv
e
Impro
v
ed DEAlgo
rithms
3.1 Fiv
e
Impr
ov
ed DEAlgorithms
The fund
am
ental DE al
gorithm
can
be de
scrib
ed as:
DE/rand/1/bin. “b
in” mea
n
s
c
r
oss
o
ver operation.
DE/x/y/z
is
us
ed
to
differe
ntiate
the othe
r DE
d
e
formati
ons.
x
define
s
wheth
e
r the
variant vecto
r
is “ra
ndo
m”
or “o
ptimal”,
y
denote
s
the
numbe
r of re
sidu
al vectors
use
d
, and
z
stands for th
e m
e
thod
of cro
s
sover op
erati
on. Belo
w a
r
e the
DE def
ormatio
n
s if we
only con
s
id
er the sele
ction
modes of ba
se
poi
nts an
d
the numbe
r of difference vectors:
DE/rand/1
12
3
()
tt
t
t
ir
r
r
vx
F
x
x
DE/best/1
12
()
tt
t
t
ib
e
s
t
r
r
vx
F
x
x
DE/rand/2
12
3
4
5
()
tt
t
t
t
t
ir
r
r
r
r
vx
F
x
x
x
x
DE/best/2
12
3
4
()
t
t
tt
tt
ib
e
s
t
r
r
r
r
vx
F
x
x
x
x
DE/rand
-to-
best/1
12
()
(
)
tt
t
t
t
t
ii
b
e
s
t
i
r
r
vx
F
x
x
F
x
x
3.2Sev
eral Benchmar
k Test Fun
ction
s
(1) Ban
ana fu
nction
22
2
21
1
(
)
100
(
)
(
1
)
f
xx
x
x
(4)
12
3,
3
xx
Global o
p
timal solutio
n
s:
1
9
.
99
8919e
01
x
,
2
9
.
998
012e
01
x
, and
(
)
0
.
00
0000
fx
.
(2) S
c
haffer f
unctio
n
22
2
2
2
2
12
12
1
2
(
)
0.
5
[
(
s
i
n
)
0
.
5
]
/
(
1
0.001
(
)
)
1
0
,
10
fx
x
x
x
x
x
x
(5)
Global o
p
timal solutio
n
s:
1
0.00
00
00
x
,
2
0.000000
x
, and
(
)
0.
5
0
00
0
0
fx
(3) Bohachev
s
k
y
func
tion
22
12
1
2
(
)
0.
3
c
os
(3
)
0
.
3
cos
(
4
)
0.
3
fx
x
x
x
x
(6)
The optimal
solution is -0.2
4, which lies
betwe
en [0,-0
.
24] and[0,0.2
4
].
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Mutli Objecti
v
e Optim
i
zation Based o
n
Im
prove
d
Differential E
v
olu
t
ion …. (Shuq
iang Wang
)
981
(4) Multim
oda
l function
2
2
0.
2
5
2
2
0.
1
2
12
1
2
1
2
(
)
((
)
)
((s
i
n(50
(
)
)
)
1
)
5.
12
,
5
.
1
2
f
x
xx
xx
x
x
(7)
The minimu
m
value here i
s
0 and there i
s
an infinite lo
cal minim
u
m.
Below a
r
e the
graph
s of the
above four te
st function
s:
(a) Ban
ana fu
nction
(b) Sch
a
ffer fu
nction
(c) Boha
chev
sky fun
c
tion
(d) Multimodal fun
c
tio
n
Figure 1. Fou
r
ben
chm
a
rk test functio
n
s
(a) Evolution
curve of ba
na
na functio
n
(b) Evolution curve of
Schaffer fun
c
tion
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 4, Dece
mb
er 201
4: 977
– 984
982
(c) Evolution curve of Boh
a
ch
evsky function
(d) Evolution cu
rve of multimod
al function
Figure 2. Evolution cu
rves
offour ben
ch
mark test fun
c
tion
s
3.3 Test
Res
u
lts
Test the five
algo
rithms u
s
ing th
e te
st function
s i
n
trodu
ce
d in
3.2.
15
NP
,
0.9
F
,
0.9
CR
, and the ma
ximum evolution gen
eratio
n is 20
0. Average th
e re
su
lts of the 30 t
i
mes of
runni
ng an
d we can get th
e evolution curves
sho
w
n
as in Figu
re 2
.
Acco
rdi
ng to
the re
sult
s an
d ev
olution
curves of the
above te
st
fu
nction
s, in th
e ca
se
of
the sa
me initi
a
l pop
ulation,
there
are no
signifi
c
ant
chang
es i
n
th
e re
sult
s of e
v
ery 30 time
s of
runni
ng. Besi
des, the five improve
d
alg
o
rithm
s
ca
n optimize fun
c
tions well an
d thus yield very
good
re
sults.
