Geoscience Reference
In-Depth Information
Theorem 9.3
X
wPar
.f
1
;:::;f
Q
/
D
[
p;q;r
X
wPar
.f
p
;f
q
;f
r
/:
2
Q
p<q<r
X
wPar
.f
1
;:::;f
Q
/ if and only if
\
q2
Q
L
<
.f
q
;
Proof
By Theorem
9.1
, x
2
f
q
.x//
D;
. Furthermore, by Helly's theorem (see Rockafellar
1970
), this
intersection is empty if and only if there exist p;q;r
2
.p<q<r/such
that L
<
.f
p
;f
p
.x//
\
L
<
.f
q
;f
q
.x//
\
L
<
.f
r
;f
r
.x//
D;
and this is equivalent
to x
2
Q
X
wPar
.f
p
;f
q
;f
r
/. Since in any case we have that
[
X
wPar
.f
p
;f
q
;f
r
/
X
wPar
.f
1
;:::;f
Q
/;
p;q;r
2
Q
p<q<r
the result follows.
Remark 9.3
This result extends previous characterizations in the literature:
-Takingf
i
.x/
Dk
x
a
i
k
with a
i
2
2
for i
D
1;:::;Qand
kk
being a strictly
convex norm or a norm derived from a scalar product, we get Proposition 1.3,
Theorem 4.3 and Corollary 4.1 in Durier and Michelot (
1986
). The set of weakly
efficient locations is the convex hull of the points a
i
with i
D
1;:::;Q.In
Example
9.3
, we illustrate this result.
-Takingf
i
.x/
Dk
x
a
i
k
with a
i
2
R
2
for i
D
1;:::;Q and
kk
being
a polyhedral gauge we get Theorem 6.1 in Durier (
1990
), where the set of
weakly efficient locations is the union of elementary convex sets, (see Durier
and Michelot
1985
for a definition). In Example
9.4
, we illustrate this result.
-Takingf
i
.x/
D
max
j2
M
w
i
j
k
x
a
j
k
with a
j
2
R
2
,
w
i
j
>0for i
D
1;:::;Q,
R
WD f
1;:::;m
g
and
kk
being the `
1
-norm, we get Theorem 6.1
in Hamacher and Nickel (
1996
), where the set of weakly efficient locations is
the union of the sets of weakly efficient locations for all pairs of functions. In
Example
9.5
, we illustrate the use of this result.
Example 9.3 (See Fig.
9.3
)
Let us consider the points a
1
D
.0;0/, a
2
D
.5;
10/,
a
3
D
.10;0/ and the functions f
i
.x/
Dk
x
a
i
k
2
for i
D
1;2;3. By Theorem
9.2
,
X
wPar
.f
1
;f
2
;f
3
/ is the dark region, which in this case is the convex hull of a
1
,
a
2
and a
3
.
j
2
M
Example 9.4 (See Fig.
9.4
)
Let us consider the points a
1
D
.0;0/, a
2
D
.8;3/,
a
3
D
.
3;5/ and the functions f
1
.x/
Dk
x
a
1
k
1
, f
2
.x/
Dk
x
a
2
k
1
and
f
3
.x/
Dk
x
a
3
k
1
. By Theorem
9.1
,
X
wPar
.f
1
;f
2
/ is the thick path joining a
1
X
wPar
.f
1
;f
3
/
is the dark rectangle with a
1
and a
3
as opposite extreme points. Therefore, by
Theorem
9.2
,
X
wPar
.f
2
;f
3
/ is the thick path joining a
2
and a
3
,and
and a
2
,
X
wPar
.f
1
;f
2
;f
3
/ is the dark region surrounded by the union of
