I
n
t
e
r
n
at
ion
al
Jou
r
n
al
of
E
lec
t
r
ical
an
d
Com
p
u
t
e
r
E
n
gin
e
e
r
in
g
(
I
JE
CE
)
Vol.
15
,
No.
1
,
F
e
br
ua
r
y
20
25
,
pp.
592
~
603
I
S
S
N:
2088
-
8708
,
DO
I
:
10
.
11591/i
jec
e
.
v
15
i
1
.
pp
5
92
-
603
592
Jou
r
n
al
h
omepage
:
ht
tp:
//
ij
e
c
e
.
iaes
c
or
e
.
c
om
D
is
c
r
e
t
e
o
p
t
imiz
at
io
n
m
o
d
e
l
f
or
m
u
lti
-
p
r
od
u
c
t
m
u
lti
-
su
p
p
li
e
r
ve
h
ic
l
e
r
o
u
t
in
g
p
r
o
b
le
m
w
ith
r
e
la
xe
d
t
im
e
w
in
d
ow
M
u
li
awan
F
irda
u
s
1
,
Her
m
an
M
awe
n
gk
an
g
2
,
T
u
lu
s
2
,
S
awalu
d
d
i
n
2
1
G
r
a
dua
te
S
c
hool
of
M
a
th
e
ma
ti
c
s
, U
ni
ve
r
s
it
a
s
S
uma
te
r
a
U
ta
r
a
,
M
e
da
n, I
ndone
s
ia
2
D
e
pa
r
tm
e
nt
of
M
a
th
e
ma
ti
c
s
, U
ni
ve
r
s
it
a
s
S
uma
te
r
a
U
ta
r
a
, M
e
d
a
n, I
ndone
s
ia
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
8,
2024
R
e
vis
e
d
Aug
12,
2024
Ac
c
e
pted
Aug
20,
2024
T
h
i
s
s
t
u
d
y
ex
ami
n
es
t
h
e
co
mp
l
i
ca
t
ed
l
o
g
i
s
t
i
c
s
o
p
t
i
m
i
za
t
i
o
n
i
s
s
u
e
k
n
o
w
n
as
t
h
e
v
e
h
i
cl
e
ro
u
t
i
n
g
p
r
o
b
l
em
fo
r
m
u
l
t
i
-
p
ro
d
u
c
t
an
d
mu
l
t
i
-
s
u
p
p
l
i
ers
(V
RP
-
MPMS),
w
h
i
c
h
d
ea
l
s
w
i
t
h
t
h
e
effec
t
i
v
e
ro
u
t
i
n
g
o
f
a
fl
eet
o
f
v
e
h
i
c
l
es
t
o
c
o
n
v
ey
n
u
mer
o
u
s
i
t
em
s
fro
m
m
u
l
t
i
p
l
e
s
u
p
p
l
i
ers
t
o
a
s
et
o
f
co
n
s
u
mers
.
I
n
t
h
i
s
p
r
o
b
l
em,
p
r
o
d
u
ct
s
fro
m
v
ar
i
o
u
s
s
u
p
p
l
i
er
s
n
ee
d
t
o
b
e
d
el
i
v
ere
d
t
o
d
i
ffere
n
t
cu
s
t
o
mers
w
h
i
l
e
c
o
n
s
i
d
eri
n
g
v
e
h
i
c
l
e
cap
ac
i
t
y
co
n
s
t
rai
n
t
s
,
t
i
me
w
i
n
d
o
w
s
,
a
n
d
mi
n
i
m
i
zi
n
g
t
ran
s
p
o
rt
a
t
i
o
n
co
s
t
s
.
W
e
p
ro
p
o
s
e
a
h
y
b
r
i
d
ap
p
r
o
ach
t
h
at
co
m
b
i
n
es
a
g
en
era
l
i
ze
d
red
u
ce
d
g
rad
i
en
t
met
h
o
d
t
o
i
d
en
t
i
f
y
feas
i
b
l
e
reg
i
o
n
s
w
i
t
h
a
feas
i
b
l
e
n
ei
g
h
b
o
r
h
o
o
d
s
earc
h
t
o
ach
i
e
v
e
o
p
t
i
ma
l
o
r
n
ear
-
o
p
t
i
mal
s
o
l
u
t
i
o
n
s
.
T
h
e
ai
m
o
f
t
h
e
ex
ac
t
met
h
o
d
i
s
t
o
g
et
t
h
e
reg
i
o
n
o
f
feas
i
b
l
e
s
o
l
u
t
i
o
n
.
T
h
e
n
w
e
ex
p
l
o
re
t
h
e
reg
i
o
n
u
s
i
n
g
fe
as
i
b
l
e
n
e
i
g
h
b
o
rh
o
o
d
s
earch
,
t
o
g
et
a
n
i
n
t
eg
er
fea
s
i
b
l
e
o
p
t
i
mal
(
s
u
b
o
p
t
i
m
al
)
s
o
l
u
t
i
o
n
.
Co
mp
u
t
a
t
i
o
n
a
l
ex
p
eri
me
n
t
s
d
emo
n
s
t
rat
e
t
h
at
o
u
r
mo
d
e
l
an
d
met
h
o
d
effect
i
v
el
y
red
u
ce
t
ra
n
s
p
o
r
t
at
i
o
n
co
s
t
s
w
h
i
l
e
s
at
i
s
f
y
i
n
g
v
e
h
i
c
l
e
cap
ac
i
t
y
co
n
s
t
ra
i
n
t
s
an
d
rel
ax
e
d
t
i
me
w
i
n
d
o
w
s
.
O
u
r
fi
n
d
i
n
g
s
p
r
o
v
i
d
e
a
v
i
ab
l
e
s
o
l
u
t
i
o
n
fo
r
i
mp
r
o
v
i
n
g
l
o
g
i
s
t
i
c
s
o
p
era
t
i
o
n
s
i
n
real
-
w
o
rl
d
s
ce
n
ari
o
s
.
K
e
y
w
o
r
d
s
:
Dis
c
r
e
te
opti
mi
z
a
ti
on
model
M
ult
i
-
pr
oduc
t
M
ult
i
-
s
uppli
e
r
T
im
e
windows
Ve
hicle
r
outi
ng
pr
ob
lem
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
:
He
r
man
M
a
we
ngka
ng
De
pa
r
tm
e
nt
of
M
a
thema
ti
c
s
,
Unive
r
s
it
a
s
S
umate
r
a
Uta
r
a
J
l.
Dr
.
T
.
M
a
ns
ur
No.
9,
P
a
da
ng
B
ulan,
M
e
da
n
B
a
r
u,
M
e
da
n
C
it
y,
Nor
th
S
umatr
a
20155
,
I
ndone
s
ia
E
mail:
maw
e
ngka
ng@us
u.
a
c
.
id
1.
I
NT
RODU
C
T
I
ON
As
a
r
e
s
ult
of
it
s
s
igni
f
ica
nc
e
to
the
e
c
onomy,
th
e
f
ield
of
logi
s
ti
c
s
a
nd
s
upply
c
ha
in
mana
ge
ment
ha
s
be
e
n
a
dva
nc
ing
c
ons
i
s
tently
a
nd
quickly
in
r
e
c
e
nt
ye
a
r
s
.
S
ince
the
major
it
y
o
f
bus
ines
s
e
s
vie
w
s
upply
c
ha
in
mana
ge
ment
a
s
a
f
unc
ti
on
that
e
nha
nc
e
s
their
mar
ke
t,
it
plays
a
c
r
uc
ial
pa
r
t
in
their
s
tr
a
tegic
d
e
c
is
ion
-
making.
Due
to
the
c
ons
tantly
inc
r
e
a
s
ing
de
mand
f
r
om
c
us
tom
e
r
s
,
bus
ines
s
e
s
ne
e
d
e
f
f
icie
nt
de
li
ve
r
y
s
e
r
vice
s
without
s
a
c
r
if
icing
the
s
tanda
r
d
of
thei
r
c
us
tom
e
r
c
a
r
e
in
or
de
r
to
r
e
main
pr
o
f
it
a
ble.
I
n
or
de
r
to
f
ul
f
il
thes
e
incr
e
a
s
ing
e
xpe
c
tations
,
logi
s
ti
c
s
ope
r
a
ti
ons
h
a
ve
be
e
n
unde
r
c
onti
nua
l
pr
e
s
s
ur
e
to
be
c
ome
mor
e
e
f
f
e
c
ti
ve
.
As
a
r
e
s
ult
,
f
o
r
e
f
f
e
c
ti
ve
mana
ge
ment,
bus
ines
s
e
s
mus
t
de
s
ign
the
r
outes
f
o
r
their
ve
hicle
s
s
o
a
s
to
s
a
ve
e
xpe
ns
e
s
while
s
ti
ll
s
a
ti
s
f
ying
c
us
tom
e
r
r
e
quir
e
ments
.
T
he
ve
hicle
r
outi
ng
p
r
oblem
(
VR
P
)
is
the
na
me
us
e
d
in
ope
r
a
ti
ons
r
e
s
e
a
r
c
h
to
de
s
c
r
ibe
th
is
method
of
r
oute
de
ter
mi
ning
[
1]
,
[
2
]
.
M
a
thema
ti
c
a
ll
y,
the
VR
P
c
a
n
be
de
s
c
r
ibed
a
s
f
oll
owing:
L
e
t
=
(
,
)
be
a
dir
e
c
ted
g
r
a
ph,
whe
r
e
=
{
0
,
.
.
.
,
}
is
a
s
e
t
of
node
s
,
with
ve
r
tex
0
r
e
p
r
e
s
e
nti
ng
the
d
e
pot
a
nd
the
r
e
maining
ve
r
ti
c
e
s
r
e
p
r
e
s
e
nti
ng
c
us
tom
e
r
s
.
=
{
(
,
)
:
,
∈
,
≠
}
r
e
pr
e
s
e
nts
a
r
c
s
in
the
gr
a
ph.
T
he
de
pot
wa
s
home
to
a
f
lee
t
of
m
identica
l
ve
hicle
s
,
e
a
c
h
ha
ving
a
c
a
pa
c
it
y.
T
he
f
lee
t
s
ize
,
is
a
de
c
is
ion
va
r
iable
that
ne
e
ds
to
be
de
ter
mi
ne
d.
E
a
c
h
c
us
tom
e
r
ha
s
a
r
e
que
s
t
,
whic
h
is
a
non
-
ne
ga
ti
ve
qua
nti
ty
r
e
p
r
e
s
e
nti
ng
their
de
mand.
Additi
ona
ll
y,
ther
e
is
a
c
os
t
matr
ix
de
f
ined
f
or
e
a
c
h
a
r
c
(
,
)
in
,
whic
h
r
e
pr
e
s
e
nts
the
c
os
t
(
or
dis
tanc
e
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
E
lec
&
C
omp
E
ng
I
S
S
N:
2088
-
8708
Dis
c
r
e
te
opti
miz
ati
on
mode
l
for
multi
-
pr
oduc
t
mul
ti
-
s
uppli
e
r
v
e
hicle
…
(
M
uli
aw
an
F
ir
daus
)
593
a
s
s
oc
iate
d
with
tr
a
ve
li
ng
f
r
om
node
to
node
.
I
t
i
s
im
por
tant
to
note
that
in
thi
s
f
or
mul
a
ti
on,
a
s
s
ume
that
dis
tanc
e
s
,
tr
a
ve
l
c
os
ts
,
a
nd
ti
mes
a
r
e
mea
s
ur
e
d
e
qu
ivale
nt.
T
h
e
V
R
P
’
s
go
a
l
is
t
o
c
r
e
a
t
e
v
e
hic
le
r
o
utes
t
ha
t
a
dh
e
r
e
to
t
he
f
ol
lo
wi
ng
r
e
s
t
r
ic
ti
ons
:
T
he
de
po
t
(
n
od
e
0
)
is
w
he
r
e
e
ve
r
y
r
o
ut
e
m
us
t
b
e
g
in
a
nd
te
r
mi
na
te
.
