Geoscience Reference
In-Depth Information
9.2.1.1
Bicriteria Case
In this section we restrict ourselves to the bicriteria case, which, as will be seen later,
is the basis for solving the Q-criteria case. To this end, we are looking for the Pareto
solutions of the vector optimization problem in
2
,
R
f
1
.x/
WD
w
i
i
.x
a
i
/
!
;
X
M
X
M
w
i
i
.x
a
i
/; f
2
.x/
WD
min
x2
R
2
iD1
iD1
where the weights
w
i
are non negative (i
D
1;:::;M; q
D
1;2). The following
theorem provides a geometric characterization of the set
X
Par
.
X
Par
f
1
;f
2
is a connected chain from
X
.f
1
/
to
X
.f
2
/
Theorem 9.4
consisting of faces or vertices of cells, or complete cells.
X
.f
q
/
¤;
for q
D
1;2 (see Puerto and Fernández
Proof
First, we note that
X
.f
q
/
¤;
for q
D
1;2. Therefore, we know that
X
Par
¤;
, so we can choose x
2
X
Par
\
2000
). Moreover,
X
Par
. There exists at least one cell C
2
with
x
2
C. We can assume without loss of generality that C is bounded. We also note
that the functions f
1
and f
2
are linear within each cell (see Rodríguez-Chía et al.
2000
). Given a set A, in what follows, conv(A), bd(A) and int(A) will denote the
convex hull, the boundary and the interior of the set A, respectively. Hence three
cases may occur:
C
X
Par
we obtain
Case 1: x
2
int.C/.Sincex
2
\
2
\
2
L
.f
q
;f
q
.x//
D
L
D
.f
q
;f
q
.x//
qD1
qD1
and by linearity of the median problem in each cell we have
\
2
\
2
L
.f
q
;f
q
.y//
D
L
D
.f
q
;f
q
.y//
8
y
2
C
qD1
qD1
X
Par
.
Case 2: x
2
ab
WD
conv.
f
a;b
g
/
bd.C/ and a;b
2
Ext.C/. We can choose
y
2
int.C/ and two cases can occur:
Case 2.1: y
2
which means y
2
X
Par
8
y
2
C, hence C
X
Par
. Hence we can continue as in Case 1.
X
Par
. Therefore using the linearity we first obtain
Case 2.2: y
…
\
2
\
2
L
.f
q
;f
q
.
z
//
¤
L
D
.f
q
;f
q
.
z
//
8
z
2
int.C/:
qD1
qD1
