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In order to prevent the incorporation of nodes associated with containers that
have not been selected for incorporation into the routes of the vehicles constraint
(52) is added to the model.
y
rk
=
i
∈N
z
ik
∀
k
∈C
(53)
r
∈
REQS
y
rk
=
i
∈N
z
+
ik
∀
k
∈C
(54)
r
∈
REQS
Finally, constraints (53) and (54) are used to prevent the visitation of con-
tainers for which a pickup and a delivery node are specified but which is not
assigned to any request.
Objective Function
·
x
VEH
ijf
d
(
i, j
)
→
min
(55)
i
∈N
j
∈N
f
∈V
The trucking company aims at minimizing the overall travel distance of their
vehicles (55).
5 Numerical Example
The presented decision model is very dicult to solve. Using appropriate pa-
rameter instantiations, well-studied NP-hard decision problems like the pickup
and delivery problem are identified as special cases of the presented model. We
cannot expect to solve this model to optimality within an acceptable time. Ini-
tial computational experiments with the commercial solver CPLEX showed that
even quite small instances of the integrated request completion, container flow
and vehicle routing problem cannot be solved within reasonable computation
time independently from the applied objective function. Often, the solver even
failed to identify a feasible solution.
Being aware of the challenging diculty of the CPDP, we focus on the assess-
ment of the suitability of the presented constraint collection for a rather small
example scenario. We instruct CPLEX to identify any feasible solution and ter-
minate the search then. We use the identified solution to verify the feasibility
with respect to the aforementioned problem constraints. In order to get further
insights into the performance of the model it is necessary to develop specific
(meta-)heuristics which is, however, beyond the scope of the present paper.
The example shown in Fig. 2 comprises a hinterland system with eight 20-
foot containers. There is a (seaside or railway) container terminal comprising
the nodes 1 to 8. Six customer locations are represented by nodes 9 to 14. Two
trucks of capacity 2 TEU are available at a vehicle depot with outbound node
15 and inbound node 16. One hinterland container depot (nodes 17 to 24) can
be used to store empty containers or to pickup empty containers needed for the
fulfillment of a provide request. Nodes with a surplus of containers are shaded