For different
function
s, the rate of
con
v
ergen
ce va
ri
es, so doe
s t
he efficien
cy
of
finding
optim
al solution
s.
The five al
g
o
rithm
s
h
a
ve
simila
r o
p
timal pe
rform
ance, but
so
me
algorith
m
s h
a
v
e longer
run
n
ing time and
some fun
c
tio
n
s have b
e
tter optimization
.
4. Application of DEAlgorithm in
Multi-objectiv
e
O
p
timiz
a
tion
Based
on th
e previou
s
d
e
scriptio
n, DE algor
ith
m
i
s
of g
r
e
a
t importa
nce in
solvin
g
compli
cate
d optimizatio
n probl
em
s.
In the
next
pa
rt,
we use
DE
algorith
m
to solve singl
e- and
multiple-obje
c
tive optimiza
t
ion probl
em
s with
equality con
s
trai
nts o
r
inequality co
nstrai
nts.
4.1Single-ob
jectiv
e Optimization Pro
b
lem
The sta
nda
rd
form of singl
e-obj
ective o
p
timization i
s
gene
rally exp
r
esse
d as:
min
12
3
(,
,
,
,
)
n
f
xx
x
x
s.
t
12
3
(,
,
,
,
)
0
in
gx
x
x
x
1,
2
,
3
,
,
im
12
3
(,
,
,
,
)
0
jn
hx
x
x
x
1,
2
,
3
,
,
jl
(8)
ii
i
ax
b
To
solve th
e ab
ove p
r
oblem
s,
we
usua
lly co
nvert con
s
trained
proble
m
s
i
n
to
uncon
strain
e
d
one
s by din
t
of t
he penalt
y
function me
thod. He
re a
r
e its ba
sic id
e
a
s: me
rging t
h
e
con
s
trai
nt fun
c
tion of
a p
r
oblem i
n
to a
n
obje
c
tive
fu
nction i
n
a
certain
way,
so that the
wh
ole
probl
em i
s
converted
to
an u
n
con
s
tra
i
nt pro
b
le
m.
To reali
z
e it,
we
can
pro
duce the
fitness
function in th
e form of
()
()
()
Wx
f
x
r
D
x
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Mutli Objecti
v
e Optim
i
zation Based o
n
Im
prove
d
Differential E
v
olu
t
ion …. (Shuq
iang Wang
)
983
In the expression, pen
alty function
()
Dx
is a continuou
s function that meet
s
0
()
0
x
X
Dx
x
X
.
()
f
x
is th
e
scale
coeffici
en
t and
0
r
.
X
is t
h
e fea
s
ible
re
gion
of the
probl
em. Besi
des, we ca
n deal with the
con
s
trai
nt co
ndition
ii
i
ax
b
as
follows
:
2
1,
2
,
3
,
1,
2
,
3
,
mi
i
i
mi
i
i
g
xb
i
m
g
ax
i
m
(9)
The structu
r
e
of penalty function d
ep
e
n
d
s on the ext
e
rio
r
point me
thod:
3
22
11
(
)
(
(
))
(
(
)
)
(
(
))
im
ji
i
ji
D
x
h
X
gX
gX
(10
)
Whe
r
e
0(
)
0
((
)
)
1(
)
0
i
i
i
gX
gX
gX
For
an
obje
c
t
i
ve functio
n
,
we
ca
n first fi
nd its minim
a
and
then
u
s
e DE
alg
o
rith
m so a
s
to find the maxima of the function. The
conv
ersion m
e
thod is the fitness fun
c
tion
below:
3
22
11
(
)
(
(
))
(
(
)
)
(
(
))
im
ji
i
ji
D
x
h
X
gX
gX
(11)
mi
n
((
)
)
(
)
f
itn
e
s
s
f
x
W
W
x
(12)
Whe
r
e
((
)
)
0
f
itne
s
s
f
x
and
mi
n
W
is a given val
ue or th
e mini
mum value in
()
Wx
. In this
way, the maximization of a
n
uncon
strain
ed pro
b
lem i
s
converte
d into a minimization pro
b
lem.
4.2 Multi-obj
ectiv
e Optimization Probl
em
Und
e
r no
rmal
conditio
n
s, the multiple
o
b
jective fun
c
tion is expressed as:
11
2
2
{
(
)
,
()
,
,
()
}
qq
M
ax
z
f
x
z
f
x
z
f
x
(13)
()
0
1
,
2
,
3
,
.