E
a
c
h
c
us
t
om
e
r
’
s
de
man
d
m
us
t
b
e
s
a
t
is
f
ied
by
ha
vi
ng
e
xa
c
t
ly
on
e
ve
h
ic
le
v
is
i
t
th
e
m
.
E
a
c
h
r
o
ut
e
’
s
o
ve
r
a
ll
de
ma
nd
c
a
nn
ot
be
m
o
r
e
th
a
n
t
he
ve
hic
le
’
s
c
a
pa
c
i
ty
.
E
a
c
h
r
o
u
te's
t
ota
l
len
g
th
(
o
r
p
r
i
c
e
)
m
us
t
no
t
go
be
yo
n
d
a
c
e
r
ta
in
th
r
e
s
h
ol
d
,
.
T
h
e
o
bje
c
t
iv
e
is
t
o
m
in
i
mi
z
e
th
e
o
ve
r
a
l
l
t
r
a
ns
p
o
r
t
a
t
io
n
c
os
t
o
r
r
o
ute
le
ng
th
wh
i
le
f
u
l
f
il
li
ng
th
e
a
f
o
r
e
me
n
ti
on
e
d
li
mi
ta
t
io
ns
b
y
o
pt
im
iz
in
g
the
a
s
s
i
gn
men
t
o
f
c
li
e
n
ts
t
o
ve
hi
c
l
e
s
a
n
d
t
he
o
r
d
e
r
in
w
h
ich
the
y
a
r
e
v
is
it
e
d
.
I
n
the
s
ym
me
t
r
i
c
c
a
s
e
,
tha
t
is
,
whe
n
=
f
o
r
a
l
l
(
,
)
∈
,
t
he
s
ol
ut
i
on
s
e
a
r
c
h
is
u
s
ua
l
ly
do
ne
us
i
ng
the
s
e
t
o
f
e
d
ge
s
,
=
{
(
,
)
:
,
∈
,
<
}
.
T
he
VR
P
wa
s
f
ir
s
t
de
ve
loped
by
Da
ntzig
a
nd
R
a
ms
e
r
[
3]
to
a
ddr
e
s
s
the
p
r
oblem
of
e
f
f
icie
ntl
y
r
outi
ng
a
f
lee
t
o
f
f
ue
l
de
li
ve
r
y
tr
uc
ks
f
r
om
a
b
ulk
ter
mi
na
l
to
nume
r
ous
s
e
r
vice
outl
e
ts
s
uppli
e
d
by
the
ter
mi
na
l.
T
his
mar
ke
d
the
or
igi
n
of
VR
P
a
s
a
mathe
matica
l
opti
mi
z
a
ti
on
pr
oblem.
Ove
r
the
ye
a
r
s
,
VR
P
ha
s
ga
ined
s
igni
f
ica
nt
a
tt
e
nti
on
in
the
r
e
s
e
a
r
c
h
c
omm
u
nit
y,
a
nd
va
r
ious
e
xtens
ions
a
nd
va
r
iations
o
f
the
pr
oblem
ha
ve
be
e
n
e
xplor
e
d
.
F
ur
the
r
li
ter
a
tur
e
on
the
VR
P
a
nd
it
s
dif
f
e
r
e
nt
a
tt
r
ibut
e
s
c
a
n
be
r
e
f
e
r
r
e
d
to
the
f
oll
owing
s
our
c
e
s
f
or
a
c
ompr
e
he
ns
ive
s
ur
ve
y
a
nd
ove
r
view
[
4]
–
[
6]
.
T
he
s
e
r
e
f
e
r
e
nc
e
s
pr
ovide
va
luable
ins
ight
s
int
o
the
VR
P
a
nd
it
s
e
volut
ion,
including
di
f
f
e
r
e
nt
pr
o
blem
va
r
iants
a
nd
s
olut
ion
a
ppr
oa
c
he
s
that
ha
ve
be
e
n
de
ve
loped
ove
r
ti
me.
R
e
s
e
a
r
c
h
on
the
VR
P
c
onti
nue
s
to
be
a
dyna
mi
c
f
ield
due
to
both
un
r
e
s
olved
theor
e
ti
c
a
l
c
ha
ll
e
nge
s
a
nd
the
ongoing
inf
lux
of
pr
a
c
ti
c
a
l
log
is
ti
c
s
da
ta
f
r
om
s
upply
c
ha
in
ope
r
a
ti
ons
.
One
of
the
mos
t
e
xt
e
ns
ively
s
tudi
e
d
va
r
iants
of
the
VR
P
is
with
t
im
e
windows
(
VR
P
T
W
)
,
whic
h
wa
s
f
ir
s
t
int
r
oduc
e
d
by
S
c
h
r
a
ge
[
7]
.
I
n
the
VR
P
T
W
,
s
pe
c
if
ic
t
im
e
c
ons
tr
a
int
s
a
r
e
plac
e
d
on
c
us
tom
e
r
vis
it
s
,
known
a
s
ti
me
windows
.
T
im
e
window
c
ons
tr
a
int
s
in
VR
P
T
W
c
a
n
be
dr
iven
by
va
r
ious
f
a
c
tor
s
,
s
uc
h
a
s
pr
oduc
t
c
ons
tr
a
int
s
(
e
.
g.
,
p
r
odu
c
t
us
a
ble
da
tes
)
,
pr
oduc
ti
on
li
mi
ts
,
o
r
r
e
qui
r
e
ments
im
pos
e
d
by
c
us
tom
e
r
s
ba
s
e
d
on
their
inventor
y
po
li
c
ies
.
I
n
a
ddit
ion
to
thes
e
t
im
e
windows
f
or
c
us
tom
e
r
vis
it
s
,
ther
e
a
r
e
a
ls
o
t
r
a
ve
l
ti
mes
be
twe
e
n
a
ll
c
us
tom
e
r
s
a
nd
be
twe
e
n
c
us
tom
e
r
s
a
nd
the
de
pot.
T
he
main
objec
ti
ve
s
in
V
R
P
T
W
a
r
e
to
plan
ve
hicle
r
outes
that
s
a
ti
s
f
y
the
f
o
ll
owing
c
r
it
e
r
ia:
E
a
c
h
ve
hicle
mus
t
s
e
r
ve
a
ll
a
s
s
igned
c
us
tom
e
r
s
withi
n
thei
r
r
e
s
pe
c
ti
ve
ti
me
windows
,
Ve
hi
c
les
a
r
e
pe
r
mi
tt
e
d
to
a
r
r
ive
a
t
the
loca
ti
on
of
a
c
li
e
nt
be
f
o
r
e
the
ti
me
window
be
gins
,
but
they
mus
t
wa
it
if
th
e
y
do
s
o
be
f
or
e
the
c
ons
umer
is
pr
e
pa
r
e
d
to
be
s
e
r
ve
d.
V
e
hicle
s
a
r
e
not
pe
r
mi
tt
e
d
to
a
r
r
ive
late
or
a
f
ter
t
he
ti
me
window
e
nds
f
or
a
ny
c
us
tom
e
r
.
T
he
p
r
im
a
r
y
goa
l
is
to
mi
nim
ize
the
tot
a
l
t
r
a
ns
por
tation
c
os
t,
tak
ing
int
o
a
c
c
ount
both
tr
a
ve
l
dis
tanc
e
s
a
nd
ti
me
-
r
e
late
d
pe
na
lt
ies
f
or
viol
a
ti
ng
the
ti
me
windows
.
VR
P
T
W
is
pa
r
ti
c
ular
ly
r
e
leva
nt
in
s
it
ua
ti
ons
whe
r
e
ti
me
-
s
e
ns
it
ive
de
li
ve
r
ies
a
r
e
c
r
uc
ial,
a
nd
it
pos
e
s
a
ddit
ional
c
ha
ll
e
nge
s
c
ompar
e
d
to
the
c
las
s
ic
VR
P
.
Due
to
i
ts
pr
a
c
ti
c
a
l
im
por
tanc
e
a
nd
c
ompl
e
xit
y
,
VR
P
T
W
ha
s
be
e
n
the
s
ubjec
t
of
e
xtens
ive
r
e
s
e
a
r
c
h
a
nd
ha
s
led
to
th
e
de
ve
lopm
e
nt
of
va
r
ious
s
olut
ion
methods
a
nd
a
l
gor
it
hms
to
f
ind
opti
mal
o
r
ne
a
r
-
opti
mal
s
olut
ions
in
r
e
a
l
-
wor
ld
logi
s
ti
c
s
a
nd
t
r
a
ns
por
tation
a
ppli
c
a
ti
ons
[
8]
–
[
10]
.
T
he
VR
P
T
W
is
indee
d
a
c
ha
ll
e
nging
p
r
oblem
due
to
it
s
c
ombi
na
tor
ial
na
tu
r
e
,
whic
h
make
s
it
dif
f
icult
to
f
ind
opti
mal
s
olut
ions
,
e
s
pe
c
ially
f
or
la
r
ge
r
ins
tanc
e
[
11]
.
Kohl
[
12
]
e
s
tablis
he
d
that
VR
P
T
W
is
a
n
NP
-
ha
r
d
pr
oblem,
indi
c
a
ti
ng
that
s
olvi
ng
it
to
opti
malit
y
be
c
omes
c
omput
a
ti
ona
ll
y
inf
e
a
s
ibl
e
a
s
the
pr
oblem
s
ize
incr
e
a
s
e
s
.
As
a
r
e
s
ult
,
r
e
s
e
a
r
c
he
r
s
ha
ve
tur
ne
d
to
he
ur
is
ti
c
a
nd
meta
he
ur
is
ti
c
a
ppr
oa
c
he
s
to
f
in
d
good
-
qua
li
ty
s
olut
ions
withi
n
r
e
a
s
ona
ble
c
omput
a
ti
on
a
l
ti
me.
Va
r
ious
meta
he
ur
is
ti
c
s
a
nd
he
ur
is
ti
c
s
h
a
ve
be
e
n
pr
opos
e
d
to
tac
kle
VR
P
T
W
,
a
im
ing
to
s
tr
ike
a
ba
lanc
e
be
twe
e
n
s
olut
ion
qua
li
ty
a
nd
c
omp
utational
e
f
f
icie
nc
y.
S
ome
o
f
thes
e
a
ppr
oa
c
he
s
include
thos
e
de
ve
loped
by
r
e
s
e
a
r
c
he
r
s
s
uc
h
a
s
[
13]
–
[
16]
.
E
f
f
icie
nt
meta
he
ur
is
ti
c
s
,
in
pa
r
ti
c
ular
,
of
ten
r
e
ly
on
loca
l
s
e
a
r
c
h
-
ba
s
e
d
r
e
f
ineme
nt
pr
oc
e
dur
e
s
a
nd
f
oc
us
much
of
their
c
omput
a
ti
ona
l
e
f
f
o
r
t
on
e
xplor
ing
ne
ighbor
hoods
of
s
olut
ions
.
T
his
a
ppr
oa
c
h
c
a
n
s
igni
f
ica
ntl
y
im
p
r
ove
the
qua
li
ty
of
s
olut
ions
f
ound,
but
it
a
ls
o
plac
e
s
a
n
e
mphas
is
on
e
va
luating
the
i
mpac
t
of
potential
s
olut
ion
c
ha
nge
s
e
f
f
icie
ntl
y.
How
e
ve
r
,
i
t's
wor
th
noti
ng
t
ha
t
e
ve
n
f
indi
ng
a
f
e
a
s
ibl
e
s
olut
ion
f
or
VR
P
T
W
,
without
ne
c
e
s
s
a
r
il
y
a
im
ing
f
or
opti
malit
y,
r
e
mains
c
omp
utationally
c
ha
ll
e
nging.
