Geoscience Reference
In-Depth Information
for some componentwise non-decreasing
F
: The simplest case is given by
F
.c
ij
.x
i
;x
j
//
i¤j
D
max
i¤j
c
ij
.x
i
;x
j
/;
(6.19)
and thus C
F
.
x
/ is calculated as the highest facility-facility interaction, i.e., the one
of the closest pairs of facilities. Hence, under (
6.19
),
C
F
.
x
/
ı if and only if
c
ij
.x
i
;x
j
/
ı
8
i;j; i
¤
j; if and only if
k
x
i
x
j
k
.'
ij
/
1
.ı/
8
i;j; i
¤
j:
Assuming all c
ij
in (
6.18
) are modeled by means of the same '
ij
function, '
ij
D
'
F
,wehave
C
F
.
x
/
ı if and only if
min
i;j
i¤j
k
x
i
x
j
k
;
(6.20)
with
D
'
F
1
.ı/. See Lei and Church (
2013
) for a discussion and extension
of (
6.19
) to so-called partial-sum criteria.
6.2.3
The Anti-covering Model
Depending on the specific problem under consideration, either one or the two
covering criteria C, C
F
are to be optimized. Pure repulsion among facilities
naturally leads to a dispersion criterion (Erkut and Neuman
1991
; Kuby
1987
;
Lei and Church
2013
). By (
6.20
), minimizing C
F
amounts to maximizing the
minimal distance among facilities. This criterion alone yields a simple geometrical
interpretation: a set of p non-overlapping circles (the location of the facilities) is
sought so that their (common) radius is maximized (Mladenovi´cetal.
2005
).
When both C and C
F
are relevant, one naturally faces a biobjective optimization
problem in which both C and C
F
are to be minimized,
C.
x
/;C
F
.
x
/
;
min
x
2
S
(6.21)
2
/
p
is the feasible region, which is assumed to be a compact subset,
and thus embedded in a box. Sensible examples for
where
S
.
R
D
S
p
; where S
S
may be
S
Df
1
gf
2
g
:::
f
k
g
S
pk
,whereS is a
polygon in the plane, and
1
;:::;
k
are fixed points in the plane, corresponding to
facilities already located.
is a polygon in the plane, or
S
