Geoscience Reference
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Fig. 20.1
An example of solution to the concentrator location problem
directly connected to the concentrators, and the backbone network connecting
the concentrators is also a star—concentrators are directly connected to a central
node. Figure 20.1 illustrates such a topology. The concentrators are represented by
diamonds, round nodes represent end-user, dashed edges are connections between
end-users and concentrators while plain edges form the backbone network.
Let I Df 1;:::;i;:::;m g be the set of potential concentrator locations and
J Df 1;:::;j;:::;n g the set of end-users. The objective is to minimize the sum
of the costs c ij incurred by establishing a link between node j and a concentrator at
node i, and the sum of costs f i for installing a concentrator at node i,linkedtothe
central node. Furthermore, we denote by d j the demand of end-user j and by q i the
capacity of concentrator i.
Using binary variables y i to indicate if concentrator i is open, and binary
variables x ij to indicate if end-user j is assigned to concentrator i, the basic version
of the Concentrator Location Problem can be formulated as
Minimize X
i2I
f i y i C X
i2I
X
c ij x ij
(20.1)
j2J
subject to X
i2I
x ij D 1
j 2 J;
(20.2)
 
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