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In-Depth Information
T
m
∑∑∑
==∈
(
t
+
t
(
t
))
x
(
t
)
=
a
,
(5)
j
=
2,
L
,
n
ij
ijd
j
t
11
d
i
P
(
j
)
T
m
∑∑∑
==∈
tx
(
t
)
=
a
+
vt
,
i
=
1
,
,
n
1
(6)
L
ijd
i
i
t
11
d
j
S
(
i
)
a
=
1
,
a n
T
(7)
1
a
>
0
,
i
=
1
,
,
n
(8)
L
i
(9)
[ LMG
˄ W
˅
ˈ
H LM
(
ˈ
G "
P
ˈ
W "
7
The objective of the TDTOP is to maximize the total collected score, as shown in
(1). In this formulation, constraint (2) and (3) are flow-conservation constraints. Con-
straint (4) ensures that every location is visited at most once. Constraint (5) and (6)
guarantees that if one edge is visited in a given tour, the arrival time of the edge fol-
lowing node is the sum of the preceding arrival time, visiting time and the edge travel
time. Constraint (7) is the start time and latest finish time constraint. Constraint (8)
and (9) are the variables constraint.
3.2 Optimal Algorithm
Definition 1.
(
i
,
a
,
k
)
is the label of node
v , where i is the subscript of node
v fol-
j
lowed by node
, and k shows what stage it is.
v
j
is the set of nodes belong to the k stage, for
,
(
)
is
Definition 2.
L
v
L
P k
j
j
k
1
the set of predecessor nodes belong to the
k
1
stage,
P
(
j
)
=
L
P
(
j
)
.
k
1
k
1
Definition 3.
U k
(
j
,
t
)
represents the optimal collected score from sourcing node to
the node
v at time t on stage k ,
v
L
.
j
k
Definition 4.
U
( i
)
represents the optimal collected score from sourcing nodes
v to
node
v . So,
U
(
i
)
=
max{
U
(
i
,
a
)
|
k
=
1
2
,
K
}
, where K is the maximum of stages
L
k
i
division.
According to the idea of network planning and dynamic programming, a novel dy-
namic node labeling algorithm is presented in the following.
Step 1 (Initialization). Given the score s and visiting time
vt on node
v ,
i
i
=
2
,
n
1
. Let
s
=
s
=
0
,
vt
=
vt
=
0
,
d
=
1
,
V
=
V
;
L
1
n
1
n
Step 2 (St ag es division). According to the arcs on the route, the time dependent net-
work
is divided into K stages;
v is the sourcing node, let
a
=
1
,
k
=
0
,
G
=
(
V
, E
)
1
U
(
=
0
;
0
 
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