As
mentioned,
S
a
ve
ls
be
r
gh
[
17]
de
mons
tr
a
ted
that
de
ter
m
ini
ng
a
f
e
a
s
ibl
e
s
olut
ion
f
o
r
VR
P
T
W
is
a
ls
o
a
n
NP
-
ha
r
d
pr
oblem
.
T
his
h
ighl
ight
s
the
inher
e
nt
c
ompl
e
xit
y
o
f
the
pr
oblem
a
nd
unde
r
s
c
or
e
s
the
ne
e
d
f
o
r
s
ophis
ti
c
a
ted
opti
mi
z
a
ti
on
tec
hn
iques
to
a
ddr
e
s
s
it
e
f
f
e
c
ti
ve
ly,
e
s
pe
c
ially
whe
n
de
a
li
ng
wit
h
r
e
a
l
-
wor
ld
ins
tanc
e
s
with
pr
a
c
ti
c
a
l
c
ons
tr
a
int
s
a
nd
lar
ge
r
number
s
of
c
us
tom
e
r
s
or
s
ubs
c
r
iber
s
[
18]
.
I
n
or
de
r
to
de
ve
lop
e
a
r
ly
s
olut
ions
f
or
the
VR
P
T
W
,
it
is
f
r
e
que
ntl
y
us
e
d
int
e
r
media
te
s
olut
ions
wit
h
f
lexible
ti
me
f
r
a
me
li
mi
tat
ions
.
How
e
ve
r
,
a
s
m
e
nti
one
d,
it
may
not
a
lwa
ys
be
f
e
a
s
ibl
e
or
opt
im
a
l
f
o
r
gua
r
a
ntee
ing
a
va
il
a
bil
it
y
of
mul
ti
ple
ini
ti
a
l
s
olut
ions
.
S
e
ve
r
a
l
r
e
laxa
ti
on
s
c
he
mes
ha
ve
be
e
n
e
xp
lor
e
d
in
pr
e
vious
r
e
s
e
a
r
c
h
to
ha
ndle
ti
me
window
c
ons
tr
a
int
s
mor
e
e
f
f
e
c
ti
ve
ly:
P
e
nalt
ies
for
L
ate
A
r
r
ivals
:
One
c
omm
on
r
e
laxa
ti
on
method
invol
ve
s
a
s
s
igni
ng
pe
na
lt
ies
f
or
late
a
r
r
ivals
a
t
c
us
tom
e
r
loca
ti
ons
.
T
hi
s
a
ll
ows
f
or
s
ome
f
lexibil
it
y
in
mee
ti
ng
ti
me
windows
while
pe
na
li
z
ing
de
viations
f
r
om
the
de
s
ir
e
d
s
c
he
dule.
T
his
a
ppr
oa
c
h
wa
s
dis
c
us
s
e
d
by
S
un
e
t
al
.
[
19]
.
E
ar
ly
and
L
ate
A
r
r
ivals
:
Anothe
r
r
e
laxa
ti
on
s
c
he
me
c
ons
ider
s
both
e
a
r
ly
a
nd
late
a
r
r
ivals
a
t
c
us
tom
e
r
s
it
e
s
.
I
t
a
ll
ows
ve
hicle
s
to
a
r
r
ive
be
f
or
e
or
a
f
ter
the
ti
me
win
dow
but
incur
s
pe
na
lt
ies
a
c
c
or
dingl
y.
T
his
a
ppr
oa
c
h
wa
s
s
tudi
e
d
by
I
ba
r
a
ki
e
t
al
.
[
20
]
.
R
e
fund
P
e
nalt
ies
on
T
i
me
:
[
21]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2088
-
8708
I
nt
J
E
lec
&
C
omp
E
ng
,
Vol
.
15
,
No.
1
,
F
e
br
ua
r
y
20
25
:
592
-
603
594
pr
opos
e
d
a
r
e
f
und
pe
na
lt
y
a
ppr
oa
c
h,
whe
r
e
ve
hi
c
les
c
ould
e
a
r
n
r
e
f
unds
f
o
r
a
r
r
ivi
ng
e
a
r
li
e
r
than
r
e
quir
e
d
while
be
ing
pe
na
li
z
e
d
f
or
a
r
r
ivi
ng
late
.
T
his
a
ppr
oa
c
h
e
nc
our
a
ge
s
e
a
r
ly
a
r
r
ivals
but
s
ti
ll
r
e
s
p
e
c
ts
ti
me
window
c
ons
tr
a
int
s
.
W
hil
e
thes
e
r
e
laxa
ti
on
s
c
he
mes
of
f
e
r
s
ome
f
lex
ibi
li
ty
in
f
indi
ng
ini
ti
a
l
s
olut
ions
,
they
c
a
n
a
ls
o
int
r
oduc
e
c
ompl
e
xit
y
in
e
va
luating
potential
s
o
lut
ion
c
ha
nge
s
a
nd
may
lea
d
to
lar
ge
r
s
e
a
r
c
h
s
pa
c
e
s
.
T
he
r
e
f
or
e
,
your
r
e
s
e
a
r
c
h
a
im
s
to
e
s
tablis
h
a
ne
w
r
e
laxa
ti
on
s
c
he
me
that
ba
lanc
e
s
f
e
a
s
ibi
li
ty
a
n
d
s
olut
ion
qua
li
ty
mor
e
e
f
f
e
c
ti
ve
ly.
De
ve
lopi
ng
innovative
r
e
laxa
ti
on
s
tr
a
tegie
s
that
s
tr
ike
the
r
ight
ba
lanc
e
be
twe
e
n
f
e
a
s
ibi
li
ty
a
nd
s
olut
ion
qua
li
ty
is
e
s
s
e
nti
a
l
in
the
c
ontext
of
VR
P
T
W
opti
mi
z
a
ti
on,
a
s
it
c
a
n
lea
d
to
im
pr
ove
d
ini
ti
a
l
s
olut
ions
a
nd
ult
im
a
tely
e
nha
nc
e
the
pe
r
f
or
manc
e
of
he
ur
is
ti
c
a
nd
meta
he
ur
is
ti
c
a
lgor
it
hms
f
o
r
s
olvi
ng
thi
s
c
ha
ll
e
nging
pr
oblem.
T
he
inf
luenc
e
of
r
e
a
l
-
wor
ld
t
r
a
f
f
ic
c
ondit
ions
on
v
e
hicle
r
outi
ng
is
indee
d
a
s
igni
f
ica
nt
c
ha
ll
e
nge
in
pr
a
c
ti
c
a
l
s
upply
c
ha
in
ope
r
a
ti
ons
.
T
r
a
dit
ional
VR
P
models
typi
c
a
ll
y
a
s
s
ume
c
ons
tant
t
r
a
ve
l
ti
mes
,
whic
h
c
a
n
lea
d
to
s
ubopti
mal
or
in
f
e
a
s
ibl
e
s
olut
ions
whe
n
f
a
c
e
d
with
the
unc
e
r
tainti
e
s
a
nd
va
r
iabili
ty
c
a
us
e
d
b
y
tr
a
f
f
ic
c
onge
s
ti
on.
T
o
a
ddr
e
s
s
thi
s
is
s
ue
,
r
e
s
e
a
r
c
he
r
s
ha
ve
int
r
oduc
e
d
mor
e
a
dva
nc
e
d
va
r
iants
of
the
pr
o
blem
that
c
ons
ider
ti
me
-
de
pe
nde
nt
tr
a
ve
l
ti
mes
.
One
s
uc
h
v
a
r
iant
is
the
time
-
de
pe
nde
nt
ve
hicle
r
outi
ng
pr
obl
e
m
with
ti
me
windows
(
T
DV
R
P
T
W
)
,
whic
h
wa
s
int
r
oduc
e
d
by
Kuma
r
a
nd
P
a
nne
e
r
s
e
lvam
[
22]
.
I
n
the
T
D
VR
P
T
W
,
the
modeling
e
xpli
c
it
ly
take
s
int
o
a
c
c
ount
the
va
r
iabili
ty
in
tr
a
ve
l
t
im
e
s
due
to
tr
a
f
f
ic
c
ondit
ions
.
T
his
is
a
c
hieve
d
by
c
ons
ider
ing
ti
me
-
de
pe
nde
nt
tr
a
ve
l
ti
mes
f
or
e
a
c
h
a
r
c
or
r
ou
te
s
e
gment,
whic
h
c
a
n
c
ha
nge
ba
s
e
d
on
f
a
c
tor
s
li
ke
t
r
a
f
f
ic
c
onge
s
ti
on,
t
im
e
o
f
da
y
,
a
nd
r
oa
d
c
ondit
ions
.
T
o
s
olve
the
T
DV
R
P
T
W
,
va
r
ious
a
ppr
oa
c
he
s
ha
ve
be
e
n
e
xplor
e
d
[
23]
–
[
25]
,
includ
ing
mi
xe
d
int
e
ge
r
pr
og
r
a
mm
ing
(
M
I
P
)
f
or
mul
a
ti
ons
[
26]
,
whic
h
pr
ovide
a
mathe
matica
l
r
e
pr
e
s
e
ntation
of
t
he
pr
oblem
with
ti
me
-
de
pe
nde
nt
c
ons
tr
a
int
s
.
Additi
ona
ll
y,
meta
he
ur
is
ti
c
a
lgor
it
hms
li
ke
ge
ne
ti
c
a
lgor
it
hms
ha
ve
be
e
n
a
ppli
e
d
to
f
ind
good
-
qua
li
ty
s
olut
io
ns
withi
n
r
e
a
s
ona
ble
c
omput
a
ti
ona
l
ti
mef
r
a
mes
.
B
y
c
ons
ider
ing
the
im
pa
c
t
of
t
r
a
f
f
ic
c
ondit
ions
,
the
T
D
VR
P
T
W
pr
ovides
a
mor
e
r
e
a
li
s
ti
c
r
e
pr
e
s
e
ntation
of
r
outi
ng
c
ha
ll
e
nge
s
f
a
c
e
d
by
ve
hicle
s
in
s
upply
c
ha
in
ope
r
a
ti
ons
.
I
t
a
ll
ows
f
or
the
opti
mi
z
a
ti
on
of
r
outes
that
a
r
e
be
tt
e
r
a
da
pted
to
r
e
a
l
-
wor
ld
s
it
ua
ti
ons
,
ul
ti
mate
ly
im
p
r
oving
the
e
f
f
icie
nc
y
a
nd
r
e
li
a
bil
i
ty
of
logi
s
ti
c
s
a
nd
t
r
a
ns
por
tation
pr
oc
e
s
s
e
s
.
E
nga
ging
with
mu
lt
ipl
e
s
uppli
e
r
s
in
a
T
DV
R
P
T
W
int
r
oduc
e
s
s
igni
f
ica
nt
c
ompl
e
xit
y.
C
oor
dinatin
g
vis
it
s
to
s
uppli
e
r
s
with
diver
s
e
ti
me
windows
,
e
ns
ur
ing
ti
mely
p
ickups
,
a
nd
a
dhe
r
ing
to
de
li
ve
r
y
pe
r
iods
a
dd
c
ha
ll
e
nge
s
.
Additi
ona
ll
y,
a
c
c
ounti
ng
f
or
r
e
a
l
-
ti
me
tr
a
f
f
ic
c
ondit
ions
is
c
r
uc
ial.
