I
AE
S
I
n
t
e
r
n
at
ion
al
Jou
r
n
al
of
Ar
t
if
icial
I
n
t
e
ll
ig
e
n
c
e
(
I
J
-
AI
)
Vol.
14
,
No.
4
,
Augus
t
2025
,
pp.
2741
~
2752
I
S
S
N:
2252
-
8938
,
DO
I
:
10
.
11591/i
jai
.
v
14
.i
4
.
pp
27
41
-
2752
2741
Jou
r
n
al
h
omepage
:
ht
tp:
//
ij
ai
.
iaes
c
or
e
.
c
om
S
ol
vi
n
g
k
-
c
ity m
u
ltip
l
e
t
r
av
e
ll
in
g sal
e
sm
an
u
si
n
g ge
n
e
t
ic
al
gor
ith
m
Alik
ap
at
i
P
r
ak
a
sh
1
,
2
,
Ur
u
t
u
r
u
B
alak
r
is
h
n
a
3
,
M
an
ogar
an
T
h
an
gar
aj
4
,
T
h
e
n
e
p
all
e
Jayan
t
h
K
u
m
ar
4
1
D
e
pa
r
tm
e
nt
of
M
a
th
e
ma
ti
c
s
,
J
a
w
a
ha
r
la
l
N
e
hr
u T
e
c
hnol
ogi
c
a
l
U
ni
ve
r
s
it
y A
na
nt
a
pur
,
A
na
nt
a
pur
a
mu, I
ndi
a
2
D
e
pa
r
tm
e
nt
of
H
uma
ni
ti
e
s
a
nd S
c
ie
n
c
e
s
, V
e
mu I
ns
ti
tu
te
of
T
e
c
hnol
ogy,
P
a
ka
la
, I
ndi
a
3
D
e
pa
r
tm
e
nt
of
S
c
ie
nc
e
a
nd
H
uma
ni
ti
e
s
, S
r
e
e
e
ni
va
s
a
I
ns
ti
tu
te
o
f
T
e
c
hnol
ogy a
nd M
a
na
ge
me
nt
S
tu
d
i
e
s
,
B
a
nga
lo
r
e
, I
ndi
a
4
D
e
pa
r
tm
e
nt
of
M
a
th
e
ma
ti
c
s
, F
a
c
ul
ty
of
E
ngi
ne
e
r
in
g a
nd T
e
c
h
nol
ogy, J
A
I
N
(
D
e
e
me
d
-
To
-
B
e
U
ni
ve
r
s
it
y)
, K
a
n
a
ka
pur
a
, I
ndi
a
Ar
t
icle
I
n
f
o
AB
S
T
RA
CT
A
r
ti
c
le
h
is
tor
y
:
R
e
c
e
ived
J
un
11,
2024
R
e
vis
e
d
M
a
r
27,
2025
Ac
c
e
pted
J
un
8,
2025
T
h
i
s
p
ap
er
ad
d
res
s
es
a
n
o
v
e
l
v
ar
i
an
t
o
f
t
h
e
c
l
as
s
i
ca
l
mu
l
t
i
p
l
e
t
r
av
e
l
i
n
g
s
al
e
s
man
p
ro
b
l
em
(M
T
SP)
i
.
e.
k
-
ci
t
y
mu
l
t
i
p
l
e
t
ra
v
el
i
n
g
s
a
l
es
ma
n
p
r
o
b
l
em
(k
-
MT
SP).
T
h
e
p
ro
b
l
em
ca
n
d
es
cr
i
b
e
a
s
f
o
l
l
o
w
s
.
L
e
t
t
h
ere
are
ci
t
i
es
,
s
al
es
ma
n
p
o
s
i
t
i
o
n
e
d
at
d
ep
o
t
ci
t
y
a
n
d
a
p
red
efi
n
ed
p
o
s
i
t
i
v
e
v
a
l
u
e
.
T
h
e
d
i
s
t
a
n
ce
b
et
w
een
each
p
a
i
r
o
f
ci
t
i
e
s
is
k
n
o
w
n
.
T
h
e
o
b
j
e
ct
i
v
e
o
f
t
h
e
k
-
M
T
SP
i
s
t
o
d
e
t
ermi
n
e
a
co
l
l
ect
i
o
n
o
f
cl
o
s
ed
t
o
u
r
s
fo
r
s
al
e
s
man
,
w
h
i
c
h
co
v
er
s
ex
act
l
y
(i
n
c
l
u
d
i
n
g
d
ep
o
t
ci
t
y
)
o
f
ci
t
i
e
s
s
u
ch
t
h
at
t
h
e
t
o
t
a
l
d
i
s
t
an
ce
co
v
ere
d
i
s
mi
n
i
m
u
m.
T
h
e
k
-
M
T
SP
can
b
e
s
een
as
a
co
mb
i
n
a
t
i
o
n
o
f
b
o
t
h
s
u
b
s
et
s
e
l
ect
i
o
n
an
d
p
erm
u
t
a
t
i
o
n
c
h
arac
t
eri
s
t
i
cs
.
Fro
m
t
h
e
t
h
ro
u
g
h
l
i
t
era
t
u
re
rev
i
e
w
,
i
t
i
s
f
o
u
n
d
t
h
a
t
t
h
i
s
s
t
u
d
y
o
n
k
-
M
T
SP
i
s
fi
r
s
t
o
f
i
t
s
k
i
n
d
t
o
t
h
e
b
es
t
o
f
au
t
h
o
r’
s
k
n
o
w
l
e
d
g
e.
T
h
e
p
ap
er
i
n
t
ro
d
u
ce
s
a
zero
-
o
n
e
i
n
t
eg
er
l
i
n
ear
p
ro
g
rammi
n
g
(0
-
1
IL
P)
fo
rmu
l
at
i
o
n
al
o
n
g
s
i
d
e
an
eff
i
ci
en
t
g
e
n
et
i
c
al
g
o
r
i
t
h
m
(G
A
),
d
es
i
g
n
ed
t
o
ad
d
res
s
k
-
M
T
SP.
N
o
c
o
mp
ara
t
i
v
e
s
t
u
d
i
es
carr
i
ed
o
u
t
d
u
e
t
o
t
h
e
ab
s
en
c
e
o
f
ex
i
s
t
i
n
g
s
t
u
d
i
e
s
o
n
k
-
M
T
SP.
H
o
w
e
v
e
r,
t
h
e
d
ev
el
o
p
e
d
G
A
i
s
t
e
s
t
e
d
o
v
er
v
ar
i
o
u
s
b
en
c
h
mark
t
es
t
cas
e
s
fro
m
T
SP
L
IB
an
d
res
u
l
t
s
are
rep
o
r
t
ed
,
w
h
i
ch
ma
y
p
o
t
en
t
i
a
l
l
y
s
erv
e
as
b
a
s
i
s
fo
r
fu
rt
h
er
co
mp
ara
t
i
v
e
s
t
u
d
i
es
.
O
v
era
l
l
fi
n
d
i
n
g
s
d
emo
n
s
t
rat
e
t
h
at
t
h
e
G
A
co
n
s
i
s
t
e
n
t
l
y
p
ro
d
u
ce
s
b
es
t
s
o
l
u
t
i
o
n
s
w
i
t
h
i
n
rea
s
o
n
a
b
l
e
co
m
p
u
t
at
i
o
n
al
t
i
me
s
fo
r
re
l
at
i
v
e
l
y
s
ma
l
l
er
a
n
d
med
i
u
m
t
es
t
cas
es
,
s
u
g
g
es
t
i
n
g
i
t
s
ro
b
u
s
t
n
es
s
an
d
effect
i
v
en
es
s
i
n
t
ac
k
l
i
n
g
t
h
e
k
-
MT
SP.
H
o
w
ev
er,
t
o
en
h
a
n
ce
co
n
s
i
s
t
en
c
y
an
d
eff
i
ci
e
n
cy
,
p
art
i
cu
l
ar
l
y
fo
r
l
arg
er
d
a
t
as
e
t
s
,
fu
r
t
h
er
a
l
g
o
ri
t
h
m
i
m
p
ro
v
eme
n
t
s
are
n
ece
s
s
ar
y
.
K
e
y
w
o
r
d
s
:
Ge
ne
ti
c
a
lgor
it
hm
M
ult
ipl
e
tr
a
ve
ll
ing
s
a
les
man
pr
oblem
T
r
a
ve
ll
ing
s
a
les
man
pr
oblem
T
S
P
L
I
B
Z
e
r
o
-
one
int
e
ge
r
li
ne
a
r
pr
ogr
a
mm
ing
p
r
oblem
Th
i
s
i
s
a
n
o
p
en
a
c
ces
s
a
r
t
i
c
l
e
u
n
d
e
r
t
h
e
CC
B
Y
-
SA
l
i
ce
n
s
e.
C
or
r
e
s
pon
din
g
A
u
th
or
:
Ur
utur
u
B
a
lakr
is
hna
De
pa
r
tm
e
nt
of
S
c
ienc
e
a
nd
Huma
nit
ies
,
S
r
e
e
nivas
a
I
ns
ti
tut
e
of
T
e
c
hnology
a
nd
M
a
na
ge
ment
S
tudi
e
s
Dr
.
Vis
we
s
wa
r
a
iah
R
oa
d,
B
a
nga
lor
e
-
T
ir
upa
thi
,
B
ypa
s
s
R
d,
M
ur
uka
mbattu,
Andhr
a
P
r
a
de
s
h
517127
,
I
ndia
E
mail
I
d
:
ba
lus
it
a
ms
.
maths
@gmail.
c
om
1.
I
NT
RODU
C
T
I
ON
T
he
tr
a
ve
li
ng
s
a
les
man
pr
oblem
(
T
S
P
)
s
tands
a
s
a
we
ll
-
known
c
ombi
na
tor
ial
opti
mi
z
a
ti
on
pr
oblem
withi
n
the
domains
of
c
omput
e
r
s
c
ienc
e
a
nd
math
e
matics
.
I
ts
c
or
e
objec
ti
ve
is
to
identif
y
the
opti
m
a
l
r
oute
that
or
igi
na
tes
a
nd
c
onc
ludes
in
the
s
a
me
c
it
y,
c
ove
r
ing
a
ll
c
it
ies
in
a
given
li
s
t
while
c
ove
r
ing
the
dis
tanc
e
s
be
twe
e
n
e
a
c
h
pa
ir
of
c
it
ies
.
