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In-Depth Information
x
VEH
ijf
=0
∀
f
∈V
,i
∈N
(27)
x
VEH
START
(
f
)
,j,f
=1
∀
f
∈V
(28)
j
∈N
x
VEH
j,STOP
=1
∀
f
∈V
(29)
(
f
)
,f
j
∈N
=
j
∈N
x
VEH
jif
x
VEH
ijf
∈V
,i
∈N \N
VDEPOT
∀
f
(30)
j
∈N
Self-referencing loops are forbidden by (27). Each vehicle leaves its home position
(28) and terminates in its designated terminus node (29). Constraint (30) ensures
the vehicle flow preservation at nodes away from the vehicle depot.
t
VEH
if
≥
0
∀
i
∈N
,f
∈V
(31)
t
VEH
if
·
x
VEH
ijf
+
d
(
i, j
)
(32)
≤
t
VEH
jf
−
x
VEH
ijf
+(1
)
·
M
∀
i, j
∈N
,f
∈V
(33)
t
VEH
if
≤
t
VEH
jf
+(1
−
w
kf
)
·
M
(34)
−
z
+
ik
−
z
jk
+(1
)
·
M
+(1
)
·
M
∀
i, j
∈N
,f
∈V
,k
∈C
(35)
We use the arrival time - based prevention of short cycles of the vehicle routes
(31)-(33). Constraint (35) guarantees that vehicle
f
visits the pickup location
i
of container
k
(equiv. to
z
+
ik
= 1) earlier than the associated delivery location
j
of
k
(equiv. to
z
jk
= 1) if vehicle
f
has been selected to move container
k
(equiv.
to
w
kf
=1).
t
VEH
if
=0
∀
i
∈N
,f
∈V
(36)
t
VEH
STOP
(
f
)
f
≤
T
max
∀
f
∈V
(37)
All vehicles depart from their depot at time 0 (36) and return there not later
than
T
max
(37).
l
START
(
f
)
f
=0
∀
f
∈V
(38)
l
STOP
(
f
)
f
=0
∀
f
∈V
(39)
δ
START
(
f
)
f
=0
∀
f
∈V
(40)
δ
STOP
(
f
)
f
=0
∀
f
∈V
(41)
Each vehicle leaves its starting node empty (38) and heads emptied towards its
terminus node (39). No load variation is effective at these two locations (40),(41).
−
z
+
ik
δ
if
+(1
−
w
kf
)
·
M
+(1
)
·
M
≥
C
(
k
)
∀
i
∈N
,k
∈C
,f
∈V
(42)
−
z
+
ik
δ
if
≤
C
(
k
)+(1
)
·
M
∀
f
∈V
,i
∈N
,k
∈C
(43)
−
z
ik
δ
if
+(1
−
w
kf
)
·
M
+(1
)
·
M
≤−
C
(
k
)
∀
i
∈N
,k
∈C
,f
∈V
(44)
−
z
ik
δ
if
≥−
C
(
k
)
−
(1
)
·
M
∀
f
∈V
,i
∈N
,k
∈C
(45)