T
o
a
ddr
e
s
s
thes
e
c
ompl
e
xit
ies
,
opti
mi
z
a
ti
on
a
ppr
oa
c
he
s
invol
ve
dyna
mi
c
s
c
he
duli
ng,
a
dva
nc
e
d
mathe
matica
l
models
,
he
ur
is
ti
c
a
lgor
it
hms
,
a
nd
r
e
a
l
-
ti
me
da
ta
int
e
gr
a
ti
on
to
f
ind
e
f
f
icie
nt
s
o
lut
ions
that
c
ons
ider
the
dyna
mi
c
na
tur
e
o
f
s
upp
ly
c
ha
in
ope
r
a
ti
ons
a
nd
tr
a
f
f
ic
c
ondit
ions
,
e
nha
nc
ing
the
r
e
li
a
bil
i
ty
a
nd
e
f
f
icie
nc
y
o
f
mul
ti
-
s
uppli
e
r
logi
s
ti
c
s
a
nd
tr
a
ns
por
tation
pr
oc
e
s
s
e
s
.
E
xpe
r
im
e
nts
invol
ving
56
ins
tanc
e
s
f
r
om
the
S
olom
on
be
nc
hmar
k
[
27]
,
e
a
c
h
c
ompr
is
ing
100
c
us
tom
e
r
node
s
,
25
ve
hicle
s
with
a
200
-
unit
c
a
pa
c
it
y
e
a
c
h,
ha
ve
yielde
d
r
e
s
ult
s
only
c
ompar
a
ble
to
e
xis
ti
ng
a
lgor
it
hms
.
C
ons
e
que
ntl
y,
thi
s
s
tudy
a
im
s
to
pr
opos
e
a
n
innovative
dis
c
r
e
te
opti
mi
z
a
ti
on
model
a
s
a
n
a
lt
e
r
na
ti
ve
a
ppr
oa
c
h
f
or
a
ddr
e
s
s
ing
the
T
DV
R
P
T
W
in
mul
t
i
-
s
uppli
e
r
s
e
tt
ings
.
Additi
ona
ll
y,
the
r
e
s
e
a
r
c
h
will
de
ve
lop
a
meta
he
ur
is
ti
c
a
lgor
i
thm
,
with
ini
ti
a
l
s
olut
ions
ge
ne
r
a
ted
thr
ough
ti
me
window
r
e
lax
a
ti
on,
to
tac
kle
thi
s
c
ompl
e
x
pr
oblem
e
f
f
e
c
ti
ve
ly.
2.
M
AT
E
R
I
AL
AN
D
M
E
T
HO
DS
2.
1.
F
or
m
at
ion
of
ve
h
icle
r
ou
t
e
s
A
r
oute
∈
will
s
ti
ll
be
f
e
a
s
ibl
e
whe
n
the
r
ou
te
s
tar
t
s
a
t
a
dif
f
e
r
e
nt
t
im
e
ins
tant.
T
hus
,
f
o
r
e
a
c
h
r
oute
,
it
is
noti
c
e
d
that
ther
e
a
r
e
mul
ti
ple
r
ou
tes
,
one
f
or
e
a
c
h
ins
tant
of
pos
s
ibl
e
de
pa
r
tur
e
s
.
T
he
dur
a
ti
on
of
r
ou
te
,
,
wi
ll
be
dif
f
e
r
e
nt
f
or
dif
f
e
r
e
nt
ins
tanc
e
s
of
de
pa
r
tu
r
e
,
due
to
the
wa
it
ing
t
im
e
f
or
s
e
r
ving
dif
f
e
r
e
nt
c
us
tom
e
r
s
.
S
uppos
e
(
1
,
…
,
|
|
)
is
the
or
de
r
of
c
li
e
nts
vis
it
e
d
on
r
oute
∈
.
T
he
ini
ti
a
l
pos
s
ibl
e
ins
tant
of
ti
me
to
e
nd
r
ou
te
is
′
−
=
|
|
+
|
|
+
|
|
0
,
whe
r
e
|
|
is
the
f
ir
s
t
ins
tanc
e
to
be
gin
p
r
ovis
ion
on
the
latter
c
li
e
nt
|
|
in
r
oute
.
T
he
n
c
a
lcula
te
′
−
,
takin
g
int
o
a
c
c
ount
that
ℎ
=
m
a
x
{
ℎ
−
1
+
ℎ
−
1
+
ℎ
−
1
ℎ
,
ℎ
}
f
or
ℎ
∈
{
1
,
…
,
|
|
}
whe
r
e
0
=
0
.
T
h
is
mea
ns
that
s
tar
ti
ng
r
oute
a
t
a
ny
ins
tanc
e
∗
≤
−
=
1
−
0
1
c
a
us
ing
it
to
s
top
ins
tantly
′
−
.
T
hus
,
s
uc
h
a
r
oute
is
s
ubjec
t
by
r
outes
s
tar
ti
ng
a
t
the
−
ins
tanc
e
,
s
o
they
do
not
ne
e
d
to
be
c
onc
e
r
ne
d.
I
n
the
s
a
me
wa
y,
las
t
e
nding
ti
me
ins
tanta
ne
ous
on
r
oute
is
′
+
=
|
|
+
|
|
+
|
|
0
,
whe
r
e
|
|
is
the
ins
tanc
e
to
be
gin
p
r
ovis
ion
on
the
c
li
e
n
t
|
|
|
in
c
our
s
e
a
nd
ℎ
=
m
in
{
ℎ
−
1
+
ℎ
−
1
+
ℎ
−
1
ℎ
,
ℎ
}
,
f
or
ℎ
∈
{
1
,
…
,
|
|
}
with
0
=
0
.
I
t
c
onc
ludes
that
s
tar
ti
ng
r
oute
in
ti
me
a
f
ter
+
=
1
−
0
1
r
e
s
ult
s
in
that
r
ou
te
be
ing
in
f
e
a
s
ibl
e
,
be
c
a
us
e
it
ign
or
e
s
a
t
mi
nim
um
of
one
c
li
e
nt
ti
me
windows
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
E
lec
&
C
omp
E
ng
I
S
S
N:
2088
-
8708
Dis
c
r
e
te
opti
miz
ati
on
mode
l
for
multi
-
pr
oduc
t
mul
ti
-
s
uppli
e
r
v
e
hicle
…
(
M
uli
aw
an
F
ir
daus
)
595
I
t
s
hould
be
noted
that
pa
th
s
tar
ts
in
ti
me
[
−
,
+
]
,
the
wa
it
ing
ti
me
is
r
e
duc
e
d
he
nc
e
it
would
ha
ve
mi
nim
ize
d
du
r
a
ti
on.
As
a
n
e
xa
mpl
e
,
e
a
c
h
∈
,
t
he
ti
me
[
−
,
+
]
,
is
c
a
lcula
ted.
T
hus
,
f
e
a
s
ibl
e
r
ou
tes
qua
nti
ty
will
e
qua
l
∑
⌈
+
−
−
+
1
⌉
∈
,
whe
r
e
is
the
unit
of
ti
me
a
s
s
umed
to
e
qua
l
1
.
2.
2.
Dis
c
r
e
t
e
m
od
e
l
d
e
ve
lop
m
e
n
t
I
nc
or
por
a
ti
ng
mul
t
i
-
s
uppli
e
r
c
ons
tr
a
int
s
a
dds
s
igni
f
ica
nt
c
ompl
e
xit
y
to
the
model
.
I
n
thi
s
r
e
s
e
a
r
c
h,
a
dir
e
c
ted
a
c
yc
li
c
gr
a
ph
=
(
,
)
is
us
e
d
to
r
e
pr
e
s
e
nt
e
a
c
h
wor
king
da
y.
T
he
ve
r
t
ice
s
,
=
{
0
,
1
,
…
,
}
,
s
igni
f
ies
dis
ti
nc
t
ti
me
o
f
0
to
the
da
y's
length,
wh
il
e
=
{
(
,
)
^
:
0
≤
<
≤
,
∈
[
_
^
−
,
_
^
+
]
,
=
+
_
,
∈
}
∪
{
(
,
)
^
0
:
0
≤
<
≤
,
=
+
1
}
r
e
pr
e
s
e
nts
a
r
c
s
.
T
he
s
e
a
r
c
s
(
unit
a
r
c
s
)
r
e
f
e
r
to
pos
s
ibl
e
r
outes
or
wa
it
ing
ti
mes
.
T
he
ve
hicle
's
ti
me
a
t
the
de
pot
ins
ide
a
wor
king
da
y
is
s
hown
by
th
e
wa
it
ing
ti
me
a
r
c
.
I
t
is
e
s
s
e
nti
a
l
to
note
that
in
the
model
unde
r
de
ve
lopm
e
nt,
the
s
tar
t
ti
me
o
f
e
a
c
h
r
oute
∈
is
a
djus
ted
to
the
pr
e
vious
ti
me
ins
tant
∑
∈
to
a
c
c
ount
f
o
r
the
ve
hicle
's
loading
ti
me.
C
ons
ider
ing
the
f
a
c
tor
s
mentioned
a
bove
,
the
mod
e
l
be
ing
de
ve
loped
will
ha
ve
c
ons
tr
a
int
s
that
gr
ow
polynom
ially
with
the
s
ize
of
,
the
nu
mber
of
va
r
iable
s
that
a
ls
o
gr
ow
polynom
ial
with
s
ize
,
a
nd
pos
s
ibl
e
r
outes
will
be
bounde
d
by
a
c
ons
tant
de
t
e
r
mi
ne
d
by
the
pa
r
a
mete
r
.
As
a
r
e
s
ult
,
a
c
oll
e
c
ti
o
n
of
ps
e
udo
-
polynom
ial
va
r
iable
s
a
nd
c
ons
tr
a
int
s
will
be
pr
e
s
e
nt
in
the
f
inal
model
.
T
he
va
r
iable
s
will
s
e
r
ve
a
s
a
r
e
pr
e
s
e
ntation
of
f
low
in
a
r
c
(
,
)
,
indi
c
a
ti
ng
the
number
of
ve
hicle
s
tr
a
ve
li
ng
a
long
r
oute
,
de
pa
r
ti
ng
f
r
o
m
s
tations
a
t
ti
me
ins
tant
,
a
r
r
ies
a
t
ti
me
withi
n
wor
kda
y.
T
he
va
r
iable
will
r
e
pr
e
s
e
nt
the
gr
a
ph
tot
a
l
f
low
a
nd
int
e
r
pr
e
ted
a
s
the
node
r
e
tur
n
f
low
ba
c
k
to
node
0.
Additi
ona
ll
y,
will
be
uti
li
z
e
d
to
de
note
the
c
os
t
a
s
s
oc
iate
d
with
r
oute
,
c
a
lcula
ted
a
s
tot
a
l
dis
tanc
e
tr
a
ve
ll
e
d
a
long
that
r
oute.
T
he
models
a
r
e
de
f
ined
a
s
(
1)
:
M
ini
mi
z
e
,
∑
(
−
∑
∈
)
(
,
)
∈
Ψ
(
1)
with
c
ons
tr
a
int
s
∑
(
,
)
∈
Ψ
|
∈
≤
1
,
∀
∈
(
2)
−
∑
(
,
)
∈
Ψ
+
∑
(
,
)
∈
Ψ
=
{
=
0
0
if
=
1
,
…
,
−
1
−
if
=
(
3)
≤
(
4)
>
0
a
nd
int
e
ge
r
s
,
∀
(
,
)
∈
Ψ
(
5)
≥
0
a
nd
int
e
ge
r
s
(
6)
In
(
1)
outl
ines
the
model's
objec
ti
ve
,
whic
h
is
to
r
e
duc
e
the
c
ove
r
e
d
dis
tanc
e
of
tot
a
l
ve
hicle
s
withi
n
a
s
ingl
e
wor
king
da
y
.
C
ons
tr
a
int
(
4)
is
in
plac
e
to
a
c
kn
owle
dge
that
vis
it
ing
a
ll
c
us
tom
e
r
s
may
not
be
f
e
a
s
ibl
e
be
c
a
us
e
of
the
a
va
il
a
ble
ve
hicle
’
s
li
mi
tation,
e
xp
r
e
s
s
e
d
withi
n
the
inequa
li
ty
c
ons
tr
a
int
s
(
2)
.