E
xpa
nding
upon
the
c
las
s
ica
l
T
S
P
,
the
m
ult
ipl
e
tr
a
ve
li
ng
s
a
les
man
pr
oblem
(
M
T
S
P
)
int
r
oduc
e
s
a
va
r
iation
whe
r
e
in
mul
ti
ple
s
a
les
men
a
r
e
a
s
s
igned
the
tas
k
of
vis
it
ing
a
s
pe
c
if
ied
s
e
t
of
c
it
ies
.
T
he
a
im
is
to
de
ter
mi
ne
the
mos
t
e
f
f
icie
nt
r
outes
f
or
a
ll
s
a
les
men,
c
oll
e
c
ti
ve
ly
mi
nim
izing
the
ove
r
a
ll
tr
a
ve
r
s
a
l
dis
tanc
e
,
while
e
ns
ur
ing
e
a
c
h
c
it
y
is
vis
it
e
d
only
onc
e
by
a
s
ingl
e
s
a
les
man.
Add
r
e
s
s
ing
th
e
M
T
S
P
pr
ove
s
to
be
a
mor
e
c
ha
ll
e
nging
c
ompar
e
d
to
the
c
on
ve
nti
ona
l
T
S
P
due
to
the
a
ddit
ional
c
ons
tr
a
int
of
invol
ving
mul
ti
ple
s
a
les
men
.
S
im
il
a
r
to
the
T
S
P
,
the
M
T
S
P
is
c
a
tegor
ize
d
a
s
NP
-
ha
r
d
pr
oblem.
T
his
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
4
,
Augus
t
2025
:
274
1
-
2752
2742
c
las
s
if
ica
ti
on
im
pli
e
s
that
a
s
the
number
of
c
it
ies
a
nd
s
a
les
men
incr
e
a
s
e
s
,
f
indi
ng
the
mos
t
e
f
f
icie
nt
s
olut
ion
be
c
omes
pr
ogr
e
s
s
ively
mor
e
c
omput
a
ti
ona
ll
y
de
manding.
Due
to
it
s
pr
a
c
ti
c
a
l
s
igni
f
ica
nc
e
,
the
c
las
s
ica
l
M
T
S
P
a
nd
it
s
va
r
iants
ha
s
be
e
n
e
xtens
ively
s
tudi
e
d
[
1]
–
[
8]
a
nd
wide
va
r
iety
of
s
olut
i
on
methods
h
a
ve
be
e
n
de
ve
loped.
R
e
view
ing
the
e
xis
ti
ng
s
tudi
e
s
on
M
T
S
P
a
nd
it
s
a
ll
ied
pr
oblems
,
the
s
tudy
[
9]
int
r
oduc
e
s
a
nove
l
ge
ne
ti
c
a
lgor
it
hm
(
GA
)
c
hr
omos
ome
a
nd
ope
r
a
t
or
s
f
or
the
M
T
S
P
,
de
mons
tr
a
ti
ng
s
u
pe
r
ior
c
omp
utational
pe
r
f
or
manc
e
a
nd
s
maller
s
e
a
r
c
h
s
pa
c
e
c
ompar
e
d
to
pr
ior
methods
thr
ough
theor
e
ti
c
a
l
a
na
ly
s
is
a
nd
c
omput
a
ti
ona
l
tes
ti
ng.
A
nove
l
gr
ouping
GA
is
de
ve
loped
by
[
10]
f
or
the
M
T
S
P
,
whic
h
opt
im
i
z
e
s
two
objec
ti
ve
s
na
mely
tr
a
ve
r
s
a
l
dis
tanc
e
a
nd
mi
ni
mi
z
ing
the
maximu
m
t
r
a
ve
l
dis
tanc
e
f
or
a
ny
s
a
les
m
a
n.
T
h
is
a
ppr
oa
c
h
s
hown
s
upe
r
ior
pe
r
f
or
manc
e
c
ompar
e
d
t
o
e
xis
ti
ng
li
ter
a
tur
e
a
ppr
oa
c
he
s
on
both
objec
ti
ve
s
.
A
nove
l
c
r
os
s
ove
r
method,
ter
med
the
"
two
-
pa
r
t
c
hr
omos
ome
c
r
os
s
ove
r
"
to
a
ddr
e
s
s
the
M
T
S
P
thr
ough
the
a
p
pli
c
a
ti
on
of
a
GA
,
a
im
ing
to
a
c
hieve
ne
a
r
-
opti
mal
s
olut
ions
is
de
ve
loped
[
11]
.
T
he
mu
lt
ipl
e
de
pot
M
T
S
P
,
whic
h
invol
ve
s
a
n
unl
im
it
e
d
number
of
s
a
les
pe
ople
vi
s
it
ing
a
s
pe
c
if
ic
gr
oup
o
f
c
it
ies
s
tudi
e
d
a
nd
de
ve
loped
polyhedr
a
l
ba
s
e
d
br
a
nc
h
-
a
nd
-
c
ut
a
lgor
it
hm
[
12]
.
An
e
f
f
icie
nt
e
volut
ionar
y
a
lgor
i
thm
that
int
e
gr
a
tes
two
r
e
vis
e
d
ve
r
s
ions
of
the
im
pe
r
ialis
t
c
ompetit
ive
a
l
gor
it
hm
a
nd
the
L
in
-
Ke
r
nighan
a
lgor
it
hm
is
de
ve
loped
f
or
s
olvi
ng
c
las
s
ica
l
M
T
S
P
[
13]
.
A
nove
l
va
r
iant
of
c
las
s
ica
l
M
T
S
P
known
a
s
t
he
c
olor
e
d
T
S
P
ha
s
be
e
n
int
r
oduc
e
d
[
14]
,
f
or
whic
h
two
hybr
id
a
lgor
it
hms
na
mely
hil
l
-
c
li
mbi
ng
ba
s
e
d
GA
a
nd
s
im
ulate
d
a
nne
a
li
ng
-
ba
s
e
d
GA
a
r
e
de
ve
loped
f
o
r
a
c
hieving
be
s
t
qu
a
li
ty
s
olut
ions
.
S
oylu
[
15
]
s
tudi
e
d
M
T
S
P
invo
l
ving
two
s
e
pa
r
a
te
objec
ti
ve
f
unc
ti
ons
:
mi
nim
izing
the
longes
t
tour
length
a
nd
the
tot
a
l
length
of
a
ll
tour
s
,
by
p
r
opos
ing
a
ge
ne
r
a
l
va
r
iable
ne
ighbor
hood
s
e
a
r
c
h.
S
ubs
e
que
ntl
y,
two
meta
he
ur
is
ti
c
tec
hniques
f
or
the
M
T
S
P
a
r
e
pr
opos
e
d:
one
ba
s
e
d
on
the
a
r
ti
f
icia
l
be
e
c
olo
ny
a
lgor
it
hm,
a
nd
the
other
uti
li
z
ing
the
invas
ive
we
e
d
opti
mi
z
a
ti
on
a
lgor
i
thm
[
16]
.
T
he
g
r
a
vit
a
ti
ona
l
e
m
ulation
loca
l
s
e
a
r
c
h
a
lgor
it
hm
[
17
]
is
de
ve
loped
t
o
tac
kle
the
s
ymm
e
tr
ic
M
T
S
P
.
T
h
is
a
lgor
it
hm
r
e
li
e
s
on
the
pr
inciples
of
loca
l
s
e
a
r
c
h,
incor
por
a
ti
ng
two
f
und
a
menta
l
phys
ics
pa
r
a
mete
r
s
:
ve
locity
a
nd
g
r
a
vit
y.
I
t
is
a
ls
o
note
that
M
T
S
P
e
xhibi
ts
a
s
tr
ong
c
onne
c
ti
on
with
va
r
ious
opti
mi
z
a
ti
on
models
,
including
the
ve
hicle
r
outi
n
g
pr
oblem
(
VR
P
)
[
18]
,
[
19]
.
Xu
e
t
al
.
[
20]
int
r
od
uc
e
d
the
two
-
pha
s
e
he
ur
is
ti
c
a
lgor
it
hm
f
or
s
olvi
ng
the
M
T
S
P
,
whic
h
in
tegr
a
tes
a
n
im
pr
ove
d
ve
r
s
ion
o
f
the
k
-
mea
ns
a
lgor
it
hm
to
s
or
t
the
c
it
ies
to
be
vis
it
e
d
ba
s
e
d
o
n
th
e
ir
unique
c
a
pa
c
it
y
c
on
s
tr
a
int
s
a
nd
s
pa
ti
a
l
pos
it
ion
s
.
Give
n
it
s
s
igni
f
ica
nc
e
a
s
a
n
opti
mi
z
a
ti
on
pr
oblem
with
pr
a
c
ti
c
a
l
a
ppli
c
a
ti
ons
,
numer
ous
r
e
s
e
a
r
c
he
r
s
ha
ve
uti
li
z
e
d
M
T
S
P
to
a
ddr
e
s
s
r
e
a
l
-
ti
me
s
c
e
na
r
ios
.
A
ne
w
pr
a
c
ti
c
a
l
va
r
iation
known
a
s
the
ope
n
-
cl
os
e
M
T
S
P
[
21]
,
whic
h
e
xtends
the
c
las
s
ica
l
M
T
S
P
.
Ove
r
ti
me,
it
ha
s
f
ound
r
e
leva
nc
e
in
va
r
ious
r
e
a
l
-
wor
ld
s
it
ua
ti
ons
.
T
o
c
it
e
f
e
w
,
wi
r
e
les
s
r
e
c
ha
r
ge
a
ble
s
e
ns
or
ne
two
r
ks
[
22]
,
na
tur
a
l
dis
a
s
ter
mana
ge
ment
[
23]
,
a
nd
opti
mal
de
li
ve
r
y
dis
tr
ibut
ion
of
L
P
G
c
yli
nde
r
s
[
24]
.
I
n
a
ddit
ion
to
the
c
it
e
d
wor
ks
,
r
e
s
e
a
r
c
he
r
s
ha
ve
a
ls
o
f
o
c
us
e
d
on
de
ve
lopi
ng
hybr
id
a
lgor
it
hms
f
or
the
M
T
S
P
a
nd
it
s
a
ll
ied
pr
oblems
to
im
pr
ove
the
s
olut
ion
qua
li
ty
[
25]
–
[
28]
.
