Information Technology Reference
In-Depth Information
x
(
o,d,rf
)
i,j
u
r
k
y
i,j
≤
0
∀
(
i, j
)
∈ A
−
(8)
(
o,d,rf
)
∈Θ
s∈S
+
t
∈T
x
t
i,j
x
(
o,d,t
)
i,j
u
d
s
y
i,j
≤
0
∀
(
i, j
)
∈ A
−
(9)
(
o,d,t
)
∈Θ
s∈S
a
(
o,d,t
)
i∈Out
(
o
)
x
(
o,d,t
)
o,i
y
o,i
≤
∀
(
o, d, t
)
∈
Θ
(10)
s∈S
i∈Out
(
o
)
−
k∈Out
(
j
)
x
(
o,d,t
)
i,j
x
(
o,d,t
)
j,k
∀
(
o, d,t
)
∈ Θ,j ∈ V
\
(
o ∪ d
)
=0
(11)
i∈In
(
j
)
x
i,j
−
k∈Out
(
j
)
∀t ∈ T,j ∈ V
\ V
t
∗
x
j,k
=0
(12)
i∈In
(
j
)
The domains of the variables are as previously described. The objective con-
sists of four components. First, objective (1) takes into account the precomputed
sailing costs on arcs between inflexible visitations and the port fees at each visi-
tation. Note that the fixed sailing cost on an arc does not only include fuel costs,
but can also include canal fees or the time-charter bonus for entering an SOS.
Second, the profit from delivering cargo (2) is computed based on the revenue
from delivering cargo minus the cost to load and unload the cargo from the
vessel. Note that the model can choose how much of a demand to deliver, even
choosing to deliver a fractional amount. We can allow this since each demand is
an aggregation of cargo between two ports, meaning at most one container be-
tween two ports will be fractional. Third, equipment profit is taken into account
in (3). Equipment is handled similarly to cargo, except that equipment can flow
from any port where it is in supply to any port where it is in demand.
Multiple vessels are prevented from visiting the same visitation in constraints
(4). The flow of each vessel from its source node to the graph sink is handled by
constraints (5), (6) and (7), where (5) starts the vessel flow, (6) ensures that all
vessels arrive at the sink, and (7) balances the flow at each node.
Arcs are assigned a capacity if a vessel utilizes the arc in constraints (8), which
assigns the reefer container capacity, and in (9), which assigns the total container
capacity, respectively. Note that constraints (8) do not take into account empty
reefer equipment, since empty containers do not need to be turned on, and
can therefore be placed anywhere on the vessel. Cargo is only allowed to flow
on arcs with a vessel in constraints (10). The flow of cargo from its source
to its destination, through intermediate nodes, is handled by constraints (11).
Constraints (12) balance the flow of equipment in to and out of nodes. Since the
amount of equipment carried is limited only by the capacity of the vessel, no
flow source/sink constraints are required.
4 Node Flow Model
When the number of demands and arcs grows, the number of variables in the arc
flow model can become too large to fit in memory, and even when the model fits
in memory, it is still often too large to solve. In both [18] and [16], the authors are
unable to solve the LSFRP to optimality on problems with more than 9 vessels.
