Int
ern
at
i
onal
Journ
al of Ele
ctrical
an
d
Co
mput
er
En
gin
eeri
ng
(IJ
E
C
E)
Vo
l.
8
,
No.
6
,
D
ece
m
ber
201
8
, pp.
5351
~
53
58
IS
S
N:
20
88
-
8708
,
DOI: 10
.11
591/
ijece
.
v
8
i
6
.
pp
5351
-
53
58
5351
Journ
al h
om
e
page
:
http:
//
ia
es
core
.c
om/
journa
ls
/i
ndex.
ph
p/IJECE
HOPX
Crosso
ver Op
erator for
the
Fi
xed Charg
e
L
og
i
sti
c
Model
wi
th Prior
ity Bas
ed
Encodi
ng
Ah
med
La
hjo
uj
i E
l
Idrissi
1
,
C
h
ak
ir
T
ajani
2
Mohame
d Sabb
an
e
3
1,3
Facul
t
y
of
Sci
enc
e
Mekne
s,
M
oulay
Ism
ai
l
Uni
ver
sit
y
,
Morocc
o
2
Pol
y
disc
ipl
in
ar
y
fac
u
lty
of
L
arache
,
Abdelmal
ek
Essaa
di
Univ
ersi
t
y
,
Moroc
co
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
Ma
r 10
, 201
8
Re
vised
Ju
l
25,
2018
Accepte
d
Aug 10
, 201
8
In
thi
s
pape
r
,
w
e
are
intere
sted
to
an
importan
t
Logi
stic
p
roblem
m
odel
ised
us
opti
m
iz
ation
proble
m
.
It
is
the
fix
ed
ch
ar
ge
tra
nsport
at
io
n
proble
m
(FCTP
)
where
the
ai
m
is
to
fin
d
the
opti
m
al
so
lut
ion
which
m
i
nimize
s
th
e
obje
c
ti
ve
func
tion
cont
ai
n
ig
two
cos
ts,
var
ia
b
l
e
costs
proportional
to
the
amount
shipped
and
fixe
d
cost
reg
ard
le
ss
of
the
quant
ity
t
ran
sported.
To
solve
thi
s
kind
of
proble
m
,
m
et
ahe
urist
ic
s
and
evol
uti
on
ar
y
m
et
hods
should
be
app
li
ed
.
G
e
net
i
c
a
lgori
thms
(GA
s)
see
m
t
o
be
on
e
of
such
hopef
u
l
a
pproa
ch
es
whi
ch
is
base
d
bot
h
on
proba
bi
li
t
y
op
era
to
rs
(Cross
over
and
m
uta
ti
on)
r
esponsible
for
widen
the
solu
t
ion
spac
e.
Th
e
diff
ere
n
t
cha
ra
cteri
sti
cs
of
those
oper
at
o
rs
infl
uence
on
the
per
form
an
ce
and
t
h
e
qual
ity
of
the
ge
net
i
c
al
gor
it
hm
.
In
orde
r
to
im
prove
the
per
fo
rm
anc
e
of
th
e
GA
to
solve
the
FC
TP,
we
propo
se
a
new
ada
pted
cro
ss
over
oper
at
or
called
HO
PX
with
the
priori
t
y
-
b
ase
d
e
ncodi
ng
b
y
h
y
br
idi
zi
ng
the
ch
aract
er
isti
c
s
of
the
two
m
ost
pe
rform
ent
oper
at
o
rs,
the
Order
Cr
oss
over
(OX
)
an
d
Pos
it
ion
-
base
d
cro
ss
over
(PX
).
Nu
m
eri
ca
l
result
s
are
p
rese
nte
d
and
di
scuss
ed
for
seve
ral
insta
nc
e
s
show
ing
the
per
form
anc
e
of
the
dev
el
oped
appr
oac
h
to
obta
in
opt
imal
soluti
on
in
red
u
c
ed
ti
m
e
in
compari
son
to
GA
s
wi
th
other
cro
ss
over
oper
ators
.
Ke
yw
or
d:
Gen
et
ic
al
gorithm
Lo
gisti
c
m
od
el
Pr
io
rity
b
ase
d enco
ding
Transp
or
ta
ti
on
pro
blem
Copyright
©
201
8
Instit
ute of
Ad
v
ance
d
Engi
ne
eri
ng
and
Sc
ie
n
ce
.
Al
l
rights re
serv
ed
.
Corres
pond
in
g
Aut
h
or
:
Ah
m
ed
La
hjou
j
i El
I
dri
ssi
,
Dep
a
rtem
ent o
f
Ma
them
at
ic
s,
Faculty
of S
ci
e
nce Me
kn
es
,
Mor
occo.
Em
a
il
:
idrissil
a
@g
m
ai
l.
co
m
1.
INTROD
U
CTION
Lo
gisti
c
m
od
el
are
t
he
m
os
t
im
po
rtant
pro
bl
e
m
that
nee
d
t
o
be
op
ti
m
iz
ed
f
or
t
he
sm
oo
t
h
op
e
rati
on
of
the
e
ntire
s
upply
chain
[1]
.
It
determ
ines
the
num
ber
and
ty
pe
of
pl
ants,
wa
re
house
s
and
distrib
utio
n
centers
(
DCs)
to
be
us
e
d.