S
ome
of
the
late
s
t
s
tudi
e
s
on
M
T
S
P
a
r
e
a
s
f
oll
o
ws
:
Ya
ng
a
nd
F
a
n
[
29]
im
pleme
nted
e
ne
r
gy
a
nd
r
e
s
our
c
e
c
ons
umpt
ion
c
ons
tr
a
int
s
to
c
r
e
a
te
the
r
e
s
tr
icte
d
mul
ti
-
de
pot
T
S
P
.
T
he
c
ons
is
tent
T
S
P
s
e
a
r
c
he
s
f
or
a
mi
nim
um
-
c
os
t
c
oll
e
c
ti
on
of
Ha
mi
lt
onian
pa
ths
de
s
c
r
ibed
by
[
30]
,
the
f
ir
e
f
ly
a
ppr
oa
c
h
is
pr
e
s
e
nted
to
s
olve
the
s
ingl
e
de
pot
M
T
S
P
us
ing
a
thr
e
s
hold
tec
hn
ique
[
31]
.
A
bi
-
objec
ti
ve
M
T
S
P
with
a
load
b
a
lanc
ing
c
ons
tr
a
int
uti
li
z
ing
GA
[
32]
.
Ve
e
r
e
s
h
e
t
al
.
[
33
]
i
ntr
oduc
e
a
meta
-
he
ur
is
ti
c
ter
med
mul
ti
c
h
r
omos
ome
-
ba
s
e
d
GA
to
s
olve
ope
n
-
c
los
e
d
M
T
S
P
.
M
oti
va
ted
by
the
a
bove
-
m
e
nti
one
d
wor
ks
,
the
c
ur
r
e
nt
r
e
s
e
a
r
c
h
f
oc
us
e
s
on
a
nove
l
va
r
iant
c
a
ll
e
d
k
-
M
T
S
P
,
a
nd
e
f
f
e
c
ti
ve
meta
he
ur
is
ti
c
in
the
f
or
m
o
f
a
GA
.
T
his
GA
incor
por
a
tes
c
ompl
e
x
mut
a
ti
on
s
t
r
a
tegie
s
,
yielding
opti
mal/nea
r
opti
mal
r
e
s
ult
s
.
Ac
c
or
din
g
to
the
a
uthor
's
unde
r
s
tandin
g,
thi
s
GA
r
e
pr
e
s
e
nts
the
pionee
r
ing
e
volut
ionar
y
tec
hnique
a
ppli
e
d
to
t
he
k
-
M
T
S
P
.
T
he
s
ubs
e
que
nt
s
e
c
ti
ons
of
the
p
a
pe
r
a
r
e
or
ga
nize
d
in
the
f
oll
owing
manne
r
.
T
he
s
e
c
ond
s
e
c
ti
on
will
pr
e
s
e
nt
the
de
f
ini
ti
on
a
nd
f
o
r
mul
a
ti
o
n
of
the
k
-
M
T
S
P
.
S
e
c
ti
on
3
will
pr
o
vide
a
de
s
c
r
ipt
ion
of
th
e
GA
a
nd
it
s
ope
r
a
tor
s
.
T
he
c
omput
a
ti
ona
l
f
indi
ng
s
will
be
s
howc
a
s
e
d
in
s
e
c
ti
on
4,
while
s
e
c
ti
on
5
will
e
nd
by
s
umm
a
r
izing
the
f
indi
ngs
a
nd
dis
c
us
s
ing
potential
dir
e
c
ti
ons
f
or
f
utur
e
r
e
s
e
a
r
c
h.
2.
M
AT
HE
M
A
T
I
CA
L
M
ODE
L
T
h
e
k
-
M
T
S
P
c
a
n
b
e
f
o
r
m
a
l
l
y
d
e
f
i
n
e
d
a
s
f
o
l
l
o
w
s
:
L
e
t
=
(
,
)
b
e
a
n
u
n
d
i
r
e
c
t
e
d
w
e
i
g
h
t
e
d
a
n
d
c
o
n
n
e
c
t
e
d
g
r
a
p
h
,
w
h
e
r
e
=
{
1
,
2
,
.
.
.
,
}
b
e
t
h
e
s
e
t
o
f
c
i
t
i
e
s
/
n
o
d
e
s
(
i
n
c
l
u
d
i
n
g
d
e
p
o
t
c
i
t
y
)
a
n
d
=
{
(
,
)
/
(
,
)
∈
,
≠
}
b
e
a
n
e
d
g
e
/
a
r
c
s
e
t
.
F
o
r
e
a
c
h
e
d
g
e
(
,
)
∈
,
a
p
o
s
i
t
i
v
e
d
i
s
t
a
n
c
e
(
>
0
,
=
∞&
=
)
is
a
s
s
igned,
whic
h
indi
c
a
te
tr
a
ve
l
dis
tanc
e
/cos
t
f
r
om
ℎ
c
it
y
to
ℎ
c
it
y.
L
e
t
=
{
1
,
2
,
.
.
.
,
}
be
the
s
e
t
of
s
a
les
men
pos
it
ioned
a
t
a
de
pot
c
it
y.
L
e
t
be
a
binar
y
va
r
iable
,
whic
h
take
s
the
va
lue
1
whe
n
s
a
les
man
p
tr
a
ve
r
s
e
s
f
r
om
c
it
y
i
to
c
it
y
j
,
a
nd
0
,
o
ther
wis
e
.
T
he
c
it
ies
o
ther
than
the
de
pot
a
r
e
known
to
be
in
ter
ve
ning
c
i
ti
e
s
.
T
he
k
-
M
T
S
P
a
im
s
to
identif
y
a
s
e
t
of
c
los
e
d
tour
s
,
e
nc
ompas
s
ing
of
th
e
c
it
ies
,
whe
r
e
e
a
c
h
in
ter
ve
ning
c
it
y
is
vis
it
e
d
by
e
xa
c
tl
y
one
s
a
les
man
with
mi
nim
a
l
d
is
tanc
e
.
T
he
f
or
mu
lation
o
f
the
k
-
M
T
S
P
model
is
ba
s
e
d
on
the
f
oll
owing
a
s
s
umpt
ions
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
Solving
k
-
c
it
y
multi
ple
tr
av
e
ll
ing
s
ales
man
us
ing
ge
ne
ti
c
algor
it
hm
(
A
li
k
apati
P
r
ak
a
sh
)
2743
−
T
he
r
e
a
r
e
number
o
f
c
it
ies
,
out
of
whic
h
c
it
ies
a
r
e
to
be
c
ove
r
e
d
by
s
a
le
s
men,
a
ll
s
tationed
a
t
the
de
pot
c
it
y.
−
All
s
a
les
men
mus
t
s
tar
t
their
r
outes
f
r
om
the
de
pot
c
it
y
a
nd
c
onc
lude
thei
r
tou
r
s
ther
e
a
s
we
ll
.
−
T
he
f
e
a
s
ibl
e
s
olut
ion
c
ompr
is
e
s
a
s
e
t
of
c
los
e
d
tour
s
−
T
he
a
ll
oc
a
ti
on
of
c
it
ies
to
e
a
c
h
s
a
les
man
is
dyna
mi
c
,
a
im
ing
to
mi
nim
ize
the
ove
r
a
ll
tr
a
ve
l
d
is
tanc
e
.
−
E
a
c
h
c
it
y,
e
xc
e
pt
f
or
the
de
pot
,
mus
t
be
vis
it
e
d
pr
e
c
is
e
ly
onc
e
by
only
one
s
a
les
man.
−
T
h
e
n
um
be
r
o
f
c
i
ti
e
s
vi
s
i
te
d
by
a
n
y
s
a
l
e
s
m
a
n
is
c
o
ns
t
r
a
i
ne
d
,
w
it
h
a
m
in
i
mu
m
o
f
1
a
n
d
a
m
a
x
im
um
o
f
−
c
i
ti
e
s
.
T
he
nomenc
latur
e
us
e
d
in
the
model
a
r
e
given
in
T
a
ble
1
a
s
f
oll
ows
:
T
a
ble
1
.
Nome
nc
latur
e
N
ot
a
ti
ons
D
e
s
c
r
ip
ti
on
=
(
,
)
An
undi
r
e
c
te
d w
e
ig
ht
e
d a
nd c
onne
c
te
d gr
a
ph
=
{
1
,
2
,
.
.
.
,
}
N
ode
s
e
t
w
it
h
c
it
ie
s
=
{
(
,
)
/
,
∈
;
≠
}
E
dge
s
e
t
∈
{
0
,
1
}
,
∀
(
,
)
∈
,
∈
A
bi
na
r
y va
r
ia
bl
e
=
[
]
×
A
s
ymm
e
tr
ic
ma
tr
ix
r
e
pr
e
s
e
nt
in
g di
s
ta
nc
e
s
be
twe
e
n p
a
ir
s
of
c
it
ie
s
;
(
=
;
=
∞
,
>
0
)
D
is
ta
nc
e
f
r
om
ℎ
c
it
y t
o
ℎ
c
it
y
A
pos
it
iv
e
i
nt
e
ge
r
r
e
pr
e
s
e
nt
in
g t
he
t
ot
a
l
numbe
r
of
c
it
ie
s
t
ha
t
a
l
l
th
e
s
a
le
s
m
e
n ne
e
d t
o c
ove
r
(
i.
e
.
out
of
)
∈
{
0
,
1
}
,
∈
A
bi
na
r
y va
r
ia
bl
e
a
s
s
oc
i
a
te
d w
it
h vi
s
it
e
d c
it
ie
s
T
he
mathe
matica
l
model
o
f
k
-
M
T
S
P
is
a
s
f
oll
ows
:
=
∑
∑
=
1
=
1
,
1
≤
≤
(
1)
S
ubjec
t
to
,
t
he
objec
ti
ve
f
unc
ti
on
(
1)
tells
that
mi
nim
iza
ti
on
of
tot
a
l
dis
tanc
e
c
ove
r
e
d
by
s
a
les
men
.
