Information Technology Reference
In-Depth Information
Phase-out and Phase-in.
Each ship is assigned a particular visitation,
σ
s
∈
V
, at which the ship
s
S
begins its repositioning. This visitation represents
the earliest allowed phase-out time for that vessel. A visitation is then created
for each subsequent port call of the ship on its phase-out slot. Each phase-
out visitation is connected to the next one with an arc. Note that phase-out
visitations do not connect to the phase-out visitations of other ships.
Vessels may leave phase-out nodes to sail to SOS opportunities or to a phase-
in slot. Thus, arcs are created from each phase-out visitation to each phase-in
visitation and SOS entry visitation such that sailing between the visitations
is temporally feasible (i.e. the starting time of the phase-in/SOS visitation is
greater than the end time of the phase-out visitation plus the sailing time).
Finally, phase-out visitations have incoming arcs from phase-in visitations in
the same trade zone, which we define as a set of ports in the same geographical
region. This allows ships to avoid sailing back and forth between ports when
transferring directly between the phase-out and phase-in.
We create visitations for each port call along a phase-in slot, and connect sub-
sequent phase-in visitations to each other. The final visitation in a slot, which
represents the time at which regular operations must begin on a service, is con-
nected to the graph sink,
τ
. Due to the node disjointness of the vessel paths,
this structure ensures that each slot on the goal service receives a single vessel.
∈
Sail-On-Service.
SOS opportunities are modeled with a special structure that
ensures each one is only used by at most a single vessel. The structure partitions
the visitations of an SOS into three sets:
entry ports
, where vessels may join the
SOS,
through ports
, in which a vessel must already be on the SOS, and
end ports
where a vessel may leave the SOS.
Sailing Cost.
The fuel consumption of a ship is a cubic function of the speed
of the vessel. We precompute the optimal cost for each arc using a linearized
bunker consumption function. All arcs in the model are assigned a sailing cost
for each ship that is the optimal sailing cost given the total duration of the arc.
Since ships have a minimum speed, if the duration of the arc is greater than the
time required to sail on the arc at a ship's minimum speed, the cost is calculated
using the minimum speed and then the ship simply waits for the remainder of
the duration. This is a common practice for shipping lines in order to add buffer
to their schedules, thus making the network more robust to disruptions.
3.2 Mixed-Integer Programming Model
We now present the MIP model from [18,16] excluding flexible node/arc compo-
nents. We use the following parameters and variables for the model.
Parameters
T
Set of equipment types.
T
=
{dc, rf }
.
S
Set of ships.
V
Set of visitations minus the graph sink,
τ
.
