Geoscience Reference
In-Depth Information
from the union of all the three criteria Pareto solution sets. The reader may notice
that all this process reduces to obtaining the bicriteria Pareto chains as proved in
Theorem
9.6
.
Theorem 9.7
The following inclusions hold:
I.
[
p;q;r
cl
encl
Par
.f
p
;f
q
;f
r
/
gen
Par
.f
1
;:::;f
Q
/
.
X
X
2
Q
p<q<r
X
Par
f
1
;:::;f
Q
II.
[
p;q;r
Par
.f
p
;f
q
;f
r
/
[
[
p;q;r
encl
X
Par
.f
p
;f
q
;f
r
/
gen
gen
X
2
Q
p<q<r
2
Q
p<q<r
X
wPar
f
1
;:::;f
Q
.
D
cl
encl
X
Par
.f
p
;f
q
;f
r
/
. This is equivalent to
(I) Let x
2
S
p;q;r
gen
Proof
2
Q
p<q<r
x
2
cl
encl
X
Par
.f
p
;f
q
;f
r
/
for some p;q;r
2
gen
Q
;p<q<r:
X
sPar
.f
p
;f
q
;f
r
/ for some p;q;r
2
Then,
by
Lemma
9.2
, x
2
Q
;p<q<r. Applying characterization (
9.4
), this is equivalent to
L
.f
p
;f
p
.x//
\
L
.f
q
;f
q
.x//
\
L
.f
r
;f
r
.x//
D
x
g
for
some
;p<q<rand since x
2
L
.f
q
;f
q
.x// for all q
2
p;q;r
2
Q
Q
that
T
qD1
L
.f
q
;f
q
.x//
D
x
g
.
it
follows
Finally,
again
by
(
9.4
),
X
sPar
f
1
;:::;f
Q
, which implies that x
2
X
Par
f
1
;:::;f
Q
.
x
2
X
Par
f
1
;:::;f
Q
then x
2
X
wPar
f
1
;:::;f
Q
and, by (
9.2
), this
(II) Let x
2
is equivalent to
T
qD1
L
<
.f
q
;f
q
.x//
D;
. By Helly's theorem, there exists
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;
. By characterization (
9.2
), this is equivalent to x
2
X
wPar
.f
p
;f
q
;f
r
/ for some p;q;r
2
Q
;p<q<rand, by Theorem 3.2 in
Rodríguez-Chía and Puerto (
2002
), this implies that x
2
Q
gen
Par
.f
p
;f
q
;f
r
/
[
X
encl
X
Par
.f
p
;f
q
;f
r
/
for some p;q;r
2
gen
Q
;p<q<r. Finally, this can
be equivalently written as
x
2
[
p;q;r
Par
.f
p
;f
q
;f
r
/
[
[
p;q;r
encl
X
Par
.f
p
;f
q
;f
r
/
:
gen
gen
X
2
Q
p<q<r
2
Q
p<q<r
In the Q-criteria case the crucial region is now given by the cells C
2
C
with
C
[
p;q;r
Par
.f
p
;f
q
;f
r
/
n
[
p;q;r
encl
X
Par
.f
p
;f
q
;f
r
/
gen
gen
X
2
Q
p<q<r
2
Q
p<q<r
D
[
p;q
wPar
.f
p
;f
q
/
n
[
p;q;r
encl
Par
.f
p
;f
q
;f
r
/
:
gen
X
X
2
Q
p<q
2
Q
p<q<r