T
he
c
ons
tr
a
int
s
e
t
(
2
)
a
nd
(
3)
c
onf
ir
ms
that
e
xa
c
tl
y
−
1
(
e
xc
ludi
ng
de
pot
c
it
y)
of
the
c
it
ies
c
ove
r
by
s
a
les
men
a
nd
a
f
e
a
s
ibl
e
s
olut
ion
mus
t
inc
lude
+
1
e
dge
s
c
onne
c
ti
ng
the
c
it
ies
.
C
ons
tr
a
int
s
e
ts
(4
)
a
nd
(
5)
pr
ovides
that
s
a
les
men
de
pa
r
t
a
nd
r
e
tur
ns
the
de
p
ot
c
it
y
.
C
ons
tr
a
int
s
e
ts
(
6
)
a
nd
(
7)
a
im
s
that
e
a
c
h
c
it
y
is
be
ing
vis
it
e
d
e
xa
c
tl
y
onc
e
by
only
one
s
a
les
ma
n.
C
ons
tr
a
int
(
8)
e
s
tablis
he
s
t
he
mi
nim
um
a
nd
maximum
number
of
c
it
ies
that
a
ny
s
a
les
man
c
a
n
vis
it
.
C
ons
tr
a
int
(
9
)
is
de
s
igned
to
pr
e
ve
nt
the
f
o
r
mation
o
f
s
ub
-
tour
s
withi
n
the
s
olut
ion.
T
he
c
ons
tr
a
int
(
10)
indi
c
a
tes
the
binar
y
va
r
iable
,
i
.
e
.
i
t
take
s
1
if
ℎ
s
a
les
man
tr
a
v
e
ls
f
r
om
ℎ
c
it
y
to
ℎ
c
it
y
a
nd
0
other
wis
e
.
F
inally,
a
not
he
r
binar
y
va
r
iable
is
int
r
oduc
e
d
in
the
c
ons
tr
a
int
(
11)
,
whic
h
take
s
1
i
f
ℎ
s
a
les
man
vis
it
s
ℎ
c
it
y
a
nd
0
ot
he
r
wis
e
.
∑
=
2
=
−
1
,
∈
(
2)
∑
∑
=
1
=
1
=
+
1
,
≠
,
∈
(
3)
∑
1
=
2
=
,
∈
(
4)
∑
1
=
2
=
,
∈
(
5)
∑
=
1
=
1
,
≠
,
∀
∈
\
{
1
}
,
∈
(
6)
∑
=
1
=
1
,
≠
,
∀
∈
\
{
1
}
,
∈
(
7)
1
≤
∑
∑
=
1
=
1
≤
−
,
∈
(
8)
+
S
u
b
to
u
r
e
l
im
in
a
ti
o
n co
ns
tr
a
in
ts
(
9)
∈
{
0
,
1
}
,
≠
,
∀
,
∈
,
∈
(
10)
∈
{
0
,
1
}
,
∀
∈
,
∈
(
11
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
4
,
Augus
t
2025
:
274
1
-
2752
2744
3.
M
E
T
HO
D
OL
OG
Y
T
he
pr
opos
e
d
GA
is
r
e
c
ognize
d
a
s
a
c
omm
only
us
e
d
meta
he
ur
is
ti
c
withi
n
the
domain
o
f
e
volut
ionar
y
c
omput
a
ti
on
s
tudi
e
s
,
s
pe
c
if
ica
ll
y
de
s
igned
f
o
r
a
dd
r
e
s
s
ing
pr
oblems
invol
v
ing
the
opt
i
mi
z
a
ti
on
of
a
c
ombi
na
ti
on
[
34]
.
T
his
population
-
ba
s
e
d
a
p
pr
oa
c
h
c
onti
nua
ll
y
r
e
f
ines
the
s
olut
ion
by
e
nha
nc
ing
it
s
f
it
ne
s
s
va
lue
thr
ough
it
e
r
a
ti
ve
upda
tes
.
F
indi
ng
the
be
s
t
s
olut
ion
to
the
k
-
M
T
S
P
by
us
ing
pr
opos
e
d
a
lgor
it
hm
include
s
e
ve
r
a
l
c
r
uc
ial
e
leme
nts
s
uc
h
a
s
r
e
pr
e
s
e
ntation
of
c
hr
om
os
omes
,
population
ge
ne
r
a
ti
on
,
f
it
ne
s
s
e
va
luation,
s
e
lec
ti
on,
mut
a
ti
on
a
n
d
the
ge
ne
r
a
l
pa
r
a
mete
r
s
of
GA
.
3.
1
.
Chrom
os
om
e
r
e
p
r
e
s
e
n
t
at
ion
F
ind
ing
op
ti
m
a
l
r
outes
f
or
m
ult
ipl
e
s
a
les
m
e
n
vis
i
ti
ng
e
a
c
h
c
it
y
o
nc
e
in
t
he
M
T
S
P
r
e
q
ui
r
e
s
e
nc
odi
ng
s
olut
i
ons
in
to
c
h
r
o
mos
omes
.
T
he
r
e
a
r
e
pr
im
a
r
il
y
t
wo
met
hods
:
the
s
in
gle
-
c
h
r
omos
o
me
[
35
]
a
pp
r
oa
c
h
a
nd
t
he
two
-
c
hr
omos
o
me
[
3
6]
a
pp
r
oa
c
h.
T
he
ini
t
ial
me
th
od
e
mpl
oys
a
c
hr
omos
o
me
whos
e
len
gth
is
(
+
−
1
)
,
with
r
e
p
r
e
s
e
n
ti
ng
t
he
tot
a
l
c
it
ies
a
nd
indi
c
a
t
ing
the
qua
nti
ty
of
s
a
les
men
,
to
di
r
e
c
t
ly
e
nc
od
e
b
oth
c
i
ti
e
s
vis
i
t
or
de
r
a
n
d
a
s
s
ig
ne
d
s
a
les
men
.
T
he
two
-
c
h
r
o
mos
o
me
s
t
r
a
teg
y
i
nvol
ve
s
u
ti
l
izin
g
a
pa
i
r
o
f
c
hr
omos
o
mes
,
e
a
c
h
with
a
le
ngt
h
o
f
.
T
he
f
ir
s
t
de
f
i
ne
s
th
e
un
ive
r
s
a
l
c
i
ty
v
is
it
or
de
r
,
whi
le
t
he
s
e
c
o
nd
a
s
s
i
gns
s
pe
c
if
ic
s
a
l
e
s
men
ba
s
e
d
on
thei
r
c
o
r
r
e
s
pond
ing
pos
i
ti
on
in
the
f
i
r
s
t
c
h
r
om
os
ome
.
E
me
r
gi
ng
i
n
t
he
f
ield
of
c
h
r
o
mos
ome
r
e
p
r
e
s
e
ntat
ion
is
a
nove
l
tec
hni
que
kno
wn
a
s
t
he
two
-
pa
r
t
c
hr
o
mos
ome
a
p
pr
o
a
c
h
[
9
]
.
E
ve
r
y
c
h
r
om
os
ome
is
divi
d
e
d
int
o
t
wo
s
e
g
ments
:
the
ini
ti
a
l
s
e
gm
e
nt
is
a
s
e
que
nc
e
of
c
i
ti
e
s
,
n
umbe
r
i
ng
−
1
,
a
r
r
a
ng
e
d
in
or
de
r
f
r
o
m
2
up
to
.
T
he
s
e
c
ond
pa
r
t
de
f
ines
the
r
o
ute
b
r
e
a
k
point
s
f
o
r
e
a
c
h
o
f
t
he
s
a
les
man
w
it
h
in
thei
r
a
s
s
i
gne
d
c
it
ies
.
T
he
two
-
pa
r
t
c
h
r
o
mos
ome
f
o
r
1
0
c
i
ty
k
-
M
T
S
P
w
it
h
3
s
a
les
me
n
g
iven
in
F
igu
r
e
1.
I
n
th
is
s
it
u
a
ti
on
,
the
s
a
les
man
1
v
is
it
s
3
c
it
i
e
s
1
→
7
→
5
→
8
,
the
s
a
les
man
2
vis
i
ts
2
c
it
ies
1
→
3
→
9
a
nd
the
s
a
les
man
3
vis
i
ts
2
c
i
ti
e
s
1
→
2
→
4
.
T
he
r
out
e
plan
f
or
3
s
a
les
men
c
ove
r
in
g
8
out
of
1
0
c
it
ies
g
iven
i
n
F
i
gur
e
2
.
F
igu
r
e
1.
T
wo
-
pa
r
t
c
h
r
omos
ome
r
e
pr
e
s
e
ntation
of
10
c
it
y
k
-
M
T
S
P
with
3
s
a
les
men
F
igur
e
2.
R
oute
plan
f
or
3
s
a
les
men
c
ove
r
ing
8
ou
t
of
10
c
it
ies
(
I
nc
ludi
ng
de
pot
c
it
y
)
3.
2
.
I
n
it
ial
p
op
u
lat
ion
e
n
c
od
in
g
T
he
s
uc
c
e
s
s
of
a
GA
is
g
r
e
a
tl
y
in
f
luenc
e
d
by
the
qua
li
ty
of
the
ini
ti
a
l
population
s
e
t,
unde
r
s
c
or
ing
the
s
igni
f
ica
nc
e
of
s
e
lec
ti
ng
the
r
ight
e
nc
oding
o
pe
r
a
tor
.
I
n
the
r
e
a
lm
o
f
M
T
S
P
a
nd
it
s
va
r
iations
,
s
tudi
e
s
i
n
di
c
a
t
e
th
a
t
p
e
r
m
uta
t
io
n
e
nc
od
in
g
s
ta
nds
o
u
t
a
s
the
o
pt
i
ma
l
c
ho
ic
e
f
o
r
c
r
e
a
t
in
g
po
te
nt
i
n
it
ia
l
po
pu
la
ti
o
ns
.
T
h
is
i
s
e
v
i
de
n
t
in
th
e
w
ides
p
r
e
a
d
a
do
pt
io
n
o
f
GA
tec
hn
iqu
e
s
e
mp
lo
yi
ng
pe
r
mu
ta
ti
on
e
n
c
o
di
ng
in
th
is
c
on
te
xt
[
20]
.
3.
3
.
F
it
n
e
s
s
f
u
n
c
t
io
n
T
he
f
it
ne
s
s
f
unc
ti
o
n
is
us
e
d
t
o
c
a
lcu
late
the
in
div
id
ua
l
c
h
r
omos
o
mes
o
f
the
p
opu
lati
on
.