It
al
so
est
ablishes
distrib
ution
c
ha
nn
el
s
a
nd
the
qu
a
ntit
y
of
pro
du
ct
s
to
be
s
hi
pp
e
d
from
su
pp
li
er
s
for
eac
h
cu
stom
er.
Log
ist
ic
m
od
el
co
ve
r
s
a
wi
de
ra
ng
e
of
f
or
m
ulati
on
s
f
ro
m
l
inea
r
determ
inist
ic
m
od
el
s to
no
nlinear s
t
och
a
sti
c com
plexes o
nes.
Fo
r
t
he
first
f
orm
ulati
on
cal
led
li
near
lo
gisti
c
m
od
el
prob
le
m
or
li
ner
tran
sp
ort
at
ion
pro
bl
e
m
.
It
is
a
netw
ork
opti
m
iz
at
ion
pro
ble
m
intro
duce
d
by
Hitc
hco
c
k
[
2]
wh
ic
h
c
on
si
st
s
to
m
ini
m
iz
e
t
he
total
cost
in
order
to
trans
port
ho
m
og
eneous
pr
oducts
f
ro
m
sever
al
s
ources
to
seve
ral
de
posit
s
sat
isfyi
ng
the
lim
i
ts
of
su
pply
and
dem
and
.
This
m
od
el
can
be
fin
d
in
in
dustry,
plan
ning,
co
m
m
un
ic
at
ion
netw
ork,
sche
du
li
ng
,
trans
portat
ion
a
nd
at
tri
bu
ti
on
.
Seve
ral
searc
h
at
ta
cks
the
li
near
lo
gisti
cs
m
od
el
s
wh
ic
h
can
be
s
olv
e
d
by
th
e
si
m
plex
m
et
hod
intr
oduce
d
by
Geo
r
ge
D
ant
zi
g
in
1947
[
3]
.
Also,
it
can
be
so
lve
d
by
ap
pro
xim
a
ti
on
m
et
hods
su
c
h us
the
m
e
thod
of Russell
and the
m
et
ho
d of V
ogel
[4
]
,
[
5].
The
sec
ond
f
orm
ula
ti
on
,
wh
ic
h
is
the
ob
j
ect
ive
of
t
his
stu
dy
,
is
the
fi
xe
d
charge
l
og
ist
ic
m
od
el
.
Th
e
li
te
ratur
e
ar
ound
t
his
first
ca
se
extensi
on
is
ver
y
ric
h.
Hir
sch
an
d
Dan
zi
g
in
[
6]
was
t
he
first
to
f
or
m
ulate
the
FCTP
us
e
xten
sion
of
the
li
ne
r
l
og
ist
ic
m
od
el
.
Ma
ny
pr
act
ic
al
transpor
ta
ti
on
a
nd
distrib
ut
ion
pro
blem
s,
su
c
h
as
the
m
ini
mu
m
cost
netw
ork
fl
ow
(tra
nsshipm
ent)
pr
ob
le
m
with
a
fixe
d
-
c
harge
for
lo
gisti
cs,
can
be
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
& C
om
p
Eng
,
V
ol.
8
, No.
6
,
Decem
ber
201
8
:
535
1
-
5358
5352
form
ulate
d
as
fixe
d
-
c
harge
l
og
ist
ic
m
od
el
.
For
insta
nce,
a
fixe
d
c
os
t
m
ay
be
inc
urr
ed
f
or
each
shi
pm
ent
betwee
n
a
gi
ve
n
plant
a
nd
a
giv
e
n
cust
om
er
an
d
a
fac
il
ity
of
a
plan
t
m
ay
resu
lt
in
a
fixe
d
am
ount
on
inv
est
m
ent.
Th
e
FCTP
ta
kes
these
fi
xed
c
ha
rg
e
i
nto
acc
ou
nt,
s
o
that
the
TP
can
be
c
on
sidere
d
as
a
n
FCTP
with e
qu
al
fixe
d
c
os
ts
of ze
r
o for all
ro
uts.
Ba
la
insk
i
in
1961
m
od
ifie
d
the
FCTP
to
a
li
near
intege
r
pro
blem
[7
]
;
he
ob
se
rv
e
d
tha
t
there
is
an
op
ti
m
al
so
luti
on
for
the
m
od
if
ie
d
ve
rsion
of
FCTP.
A
dlak
ha
in
[8
]
pro
po
s
ed
a
m
et
ho
d
w
hich
c
onsist
s
of
tw
o
par
ts;
it
gets
th
e
best
i
niti
al
so
luti
on
in
t
he
fir
st
par
t
an
d
us
e
s
te
ch
niques
t
o
im
pr
ov
e
this
s
olu
ti
on
a
nd
to
chec
k
it
s o
ptim
ality
.
The
ge
netic
al
gorithm
(G
A
)
is
an
e
vo
l
utio
nar
y
al
gorithm
that
re
pr
e
sent
s
a
fam
ou
s
m
et
aheurist
ic
pro
po
se
d
by
Jhon
Holl
and
in 1
97
5
[
9]
w
hich
is
insp
ire
d
by b
io
lo
gical
m
ec
han
ism
s
su
ch
a
s
Me
nd
el
'
s
la
w
s
an
d
the
theo
ry
of
e
vo
l
ution
pro
posed
by
Cha
rles
Dar
wi
n.
