Geoscience Reference
In-Depth Information
to aggregate the covering functions c
i
; yielding, for example, the proposal of Hwang
et al. (
2004
),
c.a;
x
/
D
1
Y
1ip
.1
c
i
.a;x
i
//;
(6.5)
which, if each c
i
has the form (
6.2
) is identical to (
6.4
). Alternatively, realizing that
the max operator used in (
6.4
) is nothing but taking one of the ordered values of
c
i
.a;x
i
/; further extensions are natural:
X
p
c.a;
x
/
D
max
.
1
;:::;
p
/2
i
c
i
.a;x
i
/
(6.6)
iD1
for a given : Taking as the set
D
(
.
1
;:::;
p
/
W
i
D
1;
i
0
8
i
)
;
p
X
iD1
one recovers (
6.4
); taking
D
(
.
1
;:::;
p
/
W
r
i
0
8
i
)
;
p
X
i
D
1;
1
iD1
for some integer r
2f
1;2;:::;p
g
; one obtains as coverage the weighted sum of
the r highest covers. These covering models belong to the class of so-called ordered
covering models (Berman et al.
2009c
), in which a weighted sum of the ordered
values of the covers are considered.
Another class of models is given by the so-called cooperative cover model,
discussedinBermanetal.(
2009a
):
c.a;
x
/
D
1; if
P
iD1
i
c
i
.a;x
i
/
0; otherwise
(6.7)
for some positive fixed scalars
i
and threshold value : Assuming that each facility
covering function c
i
follows the all-or-nothing model (
6.2
), model (
6.7
) means that
we may consider an individual to be covered if the weighted sum of 1-facility covers
yields a value above a threshold limit :
Summing up, the different proposals in the literature can be considered as
particular cases of a general model of the form
c.a;
x
/
D
c
1
.a;x
1
/;c
2
.a;x
2
/;:::;c
p
.a;x
p
/
;
(6.8)