T
h
e
s
e
lec
t
ion
s
tr
a
teg
y
i
n
GA
s
r
e
li
e
s
on
the
i
mpo
r
ta
nc
e
o
f
th
e
f
it
n
e
s
s
va
lue
,
r
e
pr
e
s
e
n
ti
n
g
a
c
r
uc
i
a
l
s
te
p.
S
pe
c
if
ica
ll
y,
a
highe
r
f
it
ne
s
s
va
l
ue
f
o
r
a
c
h
r
om
os
ome
ind
ica
tes
a
n
inc
r
e
a
s
e
d
li
ke
li
hood
o
f
be
ing
c
h
os
e
n
f
o
r
t
he
ne
xt
ge
ne
r
a
ti
on
.
I
n
our
s
tu
dy
,
the
ob
jec
ti
ve
f
u
nc
ti
on
is
r
e
ga
r
de
d
a
s
i
de
n
ti
c
a
l
wi
th
the
f
i
tnes
s
f
unc
ti
o
n
out
li
ne
d
in
(
1
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
Solving
k
-
c
it
y
multi
ple
tr
av
e
ll
ing
s
ales
man
us
ing
ge
ne
ti
c
algor
it
hm
(
A
li
k
apati
P
r
ak
a
sh
)
2745
3.
4
.
S
e
lec
t
ion
op
e
r
at
or
T
he
s
e
lec
ti
on
ope
r
a
tor
s
igni
f
ica
ntl
y
im
pa
c
ts
the
GA
e
f
f
icie
nc
y.
T
he
GA
e
mpl
oys
the
c
onve
nti
ona
l
r
oulette
whe
e
l
a
ppr
oa
c
h
a
s
it
s
s
e
lec
ti
on
mec
ha
nis
m.
T
his
a
ppr
oa
c
h
s
e
lec
ts
a
c
hr
omos
ome
f
or
the
br
e
e
ding
pool
s
tatis
ti
c
a
ll
y,
a
c
c
or
ding
to
it
s
f
it
ne
s
s
va
lue
f
r
o
m
the
population
.
3.
5
.
M
u
t
a
t
ion
o
p
e
r
at
or
M
utation
is
uti
li
z
e
d
in
the
GA
to
a
void
c
onve
r
g
e
nc
e
a
t
loca
l
opti
ma
a
nd
to
e
nha
nc
e
the
ge
ne
ti
c
va
r
iation
a
mong
the
population
.
T
his
tas
k
uti
li
z
e
s
a
c
ompl
e
x
mut
a
ti
on
ope
r
a
tor
that
include
s
va
r
ious
ope
r
a
ti
ons
:
f
li
p,
s
wa
p,
s
li
de
,
a
s
we
ll
a
s
c
ombi
na
ti
o
ns
li
ke
f
li
p
with
modi
f
y
b
r
e
a
ks
,
s
wa
p
with
modi
f
y
br
e
a
ks
,
a
nd
s
li
de
with
modi
f
y
br
e
a
ks
.
T
he
s
e
mut
a
ti
on
ope
r
a
tor
s
a
r
e
e
mpl
oye
d
to
ident
if
y
the
s
hor
tes
t
pos
s
ibl
e
dis
tanc
e
while
a
ls
o
r
e
duc
ing
the
ti
me
r
e
quir
e
d
f
or
c
omput
a
ti
on.
T
he
f
li
p
ope
r
a
tor
is
a
mu
tation
ope
r
a
ti
o
n
that
r
e
ve
r
s
e
s
the
or
de
r
of
a
s
e
gment
of
the
r
oute
be
t
we
e
n
two
ins
e
r
ti
on
point
s
.
I
n
F
igu
r
e
3,
the
f
li
p
o
pe
r
a
tor
is
a
ppli
e
d
to
the
be
s
t
r
oute
a
mong
the
thr
e
e
r
outes
ge
ne
r
a
ted
in
e
a
c
h
it
e
r
a
ti
on
of
the
loop.
T
he
s
wa
p
ope
r
a
tor
in
a
GA
is
a
mut
a
ti
on
ope
r
a
tor
t
ha
t
e
xc
ha
nge
s
the
pos
it
ions
of
two
e
leme
nts
in
a
s
olut
ion
or
indi
vi
dua
l.
I
n
F
igur
e
4,
the
s
wa
p
ope
r
a
ti
on
is
a
ppli
e
d
to
a
r
oute
,
whe
r
e
two
c
it
ies
a
r
e
c
hos
e
n,
a
nd
their
pos
it
ions
in
the
r
oute
a
r
e
s
wa
ppe
d.
T
he
s
li
de
ope
r
a
tor
is
a
mut
a
t
ion
o
pe
r
a
tor
us
e
d
in
GA
f
o
r
f
indi
ng
the
opti
mal
dis
ta
nc
e
.
T
h
is
ope
r
a
tor
invol
ve
s
movi
ng
a
s
e
gment
o
f
the
r
oute
to
a
ne
w
pos
it
ion
withi
n
the
s
a
me
r
oute
whic
h
is
given
in
F
igur
e
5
.
T
he
m
odi
f
y
b
r
e
a
ks
ope
r
a
tor
in
thi
s
c
ont
e
xt
s
e
e
ms
to
be
a
ppli
e
d
to
the
br
e
a
ks
o
r
b
r
e
a
kpoint
s
in
a
r
out
e
.
I
n
the
c
ontext
o
f
a
M
T
S
P
,
b
r
e
a
ks
or
br
e
a
k
point
s
c
ould
r
e
pr
e
s
e
nt
s
pe
c
if
ic
loca
ti
ons
whe
r
e
a
s
a
les
ma
n
make
s
a
s
top
or
pa
us
e
s
dur
ing
it
s
r
oute.
M
odif
yi
ng
br
e
a
ks
may
invol
ve
c
ha
nging
the
or
de
r
o
r
pos
it
ions
of
thes
e
br
e
a
ks
whic
h
is
given
in
F
igur
e
6
.
Fi
gur
e
3.
F
li
p
ope
r
a
tor
F
igur
e
4.
S
wa
p
ope
r
a
tor
F
igur
e
5.
S
li
de
ope
r
a
tor
F
igur
e
6.
M
odif
y
b
r
e
a
ks
ope
r
a
tor
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
4
,
Augus
t
2025
:
274
1
-
2752
2746
T
he
f
li
p
a
nd
modi
f
y
b
r
e
a
ks
ope
r
a
tor
c
ombi
ne
the
f
li
p
ope
r
a
tor
to
r
e
ve
r
s
e
a
s
e
gment
of
the
r
oute
with
the
modi
f
y
br
e
a
ks
ope
r
a
to
r
to
int
r
oduc
e
va
r
iabili
t
y
in
the
br
e
a
ks
.
T
his
c
r
e
a
tes
a
ne
w
s
olut
ion
by
a
lt
e
r
ing
the
or
de
r
o
f
c
it
ies
a
nd
the
br
e
a
ks
s
im
ult
a
ne
ous
ly,
w
hich
is
given
in
F
igur
e
7.
T
he
s
wa
p
a
nd
modi
f
y
ope
r
a
tor
c
ombi
ne
the
s
wa
p
ope
r
a
tor
with
the
modi
f
ica
ti
on
o
f
br
e
a
ks
.
I
n
F
igur
e
8,
it
s
wa
ps
the
or
de
r
of
two
c
it
i
e
s
in
the
r
oute
a
nd
s
im
ult
a
ne
ous
ly
modi
f
ies
the
br
e
a
ks
a
s
s
o
c
iate
d
with
that
r
oute.
F
igur
e
9
ge
ne
r
a
tes
a
ne
w
s
olut
ion
by
s
li
ding
a
s
e
gment
o
f
th
e
r
oute
to
the
r
ight
a
nd
then
modi
f
ying
the
br
e
a
ks
r
a
ndoml
y.
T
he
c
ombi
na
ti
on
of
thes
e
ope
r
a
ti
ons
pr
ovides
diver
s
it
y
in
the
ge
ne
r
a
ted
s
olut
ions
dur
ing
the
GA
opti
mi
z
a
ti
on
pr
oc
e
s
s
.
F
igur
e
7.
F
li
p
a
nd
modi
f
y
b
r
e
a
ks
ope
r
a
tor
F
igur
e
8.
S
wa
p
a
nd
modi
f
y
br
e
a
ks
ope
r
a
tor
F
igur
e
9.
S
li
de
a
nd
modi
f
y
br
e
a
ks
ope
r
a
tor
3.
6
.
GA
p
ar
am
e
t
e
r
s
T
he
e
f
f
e
c
ti
ve
ne
s
s
of
the
a
lgo
r
it
hm
r
e
li
e
s
on
the
v
a
lues
of
va
r
ious
pa
r
a
mete
r
s
,
including
the
s
ize
of
the
population,
the
r
a
te
of
mut
a
ti
on
p
r
oba
bil
it
y,
a
nd
the
c
r
it
e
r
ia
f
or
ter
m
ination.
I
n
thi
s
s
tudy,
the
s
ize
of
the
population
is
s
e
t
a
t
80,
r
e
pr
e
s
e
nti
ng
the
number
o
f
c
hr
omos
omes
in
a
s
ingl
e
ge
ne
r
a
ti
on.
T
he
s
tudy
doe
s
not
e
xplor
e
int
o
the
c
r
os
s
ove
r
ope
r
a
tor
,
but
it
e
mphas
ize
s
the
us
e
of
a
s
ophis
ti
c
a
ted
mut
a
ti
on
ope
r
a
tor
to
a
c
hieve
it
s
int
e
nde
d
objec
ti
ve
s
.
4.
RE
S
UL
T
S
AN
D
DI
S
CU
S
S
I
ON
T
his
s
e
c
ti
on
pr
e
s
e
nts
the
c
omput
a
ti
ona
l
r
e
s
ult
s
of
the
pr
opos
e
d
GA
.
As
ther
e
is
no
c
ompar
a
ti
ve
s
tudy
de
voted
f
or
k
-
M
T
S
P
,
diver
s
e
be
nc
hmar
k
da
tas
e
ts
s
our
c
e
d
f
r
om
T
S
P
L
I
B
(
htt
p:/
/como
pt.
if
i
.
un
i
-
he
idelbe
r
g.
de
/s
of
twa
r
e
/T
S
P
L
I
B
95/
)
we
r
e
uti
li
z
e
d
t
o
e
xe
c
ute
the
a
lgor
it
hm
.