It
us
e
s
the
sa
m
e
vo
cabu
la
ry
as
in
bio
log
y
an
d
cl
assic
a
l
gen
et
ic
s; s
o we
sp
ea
k of ge
ne, ch
ro
m
os
om
e, p
op
ulati
on
[
10]
-
[13
]
.
The
ge
netic
al
gorithm
has
be
en
us
e
d
t
o
s
ol
ve
m
any
com
bin
at
or
ia
l
pro
ble
m
s
includi
ng
FCTP
[14]
.
Its
m
ai
n
adv
a
ntage
is
th
at
it
al
lows
a
good
c
om
bin
at
io
n
bet
wee
n
the
exp
l
oitat
ion
of
s
olu
ti
on
s
a
nd
t
he
exp
l
or
at
io
n
of
the
resea
rch
sp
ace.
T
his
is
est
ablishe
d
a
s
a
functi
on
of
the
GA
pa
ra
m
et
ers
res
pect
ively
.
Howe
ver,
it
s
disad
va
ntage
li
es
in
two
po
i
nts;
a
com
pu
ta
ti
on
al
tim
e
lar
ge
e
nough
to
be
able
to
co
nv
e
r
ge
towa
rd
s
the
optim
a
l
so
luti
on
and
the
c
onvergen
ce
that
is
a
big
pro
blem
fo
r
G
As
.
In
a
ddit
ion
,
the
different
char
act
e
risti
cs
of
the
ge
netic
op
e
rato
r
s
in
fluen
ce
on
the
perform
ance
and
t
he
qual
it
y
of
the
GA
.
F
or
this
reason
a
nd
in
order
to
im
pr
ove
the
pe
rfor
m
ance
of
t
he
G
A
to
so
l
ve
the
FCTP,
we
pr
opos
e
a
ne
w
a
dap
te
d
cro
ss
over
oper
at
or
cal
le
d
H
O
PX
with
t
he
pr
iority
-
base
d
e
nc
od
i
ng
by
hybr
idizi
ng
t
he
c
ha
ra
ct
erist
ic
of
th
e
two
m
os
t per
f
or
m
e
nt ope
rators, t
he
Ord
e
r
Cr
os
s
over
(OX
)
a
nd P
os
it
ion
-
ba
sed
c
ro
ss
over
(
P
X
).
This
pa
per
is
orga
nized
as
f
ollow
s:
The
sec
ond
sect
io
n
give
s
an
ove
rv
ie
w
of
th
e
f
orm
ula
ti
on
of
the
fixe
d
cha
r
ge
l
og
ist
ic
m
od
el
.
The
thi
rd
pa
rt
pr
ese
nts
the
G
A
as
a
m
et
ho
d
of
res
olu
ti
on
and
these
para
m
et
ers
wh
ic
h
ha
ve
inf
luence
d
the
qu
al
it
y
of
the
resu
lt
s.
In
a
ddit
ion
to
the
pr
opose
d
hybri
d
c
ro
s
so
ve
r
ope
rato
r
cal
le
d
HOPX
.
Th
e
f
ourt
h
sect
io
n
de
al
s
with
the
de
velo
pm
ent
and
i
m
ple
m
entat
io
n
of
th
e
G
A
w
it
h
the
ne
w
operato
r
wh
e
re
se
veral
nu
m
erical
resu
lt
s
ar
e
pr
ese
nted
sh
owin
g
his
pe
rfor
m
ance
in
co
m
par
ison
with
var
ia
nts oper
at
or
s
.
2.
LOGISTI
CS
MO
DEL DE
S
C
RI
PTIO
N A
ND FO
R
MU
L
AT
IO
N
Lo
gisti
c
is
a
profo
und
facto
r
on
the
ad
de
d
va
lue
of
each
business
.
E
xam
i
n
es
eac
h
of
thi
s
act
ivit
y
is
done
at
seve
ral
le
vels:
log
ist
ic
eng
i
neer
i
ng,
te
chn
ic
al
pu
blica
ti
on
s,
proc
ur
e
m
ent...
S
ugges
ts
to
each
el
em
ent
a
separ
at
e
res
pons
ibil
it
y
towa
r
ds
t
he
produc
t
arr
ive
d
at
the
c
us
tom
er
or
the
c
onsum
er
in
the
c
om
plex
netw
ork [1
5
]
.
C
on
si
der
i
ng
t
he
fo
ll
owin
g
gr
a
ph
G
=
(
N,
A)
,
wh
ic
h
co
ns
ist
s
of
a
finite
set
of
nodes
N
=
{
1,2,
.
.
.
,
n}
and
a
set
of
directed
a
rcs
A
=
{(i
,
j)
,
(
k,
l
)
,
.
.
.
,
(
s,
t
)}
,
wh
ic
h
j
oi
ns
pa
irs
of
no
des
in
N
.
It
is
gra
phic
al
ly
il
lustrate
d
in
Fi
gure
1.
Figure
1. Tra
nsporta
ti
on p
la
n
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
& C
om
p
Eng
IS
S
N:
20
88
-
8708
HO
P
X Cr
os
s
ov
er Oper
ato
r
for
t
he
Fix
ed
C
ha
r
ge
Logist
ic
Mo
del wi
th Pri
or
it
y
...
(
Ah
me
d
L
ahjo
uji E
l
I
dr
issi
)
5353
2.1.
