Geoscience Reference
In-Depth Information
now .y
.v/;
w
.v// be
Let
an
optimal
solution
to
(COVLR(v)).
Prob-
lem (COVLR(v)) splits into
.COVLRy.v// min
P
i2I
f
i
C
P
j2J
v
j
a
ij
y
i
s.t.
(
5.2
); (
5.4
);
and
.COVLRw.v// min
P
j2J
P
k2K
g
jk
v
j
w
jk
s.t.
(
5.5
):
(COVLRw(v)) can be easily solved by inspection:
w
jk
.v/
D
1
,
g
jk
v
j
8
j
2
J;
8
k
2
K:
If, as in most of the models that we considered, g
jk
-values are sorted in increasing
order for each j
2
J, and assuming that v
j
2
.g
j`
j
;g
j;`
j
C1
, then the optimal
solution to (COVLRw(v)) will look like as follows:
w
j1
.v/
D
:::
D
w
j`
j
.v/
D
1;
w
j;`
j
C1
.v/
D
:::
D
w
jh
.v/
D
0:
The corresponding optimal value will be v.COVLRw.v//
D
P
j2J
.
P
`
kD1
g
jk
`
j
v
j
/.
Regarding (COVLRy(v)), we define f
0
i
WD
f
i
C
P
j2J
v
j
a
ij
8
i
2
I and we sort
these values in increasing order:
f
.1/
:::
f
.t/
0
f
.tC1/
:::
f
.n/
:
An optimal solution to (COVLRy(v)) is recursively obtained by taking
y
.i/
.v/
D
(
e
.i/
if
P
i1
`D1
y
.`/
.v/
p
e
.i/
;
p
P
i1
`D1
y
.`/
.v/ if
P
i1
`D1
y
.`/
.v/ > p
e
.i/
;
i
D
1;:::t;and y
.i/
.v/
D
0; i
D
t
C
1;:::;n. Assuming that
P
i
`D1
e
.`/
p<
P
i
0
C1
`D1
e
.`/
, with i
0
t, we then have that
0
@
f
.i/
C
X
j2J
1
i
0
1
X
A
v(COVLRy(v))
D
e
.i/
v
j
a
.i/j
iD1