T
he
pe
r
f
o
r
manc
e
of
the
a
l
gor
it
hm
wa
s
mea
s
ur
e
d
ba
s
e
d
on
the
be
s
t,
wo
r
s
t,
a
nd
a
ve
r
a
ge
r
e
s
ult
s
f
or
e
a
c
h
tes
t
c
a
s
e
,
a
s
we
ll
a
s
the
a
ve
r
a
ge
C
P
U
r
unti
mes
.
T
he
M
AT
L
AB
R
2023b
wa
s
uti
li
z
e
d
t
o
c
ode
the
a
lgor
it
hm,
whic
h
wa
s
e
xe
c
uted
on
a
pe
r
s
ona
l
c
omput
e
r
f
e
a
tur
ing
a
n
I
ntel(
R
)
C
or
e
(
T
M
)
i3
-
10110U
C
P
U
r
unning
a
t
2
.
10
GH
z
a
nd
e
quipped
with
8
GB
of
R
AM
.
T
he
GA
is
c
a
r
r
ied
out
indepe
nde
ntl
y
ten
ti
mes
f
or
e
a
c
h
tes
t
c
a
s
e
,
a
nd
the
be
s
t,
wor
s
t,
a
nd
me
a
n
r
e
s
ult
s
a
r
e
doc
umente
d
f
oll
owing
e
ve
r
y
c
yc
le.
I
n
to
tal,
1
0
ins
tanc
e
s
f
r
om
T
S
P
L
I
B
ha
ve
be
e
n
e
xa
mi
ne
d.
E
a
c
h
tes
t
c
a
s
e
is
r
un
indepe
nde
ntl
y
f
or
10
ti
mes
.
F
o
r
e
ve
r
y
c
a
s
e
,
whe
r
e
e
a
c
h
ins
tanc
e
invol
ve
s
f
our
dif
f
e
r
e
nt
s
a
les
man
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
Solving
k
-
c
it
y
multi
ple
tr
av
e
ll
ing
s
ales
man
us
ing
ge
ne
ti
c
algor
it
hm
(
A
li
k
apati
P
r
ak
a
sh
)
2747
s
ize
s
,
de
ter
mi
ne
the
be
s
t,
wor
s
t
a
nd
a
ve
r
a
ge
r
e
s
ult
s
,
a
nd
a
ls
o
r
e
c
or
d
the
a
ve
r
a
ge
C
P
U
r
unt
im
e
s
.
T
he
s
e
tes
t
s
c
e
na
r
ios
a
r
e
c
ha
r
a
c
ter
ize
d
by
E
uc
li
de
a
n,
two
-
dim
e
ns
ional
s
ymm
e
tr
y
a
nd
dis
ti
nc
t
node
s
c
a
les
,
r
a
ngi
ng
f
r
om
48
to
318
c
it
ies
.
T
he
ini
ti
a
l
c
it
y
in
e
a
c
h
in
s
tanc
e
is
de
s
ignat
e
d
a
s
the
home
c
it
y.
T
hr
e
e
dis
ti
nc
t
s
c
e
na
r
ios
(
⌊
2
⌋
,
⌊
4
⌋
,
⌊
6
⌋
)
a
r
e
c
ontemplate
d
f
or
e
a
c
h
of
thes
e
10
in
s
tanc
e
s
.
T
a
bles
2
to
4
pr
e
s
e
nt
c
omput
a
ti
ona
l
r
e
s
ult
s
f
or
ins
tanc
e
s
invol
ving
(
⌊
2
⌋
,
⌊
4
⌋
,
⌊
6
⌋
)
.
I
n
e
a
c
h
o
f
thes
e
T
a
ble
s
2
to
4
,
the
ini
ti
a
l
c
olum
n
dis
plays
s
e
r
ial
number
s
,
indi
c
a
ti
ng
40
dis
ti
nc
t
nu
mer
ica
l
ins
tanc
e
s
,
the
s
e
c
ond
c
olum
n
indi
c
a
tes
na
me
of
the
tes
t
ins
tanc
e
a
longs
ide
the
a
s
s
oc
iate
d
c
ount
of
c
it
ies
,
s
it
ua
ted
in
the
thi
r
d
c
olum
n.
T
he
ne
xt
two
c
olum
ns
r
e
pr
e
s
e
nt
the
qua
nti
ty
of
s
a
les
man
a
nd
the
c
or
r
e
s
ponding
be
s
t,
wo
r
s
t,
a
nd
a
ve
r
a
ge
r
e
s
ult
s
,
de
noted
in
s
ize
.
T
he
p
a
r
ti
c
ular
s
e
tt
ings
f
or
the
s
ugge
s
ted
a
lgor
it
hm
a
r
e
de
tailed
in
T
a
ble
2
.
T
he
C
P
U
r
unti
mes
f
o
r
e
a
c
h
c
a
s
e
a
r
e
p
r
o
vided
in
T
a
ble
s
2
to
4
,
s
a
me
is
s
hown
in
F
igur
e
10.
T
he
r
e
s
ult
s
in
T
a
ble
2
ind
ica
te
s
igni
f
ica
nt
va
r
iabili
ty
in
the
a
lgor
it
hm's
pe
r
f
o
r
manc
e
a
c
r
os
s
dif
f
e
r
e
nt
be
nc
hmar
k
ins
tanc
e
s
,
pa
r
ti
c
ular
ly
highl
ight
ing
the
im
pa
c
t
of
ins
tanc
e
c
ompl
e
xit
y
on
r
un
ti
mes
.
I
ns
tanc
e
s
with
mor
e
node
s
,
s
uc
h
a
s
bier
127
a
nd
gil
262,
e
xhibi
t
notably
longer
r
un
ti
mes
,
s
ugge
s
ti
ng
c
ha
ll
e
nge
s
in
ha
ndli
ng
lar
ge
r
da
tas
e
ts
.
W
hil
e
s
ome
ins
tanc
e
s
,
li
ke
e
il
51,
s
how
c
ons
is
tent
pe
r
f
or
manc
e
with
c
los
e
r
un
ti
mes
a
c
r
os
s
s
olut
ions
,
other
s
dis
play
ins
tabili
ty,
a
s
s
e
e
n
in
bier
127,
whe
r
e
the
ga
p
be
twe
e
n
be
s
t
a
nd
wor
s
t
ti
mes
is
s
ubs
tantial.
I
nter
e
s
ti
ng
ly,
be
s
t
s
olut
ions
do
not
c
ons
is
tently
im
pr
ove
with
mo
r
e
s
olut
ions
,
r
e
f
lec
ti
ng
potential
inef
f
icie
nc
ies
.
Ove
r
a
ll
,
the
f
indi
ngs
s
ugge
s
t
that
while
the
a
lgor
i
thm
pe
r
f
or
ms
we
ll
on
s
maller
ins
t
a
nc
e
s
,
it
s
s
c
a
labili
ty
a
nd
r
e
li
a
bil
it
y
on
lar
ge
r
pr
oblems
wa
r
r
a
nt
f
ur
ther
tuni
ng
a
nd
opti
mi
z
a
ti
on
to
e
nha
nc
e
e
f
f
icie
nc
y
a
nd
pe
r
f
or
manc
e
.
T
a
ble
2
.
R
e
s
ult
s
of
pr
opos
e
d
a
lgor
it
h
m
on
be
nc
hmar
k
ins
tanc
e
s
with
⌊
2
⌋
SN
I
ns
ta
nc
e
s
n
m
S
ol
ut
io
n
A
vg. C
P
U
r
un t
im
e
(
i
n s
e
c
onds
)
B
e
s
t
w
or
s
t
A
ve
r
a
ge
1
a
tt
48
48
2
18,107
25,347
22,778.7
2.850
2
3
21,824
28,411
25,423.4
3.594
3
4
25,822
32,505
29,605.2
3.115
4
5
28,581
33,887
30,805.8
3.766
5
6
7
8
e
il
51
51
2
282
302
294.8
2.914
3
291
334
317.9
3.052
4
336
367
355.1
3.074
5
345
402
382.6
3.176
9
10
11
12
be
r
li
n52
52
2
4,210
5,716
5,022
3.345
3
4,299
5,733
5,122.2
3.247
4
5,001
6,223
5,697.3
3.515
5
5,143
6,737
5,781.8
3.658
13
14
15
16
s
t7
0
70
2
461
523
503
3.537
3
492
623
553.5
3.771
4
550
699
635
4.246
5
630
829
731.6
4.399
17
18
19
20
r
a
t9
9
99
2
661
709
691.9
3.366
3
730
844
779
3.739
4
833
1,003
873.2
3.987
5
931
1,052
979.8
4.317
21
22
23
24
e
il
101
101
2
473
497
487
4.739
3
505
554
525.7
4.309
4
543
586
564.3
4.815
5
578
639
606
5.053
25
26
27
28
bi
e
r
127
127
2
49,884
51,414
50,633
5.120
3
51,982
55,291
53,739.6
5.686
4
53,899
57,719
55,679.4
5.545
5
56,851
61,791
58,710
8.705
29
30
31
32
pr
152
152
2
38,871
39,956
39,427.5
4.405
3
45,582
60,453
51,876.6
4.283
4
63,117
69,944
66,374.5
5.039
5
68,523
91,056
76,467
5.092
33
34
35
36
gi
l2
62
262
2
2,191
2,522
2,305.3
8.423
3
2,458
3,071
2,738.1
9.119
4
2,909
3,430
3,188.1
9.690
5
3,187
3,712
3,442.7
10.506
37
38
39
40
li
n318
318
2
28,445
29,477
28,829
6.011
3
29,474
31,986
30,919.6
6.521
4
30471
35457
32954.2
6.875
5
32941
41480
35740.5
7.242
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
4
,
Augus
t
2025
:
274
1
-
2752
2748
T
he
r
e
s
ult
s
in
T
a
ble
3
r
e
ve
a
l
notable
tr
e
nds
in
the
a
lgor
it
hm's
pe
r
f
o
r
manc
e
a
c
r
os
s
va
r
ious
be
nc
hmar
k
ins
tanc
e
s
,
highl
ight
ing
both
s
tr
e
ngths
a
nd
c
ha
ll
e
nge
s
.