Fixe
d
-
ch
arg
e
Lo
gistic
Model
The
FCTP
is
m
uch
m
or
e
diff
ic
ult
to
sol
ve
due
to
t
he
prese
nce
of
fixe
d
c
os
ts,
wh
ic
h
ca
us
e
disco
ntin
uiti
es
in
the
ob
j
ect
iv
e
functi
on.
I
n
the
fixe
d
-
cha
r
ge
log
ist
ic
m
odel
,
two
ty
pes
of
costs
ar
e
c
on
sidere
d
si
m
ultaneou
sly
w
hen
the
best
course
of
act
io
n
is
sel
ect
e
d:
(
1)
va
riable
c
ost
s
pro
portio
nal
to
the
act
ivit
y
le
vel;
and
(2)
fixe
d
costs.
The
F
CTP
seeks
th
e
determ
inati
o
n
of
a
m
ini
m
um
cost
transp
ort
at
ion
plan
fo
r
a
ho
m
og
e
neous
com
m
od
it
y
fr
om
a
nu
m
ber
of
pla
nts
to
a
num
ber
of
cust
om
ers.
It
re
qu
i
r
es
the
sp
eci
ficat
ion
of
the
le
vel
of
s
upply
at
each
pl
ant,
the
am
ou
nt
of
dem
and
at
each
custo
m
er,
and
the
t
ran
s
portat
io
n
cost
an
d
fixe
d
cost
fro
m
each
plant
t
o
each
cust
ome
r.
The
go
al
is
to
al
locat
e
the
su
p
ply
avail
abl
e
at
each
plant
so
as
to
op
ti
m
iz
e
a
crit
erion
w
hile
s
at
isfyi
ng
the
dem
and
at
ea
ch
c
us
tom
er.
The
usual
ob
je
ct
ive
f
un
ct
io
n
is
to
m
ini
m
iz
e
the
total
var
ia
ble
cost
an
d
fi
xed
c
os
ts
f
ro
m
the
al
locat
ion
.
It
is
one
of
the
si
m
plest
co
m
bin
at
or
ia
l
pro
blem
s
inv
ol
vin
g
c
onstrai
nts.
T
h
e
fi
xedchar
ge
lo
gisti
c
m
od
el
with
I
pla
nts
an
d
J
custom
ers
can
be
form
ulate
d
as foll
ows:
Mi
n
Z
=
∑
∑
(
+
)
=
1
=
1
(1)
∑
=
1
=
=
1
,
2
,
…
.
,
(2)
∑
=
1
=
=
1
,
2
,
…
.
(3)
≥
0
=
1
,
2
,
…
,
=
1
,
2
,
…
,
=
0
=
0
=
1
>
0
x
ij
: U
nknow
n qu
a
ntit
y t
o be t
ran
s
porte
d
f
r
om
so
ur
ce
i
to
dest
inati
on
j
;
c
ij
: Varia
ble tr
ans
portat
ion
co
st from
so
urce
i t
o
de
sti
nation
j
;
f
ij
: Fixe
d
tra
nsporta
ti
on co
st
associat
ed wit
h r
oad
(
i,j)
;
y
ij
: A bina
ry va
riable
wh
ic
h
i
s
1
if
x
ij
> 0
an
d
0 i
f x
ij
= 0
;
S
i
:
Am
ou
nt
of
su
pply
at s
ourc
e
i
;
D
j
:
Am
ou
nt
of d
em
and
at
des
ti
nation
j
.
3.
PRIORIT
Y
-
B
AS
ED
GE
NETIC
ALGO
RI
THM F
OR
FI
X
E
D CH
A
RGE
LOGISTI
C MO
DEL
In
orde
r
to
a
pp
ly
the
gen
et
ic
a
lgorit
hm
(G
A),
it
is
necessary
to
ch
os
e
a
nd
a
dap
ta
te
the
re
presentat
io
n
m
et
ho
d
f
or
t
he
so
l
ution
of
t
he
pro
blem
.
Th
ere
are
se
ver
al
m
e
tho
ds
of
re
pr
ese
ntati
on
th
at
are
us
e
d
to
so
lve
log
ist
ic
s
m
od
el
s
us
in
g
GA
,
t
her
e
is
t
he
m
a
trix
re
pr
e
senta
ti
on
[
1
5
]
,
re
pr
esentat
ion
by
t
he
nu
m
ber
of
prüf
e
r
[1
6
]
,
an
d
th
ere
is
al
so
the
pr
i
or
it
y
base
d
re
pr
ese
ntati
on
w
hich
is
a
new
and
m
or
e
a
dapt
ed
f
or
e
nc
od
i
ng
a
nd
decodin
g
t
he
log
ist
ic
s
m
od
el
s.
It
is
fir
st
use
d
by
Ge
n
an
d
Alti
par
m
ak
[1
7
]
to
co
de
a
nd
decode
a
t
wo
-
sta
ge
trans
port
pro
bl
e
m
.
Then
,
th
e
GA
co
ns
ist
s
to
sever
al
s
te
ps
:
I
niti
al
iz
at
ion
proces
ses
(G
ene
rati
on
of
an
p
o
pula
ti
on
of
so
luti
ons),
evaluati
on,
sel
ect
ion
,
crosso
ver
an
d
m
utati
on
ope
rator
s
to
create
a
new
popula
ti
on
of s
olu
ti
ons.
3.1.
