Geoscience Reference
In-Depth Information
where should take values in Œ0;1 and should be componentwise non-decreasing,
so that the higher each individual-facility cover, the higher the cover of individual
location a by the p facilities.
So far we have modeled the interaction between an affected individual at a and
the facilities at
x
D
.x
1
;:::;x
p
/. Now we address the problem of defining a global
individuals-facilities covering measure C.
x
/.
If the main concern is how much the highest coverage is, a worst-case perfor-
mance measure is suitable:
C.
x
/
D
sup
a2A
c.a;
x
/:
(6.9)
Under (
6.9
) as criterion, searching locations
x
for the facilities such that C.
x
/
Ǜ
means that no individual at all suffers a coverage of more than Ǜ:
The (safe) worst-case approach (
6.9
) may be unfeasible for densely populated
regions, and, instead of searching locations not affecting individuals, the
average
coverage may be a suitable choice. Formally, assume that affected individuals are
distributed along the plane, following a distribution given by a probability measure
on a set A
2
; and the individuals-facilities coverages are aggregated into one
single measure, namely, the
expected coverage
,givenby
C.
x
/
D
Z
R
c.a;
x
/d.a/:
(6.10)
A
Assuming, as in (
6.10
), an arbitrary probability measure for the distribution
of affected individual locations gives us full freedom to accommodate different
important models. Obviously, for a finite set A of affected individual locations,
A
Df
a
1
;:::;a
n
g
; denoting
a
D
.
f
a
g
/; we recover the basic covering model,
C.
x
/
D
X
a2A
a
c.a;
x
/;
(6.11)
in which the covering is given by the weighted sum of the covers of the different
points a: However, we can consider absolutely continuous distributions, in which
has associated a probability density function f in the plane, and now (
6.10
) becomes
C.
x
/
D
Z
c.a;
x
/f.a/
da
:
(6.12)
A
Several types of density functions f are worthy to be considered. One can take,
for instance, f as the uniform density on a region A
2
(a polygon, a disc), and
R
thus f is given as
f.a/
D
(
1
ar
.A/
; if a
2
A
0; otherwise;
(6.13)