F
o
r
s
maller
ins
tanc
e
s
li
ke
a
tt
48
a
nd
e
il
51
,
the
a
lgor
it
hm
de
mons
tr
a
tes
c
ons
is
tent
a
nd
e
f
f
icie
nt
r
un
ti
m
e
s
,
with
a
ve
r
a
ge
ti
mes
r
e
maining
r
e
lati
ve
ly
low,
indi
c
a
ti
ng
e
f
f
e
c
ti
ve
opti
mi
z
a
ti
on
s
tr
a
tegie
s
.
How
e
ve
r
,
a
s
the
pr
oblem
s
ize
incr
e
a
s
e
s
,
pa
r
ti
c
ular
ly
in
i
ns
tanc
e
s
li
ke
bier
127
a
nd
pr
152
,
r
un
ti
mes
e
s
c
a
late
s
igni
f
ica
ntl
y,
h
ighl
ight
ing
the
a
lgo
r
it
hm's
s
tr
uggle
wi
th
lar
ge
r
da
tas
e
ts
.
T
he
va
r
iabili
ty
in
be
s
t
a
nd
wo
r
s
t
r
un
ti
mes
,
e
s
pe
c
ially
in
bier
127
,
s
ugge
s
ts
that
c
e
r
tain
ins
tanc
e
s
may
r
e
quir
e
tailo
r
e
d
opti
mi
z
a
ti
on
tec
hniques
.
I
nte
r
e
s
ti
ngly,
s
ome
ins
tanc
e
s
,
s
uc
h
a
s
gil
262
,
s
how
im
pr
e
s
s
ive
be
s
t
r
un
t
im
e
s
de
s
pit
e
t
he
ir
s
ize
,
indi
c
a
ti
ng
potenti
a
l
f
or
im
p
r
ove
ment
in
a
lgo
r
it
hm
e
f
f
icie
nc
y.
Ove
r
a
ll
,
while
the
a
lgor
it
hm
pe
r
f
o
r
ms
we
ll
on
s
maller
ins
tanc
e
s
,
it
s
s
c
a
labili
ty
a
nd
s
tabili
ty
a
c
r
os
s
lar
ge
r
pr
oblems
c
a
ll
f
or
f
ur
ther
r
e
f
ineme
nt
to
e
nha
nc
e
ove
r
a
ll
pe
r
f
or
manc
e
.
T
a
ble
3
.
R
e
s
ult
s
of
pr
opos
e
d
a
lgor
it
h
m
on
be
nc
hmar
k
ins
tanc
e
s
with
⌊
4
⌋
SN
I
ns
ta
nc
e
s
n
m
S
ol
ut
io
n
A
vg. C
P
U
r
un t
im
e
(
i
n s
e
c
onds
)
B
e
s
t
w
or
s
t
A
ve
r
a
ge
1
2
3
4
a
tt
48
48
2
13,475
16,829
15,121.5
2.217
3
14,969
22,509
18,154.1
2.121
4
15,624
24,134
20,280.6
2.164
5
18,766
27,325
24,550.3
2.164
5
6
7
8
e
il
51
51
2
165
242
196.5
2.161
3
186
235
213.3
2.396
4
217
262
242.2
2.534
5
239
303
272.4
2.484
9
10
11
12
be
r
li
n52
52
2
2,111
3,435
2,871.4
2.161
3
2,122
3,450
2,999.7
2.432
4
2,589
4,431
3,251.7
2.359
5
2,825
4,162
3,543.7
2.258
13
14
15
16
s
t7
0
70
2
304
372
345.9
2.509
3
366
462
409.2
2.574
4
371
494
439
2.894
5
453
662
519.1
2.838
17
18
19
20
r
a
t9
9
99
2
357
361
357.6
2.584
3
404
429
409.2
2.892
4
459
526
474.4
3.194
5
532
623
578.7
3.094
21
22
23
24
e
il
101
101
2
238
333
297.1
2.956
3
283
364
338.5
2.932
4
315
421
385.1
2.967
5
358
449
406.5
3.147
25
26
27
28
bi
e
r
127
127
2
19,218
19,453
19,312.4
4.005
3
20,304
21,338
20,747.6
5.074
4
22,087
23,450
22,661.6
4.750
5
23,871
25,946
25,020
5.498
29
30
31
32
pr
152
152
2
21,516
21,516
21,516
2.948
3
22,993
31,983
27,076
3.739
4
26,387
40,744
33,000.5
4.371
5
31,494
47,052
38,439.2
4.283
33
34
35
36
gi
l2
62
262
2
1,386
1,435
1,406.9
5.302
3
1,550
1,895
1,673.7
5.589
4
1,931
2,229
2,065.1
6.214
5
1,843
2,505
2,293.8
6.584
37
38
39
40
li
n318
318
2
11,466
12,429
12,002.9
4.566
3
12,075
14,593
13,317.6
4.761
4
13,009
16,182
14,436.6
4.945
5
15,484
17,537
16,496.9
5.212
F
inally,
the
r
e
s
ult
s
in
T
a
ble
4
de
mons
tr
a
te
im
po
r
tant
ins
ight
s
a
bout
the
a
lgor
it
hm's
pe
r
f
or
manc
e
.
I
t
pe
r
f
o
r
ms
we
ll
on
s
maller
ins
tanc
e
s
li
ke
a
tt
48
a
nd
e
il
51
,
wi
th
c
ons
is
tently
low
a
ve
r
a
ge
r
u
n
ti
mes
.
How
e
ve
r
,
lar
ge
r
ins
tanc
e
s
,
s
uc
h
a
s
bier
127
a
nd
p
r
152
,
s
how
a
s
igni
f
ica
nt
inc
r
e
a
s
e
in
r
un
ti
mes
,
in
dica
ti
ng
s
c
a
labili
ty
is
s
ue
s
.
Va
r
iabili
ty
in
be
s
t
a
nd
wor
s
t
r
u
n
ti
mes
,
e
s
pe
c
ially
in
bier
127
,
s
ugge
s
ts
that
the
a
lg
or
it
hm
may
s
tr
uggle
with
c
e
r
tain
c
onf
igur
a
ti
ons
.
I
n
c
o
ntr
a
s
t,
r
a
t99
de
mons
tr
a
tes
s
tabili
ty
with
e
qua
l
be
s
t
a
nd
wor
s
t
ti
mes
a
c
r
os
s
s
olut
ions
.
Ove
r
a
ll
,
the
a
lgor
it
hm
is
e
f
f
e
c
ti
ve
f
or
s
maller
pr
ob
lems
in
thi
s
c
ontext,
im
pr
ove
ments
a
r
e
ne
e
de
d
to
e
nha
nc
e
pe
r
f
o
r
manc
e
f
or
lar
ge
r
da
tas
e
ts
a
nd
e
ns
ur
e
mor
e
r
e
s
ult
s
that
a
r
e
c
ons
is
tent.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
Solving
k
-
c
it
y
multi
ple
tr
av
e
ll
ing
s
ales
man
us
ing
ge
ne
ti
c
algor
it
hm
(
A
li
k
apati
P
r
ak
a
sh
)
2749
T
a
ble
4
.
R
e
s
ult
s
of
pr
opos
e
d
a
lgor
it
h
m
on
be
nc
hmar
k
ins
tanc
e
s
with
⌊
6
⌋
SN
I
ns
ta
nc
e
s
n
m
S
ol
ut
io
n
A
vg. C
P
U
r
un t
im
e
(
in
s
e
c
onds
)
B
e
s
t
w
or
s
t
A
ve
r
a
ge
1
2
3
4
a
tt
48
48
2
7,841
11,818
10,067.7
1.849
3
12,236
16,556
14,677.8
1.977
4
11,033
17,862
13,286.5
1.821
5
13,022
18,671
16,335.3
1.845
5
6
7
8
e
il
51
51
2
124
163
144.4
2.113
3
144
185
163.7
2.260
4
155
224
194.3
2.369
5
176
242
200.5
2.362
9
10
11
12
be
r
li
n52
52
2
1,363
2,373
2,122.9
1.893
3
1,796
2,477
2,188
1.976
4
2,030
2,848
2,403.6
2.258
5
1,956
2,680
2,241.8
1.921
13
14
15
16
s
t7
0
70
2
223
348
282.8
2.186
3
250
380
325.2
2.311
4
328
426
376.2
2.261
5
385
552
481.7
2.301
17
18
19
20
r
a
t9
9
99
2
257
257
257
2.211
3
305
305
305
2.499
4
366
376
367
2.522
5
449
449
449
2.520
21
22
23
24
e
il
101
101
2
203
245
217.1
2.309
3
202
269
244.4
2.776
4
253
314
282.6
2.887
5
273
332
309.4
2.809
25
26
27
28
bi
e
r
127
127
2
12,270
12,405
12,293.1
4.075
3
13,329
13,885
13,544.7
3.007
4
14,815
15,547
15,171.6
3.894
5
16,513
17,689
16,903.6
3.359
29
30
31
32
pr
152
152
2
21,909
21,909
21,909
2.432
3
27,054
32,856
2,8981.1
3.296
4
34,512
41,890
38,073.4
3.618
5
44,688
54,987
48,811.8
3.383
33
34
35
36
gi
l2
62
262
2
1,147
1,329
1,231.4
4.104
3
1,277
1,494
1,375
4.243
4
1,480
1,808
1,685.6
4.632
5
1,796
2,243
1,968.3
4.843
37
38
39
40
li
n318
318
2
7,645
8,142
7,841
3.938
3
8,445
9,418
9,108.6
3.969
4
9,379
10,909
10,057.1
4.148
5
10,633
12,922
11,901.2
4.284
F
igur
e
10.
Ave
r
a
ge
C
P
U
r
un
ti
mes
of
a
ll
ins
tanc
e
s
Ove
r
a
ll
,
the
r
e
s
ult
s
f
r
om
T
a
ble
2
r
e
ve
a
l
c
ons
is
tent
pe
r
f
or
manc
e
on
s
maller
da
tas
e
ts
li
ke
e
il
51
,
whe
r
e
a
ve
r
a
ge
r
unti
mes
r
e
main
s
table
.
