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We remedy this problem by splitting each multiple destination demand into
a demand for each origin-destination pair, and add a constraint in the node flow
model to ensure that the amount of containers delivered to all of the destinations
is not greater than the amount of demand available. In the following models, we
use the set Θ to represent the set of single origin-destination pairs of demands
in the following node flow models. Formally, let Θ = ( o,d,t ) ∈Θ d ∈d ( o, d ,t ).
4.1 Preprocessing
We base our node based model on the same graph and notation as in Section 3.
In order to model the flows based on nodes, we must determine which nodes a
particular demand can traverse. We do this using a simple reachability analysis
based on the transitive closure of the graph. Let G T =( V T ,A T ) be the transitive
closure of the graph G =( V,A ), and
d s.t. ( i,d )
i = d }
Θ Vis
i
Θ |
A T
d
A T
=
{
( o, d, t )
( o, i )
∧∃
.
Each node is thereby assigned a set of demands ( Θ Vis i ) based on whether the
node is reachable from the demand origin and at least one of the destinations
of the demand. We further define Θ Vis
i,rf
Θ Vis
i
{
( o, d, t )
|
t = rf
}
=
to be the
set of reefer demands at node i
V . Using these sets of demands, we can now
model demand flows on the nodes of the IVLSFRP graph.
4.2 Equipment as Flows
We extend the parameters in Section 3.2 with the following three parameters:
Θ , the set of single origin single destination demands, Θ Vis i ,whichisthesetof
demands (dry and reefer) that could traverse node i ,and Θ Vis
i,rf , the set of reefer
demands that could traverse node i .
Variables
x i,j [0 , max s∈S u d s ] Amount of equipment of type t ∈ T flowing on ( i,j ) ∈ A .
y i,j ∈{ 0 , 1 }
Indicates whether vessel s is sailing on arc ( i,j ) ∈ A .
z ( o,d,t )
[0 ,a ( o,d,t ) ] Amount of demand triplet ( o, d, t ) ∈ Θ carried on ship s ∈
S .
Objective and Constraints
max
s∈S
s
r ( o,d,t ) − c Mv
o
z ( o,d,t )
s
− c Mv
d
(1) + (3)
(13)
( o,d,t ) ∈Θ
s . t . (4); (5); (6); (7); (12);
z ( o,d,t )
s
≤ a ( o,d,t )
i∈Out ( o )
y o,i
( o,d, t ) ∈ Θ ,s∈ S
(14)
≤ a ( o,d,t )
d ∈d
z ( o,d,t )
s
y id
( o,d, t ) ∈ Θ ,s∈ S
(15)
i∈In ( d )
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