Geoscience Reference
In-Depth Information
X
w
−
Par
(f
1
f
3
)
a
3
a
2
a
1
X
w
−
Par
(f
1
f
2
)
Fig. 9.1
Illustration of Example
9.1
2
we define the enclosure of S by
encl .S/
WD
˚
x
2
For a set S
R
2
W9
">0with B.x;"/
\
S
D;
;
9
t
'
2
Œ0;
1
/ with
R
2
/ with '.0/
D
x
;
R
0
;
'.t
'
/
2
S for all'
2
C
1
.
R
2
Wk
y
x
k
2
"
g
. Note that S
\
encl .S/
D;
.
Informally, encl.S/ contains all the points which are surrounded by S, but do not
belong themselves to S.
We denote the union of the bicriteria chains of weak-Pareto solutions by
where B.x;"/
Df
y
2
R
[
2
[
qDpC1
X
3
wPar
f
1
;f
2
;f
3
WD
gen
wPar
.f
p
;f
q
/:
X
pD1
X
wPar
f
1
;f
2
;f
3
. The next
theorem provides useful geometric information to build
We use “gen” since this set will generate the set
X
wPar
f
1
;f
2
;f
3
.Its
proof can be found in Rodríguez-Chía and Puerto (
2002
).
Theorem 9.2
wPar
.f
1
;f
2
;f
3
/
D
encl
X
wPar
f
1
;f
2
;f
3
[
wPar
f
1
;f
2
;f
3
:
gen
gen
X
X
