Geoscience Reference
In-Depth Information
to clients may differ a lot. The same scenario occurs if one location for different
types of goods has to be found.
Multicriteria analysis of location problems has received considerable attention
within the scope of continuous, network, and discrete models in the last years. For
an overview of general methods as well as for a more bibliographic overview of
the related location literature the reader is referred to Ehrgott (
2005
) and Nickel
et al. (
2005a
). Presently, there are several problems that are accepted as classical
ones: the point-objective problem (see, e.g., Wendell and Hurter
1973
;Hansenetal.
1980
; Carrizosa et al.
1993
), the continuous multicriteria min-sum facility location
problem (see, e.g., Hamacher and Nickel
1996
; Puerto and Fernández
1999
), the
network multicriteria median location problem (see, for instance, Hamacher et al.
1999
; Wendell et al.
1977
) and the multicriteria discrete location problem (see, e.g.,
Fernández and Puerto
2003
), among others.
In contrast to problems with only one objective, we do not have a natural ordering
in higher dimensional objective spaces. Therefore, in multicriteria optimization one
has to decide which concept of “optimality” to choose.
The goal in a multicriteria location problem is to optimize simultaneously a set
of objective functions (f
1
;:::;f
Q
). Therefore, the formulation of the problem is:
.f
1
.x/;:::;f
Q
.x//;
v
min
x2X
R
(9.1)
d
where v
min stands for vectorial optimization. Observe that we get points in a
Q-dimensional objective space where we do not have the canonical order of
R
anymore. Accordingly, for this type of problems, different concepts of solution have
been proposed in the literature (the reader is referred to Ehrgott (
2005
) as a general
reference in multicriteria optimization). A point x
2
R
d
is called a Pareto location
(or Pareto-optimal) if there exists no y
2
R
d
such that f
q
.y/
f
q
.x/
8
q
2
Q
WD f
1;:::;Q
g
and f
p
.y/ < f
p
.x/ for some p
2
Q
. We denote the set of
X
Par
f
1
;:::;f
Q
or simply by
Pareto solutions by
X
Par
if this is possible without
causing confusion. If f
q
.x/
f
q
.x
0
/
8
q
2
W
f
q
.x/ < f
q
.x
0
/
we say that x dominates x
0
in the decision space and f.x/dominates f.x
0
/ in the
objective space.
Alternative solution concepts are weak Pareto-optimality and strict Pareto-opti-
mality. A point x
2
Q
and
9
q
2
Q
d
is called a weak Pareto location (or weakly Pareto-optimal)
if there exists no y
2
R
R
d
, such that f
q
.y/ < f
q
.x/
8
q
2
Q
: We denote the
X
wPar
f
1
;:::;f
Q
or simply by
set of weak Pareto solutions by
X
wPar
if this
is possible without causing confusion. A point x
2
d
is called a strict Pareto
location (or strictly Pareto-optimal) if there exists no y
2
R
R
d
, y
¤
x, such that
f
q
.y/
f
q
.x/
8
q
2
Q
: Analogously, the set of strict Pareto solutions is denoted
X
sPar
f
1
;:::;f
Q
,orsimplyby
X
sPar
by
if this is possible without causing
X
wPar
and in case we are considering
strictly convex functions these three sets coincide. Finally, we recall that Warburton
(
1983
) proved the connectedness of the set
X
sPar
X
Par
confusion. Note that
X
Par
when the functions are convex.