En
co
ding
chrom
osome
Fo
r
t
he
pri
ori
ty
based
repres
entat
ion
,
t
he
s
olu
ti
on
(c
hrom
os
om
e)
is
repr
esented
by
an
i
nteg
ral
chai
n
of
le
ng
t
h
eq
ual
ing
the
num
ber
of
s
ources
plus
the
num
ber
of
cl
i
ents.
Eac
h
gen
e
i
n
this
c
hrom
os
om
e
ind
ic
at
es
the ide
ntific
at
ion o
f
a
no
de
(
num
ber
).
In this
represe
ntati
on
,
a
ge
ne i
n
a c
hrom
os
om
e con
ta
ins
tw
o
ty
pes
of in
f
orm
at
ion
:
a)
The p
os
it
ion
of
a g
e
ne
t
o rep
r
esent the
no
des
(
s
ource /
destinat
ion)
b)
The
value
of
a
gen
e
w
hich
represe
nts
th
e
pr
i
or
it
y
of
t
he
no
de
f
or
the
co
ns
tr
uctio
n
of
a
trans
port tree.
A
ch
r
om
os
ome
co
ns
ist
s
of
th
e
pr
i
or
it
ie
s
of
s
ources
a
nd
de
posit
s
to
ob
ta
in
a
trans
port
tree
,
it
s
le
ng
t
h
is equ
al
to
the t
otal nu
m
ber
of
so
urces
m
an
d deposit
s
n
, ie.
(
m+
n)
.
Each c
hrom
os
om
e ca
n
be reco
ns
tr
uc
te
d
in
a ra
ndom
w
ay
an
d t
he
re is
no
need f
or a c
orr
ect
ion
al
gorith
m
after
the g
e
ne
rati
on of a
po
pu
la
ti
on
[18].
3
.
2
.
Sele
ctio
n
an
d
evalu
at
i
on met
hod
Sele
ct
ion
a
nd
evaluati
on
a
re
gen
et
ic
proces
ses
to
eval
uate
pro
duct
s
olu
ti
on
s
a
nd
com
pa
re
them
wit
h
existi
ng
ch
r
omoso
m
es
in
ord
er
to
ch
oose
t
he
best
to
pre
serv
e
them
in
m
e
m
or
y.
The
r
efore;
the
eval
uation
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
& C
om
p
Eng
,
V
ol.
8
, No.
6
,
Decem
ber
201
8
:
535
1
-
5358
5354
functi
on
will
s
el
ect
or
r
ef
us
e
an
in
div
i
du
al
t
o
retai
n
only
tho
s
e
in
div
id
ua
ls
with
the
best
cost
acc
ordin
g
to
the
current
po
pu
l
a
ti
on
.
I
n
pract
ic
e
there
are
sev
eral
ty
pes
of
s
el
ect
ion
ap
plied
in
ge
netic
al
gorithm
s,
especial
ly
we
em
plo
y
the
el
it
ist
m
echan
ism
;
In
fact,
t
he
best
s
olu
ti
on
of
the
c
urren
t
popula
ti
on
is
sel
ect
ed
an
d
pr
es
erve
d
in m
e
m
or
y [1
9]
.
Figure
2
.
Pr
i
or
i
ty
-
based
ch
r
om
os
o
m
es and
t
ran
s
portat
io
n
tr
ees
3
.
3
.
Adap
ted mut
at
i
on
Muta
ti
on
op
e
r
at
or
op
e
rates
by
exch
a
ngin
g
i
nfor
m
at
ion
wit
hin
a
c
hrom
os
om
e.
Howev
e
r
,
instea
d
of
us
in
g
this
op
e
r
at
or
betwee
n
two
pa
ren
ts,
we
us
e
betwee
n
two
se
gm
ents
of
a
pa
ren
t.
W
e
hav
e
c
hose
n
a
sw
a
p
m
uta
ti
on
ope
r
at
or
that
is
pr
opos
es
by
(Mic
halewicz
,
19
92)
wh
ic
h
perm
utes
the
values
of
t
wo
ra
ndom
l
y
sel
ect
ed
posit
ion
s
f
or
eac
h
c
hrom
os
om
e
to
re
du
ce
the
risk
of
re
pro
du
c
ing
a
ch
r
om
os
om
e
with
the
sam
e
so
luti
on
[20],
[
21
]
.
Th
e
pr
oce
dure
of swa
p
m
uta
ti
on
i
ll
us
tr
at
ed
bel
ow wo
rk
s
as
fo
ll
ows:
Proced
ure
of
t
he SW
AP
mutat
i
on
Figure
3
.
Ex
a
m
ple o
f
the
S
WAP m
utati
on
operat
or
3
.
4
.
Pro
po
se
d
cross
ov
er
ope
rator
fo
r
F
CTP
The
c
ro
ss
over
op
e
rato
rs
a
re
ge
netic
process
es.
They
w
ork
on
tw
o
dif
fe
re
nt
pa
ren
ts
(chr
om
os
om
es)
by
com
bin
ing
t
he
cha
racteri
sti
cs
of
the
tw
o
c
hrom
os
om
es,
t
hey
consi
st
in
app
ly
in
g
proce
dures
with
a
ce
rtai
n
pro
bab
il
it
y
cal
le
d
the
c
ro
s
sover
rate
[P
c
∈
(
0
,
1)
]
on
t
he
in
div
id
uals
sel
ec
te
d
to
giv
e
bir
th
to
on
e
or
m
ore
(u
s
ually
tw
o)
offsprin
g.
