Geoscience Reference
In-Depth Information
once), and e
i
D
1
8
i
2
I. Since we are also interested in knowing whether
each demand point j
2
J is covered or not by a second center (disregarding
the number of additional facilities which cover j), only variables
w
j1
would be
necessary if equalities (
5.3
) were replaced by inequalities (
5.6
)asintheSCP
discussed above. Alternatively, the RCLP can be obtained as a particular case
of (COV) by taking h
D
m
1, g
jk
D
0
8
j
2
J, k
2,andg
j1
D
1
8
j
2
J.
In order to prioritize the minimization of the number of open facilities, we define
f
i
D
n
C
1
8
i
2
I as a cost large enough.
Hierarchical Covering Location Problem (HCLP): An objective function which
allows the simultaneous minimization of the number of facilities that are opened
and the maximization of the number of previously existing facilities that are kept
(within the minimum total number of facilities) was introduced in Plane and
Hendrick (
1977
) in a paper devoted to the location of fire stations. Values a
ij
are
equal to one if and only if focal point i can be served by a pumper company at
location j in less than the response time specified for site i. They found a major
difficulty when using the SCP: this model does not differentiate between those
sites that have existing fire stations and those that require the construction of a
station. This drawback was fixed by modifying the objective function of the SCP
as follows: consider a partition of the set of facilities I
D
I
0
[
I
1
,whereI
0
is
the set of existing facilities and I
1
is the set of potential new facilities. Then,
define f
i
D
1
8
i
2
I
1
and f
i
D
1
">0
8
i
2
I
0
with " a small positive
amount. This way, the slightly lower cost of the already existing centers makes
them more interesting when minimizing the total cost.
Maximal Covering Location Problem: The Maximal (or Maximum) Covering
Location Problem (MCLP) was introduced in Church and ReVelle (
1974
) and, as
it has been explained in the previous section, it entails an important change with
regard to the goal of the previous models listed in this section because, since
now the number of facilities to be located is limited to a given value p<m,
we do not require to cover all the demand but to maximize the covered demand.
Then, h
D
p and b
j
D
0
8
j
2
J.Again,e
i
D
1
8
i
2
I and values a
ij
are
defined as usual. Since we need to know whether a demand point is covered
or not without minding about the number of different facilities that cover it,
we avoid that variables y
i
and variables
w
jk
with k
ยค
1 contribute to the
objective function (
5.1
) by fixing their corresponding coefficients to zero, i.e.,
f
i
D
0
8
i
2
I and g
jk
D
0
8
j
2
J,
8
k
2. Besides, we set g
j1
D
1 in order
to maximize the number of demand points covered by the open facilities.
An alternative to this model that was proposed in Church and ReVelle (
1974
)is
to combine mandatory covering of some demand points (assume these points
are indexed by means of J
1
J) and maximization of the coverage of the
remaining points (those in J
n
J
1
). This situation can also be approached by
means of model (COV) by taking h
D
p, b
j
D
1
8
j
2
J
1
, b
j
D
0
8
j
2
J
n
J
1
,
e
i
D
1
8
i
2
I,andf
i
D
0
8
i
2
I.Theg-coefficients are defined as follows:
g
j1
D
1
8
j
2
J
n
J
1
, g
jk
D
0
8
j
2
J
n
J
1
,
8
k
2,andg
jk
D
0
8
j
2
J
1
,
8
k
2
K. We call this model MCLP'.
