Geoscience Reference
In-Depth Information
Theorem 3.12 (Cho et al.
1983b
)
Let
X
X
ij
x
ij
C
X
i2P
i
N
y
i
0
(3.68)
i2P
j2D
be a facet-defining inequality of
P
m
n
. Then, (
3.68
) is also a facet-defining
inequality of
P
mn
for
m
m
,
n
n
.
Cho et al. (
1983b
) also give a constructive procedure for obtaining facets of P
mn
from cyclic adjacency matrices which do not define facets themselves.
Theorem 3.13 (Cho et al.
1983b
)
Consider
P
I
,
D
J
, such that
j
P
jD
j
D
jD
q
,
q
3
. Consider the facet-defining inequality of
P
qq
given by
X
X
x
ij
C
X
i2P
N
y
i
2q
2
i2P
j2D
i
where the sets
D
i
are all the different subsets of
D
with
j
D
i
jD
q
1
. Suppose we
add
j
S
jCj
T
j
facilities of
I
to
P
in such a way that each facility in
S
covers
q
1
destinations and each facility in
T
covers all the
q
destinations. Let
j
S
jD
s
and
j
T
jD
t
. Then,
X
X
ij
x
ij
C
X
i2I[S[T
i
N
y
i
.2q
C
s
2/.q
1/
C
t.q
2/
i2I[S[T
j2D
i
is a facet-defining inequality of
P
.qCsCt/q
,where
I.
ij
D
i
D
q
1; i
2
P
[
S;j
2
D
i
,
II.
ij
D
i
D
q
2; i
2
T;j
2
D
i
.
3.6
Conclusions
Fixed-Charge Facility Location Problems capture the main issues arising in fixed-
charge location, so they are an excellent workbench for reviewing relevant aspects
in this field. This was the aim of this chapter where we have covered a broad
range of possibilities related to the modeling and the solution process of FLPs.
Indeed the problems studied in this chapter can be seen as simplifications of
more realistic models that take into account additional issues. We have studied
deterministic static problems, without taking uncertainty into account (see, for
instance, Lin
2009
; Albareda-Sambola et al.
2011
,
2013
;Gao
2012
) or temporal
aspects (see, for instance, Albareda-Sambola et al.
2009a
,
2010
,
2012
). Also, the
way we have considered capacity constraints on the facilities may seem simplistic,
since modular capacities (incurring their corresponding costs) can be more realistic
(see, for instance, Gouveia and Saldanha da Gama
2006
; Gourdin and Klopfenstein
