Geoscience Reference
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An early example of a three-index vehicle flow formulation is that of Perl and
Daskin ( 1985 ). In fact, this reference defines a three-layer problem with suppliers,
distribution centers and customers where, in addition to the characteristics of the
basic LRP, the authors consider variable costs associated with the throughput at
each distribution center, and extra constraints limiting the length of the routes.
The proposed formulation, simplified by excluding these extra considerations, is
described next. To this end, the following binary variables will be used:
￿ For each i 2 I, y i indicates whether a facility is established at i.
￿ For each i 2 I;j 2 J, x ij indicates whether customer j is served from facility i
￿ For each .i;j/ 2 E and k 2 K, z ijk indicates whether vehicle k uses arc .i;j/.
Using the above variables, a three index vehicle flow formulation for the LRP is
detailed next:
.LRP1/ minimize X
i2I
f i y i C X
k2K
X
` ij z ijk
(15.1)
.i;j/2E
subject to X
k2K
X
z ijk D 1
j 2 J
(15.2)
i2V
X
w j X
i2V
z ijk Q
k 2 K
(15.3)
j2J
X
w j x ij q i y i 0
i 2 I
(15.4)
j2J
X
X
X
z ijk 1
I S V
(15.5)
k2K
i2S
j2V nS
X
z ijk X
j2V
z jik D 0
k 2 K;i 2 V
(15.6)
j2V
X
X
x ijk 1
k 2 K
(15.7)
i2I
j2J
X
z itk C X
t2V
z jtk x ij 1
i 2 I;j 2 J;k 2 K (15.8)
t2J
y i 2f 0;1 g
i 2 I
(15.9)
x ij 2f 0;1 g
i 2 I;j 2 J
(15.10)
z ijk 2f 0;1 g
.i;j/ 2 E;k 2 K: (15.11)
Constraints ( 15.2 ) mean that each customer is reached by one vehicle route, while
constraints ( 15.3 )and( 15.4 ) are vehicle and plant capacity constraints, respectively.
Additionally, constraints ( 15.4 ) guarantee that customers will be served from opened
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