Ne
ve
r
thele
s
s
,
incr
e
a
s
ing
the
number
of
c
us
tom
e
r
s
is
the
objec
ti
v
e
,
whic
h
is
a
f
a
vor
a
ble
outcome
.
C
ons
tr
a
int
(
3
)
r
e
pr
e
s
e
nts
a
f
unda
menta
l
f
low
c
ons
e
r
va
ti
on
c
ons
tr
a
int
withi
n
t
he
ne
twor
k,
e
ns
ur
ing
f
low
e
nter
ing
a
node
is
a
t
e
q
uil
ibr
ium
with
the
f
low
e
xit
ing
that
node
.
T
he
s
e
c
ons
tr
a
int
s
c
oll
e
c
ti
ve
ly
de
f
ine
the
opti
mi
z
a
ti
on
p
r
oblem
a
nd
g
uide
the
de
c
is
ion
-
making
pr
oc
e
s
s
f
or
e
f
f
icie
nt
ve
hicle
r
ou
ti
ng.
2.
3.
T
im
e
wi
n
d
ow
r
e
laxat
io
n
s
c
h
e
m
e
d
e
ve
lop
m
e
n
t
In
(
1
)
-
(
6)
model
node
s
r
e
pr
e
s
e
nt
ins
tants
of
ti
me.
Dis
tanc
e
a
nd
ti
me
a
r
e
us
ua
ll
y
not
in
int
e
ge
r
f
or
m
,
a
nd
thus
ther
e
a
r
e
two
a
lt
e
r
na
ti
ve
s
:
us
ing
a
s
mo
oth
dis
c
r
e
ti
z
a
ti
on
(
e
a
c
h
unit
of
ti
me
will
be
0.
0
1)
,
us
ing
r
ounding
of
f
pr
oc
e
dur
e
to
u
ti
li
z
e
ti
me
unit
s
.
T
he
f
ir
s
t
opti
on
would
pr
ovide
a
ne
two
r
k
f
low
mod
e
l
with
a
lar
ge
number
o
f
r
e
s
tr
ictions
a
nd
va
r
iable
s
,
making
it
i
mpr
a
c
ti
c
a
l
f
o
r
a
n
im
media
te
s
olut
ion.
T
he
r
e
f
or
e
,
in
thi
s
s
tudy,
the
s
e
c
ond
a
lt
e
r
na
ti
ve
is
us
e
d.
S
ince
the
s
olut
ion
a
ppr
oa
c
h
is
int
e
nde
d
to
obtain
a
n
e
xa
c
t
s
olut
ion,
a
lgor
it
hms
a
r
e
de
ve
loped
that
it
e
r
a
ti
ve
ly
r
e
f
ine
the
dis
c
r
e
ti
z
a
ti
on.
2.
3.
1.
I
n
it
ial
r
ou
n
d
i
n
g
s
t
r
at
e
gy
Notice
that
the
a
r
c
(
,
)
in
model
(
1)
-
(
6)
c
or
r
e
s
ponds
to
a
r
oute
that
s
tar
ts
a
t
ti
me
a
nd
e
nds
a
t
ti
me
.
Know
ing
that
ve
r
tex
o
f
gr
a
ph
Π
is
de
mar
c
a
te
d
the
s
e
t
o
f
va
lues
∆
=
{
0
,
…
,
}
,
it
is
e
s
s
e
nti
a
l
f
or
a
r
c
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2088
-
8708
I
nt
J
E
lec
&
C
omp
E
ng
,
Vol
.
15
,
No.
1
,
F
e
br
ua
r
y
20
25
:
592
-
603
596
(
,
)
∈
Ψ
,
r
ounding
a
nd
to
va
lues
that
a
r
e
in
the
s
e
t
∆
.
A
s
pr
e
vious
ly
mentioned,
f
ir
s
t
c
ons
ider
the
unit
s
of
ti
me
e
qua
l
to
1,
na
mely
∆
=
{
0
,
1
,
2
,
3
,
…
,
−
1
,
}
.
S
ome
pos
s
ibl
e
r
ounding
pr
oc
e
dur
e
s
a
r
e
a
s
:
−
=
⌊
⌋
a
nd
=
⌈
⌉
.
I
n
thi
s
model,
r
outes
a
r
e
c
ons
tr
uc
ted
to
s
tar
t
s
li
ghtl
y
be
f
or
e
a
nd
e
nd
s
li
ghtl
y
a
f
ter
their
a
c
tual
oc
c
ur
r
e
nc
e
.
T
his
im
pli
e
s
that
s
ome
potential
s
olut
ions
may
not
be
c
ons
ider
e
d,
but
a
ny
f
e
a
s
ibl
e
s
olut
ion
obtaine
d
by
r
e
laxing
the
c
ons
tr
a
int
s
(
1
)
-
(
6)
in
thi
s
manne
r
will
a
ls
o
be
viable
f
or
or
i
ginal
pr
oblem.
−
=
⌈
⌉
a
nd
=
⌊
⌋
.
I
n
the
c
ur
r
e
nt
s
c
e
na
r
io,
r
e
laxa
ti
on
is
f
or
the
model,
whic
h
mea
ns
the
s
olut
ions
ini
ti
a
te
may
not
a
lwa
ys
be
r
e
a
s
ona
ble.
T
hough
,
thi
s
r
e
laxa
ti
on
s
e
r
ve
s
the
pur
pos
e
o
f
e
s
tablis
hi
ng
a
n
e
f
f
e
c
ti
ve
lowe
r
bound
f
o
r
the
p
r
oblem.
−
=
⌈
⌉
a
nd
=
⌈
⌉
or
=
⌊
⌋
a
nd
=
⌊
⌋
.
I
n
the
c
ur
r
e
nt
c
a
s
e
,
a
lowe
r
bound
will
a
ls
o
be
f
ound
.
I
n
th
is
s
tudy,
the
p
r
opos
e
d
a
lgor
it
hm
us
e
s
the
s
e
c
ond
r
ounding
p
r
oc
e
dur
e
,
na
mely
by
c
ons
ider
ing
=
⌈
⌉
a
nd
=
⌊
⌋
.
T
he
r
e
a
s
ons
f
or
c
hoos
ing
thi
s
pr
oc
e
dur
e
we
r
e
:
i
)
r
e
laxa
ti
on
is
e
xpe
c
ted
to
be
ti
ght
a
nd
ii
)
the
inappr
opr
iate
ne
s
s
is
loca
l
a
nd
c
a
n
be
c
or
r
e
c
t
e
d.
C
ons
ider
ing
pa
ths
in
the
f
low
model
that
invol
ve
t
wo
s
uc
c
e
s
s
iv
e
r
outes
,
de
s
ignate
d
a
s
a
nd
′
,
with
a
ga
p
les
s
unit
s
of
ti
me
wa
it
ing
a
mong
them
,
r
e
s
ult
s
in
inf
e
a
s
ibl
e
s
olut
ions
,
it
is
c
r
uc
ial
to
h
ighl
ight
.
T
o
a
ddr
e
s
s
thi
s
is
s
ue
,
one
ini
ti
a
l
a
ppr
oa
c
h
is
r
e
c
ti
f
y
ing
s
olut
ion
e
it
he
r
movi
ng
r
oute
r
e
a
r
wa
r
d
or
r
oute
′
onwa
r
d
to
e
li
mi
na
te
c
onf
li
c
t
.
I
f
thi
s
a
djus
tm
e
nt
r
e
s
ult
s
in
a
f
e
a
s
ibl
e
s
olut
ion,
it
not
only
r
e
s
olves
the
inf
e
a
s
ibi
li
ty
but
a
ls
o
pr
ove
s
opti
malit
y,
a
s
f
e
a
s
ibl
e
s
olut
ions
s
ha
r
e
s
a
me
r
outes
a
nd
ha
ve
lowe
r
bound
e
quivale
nt
c
os
t.
I
n
e
s
s
e
nc
e
,
a
f
ter
obtaining
the
∗
r
e
s
ult
,
e
f
f
or
ts
a
r
e
made
to
c
ons
tr
uc
t
on
s
a
me
r
outs
f
a
vor
a
ble
s
olut
ion
identi
f
ied
in
the
∗
s
olut
ion.
T
he
pr
opos
e
d
a
lgor
it
hms
ope
r
a
te
in
the
f
oll
owin
g
manne
r
:
F
o
r
e
a
c
h
wor
king
da
y,
it
a
tt
e
mpt
s
to
c
r
e
a
te
a
ne
w
r
oute
while
pr
e
s
e
r
ving
the
e
xis
ti
ng
r
outes
s
e
que
nc
e
in
the
s
olut
ion
.
S
uppos
e
(
1
,
…
,
)
is
the
li
ne
of
r
ou
tes
in
we
e
kda
ys
,
a
nd
is
the
s
tar
ti
ng
r
oute
a
nd
′
is
the
e
nding
r
oute
,
∀
∈
{
1
,
…
,
}
.
S
e
t
=
m
a
x
(
−
,
−
1
′
)
.
I
f
≤
+
,
∀
∈
1
,
…
,
,
is
v
iable
s
olut
ion.
other
wis
e
,
then
f
e
a
s
ibi
li
t
y
is
not
pr
ove
n
,
a
nd
a
ddit
ional
a
lgor
it
hm
mus
t
be
c
a
s
t
-
of
f
.
2.
3.
2.
P
e
r
f
e
c
t
i
n
g
i
t
e
r
at
ive
d
is
c
r
e
t
izat
ion
T
he
s
olut
ion
a
ppr
oa
c
h
us
e
d
invol
ve
s
a
n
it
e
r
a
t
ive
c
or
r
e
c
ti
on
of
inf
e
a
s
ibi
li
ti
e
s
r
e
s
ult
ing
f
r
om
dis
c
r
e
ti
z
a
ti
on
pr
oblems
.
F
or
a
lgor
it
hm
e
a
c
h
s
tep,
ins
tanc
e
s
of
inf
e
a
s
ibi
li
ty
a
r
e
identif
ied
.
F
o
r
e
a
c
h
of
thes
e
ins
tanc
e
s
,
the
dis
c
r
e
ti
z
a
ti
on
is
loca
ll
y
a
djus
ted
by
a
ddit
ion
f
r
a
c
ti
ona
l
va
lues
r
e
quir
e
d
to
c
ompl
e
te
r
e
laxa
ti
on
ini
ti
a
ll
y
im
pos
e
d
by
or
igi
na
l
dis
c
r
e
ti
z
a
ti
on.
S
e
ve
r
a
l
ti
me
ins
tants
c
a
n
be
c
ombi
ne
d
int
o
a
s
ingl
e
int
e
g
e
r
dur
ing
the
f
ir
s
t
r
e
laxa
ti
on.
T
he
e
xis
ti
ng
g
r
a
ph
would
be
divi
de
d
int
o
s
e
ve
r
a
l
node
s
us
ing
thi
s
r
e
f
ineme
nt
method.
c
onf
li
c
ti
ng
a
r
c
s
s
e
t
(
,
)
a
nd
0
,
f
r
a
c
ti
ona
l
va
lues
f
or
a
nd
0
a
r
e
c
ons
ider
e
d
to
e
nha
nc
e
the
s
olut
ion
.
3.
RE
S
UL
T
S
AN
D
DI
S
CU
S
S
I
ON
3.
1.
P
r
ob
lem
d
e
s
c
r
ip
t
ion
3.
1.
1.
P
r
ob
lem
d
e
f
in
it
io
n
an
d
n
o
t
at
ion
I
n
M
VR
P
T
W
,
the
r
e
is
a
s
ingl
e
de
pot
r
e
pr
e
s
e
nted
a
s
a
nd
it
s
e
r
ve
s
a
s
both
s
tar
t
a
nd
e
nd
poin
t
f
or
a
ll
ve
hicle
r
outes
.