I
n
t
his
c
on
te
xt,
i
n
order
to
am
eli
or
at
e
t
he
perf
orm
ance
of
the
GA
to
so
l
ve
t
he
fixe
d
charge
lo
gisti
c
m
od
el
,
we
ha
ve
ch
os
e
n
the
t
wo
m
or
e
pe
f
orm
ante
cro
ss
ov
er
operat
or
s
(OX
an
d
P
X)
fo
ll
ow
i
ng
a
com
par
at
i
ve
stud
y
car
ried
ou
t
f
or
seve
r
al
on
es
[18].
The
n,
w
e
pro
po
s
ed
a
n
hy
bri
dizat
ion
of
th
is
tw
o
In
put
:
One
pa
rent
Step
1
:
Se
lect t
ow
element
at
ra
ndom
Step
2
:
Swap
t
he
e
le
men
t
on
th
ese
posit
ions
Ou
tput
:
One
of
f
spr
ing
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
& C
om
p
Eng
IS
S
N:
20
88
-
8708
HO
P
X Cr
os
s
ov
er Oper
ato
r
for
t
he
Fix
ed
C
ha
r
ge
Logist
ic
Mo
del wi
th Pri
or
it
y
...
(
Ah
me
d
L
ahjo
uji E
l
I
dr
issi
)
5355
op
e
rato
rs
by
hi
gh
li
ghti
ng
the
char
act
e
risti
cs
of
each
of
t
hem
.
The
pr
oc
edure
of
the
pro
po
se
d
oper
at
or
is
pr
ese
nted
b
el
l
ow a
nd the
Fig
ure
4.
il
lustrate
t
he
proce
dure
as
an
ex
am
ple.
Proced
ure
of
t
he HO
P
X
ope
rator
:
Figure
4
.
Ex
a
m
ple o
f
the
H
OPX
op
e
rato
r
4.
COMP
UTAT
IONAL
RES
ULT
S
The
m
ai
n
ob
j
e
ct
ives
of
t
his
work
a
re
to
im
pro
ve
the
be
ha
vior
of
the
ge
ne
ti
c
al
go
rithm
with
res
pec
t
to
the
c
r
os
s
ov
e
r
ope
rato
rs
bec
ause
t
hey
ha
ve
an
i
nf
l
uen
ce
on
the
qual
it
y
and
pe
rfo
rm
ance
of
these
al
go
rithm
s
and
to
est
a
blish
to
w
hat
ex
te
nt
the
propo
sed
al
g
or
it
hm
so
lves
the
di
ff
e
ren
t
ty
pes
of
lo
gisti
c
pro
blem
s
com
par
ed
t
o
th
e already
us
e
d m
et
ho
ds.
In
this
sect
io
n,
we
ha
ve
ap
plied
the
de
velo
pe
d
H
OPX
oper
at
or
to
L
og
ist
i
c
m
od
el
with
f
ixed
c
os
t
in
order
to
c
om
par
e
the
res
ults
with
th
os
e
obt
ai
ned
us
i
ng
ot
her
ada
pted
op
erators
nam
ely
OPEX,
OX
,
PX
.
In
this
stud
y,
we
hav
e
ch
os
e
n
six
instances
4x5,
5x10,
10x10,
10x2
0,
20
x30,
30x50
al
r
eady
us
ed
in
s
ever
a
l
arti
cl
es
[22],
knowin
g
t
hat
th
e
optim
al
so
luti
on
is
know
n
f
or
sm
all
instances.
T
her
e
fore,
we
ha
ve
cha
nged
th
e
gen
e
ti
c
al
gorit
hm
par
a
m
et
ers,
su
c
h
as
the
num
ber
of
it
erat
ion
s
to
see
t
he
influ
e
nce
of
th
e
la
tt
er
on
the
resu
lt
s
and on t
he qu
al
it
y of
the
so
l
ution
s
. T
his al
gor
it
h
m
is cod
e
d
i
n
J
AVA
la
ngua
ge
in
the
Net
Be
ans 8.
0.2 ID
E.
Table
1.
pr
ese
nts
the
sim
ulatio
n
res
ults
obta
ined
f
ro
m
20
tim
es
by
the
ge
netic
al
go
rit
hm
based
on
each
op
e
rato
r
us
e
d
to
so
l
ve
t
he
lo
gisti
c
pr
oble
m
with
fixe
d
c
os
t.
For
the
res
ults
dis
play
ed
in
t
his
ta
bl
e,
w
e
no
ti
ce
that
the
gen
et
ic
al
gorithm
with
the
HO
P
X
operat
or
al
lowed
ob
ta
i
ni
ng
the
op
ti
m
a
l
so
luti
on
known
for
sm
a
ll
instances
as
G
As
with
oth
er
cr
os
s
over
operat
or
s
.
H
ow
e
ver,
f
or
la
rg
e
r
insta
nc
es
a
bette
r
optim
al
so
luti
on is ac
hi
eved with
an i
m
pr
ov
em
ent in co
m
pu
ta
ti
on
ti
m
e. I
ndee
d,
for
sm
all instances,
4*
5
a
nd 5
*
10, t
he
ob
ta
ine
d
res
ults
show
that
GA
with
al
l
cro
ss
over
ope
ra
tors
us
e
d
al
lo
w
to
obta
in
t
he
best
c
hrom
os
om
e
sat
isfyi
ng
t
he
op
ti
m
al
so
luti
on
.
