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In-Depth Information
demand is equal to the sum of the capacities of all ships, and the revenue per
TEU delivered is equal to the equipment revenue for each equipment type. Our
model uses the same parameters as the arc flow model in Section 3 and the
equipment as flow model in Section 4.2. The majority of the model is the same as
the previous two models, however we include all of the objectives and constraints
for completeness.
Objective and Constraints.
We use the variables
y
i,j
and
z
(
o,d,t
s
from the
equipment as flows model and require no additional variables. The model is as
follows:
(1) + (13) (20)
subject to constraints (4), (5), (6), (7), from the arc flow model in Section 3.2,
and (14), (15), (17), and (19) from the equipment as flows model in Section 4.2.
Instead of the dry capacity constraint in the equipment as flows model, we impose
the following constraint:
max
−
z
(
o,d,t
)
s
u
dc
s
V
.
≤
∀
s
∈
S, i
∈
(21)
(
o,d,t
)
∈Θ
Vis
i
The objective, (20), combines the sailing costs and port fees from the arc flow
model with the cargo demand objective from the equipment as demands model.
Note the lack of any objective relating to equipment, as it is now a part of the
demand structure.
As in the equipment as flows model, we include several constraints from the
arc flow model to enforce node disjointness along vessel paths and control the
vessel flows. However, we omit the equipment flow balance constraints (12). We
also include the node demand constraints from the equipment as flows model,
along with the reefer capacity constraint, as they are unaffected by modeling
equipment as demands. We modify the dry capacity constraints (16) to produce
constraints (21), in which the sum of the demands carried at a particular node
must respect the vessel capacity.
5 Computational Evaluation
We evaluated our node flow model, with both versions of equipment handling,
on the 37 confidential and 37 public LSFRP instances where only inflexible
visitations are present [16]. The instances include two real-world repositioning
scenarios as well as a number of crafted scenarios based on data from Maersk
Line. Table 1 shows the results of running the arc flow model (AF), equipment
as flows (EAF), and equipment as demands (EAD) models on the confidential
instance set
3
with a timeout of 5 hours of CPU time and a maximum of 10 GB
of memory. We used CPLEX 12.4 on AMD Opteron 2425 HE processors. The
table's columns describe the following instance information. Each instance has
3
The missing instance IDs correspond to instances with flexible arcs, which the EAD
and EAF models do not solve.