T
he
f
lee
t
c
ons
is
ts
of
homogene
ous
ve
hicle
s
,
mea
ning
that
a
ll
ve
hicle
s
a
r
e
identica
l
in
ter
ms
of
c
a
pa
c
it
y
a
nd
other
a
tt
r
ibut
e
s
.
E
a
c
h
ve
hicle
in
the
f
lee
t
ha
s
a
c
a
pa
c
it
y
unit
s
.
f
ur
ther
e
xpe
c
ted
a
r
e
a
tot
a
l
of
ve
hicle
s
a
c
c
e
s
s
ibl
e
in
thi
s
f
lee
t
f
o
r
c
a
r
r
ying
out
r
ou
ti
ng
tas
ks
.
I
n
M
VR
P
T
W
:
T
he
s
e
t
of
c
us
tom
e
r
s
is
de
noted
a
s
=
{
1
,
…
,
}
.
E
a
c
h
pa
ir
of
loca
ti
ons
,
including
c
us
tom
e
r
s
a
nd
the
de
pot,
ha
s
a
n
a
s
s
oc
iate
d
dis
tanc
e
a
nd
tr
a
ve
l
ti
me
.
E
a
c
h
c
li
e
nt
ha
s
a
s
pe
c
if
ic
r
e
que
s
t
or
r
e
que
s
t
,
a
s
e
r
vice
ti
me
,
a
r
e
ve
nue
,
a
nd
a
ti
m
e
window
[
,
]
.
T
he
ti
me
window
s
pe
c
if
ies
the
e
a
r
li
e
s
t
ti
me
a
nd
the
late
s
t
ti
me
a
t
whic
h
p
r
ovis
ion
c
a
n
be
gin
a
t
c
li
e
nt
.
T
he
windows
mus
t
be
op
e
n
f
or
ve
hicle
a
r
r
ives
e
a
r
li
e
r
than
.
I
t
is
a
s
s
umed
that,
by
de
f
a
ult
,
the
ve
hicle
s
tar
ts
s
e
r
ving
a
c
us
tom
e
r
a
s
s
oon
a
s
it
a
r
r
ives
.
T
he
=
0
,
indi
c
a
ti
ng
that
the
r
e
is
no
s
e
r
vice
ti
me
r
e
quir
e
d
a
t
the
de
pot.
T
he
t
im
e
window
s
igni
f
i
e
s
the
tot
a
l
ti
me
of
a
wor
kda
y,
whic
h
s
e
ts
the
ti
me
c
ons
tr
a
int
s
f
or
the
e
nti
r
e
r
outi
ng
pr
oblem.
I
t
is
a
s
s
umed
that
+
+
≤
,
∀
∈
.
T
hr
oughout
the
wor
king
da
y
,
e
a
c
h
ve
hicle
c
a
n
go
on
a
number
of
r
outes
.
Until
the
e
nd
of
the
wor
kda
y,
thi
s
e
ntails
be
ing
a
ble
to
c
ompl
e
te
one
r
oute,
r
e
load
a
t
the
de
pot,
a
nd
he
a
d
out
f
or
the
s
ubs
e
que
nt
r
oute.
R
oute
is
de
f
ined
by
the
or
de
r
of
vis
it
s
to
a
s
ubs
e
t
of
c
us
tom
e
r
s
⊆
.
I
t
is
pr
a
c
ti
c
a
l
if
the
to
tal
number
of
r
e
que
s
ts
f
r
om
e
ve
r
y
c
li
e
nt
in
doe
s
not
e
xc
e
e
d
the
ve
hicle
's
c
a
pa
c
it
y
a
nd
if
the
or
de
r
of
the
vis
it
s
a
ll
ows
f
or
the
vis
it
a
ti
on
of
e
ve
r
y
c
us
tom
e
r
withi
n
a
pr
e
de
ter
mi
ne
d
window
of
ti
me.
I
n
thi
s
model
i
t
is
a
ls
o
c
ons
ider
e
d
that
the
s
e
r
vice
of
a
ll
c
us
tom
e
r
s
on
the
r
oute
c
a
nnot
be
s
tar
ted
longer
than
the
maximum
ti
me
unit
a
f
ter
the
r
oute
is
s
tar
ted.
T
he
c
oll
e
c
ti
on
of
a
ll
pos
s
ibl
e
r
outes
is
de
noted
by
.
T
he
r
e
is
a
ls
o
a
s
e
tup
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
E
lec
&
C
omp
E
ng
I
S
S
N:
2088
-
8708
Dis
c
r
e
te
opti
miz
ati
on
mode
l
for
multi
-
pr
oduc
t
mul
ti
-
s
uppli
e
r
v
e
hicle
…
(
M
uli
aw
an
F
ir
daus
)
597
ti
me
to
take
int
o
a
c
c
ount
f
or
e
a
c
h
r
ou
te.
P
r
io
r
to
de
pa
r
ti
ng
the
de
pot
to
t
r
a
ve
l
r
oute
,
the
ve
hicle
∑
∈
ti
me
unit
s
to
load,
with
∈
ℝ
+
.
Due
to
the
li
m
it
e
d
numb
e
r
of
a
va
il
a
ble
ve
hicle
s
,
it
mi
ght
not
be
pos
s
ibl
e
t
o
vis
it
e
ve
r
y
c
li
e
nt.
Ne
ve
r
thele
s
s
,
it
is
us
ua
ll
y
p
r
e
f
e
r
a
ble
to
vis
it
a
s
many
c
li
e
nts
a
s
pos
s
ibl
e
.
3.
1.
2.
M
at
h
e
m
at
ical
f
or
m
u
la
t
ion
f
or
M
VR
P
T
W
T
he
de
s
c
r
ipt
ion
e
xpr
e
s
s
e
d
in
a
gr
a
ph
=
(
,
)
,
with
=
∪
{
}
a
s
e
t
of
ve
r
ti
c
e
s
a
nd
=
{
(
,
)
∶
,
∈
}
a
s
e
t
of
a
r
c
s
.
I
n
thi
s
de
s
c
r
ipt
ion,
ther
e
is
a
binar
y
va
r
iable
r
e
pr
e
s
e
nti
ng
the
s
ubs
c
r
iber
to
the
r
oute
a
nd
de
ter
mi
ne
s
the
s
e
que
nti
a
l
pa
ir
of
r
ou
tes
.
T
he
bina
r
y
va
r
iable
s
a
nd
r
e
s
pe
c
ti
ve
ly
de
f
ine,
if
a
r
c
(
,
)
a
nd
c
li
e
nt
a
s
s
oc
iate
d
to
r
oute
,
while
binar
y
v
a
r
iable
de
ter
mi
ne
s
if
a
ny
ve
hicle
t
r
a
ve
li
ng
r
oute
is
f
oll
owe
d
by
r
oute
s
with
in
we
e
kda
ys
.
T
he
no
tation
<
s
igni
f
ies
identica
l
ve
hicle
is
a
ll
oc
a
ted
to
d
o
r
oute
a
f
ter
wa
r
d
doing
r
ou
te
.
T
he
va
r
iable
r
e
pr
e
s
e
nts
the
s
tar
t
ti
me
on
c
li
e
nt
,
if
a
s
s
oc
iate
d
with
pa
t
h
,
a
nd
a
nd
′
c
ha
r
a
c
ter
ize
the
s
tar
t
a
nd
e
nd
du
r
a
ti
on
of
r
oute
.
S
uppos
e
is
a
lar
ge
e
nough
number
.
T
h
e
c
onc
is
e
f
or
mul
a
ti
on
f
o
r
M
VR
P
T
W
is
s
tate
d
a
s
(
7
)
-
(
26)
.
M
ini
mi
z
e
,
∑
∑
(
,
)
∈
∈
−
∑
∑
∈
∈
(
7)
W
it
h
c
ons
tr
a
int
s
∑
∈
=
,
∀
∈
,
∀
∈
(
8)
∑
∈
≤
1
,
∀
∈
(
9)
∑
ℎ
∈
−
∑
ℎ
∈
=
0
,
∀
ℎ
∈
,
∀
∈
(
10)
∑
∈
=
1
,
∀
∈
(
11)
∑
∈
=
1
,
∀
∈
(
12)
∑
=
1
,
∈
,
≠
0
,
≠
∈
(
13)
∑
=
1
,
∈
,
≠
0
,
≠
∈
(
14)
∑
∈
≤
,
∀
∈
(1
5
)
≤
∑
,
∈
∈
(1
6
)
+
+
−
(
1
−
)
≤
,
∀
(
,
)
∈
,
≠
,
∀
∈
(1
7
)
≤
≤
,
∀
∈
,
∀
∈
(1
8
)
≥
∑
∈
,
∀
∈
(
19
)
≤
+
,
∀
∈
,
∀
∈
(2
0
)
+
(
1
−
)
≥
′
+
∑
∈
,
∀
,
∈
,
<
(2
1
)
∑
∑
∈
|
<
∈
≥
|
|
−
(2
2
)
∈
{
0
,
1
}
,
∀
(
,
)
∈
,
∀
∈
(2
3
)
∈
{
0
,
1
}
,
∀
∈
,
∀
∈
(2
4
)
∈
{
0
,
1
}
,
∀
,
∈
,
<
(
25)
∈
{
0
,
1
}
,
∀
,
∈
,
<
(
26)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2088
-
8708
I
nt
J
E
lec
&
C
omp
E
ng
,
Vol
.
15
,
No.
1
,
F
e
br
ua
r
y
20
25
:
592
-
603
598
I
t
is
a
lwa
ys
pr
e
f
e
r
a
ble
to
vis
it
a
s
many
c
ons
umer
s
a
s
pos
s
ibl
e
,
a
c
c
or
ding
to
objec
ti
ve
f
unc
ti
o
n
(
7
)
.
B
e
mi
ndf
ul
that
the
c
ons
tant
mus
t
be
s
e
t
to
a
va
lue
that
s
uppor
ts
the
model
in
o
r
de
r
f
or
i
t
to
be
c
on
s
ider
e
d
va
li
d.
T
he
c
ons
tr
a
int
s
(
13
)
a
nd
(
19
)
de
ter
m
ine
t
he
f
lee
t
s
ize
a
nd
the
ve
hicle
c
a
pa
c
it
y,
r
e
s
pe
c
ti
v
e
ly.
T
he
r
e
s
tr
ictions
(
10
)
–
(
12)
a
r
e
f
low
c
ons
e
r
va
ti
on
li
mi
tat
ions
.
Vis
it
s
to
c
li
e
nts
mus
t
a
dhe
r
e
to
their
ti
me
wi
ndow,
a
s
indi
c
a
ted
in
(
15)
.
E
ve
r
y
two
c
li
e
nts
who
make
c
ons
e
c
uti
ve
tr
ips
on
the
s
a
me
r
oute
mus
t
ha
ve
a
matc
h
ing
vis
it
ti
me
(
14)
,
a
nd
the
s
a
me
is
tr
ue
f
o
r
tr
ips
take
n
by
the
s
a
me
ve
hicle
on
two
s
e
pa
r
a
te
oc
c
a
s
ions
(
18)
.
F
inally,
e
a
c
h
r
oute's
s
e
tup
ti
me
mus
t
a
lwa
ys
be
take
n
int
o
a
c
c
ount
(
16)
,
(
18
)
.
3.
2.
M
e
t
h
od
s
f
or
op
t
im
izat
io
n
b
as
e
d
-
on
ac
t
ive
c
on
s
t
r
ain
t
s
T
his
s
tudy
looked
a
t
a
s
e
t
of
tec
hniques
whe
r
e
the
s
e
a
r
c
h
dir
e
c
ti
on
of
the
a
c
ti
ve
c
ons
tr
a
int
c
oa
t
is
s
e
t
to
f
a
ll
be
twe
e
n
a
n
or
thogonal
matr
ix
a
nd
a
c
onve
nti
ona
l
c
ons
tr
a
int
matr
ix.