H
ow
e
ve
r,
for
pr
ob
le
m
s
with
la
r
ge
siz
e,
w
it
h
the
pro
po
se
d
c
ro
ss
over
op
erat
or
HOPX
,
w
e
are
optim
iz
ing
th
e
obj
ect
ive
f
un
ct
ion
with
a
sl
igh
t
pr
e
ci
sio
n
in
te
rm
s
of
op
t
i
m
al
so
luti
on
.
The
n,
the
la
tt
er
is
m
or
e
a
dvanta
ge
ou
s
t
o
the
le
ve
l
of
nu
m
ber
of
it
erati
ons
w
it
h
a
reduce
d
execu
ti
on
ti
m
e
.
This
m
eans
that
H
O
PX
is
m
or
e
ap
pro
pr
ia
te
tha
n
oth
e
r
ope
rators
that
ha
ve
al
re
ady
us
e
d
to
sol
ve
the
fi
xed
c
harge
lo
gisti
c m
od
el
see Fig
ur
e
5
.
a
nd Fig
ure
6.
In
put:
Tow
parent
s;
Step
1
:
S
elec
t
1/
3
subs
tring
fro
m one
paren
t
at
random
;
Step
2
:
S
elec
t
1/
4
subs
tring
and
1/4
of
set
of
posi
ti
ons f
rom
sam
e par
ent
at
random
.
Step
3
:
Produc
e
a
proto
-
ch
il
d
by
copying
the
nod
es
on
th
ese
posi
t
ions i
nto
the c
or
res
ponding
posit
ions
of
it;
Step
4
:
Del
ete
t
he
nodes
wh
ic
h
are
already
sel
e
ct
ed
from t
h
e
se
c
ond
parent;
The
result
ed
seq
uenc
e
of
nodes c
ontai
ns
th
e
node
s of
th
e
proto
-
ch
il
d
n
ee
ds;
Step
5
:
P
lac
e
th
e
nodes into
the
unfi
x
ed
posit
ions o
f the
proto
-
chil
d
from l
e
ft to
rig
ht
ac
cording
to
t
he
order of
th
e
se
qu
enc
e
to
produce
one
of
fspring;
Ou
tput:
Tow
offs
pring.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
& C
om
p
Eng
,
V
ol.
8
, No.
6
,
Decem
ber
201
8
:
535
1
-
5358
5356
It
shou
l
d
be
no
te
d
that
the g
e
netic
al
gorithm
with
the d
evel
op
e
d
c
ro
s
sover
op
e
rato
r
is
al
so
ap
plied
t
o
the
li
near
lo
gi
sti
c
m
od
el
wh
ic
h
al
so
s
ho
ws
it
s
perfor
m
ance
with
r
espect
to
the
GA
s
with
th
e
oth
er
m
entioned
op
e
rators.
Table
1.
Best
a
nd av
e
ra
ge
re
s
ults by
dif
fer
e
nt
o
pe
rato
r for t
he
te
st
prob
le
m
s o
f
Fixe
d
-
c
ha
rg
e
Logist
ic
Mod
el
Prob
le
m
Para
m
eter
OPEX
OX
PX
HOPX
Size
m
x n
Po
p
size
Maxg
en
Bes
t
Av
rg
Bes
t
Av
rg
Bes
t
Av
rg
Bes
t
Av
rg
4x5
20
300
9291
9291
9291
9291
9291
9291
9291
9291
30
500
9291
9291
9291
9291
9291
9291
9291
9291
5x10
20
300
1
2
7
1
8
1
2
7
5
1
1
2
7
1
8
1
2
7
5
1
1
2
7
1
8
1
2
7
5
1
1
2
7
1
8
1
2
7
1
8
30
500
1
2
7
1
8
1
2
7
3
4
1
2
7
1
8
1
2
7
3
4
1
2
7
1
8
1
2
7
3
4
1
2
7
1
8
1
2
7
1
8
1
0
x
1
0
20
500
1
3
9
8
7
1
4
0
7
4
1
3
9
8
7
1
4
1
3
9
1
4
0
6
5
1
4
1
3
3
1
3
9
8
7
1
4
0
4
7
30
700
1
3
9
3
4
1
4
0
7
4
1
3
9
8
7
1
4
0
7
4
1
3
9
3
4
1
4
0
6
5
1
3
9
3
4
1
3
9
8
7
1
0
x
2
0
20
500
2
2
2
5
8
2
2
4
8
4
2
2
3
7
6
2
2
5
3
1
2
2
1
9
8
2
2
8
3
4
2
2
1
5
0
2
2
2
5
8
30
700
2
2
0
9
5
2
2
4
8
4
2
2
0
9
5
2
2
1
9
8
2
2
0
9
5
2
2
5
3
2
2
2
0
9
5
2
2
1
9
8
2
0
x
3
0
20
500
3
2
8
4
0
3
4
1
1
9
3
3
1
9
2
3
4
5
8
1
3
2
6
8
3
3
4
4
1
4
3
2
4
9
2
3
3
1
4
2
30
700
3
2
5
2
6
3
3
9
1
7
3
3
9
1
7
3
3
2
3
6
3
2
4
9
2
3
3
2
3
4
3
2
4
7
1
3
2
9
3
6
3
0
x
5
0
20
700
5
5
6
1
1
5
6
3
9
9
5
6
3
9
9
5
6
7
0
5
5
5
4
5
0
5
6
0
0
7
5
5
2
6
9
5
5
4
5
0
30
1000
5
5
1
4
3
5
5
9
1
2
5
5
9
1
2
5
5
9
1
2
5
5
1
0
6
5
5
4
0
7
5
4
1
1
4
5
5
1
0
6
Figure
5
.