()
0
1
,
2
,
3
,
i
j
g
xi
m
st
hx
j
l
Below is the
evolution curve of DE algor
ithm ba
sed o
n
the simulati
on re
sults:
Figure 3. Evolution cu
rve o
f
multiple objective functio
n
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 12, No. 4, Dece
mb
er 201
4: 977
– 984
984
From the evo
l
ution cu
rves
and ru
nnin
g
result
s,
we kn
ow that all the five improved DE
algorith
m
s
can find thei
r corre
s
p
ondi
ng optimal
solution
s in the 30 time
s of runni
ng.
The
algorith
m
s re
main q
u
ite
st
able.
Ho
wev
e
r, they
have
differe
nt rate
of
conve
r
ge
nce
an
d
run
n
i
ng
time.
5 Conclu
sion
This pa
pe
r descri
b
e
s
the desi
gn ide
a
s of
DE al
gorithm a
n
d
further imp
r
oves the
para
m
eters o
f
the al
gorith
m
. After that,
it com
par
es t
he
re
sults of t
he n
u
me
rical
experim
ent a
nd
then analy
z
e
s
the pe
rform
ance of
the improve
d
DE
algorith
m
s, th
us p
r
ovidin
g the ba
sis fo
r the
appli
c
ation
of the alg
o
rith
ms. In the
en
d, the
pa
pe
r
adopt
s the
p
enalty fun
c
tio
n
metho
d
a
n
d
the
weig
hted
strategy to de
al with the
con
s
trai
nt
co
ndition
s of
multiple-obje
c
tive optimization
probl
em
s an
d uses th
e i
m
prove
d
DE
algorith
m
s
t
o
solve co
nstrained optim
ization pro
b
le
ms.
Thereby, it helps expa
nd th
e appli
c
ation
area
sof the DE algorithm.
Referen
ces
[1]
Matthias E
h
rgott, Jonas
Ide,
Anita
Schöbel. Minmax
r
o
bustness
for multi-objectiv
e
optim
ization
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lems.
Euro
pea
n Jour
nal o
f
Operationa
l R
e
searc
h
. 201
4; 239(1): 17-
31.
[2]
Marko
Kova
č
evi
ć
, Miloš
Mad
i
ć
, Miroslav
Ra
dova
novi
ć
, Dej
an
R
a
n
č
i
ć
.Softw
a
r
e
protot
yp
e
for solv
in
g
multi-ob
jectiv
e
machin
ing optimiz
ation p
r
obl
em
s: Appl
icatio
n in n
o
n
-conv
entio
na
l
machin
ing
process
e
s.
Expert Systems
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i
th Applicati
o
ns
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1
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3):565
7-5
668.
[3]
Jan H
e
ttenh
au
sen, Andr
e
w
L
e
w
i
s, T
i
moleo
n
Kipo
uros. A W
eb-b
a
sed
S
y
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alis
ation-
drive
n
Interactive Mult
i-obj
ec
tive Optimisation.
Procedi
a Co
mp
uter
Science
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14
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[4]
Bana
ja Mo
ha
n
t
y
,
Sidh
artha
Pand
a, P.K. Hota
. Differe
ntial ev
ol
ution
a
l
gorit
hm bas
e
d
autom
atic
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eratio
n co
ntrol for interc
onn
ected
p
o
w
e
r s
y
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w
i
th non-l
i
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arit
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Alexa
ndr
ia Engi
neer
in
g
Journ
a
l
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14; 53(3): 53
7-5
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2
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Piotr
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ę
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o
w
i
cz, Al
eksan
der Sk
akovski
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bas
ed
Differe
ntial
E
v
oluti
on A
l
g
o
ri
thm for th
e
Discrete-co
ntin
uous
Sch
edu
li
ng
w
i
th
Co
nti
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us R
e
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ur
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M. Elsa
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uhu
l A.
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a
r
y
l
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Essam. A s
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mbin
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o
rit
h
m
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i
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optimiz
atio
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u
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ati
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o
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Coel
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a
r
y
Multi-Objectiv
e Optimizatio
n
A Historical
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w
of the
F
i
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IEEE
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mp
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a
l Intelli
genc
e
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Laum
anns M, T
h
iele L, Deb
K, Z
i
tzler E. Co
mb
ini
ng C
onve
r
genc
e an
d Div
ersit
y
i
n
Evol
uti
onar
y Mu
lti-
Objective Optimization.
Evolu
t
ionary C
o
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Deb K, Pratap
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w
a
l S
,
Me
y
a
r
i
van T
.
A F
a
st and Elitist Multi-o
b
j
e
ctive Gen
e
tic
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:
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on Evol
utionar
y Computation
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8
2
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i
tzler E, Deb K,
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h
iele L.
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