As
a
r
e
s
ult
,
i
f
̂
=
̂
is
the
late
s
t
a
c
ti
ve
c
ons
tr
a
int
s
−
,
is
a
×
matr
ix
that
looks
li
k
e
thi
s
:
̂
=
0
(
27)
T
he
ke
y
tas
ks
that
ne
e
d
to
be
a
c
c
ompl
is
he
d
in
e
a
c
h
it
e
r
a
ti
on
(
by
ge
ne
r
a
ti
ng
a
n
a
ppr
opr
iate
de
s
c
e
nt
d
ir
e
c
ti
on,
)
a
r
e
a
s
:
a.
C
a
lcula
te
the
r
e
duc
e
d
gr
a
dient
=
.
b.
De
ve
lop
a
ppr
oxim
a
ti
ons
f
or
the
r
e
duc
ti
on
of
He
s
s
ian
=
.
c.
Ac
quir
e
a
ppr
oxim
a
ti
ons
f
o
r
s
ys
tems
of
e
qua
ti
ons
:
=
−
=
−
(
28)
d.
I
de
nti
f
y
the
dir
e
c
ti
on
to
ge
t
=
.
e.
F
ind
the
c
los
e
s
t
a
ppr
oxim
a
ti
on
to
∗
us
ing
a
li
ne
s
e
a
r
c
h
whe
r
e
:
(
+
∗
)
=
m
in
{
+
f
e
a
s
i
b
le
}
(
+
)
Along
with
ha
ving
f
ull
c
olum
n
r
a
nks
,
is
only
(
a
lgebr
a
ica
ll
y)
c
ons
tr
a
ined
by
(
27)
in
the
e
xa
mpl
e
a
bove
,
ther
e
f
or
e
c
a
n
take
on
a
f
e
w
f
or
ms
.
I
n
s
pe
c
if
ica
l
ly,
the
pa
r
a
ll
e
l
to
the
method
it
s
e
lf
ha
s
the
f
oll
o
wing
f
or
m:
=
[
−
0
]
=
[
−
−
1
0
]
}
}
}
−
−
(
29)
T
his
is
a
s
tr
a
ight
f
or
wa
r
d
e
xplana
ti
on
that
will
be
uti
li
z
e
d
f
or
e
xpos
it
ion
in
the
f
oll
owing
s
e
c
ti
on,
ho
we
ve
r
it
s
hould
be
noted
that
it
is
c
omput
a
ti
ona
ll
y
li
mi
ted
to
the
f
a
c
tor
iza
ti
ons
of
that
a
r
e
tr
iangula
r
(
L
U)
a
nd
.
T
he
r
e
is
undoubtedly
s
ome
incomplete
ne
s
s
in
the
matr
ix
c
a
lcula
ti
on.
T
he
r
e
is
a
good
r
e
a
s
on
why
,
whos
e
c
olum
n
is
or
thono
r
mal
(
=
)
,
is
s
ugge
s
ted.
T
he
tr
a
ns
f
or
mation
i
s
ke
y
be
ne
f
it
is
that
it
doe
s
n't
int
r
o
duc
e
r
e
dunda
nt
c
ondit
ions
int
o
the
p
r
oblem
r
e
duc
t
ion
(
s
e
e
the
a
f
or
e
mentioned
s
teps
a
to
d
,
in
pa
r
ti
c
ular
(
28
)
)
.
I
n
pr
og
r
a
ms
whe
r
e
is
a
c
c
umul
a
ted
a
s
a
de
ns
e
matr
ix,
thi
s
tec
hnique
ha
s
be
e
n
a
ppli
e
d
.
T
he
matr
ix
[
]
c
a
n
be
e
xpa
nde
d
to
the
e
xpa
ns
ively
dis
tr
ibut
e
d/s
pa
r
s
e
li
ne
a
r
c
ons
tr
a
int
s
us
ing
the
L
DV
f
a
c
tor
iza
ti
on:
[
]
=
[
]
whe
r
e
is
a
tr
iangle
,
is
a
diagona
l,
a
nd
1
2
⁄
is
nor
mal,
a
nd
a
nd
a
r
e
a
c
c
umul
a
ted
a
s
pr
oduc
ts
.
De
s
pit
e
thi
s
,
thi
s
f
a
c
tor
iza
ti
on
will
a
lwa
ys
be
s
ig
nif
ica
ntl
y
de
ns
e
r
than
B
's
L
U
f
a
c
to
r
iza
ti
on
i
f
S
ha
s
a
lar
ge
number
of
c
olum
ns
.
As
a
r
e
s
ult
,
it
is
de
pe
nde
nt
on
how
is
us
e
d
in
(
29)
.
B
e
a
dvis
e
d
that
mus
t
be
tr
e
a
ted
with
the
utm
os
t
c
a
r
e
be
c
a
us
e
to
−
1
unwa
nted
a
ppe
a
r
a
nc
e
.
3.
3.
S
u
m
m
ar
y
o
f
t
h
e
p
r
oc
e
d
u
r
e
B
uil
ding
upon
the
pr
e
vious
dis
c
us
s
ions
r
e
ga
r
ding
the
opti
mi
z
a
ti
on
c
ha
ll
e
nge
s
a
s
s
oc
iate
d
with
the
ve
hicle
r
outi
ng
p
r
oblem
f
or
mul
ti
-
pr
oduc
t
a
nd
mul
ti
-
s
uppli
e
r
(
VR
P
-
M
P
M
S
)
,
we
int
r
oduc
e
a
n
e
f
f
e
c
ti
ve
a
lgor
it
hm
de
s
igned
to
a
ddr
e
s
s
thes
e
c
ompl
e
xi
ti
e
s
.
T
his
a
lgor
it
hm
int
e
gr
a
tes
a
dva
nc
e
d
mathe
matica
l
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
E
lec
&
C
omp
E
ng
I
S
S
N:
2088
-
8708
Dis
c
r
e
te
opti
miz
ati
on
mode
l
for
multi
-
pr
oduc
t
mul
ti
-
s
uppli
e
r
v
e
hicle
…
(
M
uli
aw
an
F
ir
daus
)
599
modelli
ng
tec
hniques
with
c
omput
a
ti
ona
l
metho
ds
to
opti
mi
z
e
ve
hicle
r
outi
ng
while
a
dhe
r
ing
t
o
c
r
it
ica
l
c
ons
tr
a
int
s
s
uc
h
a
s
c
a
pa
c
it
y
li
mi
ts
a
nd
ti
me
wi
ndows
.
B
y
e
s
tablis
hing
a
c
lea
r
f
r
a
mew
or
k
that
de
f
ines
e
s
s
e
nti
a
l
c
omponents
,
including
de
c
is
ion
va
r
ia
bles
,
objec
ti
ve
f
unc
ti
ons
,
a
nd
c
ons
tr
a
int
,
thi
s
a
ppr
oa
c
h
f
a
c
il
it
a
tes
a
s
ys
tema
ti
c
s
olut
ion
pr
oc
e
s
s
.
W
it
h
the
a
s
s
umpt
ion
of
:
a.
[
]
=
,
≤
≤
is
c
ontent
.
b.
T
he
f
unc
ti
on
(
)
a
nd
ve
c
tor
(
)
=
[
]
.
c.
T
he
number
of
s
upe
r
ba
s
is
va
r
iable
s
,
(
0
≤
≤
−
)
.
d.
F
a
c
tor
iza
ti
on,
L
U,
on
the
ba
s
e
mat
r
ix
×
.
e.
T
he
qua
s
i
-
Ne
wton
method
to
the
×
matr
ix
is
.
f.
T
he
ve
c
tor
g
r
a
dient
ℎ
=
−
.
g.
A
ve
c
tor
mee
ts
=
.
h.
T
he
pos
it
ive
c
onve
r
ge
nc
e
tol
e
r
a
nc
e
s
f
or
T
OL
DJ
a
nd
T
OL
R
G
a
r
e
both
modes
t.
T
he
model
is
s
olved
via
the
ge
ne
r
a
li
z
e
d
r
e
duc
e
d
gr
a
dient
method,
s
tar
ti
ng
with
the
L
a
gr
a
nge
f
unc
ti
on
a
nd
p
r
oc
e
e
ding
a
c
c
or
ding
to
the
pr
oc
e
dur
e
.
Af
ter
that,
the
a
lgo
r
it
hm
wil
l
wor
k
a
s
f
oll
ows
:
S
tep
1.
If
‖
ℎ
‖
>
T
OL
R
G
,
s
tep
3
.
S
tep
2.
(
"
P
R
I
C
E
"
,
i.
e
.
,
a
dd
one
s
upe
r
ba
s
e
a
nd
L
a
gr
a
nge
m
ult
ipl
ier
c
a
lcula
ti
on)
.
a.
Gove
r
n
=
−
.
b.
C
hoos
e
1
<
−
T
OL
DJ
(
2
>
+
T
OL
DJ
)
,
the
's
is
the
gr
e
a
tes
t
e
leme
nt
whos
e
higher
bounds
c
or
r
e
s
pond
to
the
va
r
iable
s
.
I
f
not,
S
T
OP;
the
e
s
s
e
nti
a
l
c
ondit
ions
f
or
a
n
idea
l
s
olut
ion
ha
ve
be
e
n
s
a
ti
s
f
ied
a
c
c
or
ding
to
Kuhn
-
T
uc
ke
r
.
I
f
thi
s
is
not
t
he
c
a
s
e
;
−
Additi
on
a
s
the
ne
w
c
olum
n
.
−
C
hoice
=
1
or
2
on
the
ba
s
is
of
|
1
|
=
max
(
|
1
|
,
|
2
|
)
.
−
Add
a
ne
w,
pe
r
ti
ne
nt
c
olum
n
to
R
.
−
I
ns
e
r
t
1
a
s
a
ne
w
ℎ
e
leme
nt.
c.
S
is
mul
ti
pli
e
d
by
1
.
S
tep
3.
(
Dir
e
c
ti
on
of
s
e
a
r
c
h,
=
).
a.
F
ini
s
h
=
−
ℎ
.
b.
F
ini
s
h
L
U
=
−
.
c.
M
a
ke
=
[
0
]
.
S
tep
4.
(
T
e
s
t
R
a
ti
o,
"
C
HU
Z
R
"
)
.
a.
If
m
a
x
=
0
,
go
to
s
tep
7
.
b.
If
m
a
x
≥
0
,
maximi
s
e
va
lue
of
+
is
viable
.
S
tep
5.
(
L
ine
s
e
a
r
c
h)
.
a.
F
ind
,
a
n
∗
f
or
whic
h
(
+
∗
)
=
min
0
<
≤
max
(
+
)
b.
C
onve
r
t
to
+
a
nd
a
nd
to
thei
r
ne
w
va
lues
.
S
tep
6.
(
R
e
duc
e
d
s
lope
c
a
lcula
ti
on,
ℎ
̅
=
).
a.
P
r
oc
e
s
s
=
.
b.
Ne
w
s
lope
de
ter
mi
na
ti
on,
ℎ
̅
=
−
.
c.
Util
izing
,
a
nd
,
a
djus
t
f
or
g
r
a
dient
ℎ
̅
−
ℎ
.
d.
S
e
t
ℎ
̅
−
ℎ
.
e.
If
m
a
x
=
0
,
p
r
oc
e
e
d
to
s
tep
7
.
S
tep
7.
He
r
e
,
<
m
a
x
r
e
a
c
he
s
li
mi
ts
a
nd
f
or
(
0
<
≤
+
)
,
the
c
olum
n
va
r
i
a
ble
of
[
]
a
ls
o
r
e
a
c
he
s
li
mi
ts
.
a.