Aver
age c
om
pu
ta
ti
on
ti
m
e fo
r dif
fe
ren
t
operato
rs f
or
fixe
d
c
ha
rg
e
Lo
gisti
c Mo
de
l
s
Table
2.
gi
ves
the
best
c
hrom
os
om
e
for
the
diff
e
re
nt
insta
nces
wh
ic
h
re
present
the
so
l
ut
ion
f
or
t
he
fixe
d
c
harge l
ogist
ic
p
r
oble
m
.
Table
2
. Best c
hrom
os
om
e fo
r
the test
prob
le
m
s
Ins
tan
ce
Bes
t chro
m
o
so
m
e
4x5
1
6 8
2 4
3 9
7 5
5x10
1
1
1 7
3 1
3
14
2 5
6
12
4 1
0
9 1
5
8
1
0
x
1
0
8
5 1
4
12
19
2 1
6
1
5
20
9 1
0
7 1
1
3 1
6
4 1
3
18
1
7
1
0
x
2
0
2
4
5 2
1
10
9 1
7
13
1
9
12
6 2
9
1 2
8
3
1
8
20
15
4 7
1
1
2
5
23
26
2
7
16
30
2
2
2 8
14
2
0
x
3
0
5
4
2
13
31
3
3
30
2
3
41
48
9 2
7
5
0
11
2
9
40
16
20
2
5
3
22
28
43
3
2
44
3
8
6 2
6
37
49
2 2
4
36
7 1
8
35
1
4
8
1
46
34
17
1
0
15
4
12
21
45
39
4
7
19
3
0
x
5
0
5
9
50
66
4
7
8 2
9
2
51
74
56
35
6
5
27
5
28
68
25
21
1
9
40
38
7
6
53
46
7
1
42
14
5
5
61
7 4
1
22
6 1
3
72
3
6
1
5
11
3
3
7
9
57
4 4
9
30
43
73
6
0
52
4
4
58
26
48
6
4
2
3
9 7
5
20
6
7
24
7
0
32
3 6
2
39
1
0
45
5
4
12
17
78
1
6
1
69
63
77
1
8
34
3
7
80
31
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
Elec
& C
om
p
Eng
IS
S
N:
20
88
-
8708
HO
P
X Cr
os
s
ov
er Oper
ato
r
for
t
he
Fix
ed
C
ha
r
ge
Logist
ic
Mo
del wi
th Pri
or
it
y
...
(
Ah
me
d
L
ahjo
uji E
l
I
dr
issi
)
5357
Figure
6
.
Op
ti
m
u
m
so
luti
on
with
diff
e
re
nt
cro
ss
over
ope
r
at
or
s
OPEX
, OX,
PX,
HOPX
with
diff
e
re
nt
instances
f
or
fi
xed cha
r
ge
L
ogist
ic
Mod
el
s
5.
CONCL
US
I
O
N
In
this
w
ork,
we
a
re
inte
res
te
d
in
so
l
ving
log
ist
ic
s
m
odel
s
w
hich
a
re
cl
assifi
ed
as
NP
-
c
om
plet
com
bin
at
or
ia
l
pro
blem
ca
ll
ed
fixe
d
cha
rg
e
t
ran
s
portat
io
n
pro
blem
(F
CTP)
.
Its
reso
l
utio
n
by
exact
m
et
h
od
s
i
s
ver
y
diff
ic
ult,
conseq
ue
ntly
t
he
m
et
aheu
risti
cs
can
be
ex
pl
oited
su
c
h
us
gen
et
ic
al
go
rithm
s.
Then
,
w
e
are
interest
ed
in
i
m
pr
ov
in
g
t
he
perform
ance
of
G
A
s
th
rou
gh
the
c
rosso
ve
r
op
e
rato
r
t
hat
ha
s
a
gr
eat
e
ff
e
ct
.
A
fte
r
pr
ese
ntin
g
a
com
par
ison
with
three
c
ro
s
s
over
op
e
rato
rs
(
OX,
OPEX
,
and
P
X
)
ada
pted
to
our
pro
ble
m
wit
h
pr
i
or
it
y
base
b
encodin
g
,
we
pro
po
se
d
a
ne
w
hybr
i
d
op
e
r
at
or
with
the
two
operat
or
s
OX
an
d
P
X
t
hat
we
cal
le
d
(HOP
X
).
Nu
m
erical
resu
lt
s
was
de
velo
ped
sup
portin
g
the
c
on
cl
us
io
n
that
the
pr
opos
e
d
HOPX
op
e
rato
r
is
m
or
e
ef
fici
ent
t
han
the
ot
her
pr
ese
nted
opera
tors
ei
ther
f
or
the
optim
al
so
luti
on
le
vel
or
the
execu
ti
on ti
m
e
, es
pecial
ly
f
or large
r
in
sta
nce
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
87
08
In
t J
Elec
& C
om
p
Eng
,
V
ol.
8
, No.
6
,
Decem
ber
201
8
:
535
1
-
5358
5358
REFERE
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