Information Technology Reference
In-Depth Information
+
t∈T
z
(
o,d,t
)
s
x
i,j
≤ u
dc
∀s ∈ S, i ∈ V
(16)
s
(
o,d,t
)
∈Θ
Vis
i
j∈In
(
i
)
z
(
o,d,t
)
s
≤ u
r
s
∀s ∈ S, i ∈ V
(17)
(
o,d,t
)
∈Θ
Vis
i,rf
x
i,j
≤
u
d
s
y
i,j
∀
(
i,j
)
∈ A
(18)
t∈T
s∈S
z
(
o,d
,t
)
s
≤ a
(
o,d,t
)
∀s ∈ S,
(
o, d, t
)
∈ Θ
.
(19)
d∈d
The objective (13) contains the same calculation of sailing costs, equipment
profits, and port fees as in the arc flow model. However, the demand profit is now
computed using the demand flow variables. Note that unlike in
Θ
, all (
o, d, t
)
Θ
∈
= 1. Thus, the constant
c
M
d
refers to the cost at a particular visitation.
We copy constraints (4) through (7) and (12) directly from the arc flow model
in order to enforce node disjointness along vessel paths and create the vessel and
equipment flows. We refer readers to Section 3.2 for a full description of these
constraints.
Constraints (14) and (15) allow a demand to be carried only if a particular
vessel visits both the origin and a destination of the demand. Note that we do
not need to limit the demands to be taken only by a single vessel because of the
node disjointness enforced by constraints (4). Only a single vessel can enter the
origin node of a demand, ensuring that only one vessel can carry a demand.
In constraints (16) and (17) we ensure that the capacity of the vessel is not
exceeded at any node in the graph in terms of all containers and reefer containers,
respectively. Equipment flows are handled in the dry capacity constraints (16).
Due to the equipment balance constraints, ensuring that the equipment capacity
is not exceeded at a node is sucient for ensuring that the vessel is not overloaded.
Constraints (18) prevent equipment from flowing on any arc that does not
have a ship sailing on it. When a vessel utilizes an arc, the constraint allows
as much equipment to flow as the capacity of the vessel. When an arc has no
vessel, the corresponding equipment flow variables have an upper bound of 0.
And, finally, constraints (19) ensure that the amount of demand carried for each
single origin-destination demand does not exceed the amount of containers that
are actually available.
|
d
|
have
4.3 Equipment as Demands
As an alternative to modeling equipment as flows, we present a model that
creates demands for each equipment pair and adds them to
Θ
.Welet
Θ
E
t
=
{
V
t
+
V
t−
}
be the set of demands corresponding to every
pair of visitations with an equipment surplus/deficit for type
t
(
o,
{
d
}
, dc
)
|
o
∈
∧
d
∈
T
.Notethat
we set the demand type of all of the equipment demands to be dry (
dc
). We
then append the equipment demands to
Θ
as follows:
Θ
←
∈
Θ
∪
t∈T
Θ
E
t
.In
addition, let
a
(
o,d,dc
)
=
s∈S
u
dc
s
and
r
(
o,d,dc
)
=
r
Eqp
∈
Θ
E
t
. Thus, the maximum amount of equipment available for each equipment
for all
t
∈
T,
(
o, d, dc
)
dc