I
f
li
mi
t
is
r
e
a
c
he
d
by
ba
s
e
va
r
iable
(
0
<
≤
)
,
−
the
-
th
c
olum
n
r
e
plac
e
d
with
-
th
c
olum
n
of
[
T
]
a
nd
[
]
−
=
−
C
ha
nge
s
to
,
,
a
nd
a
ls
o
va
r
iation
in
−
de
ter
mi
ne
gr
a
dient
ℎ
=
−
;
−
Go
to
(
c
)
.
b.
Othe
r
wis
e
s
upe
r
ba
s
e
li
mi
t
is
r
e
a
c
he
d
(
<
≤
+
)
.
De
ter
mi
ne
=
−
.
c.
C
r
e
a
te
the
-
th
va
r
iable
in
nonba
s
is
a
t
the
a
ppr
op
r
ia
te
li
mi
t
a
s
f
oll
ows
:
−
E
li
mi
na
te
th
c
olum
n
[
]
a
nd
[
ℎ
]
;
−
Add
to
the
tr
iangula
r
matr
ix.
S
ubtr
a
c
t
by
one
a
nd
r
e
tur
n
to
s
tep
1
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2088
-
8708
I
nt
J
E
lec
&
C
omp
E
ng
,
Vol
.
15
,
No.
1
,
F
e
br
ua
r
y
20
25
:
592
-
603
600
3.
4.
S
i
m
u
lat
io
n
I
n
th
is
s
e
c
ti
on,
we
pr
e
s
e
nt
the
s
im
ulation
r
e
s
ult
s
obtaine
d
f
r
om
our
pr
opos
e
d
hybr
id
a
ppr
oa
c
h
f
o
r
s
olvi
ng
the
ve
hicle
r
outi
ng
pr
oblem
f
or
mul
ti
-
pr
oduc
t
a
nd
mul
ti
-
s
uppli
e
r
(
VR
P
-
M
P
M
S
)
with
r
e
laxe
d
ti
me
windows
.
T
he
s
im
ulations
we
r
e
c
onduc
ted
us
ing
F
or
tr
a
n
langua
ge
in
mathe
matica
l
pr
og
r
a
mm
ing
s
ys
tem
(
M
P
S
)
f
or
mat
,
whic
h
a
ll
ows
f
o
r
e
f
f
icie
nt
modell
ing
a
nd
s
olvi
ng
of
opti
mi
z
a
ti
on
p
r
oblems
.
T
he
a
im
is
to
e
va
luate
the
e
f
f
e
c
ti
ve
ne
s
s
of
our
model
by
c
ompar
ing
the
outcome
s
with
e
s
tablis
he
d
be
nc
hmar
ks
.
T
h
r
ough
a
s
e
r
ies
of
i
ter
a
ti
ve
tes
ts
,
we
a
s
s
e
s
s
va
r
ious
pe
r
f
or
manc
e
metr
ics
,
including
t
r
a
ns
por
tation
c
os
ts
,
ve
hicle
uti
li
z
a
ti
on,
a
nd
a
dhe
r
e
nc
e
to
c
a
pa
c
it
y
c
ons
tr
a
int
s
.
T
he
r
e
s
ult
s
obtaine
d
f
r
om
thes
e
s
im
ulations
will
pr
ovide
ins
ight
s
int
o
the
pr
a
c
ti
c
a
l
a
ppli
c
a
bil
i
ty
of
our
pr
op
os
e
d
s
olut
ion
in
r
e
a
l
wo
r
ld
s
c
e
na
r
ios
.
E
XI
T
--
OPT
I
M
AL
S
OL
UT
I
ON
F
OU
ND
.
NO
.
OF
I
T
E
R
AT
I
ON
S
238
OB
J
E
C
T
I
VE
VA
L
UE
5
.
1600000000000E
+
02
NO
R
M
OF
X
2
.
395E
+
03
NO
R
M
OF
P
I
4
.
081E
+
01
P
R
OB
L
E
M
NA
M
E
VR
P
OB
J
E
C
T
I
V
E
VA
L
UE
5.
1600000000
E
+
02
S
T
AT
US
OPT
I
M
AL
S
OL
N
I
T
E
R
AT
I
O
N
238
T
a
ble
1
il
lus
tr
a
tes
the
r
oute
f
or
ve
hicle
1,
whic
h
de
pa
r
ts
f
r
om
the
de
pot
a
nd
s
e
que
nti
a
ll
y
vis
it
s
c
li
e
nt
1,
c
li
e
nt
2,
c
li
e
nt
3
,
a
nd
c
li
e
nt
4
,
c
onti
nuin
g
in
thi
s
manne
r
.
S
im
il
a
r
ly
,
f
or
r
oute
1
,
ve
hicle
2
de
pa
r
ts
f
r
om
the
de
pot
a
nd
f
oll
ows
a
pa
th
to
c
li
e
nt
3,
the
n
f
r
om
c
li
e
nt
1
to
c
li
e
nt
2
,
a
nd
s
o
f
or
th.
T
a
ble
2
d
e
tails
the
tr
a
ve
l
r
outes
f
or
ve
hicle
s
us
ing
r
oute
1,
s
tar
ti
ng
f
r
om
the
de
pot
to
c
us
tom
e
r
3.
F
r
om
c
us
tom
e
r
1,
th
e
ve
hicle
r
e
tur
ns
to
the
de
pot
via
r
out
e
2.
Us
ing
r
oute
3
,
th
e
ve
hicle
tr
a
ve
ls
f
r
om
c
us
tom
e
r
4
to
c
us
tom
e
r
7
.
F
inally,
r
oute
4
de
picts
the
ve
hicle
's
jour
ne
y
f
r
om
c
us
tom
e
r
6
to
c
us
tom
e
r
5
.
T
a
ble
3
pr
e
s
e
nts
the
s
tar
ti
ng
t
im
e
s
f
or
e
a
c
h
node
(
c
us
tom
e
r
)
.
T
a
ble
1.
R
e
s
ult
of
bina
r
y
va
r
iable
s
C
us
to
me
r
V
e
hi
c
le
C
us
to
me
r
0
1
2
3
4
5
6
7
8
1
0
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
4
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
5
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
6
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
7
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
8
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
2
0
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
0.00000
1.00000
0.00000
1.00000
0.00000
0.00000
4
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
5
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
6
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
7
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
8
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
3
0
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
1
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
4
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
5
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
6
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
7
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
4
0
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
0.00000
1.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
2
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
4
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
5
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
6
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
7
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
E
lec
&
C
omp
E
ng
I
S
S
N:
2088
-
8708
Dis
c
r
e
te
opti
miz
ati
on
mode
l
for
multi
-
pr
oduc
t
mul
ti
-
s
uppli
e
r
v
e
hicle
…
(
M
uli
aw
an
F
ir
daus
)
601
Our
c
omput
a
ti
ona
l
e
xpe
r
im
e
nts
de
mons
tr
a
te
that
the
pr
opos
e
d
hybr
id
a
ppr
oa
c
h
e
f
f
e
c
ti
ve
ly
r
e
duc
e
s
tr
a
ns
por
tation
c
os
ts
while
s
a
ti
s
f
ying
ve
hicle
c
a
pa
c
it
y
c
ons
tr
a
int
s
a
nd
r
e
laxe
d
ti
me
windows
.
S
pe
c
if
ica
ll
y,
our
r
e
s
ult
s
s
how
that
thi
s
method
outper
f
or
ms
t
r
a
dit
i
ona
l
r
outi
ng
a
ppr
oa
c
he
s
by
a
c
hieving
a
mo
r
e
s
i
gnif
ica
nt
r
e
duc
ti
on
in
ove
r
a
ll
tr
a
ns
por
tation
e
xpe
ns
e
s
.
M
or
e
ove
r
,
the
hyb
r
id
model
not
only
e
ns
ur
e
s
a
dhe
r
e
nc
e
to
ve
hicle
c
a
pa
c
it
y
li
mi
ts
but
a
ls
o
a
ll
ows
f
or
f
lexibil
i
t
y
in
s
c
he
duli
ng,
making
i
t
a
viable
s
olut
ion
f
or
s
ol
ving
the
VR
P
-
M
P
M
S
with
r
e
laxe
d
t
im
e
windows
.
T
he
s
e
f
indi
ngs
unde
r
s
c
or
e
the
a
dva
ntage
s
of
ou
r
a
pp
r
oa
c
h
in
e
nha
nc
ing
logi
s
ti
c
s
e
f
f
icie
nc
y
a
nd
im
pr
oving
s
e
r
vi
c
e
de
li
ve
r
y
in
c
ompl
e
x
s
upply
c
ha
in
e
nvir
onments
.
T
a
ble
2.
R
e
s
ult
of
bina
r
y
va
r
iable
s
R
out
e
V
e
hi
c
le
R
out
e
0
1
2
3
4
5
6
7
8
1
0
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
4
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
5
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
6
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
7
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
8
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
2
0
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
0.00000
0.00000
1.00000
1.00000
0.00000
0.00000
4
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
5
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
6
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
7
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
8
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
3
0
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
2
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
4
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
5
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
6
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
7
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
4
0
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1
0.00000
1.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
2
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
3
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
4
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
0.00000
0.00000
5
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
6
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
0.00000
7
1.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
8
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.00000
0.00000
T
a
ble
3.
S
tar
t
ti
me
f
r
om
e
a
c
h
node
va
r
iable
s
C
us
to
me
r
V
e
hi
c
le
1
2
3
4
5
6
7
8
1
39.74026
69.48052
40.00000
20.64935
20.00000
10.25974
10.00000
19.87013
2
40.00000
40.00000
60.00000
20.00000
10.00000
30.00000
10.00000
20.00000
3
0.00000
0.00000
30.00000
0.00000
0.00000
59.87013
110.00000
100.12987
4
60.25974
10.51948
0.00000
99.35065
110.00000
19.87013
0.00000
0.00000
4.
CONC
L
USI
ON
T
his
pa
pe
r
c
ons
ider
s
a
c
ompany
whic
h
ope
r
a
tes
a
f
lee
t
of
ve
hicle
s
s
o
a
s
to
de
li
ve
r
mul
ti
ple
pr
oduc
ts
f
r
om
va
r
ious
s
uppli
e
r
s
to
a
s
e
t
of
c
us
tom
e
r
s
with
no
s
tr
ict
ti
me
to
be
f
ul
f
il
led
in
de
li
ve
r
ies
.
T
he
objec
t
ive
is
to
opti
mi
z
e
the
r
outi
ng
of
thes
e
ve
hicle
s
to
mi
ni
mi
z
e
the
tot
a
l
t
r
a
ns
por
tation
c
os
t,
whic
h
includ
e
s
tr
a
ve
l
dis
tanc
e
,
ve
hicle
uti
li
z
a
ti
on,
a
nd
de
li
ve
r
y
ti
me
de
viations
,
while
e
ns
ur
ing
that
c
us
tom
e
r
de
mand
is
met
a
nd
r
e
laxe
d
ti
me
windows
a
r
e
r
e
s
pe
c
ted.
T
he
model
o
f
the
pr
oblem
wa
s
f
o
r
mul
a
ted
a
s
a
c
ombi
na
tor
ial
pr
oblem.
A
hybr
idi
z
a
ti
on
a
ppr
oa
c
h
wa
s
pr
opos
e
d
f
or
the
e
xa
c
t
pa
r
t,
a
ge
ne
r
a
li
z
e
d
r
e
duc
e
d
gr
a
dient
met
hod
wa
s
de
ve
loped
in
a
wa
y
to
ge
t
“
ne
a
r
”
int
e
ge
r
f
e
a
s
ibl
e
s
olut
ion.
T
he
n
a
f
e
a
s
ibl
e
ne
ighbor
hood
s
e
a
r
c
h
wa
s
pr
opos
e
d,
ba
s
e
d
on
mi
nim
izing
the
de
ter
io
r
a
ti
on
o
f
the
objec
ti
ve
f
unc
ti
on.
Evaluation Warning : The document was created with Spire.PDF for Python.