Geoscience Reference
In-Depth Information
Assume that the weight vectors are
w
1
D
1
2
!
;
w
2
D
2
1
!
;
w
3
D
1
2
!
;
w
4
D
2
2
!
;
w
5
D
2
2
!
;
w
6
D
2
1
!
:
Using this information we get
v
1
v
2
v
3
v
4
v
5
v
6
f.
/
2
19
1
21
2
17
2
29
2
27
1
21
By pairwise comparison we get
f
1
.V/
[
f
2
.V/
:
par
.V/
Df
v
3
g[f
v
6
gD
X
X
X
Now we look at the points on the edges and get (by using concavity in the objective
functions):
v
3
dominates all points on the edges
f
v
3
;v
5
g
;
f
v
3
;v
4
g
;
f
v
3
;v
1
g
v
6
dominates all points on the edges
f
v
6
;v
2
g
;
f
v
6
;v
5
g
;
f
v
6
;v
4
g
v
2
dominates all points on the edge
f
v
2
;v
4
g
v
1
dominates all points on the edge
f
v
1
;v
5
g
We also observe that no vertex can dominate a point with both objective functions
smaller than 21. The only edge left is now
f
v
1
;v
2
g
(Fig.
9.14
).
We s e e t h a t
I. For all points x
2
P.
f
v
1
;v
2
g
/ with x
6D
v
1
;x
6D
v
2
we have f
1
.x/ < 21;
f
2
.x/ < 21.
II. No point on
f
v
1
;v
2
g
dominates another point on
f
v
1
;v
2
g
par
Df
v
3
g[f
v
6
g[
.
f
v
1
;v
2
g
; .0;1//:
)
X
22
22
21
21
f
1
20
20
f
2
19
19
Fig. 9.14
Objective
functions on the edge fv
1
;v
2
g
in Example
9.7
0
1