I
n
c
ontr
a
s
t,
T
a
ble
3
s
hows
that
ins
tanc
e
s
li
ke
bie
r
127
a
nd
pr
152
e
xpe
r
ienc
e
higher
va
r
iabili
ty,
whic
h
may
be
due
to
the
c
ompl
e
xit
y
of
thes
e
lar
ge
r
pr
oblem
s
ize
s
.
T
he
incr
e
a
s
e
d
r
unti
me
f
or
thes
e
ins
tanc
e
s
in
dica
tes
c
ha
ll
e
nge
s
in
s
c
a
labi
li
ty
f
or
the
GA
a
lgo
r
it
hm.
T
he
ove
r
a
ll
tr
e
nd
indi
c
a
tes
that
the
a
lgor
it
hm
pe
r
f
o
r
ms
we
ll
with
s
maller
da
tas
e
ts
,
a
s
s
hown
by
the
c
ons
i
s
tent
r
e
s
ult
s
f
or
ins
tanc
e
s
li
ke
e
il
51
a
nd
a
tt
48
in
T
a
ble
2.
How
e
ve
r
,
a
s
pr
oblem
s
ize
i
nc
r
e
a
s
e
s
s
hown
in
T
a
bles
3
to
4,
pe
r
f
o
r
manc
e
va
r
iabili
ty
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
4
,
Augus
t
2025
:
274
1
-
2752
2750
be
c
omes
mor
e
e
vident,
pa
r
ti
c
ular
ly
in
lar
ge
r
ins
tanc
e
s
li
ke
bier
127.
T
his
s
ugge
s
ts
that
the
a
lgor
it
hm
ne
e
ds
f
ur
ther
opti
mi
z
a
ti
on
f
o
r
ha
ndli
ng
s
c
a
labili
ty
is
s
ue
s
e
f
f
e
c
ti
ve
ly.
I
n
c
onc
lus
ion,
while
the
pr
opo
s
e
d
GA
a
lgor
it
hm
pe
r
f
or
ms
e
f
f
icie
ntl
y
on
s
maller
da
tas
e
ts
,
it
f
a
c
e
s
c
ha
ll
e
nge
s
with
s
c
a
labili
ty
a
s
the
pr
ob
lem
s
ize
incr
e
a
s
e
s
.
T
he
va
r
iabili
ty
in
r
e
s
ult
s
a
c
r
os
s
lar
ge
r
ins
tanc
e
s
s
ugge
s
t
s
that
the
a
lgor
it
hm
may
be
ne
f
it
f
r
om
f
ur
ther
op
ti
mi
z
a
ti
on,
s
uc
h
a
s
hybr
id
iza
ti
on
wi
t
h
other
tec
hniques
or
pa
r
a
ll
e
li
z
a
ti
on
f
or
lar
ge
r
da
tas
e
ts
[
28]
,
[
37
]
.
F
utur
e
wor
k
c
ould
f
oc
us
on
im
p
r
ov
ing
the
a
lgo
r
it
hm's
r
obus
tnes
s
a
nd
e
f
f
icie
nc
y
to
ha
ndle
c
ompl
e
x,
lar
ge
-
s
c
a
le
k
-
M
T
S
P
ins
tanc
e
s
mor
e
e
f
f
e
c
ti
ve
ly
.
5.
CONC
L
USI
ON
I
n
th
is
s
tudy,
we
a
ddr
e
s
s
e
d
a
unique
va
r
iant
of
c
las
s
ica
l
M
T
S
P
,
s
pe
c
if
ica
ll
y
the
k
-
M
T
S
P
,
uti
li
z
e
d
wide
ly
in
outs
our
c
ing,
tr
a
ns
por
tation
a
nd
logi
s
ti
c
s
dis
tr
ibut
ion.
T
he
a
im
o
f
thi
s
pr
oblem
is
to
f
ind
a
s
e
t
of
c
ompl
e
te
tour
s
f
or
m
s
a
les
man,
c
ove
r
ing
pr
e
c
is
e
l
y
k
out
of
the
n
c
it
ies
,
with
the
a
im
of
mi
nim
izing
th
e
ove
r
a
ll
tr
a
ve
r
s
a
l
dis
tanc
e
or
c
os
t.
Ac
c
or
ding
to
the
a
uthor
's
knowle
dge
,
thi
s
is
the
f
ir
s
t
GA
de
ve
loped
f
or
the
k
-
M
T
S
P
.
Give
n
the
a
bs
e
nc
e
of
k
-
M
T
S
P
s
tudi
e
s
,
no
c
ompar
a
ti
ve
s
tudi
e
s
a
r
e
c
a
r
r
ied
out.
Ho
we
ve
r
,
va
r
ious
be
nc
hmar
k
tes
t
ins
tanc
e
s
f
r
om
the
T
S
P
L
I
B
ha
ve
be
e
n
us
e
d
to
e
va
luate
the
e
f
f
e
c
ti
ve
ne
s
s
of
the
GA
.
T
he
c
omput
a
ti
ona
l
r
e
s
ult
s
of
the
pr
opos
e
d
a
lgor
it
hm
e
xhibi
t
s
igni
f
ica
nt
potential
in
a
tt
a
ini
ng
opti
mal/nea
r
opti
mal
r
e
s
ult
s
f
or
the
k
-
M
T
S
P
.
B
e
in
g
the
f
ir
s
t
e
volut
iona
r
y
a
lgor
it
hm
f
or
the
k
-
M
T
S
P
,
ou
r
pr
opos
e
d
GA
a
ppr
oa
c
h
will
s
e
r
ve
a
s
a
r
e
f
e
r
e
nc
e
po
int
f
or
s
ubs
e
que
nt
r
e
s
e
a
r
c
h
on
the
k
-
M
T
S
P
.
Ove
r
a
ll
,
the
a
lgor
it
h
m
e
f
f
e
c
ti
ve
ly
f
inds
be
s
t
s
olut
ions
,
but
e
nha
nc
e
ments
a
r
e
ne
e
de
d
to
im
pr
ove
c
ons
is
tenc
y
a
nd
e
f
f
icie
nc
y
,
e
s
pe
c
ially
f
or
lar
ge
r
da
tas
e
ts
.
F
or
f
utu
r
e
wor
k
,
we
r
e
c
omm
e
nd
e
xplor
in
g
hybr
id
a
lgo
r
it
hms
that
c
ombi
ne
GA
s
with
tec
hni
que
s
li
ke
s
im
ulate
d
a
nne
a
li
ng,
a
nt
c
olony
opti
mi
z
a
ti
on,
a
nd
mac
hine
lea
r
ning
methods
,
whic
h
c
ould
he
lp
a
dd
r
e
s
s
the
c
ha
ll
e
ng
e
s
pos
e
d
by
lar
ge
da
tas
e
t
s
.
Additi
ona
ll
y,
i
nc
or
por
a
ti
ng
pa
r
a
ll
e
l
a
nd
dis
tr
ibut
e
d
c
omput
ing
s
t
r
a
tegie
s
c
ould
im
pr
ove
s
c
a
labili
ty.
Othe
r
pos
s
ibl
e
e
nha
nc
e
ments
to
the
k
-
M
T
S
P
model
include
the
int
e
gr
a
ti
on
of
e
leme
nts
s
uc
h
a
s
ti
me
windows
,
mul
ti
p
le
de
pots
,
a
nd
other
r
e
a
l
-
wor
ld
va
r
iations
to
incr
e
a
s
e
the
model's
a
ppli
c
a
bil
it
y
in
p
r
a
c
ti
c
a
l
s
c
e
na
r
ios
.
AC
KNOWL
E
DGM
E
N
T
S
W
e
would
li
ke
to
thank
to
our
guide
f
or
his
unwa
ve
r
ing
guidanc
e
,
invalua
ble
ins
ight
s
,
a
nd
e
nc
our
a
ge
ment
thr
oughout
the
r
e
s
e
a
r
c
h
pr
oc
e
s
s
.
F
UN
DI
NG
I
NF
ORM
AT
I
ON
No
f
unding
is
r
a
is
e
d
f
or
thi
s
r
e
s
e
a
r
c
h.
AU
T
HO
R
CONT
RI
B
U
T
I
ONS
S
T
AT
E
M
E
N
T
T
his
jour
na
l
us
e
s
the
C
ontr
ibut
o
r
R
oles
T
a
xo
nomy
(
C
R
e
diT
)
to
r
e
c
ognize
indi
vidual
a
uthor
c
ontr
ibut
ions
,
r
e
duc
e
a
utho
r
s
hip
dis
putes
,
a
nd
f
a
c
il
it
a
te
c
oll
a
bor
a
ti
on.
Nam
e
of
Au
t
h
or
C
M
So
Va
Fo
I
R
D
O
E
Vi
Su
P
Fu
Alikapa
ti
P
r
a
k
a
sh
✓
✓
✓
✓
✓
Ur
utur
u
B
a
lakr
is
hna
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
M
a
noga
r
a
n
T
ha
nga
r
a
j
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
T
he
ne
pa
ll
e
J
a
ya
nth
Kuma
r
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
C
:
C
onc
e
pt
ua
li
z
a
ti
on
M
:
M
e
th
odol
ogy
So
:
So
f
twa
r
e
Va
:
Va
li
da
ti
on
Fo
:
Fo
r
ma
l
a
na
ly
s
is
I
:
I
nve
s
ti
ga
ti
on
R
:
R
e
s
our
c
e
s
D
:
D
a
ta
C
ur
a
ti
on
O
:
W
r
it
in
g
-
O
r
ig
in
a
l
D
r
a
f
t
E
:
W
r
it
in
g
-
R
e
vi
e
w
&
E
di
ti
ng
Vi
:
Vi
s
ua
li
z
a
ti
on
Su
:
Su
pe
r
vi
s
io
n
P
:
P
r
oj
e
c
t
a
dmi
ni
s
tr
a
ti
on
Fu
:
Fu
ndi
ng a
c
qui
s
it
io
n
CONF
L
I
CT
OF
I
NT
E
RE
S
T
S
T
AT
E
M
E
N
T
Author
s
s
tate
no
c
onf
li
c
t
of
int
e
r
e
s
t.
I
NF
ORM
E
D
CONSE
NT
W
e
ha
ve
obtaine
d
inf
or
med
c
ons
e
nt
f
r
om
a
ll
ind
ivi
dua
ls
include
d
in
thi
s
s
tudy.
Evaluation Warning : The document was created with Spire.PDF for Python.