Geoscience Reference
In-Depth Information
treated in the literature, but point out common characteristics and common solution
techniques which are used for many different types of such location problems.
Applications in the location of dimensional facilities are various: These range
from real-world applications in location theory and operations research to appli-
cations in robust statistics and computational geometry. Particular applications are
mentioned at the beginning of the respective sections.
The chapter is organized as follows. We start with a general introduction into the
topic in Sect.
7.2
where we introduce the basic notation, define the problems to be
considered and mention the properties on which we will put some focus later on. We
then discuss the two most extensively researched structures in dimensional facility
location: The location of lines and hyperplanes in Sect.
7.3
and the location of circles
and hyperspheres in Sect.
7.4
. We finally review other interesting extensions and
problem variations in Sect.
7.5
. The chapter is ended by some conclusion in Sect.
7.6
summarizing the findings and pointing out lines for further research.
7.2
Location of Dimensional Facilities
The location of dimensional facilities is a natural generalization of locating one or
more points. As in classical location problems we have given
D
of existing facilities or existing points with
positive weights
w
j
>0;j
D
1;:::;n,and
a finite set V
Df
v
1
;:::;v
n
g
R
D
D
!
a distance measure d
W
R
R
R
evaluating the distance for each pair of
D
. We mostly consider distances derived from norms or gauges.
points in
R
We look for a new facility X which minimizes a function of the weighted distances
to the existing points
0
1
w
1
d.X;v
1
/
w
2
d.X;v
2
/
:
:
:
w
n
d.X;v
n
/
@
A
minimize f.X/
D
g
;
(7.1)
where the most common functions used for f are the minsum (or median) function,
i.e., g
1
.y
1
;:::;y
n
/
D
P
jD1
y
j
or the minmax (or center) function given as
g
max
.y
1
;:::;y
n
/
D
max
jD1;:::;n
y
j
. Also, other objective functions such as the
centdian, or more general, ordered median objective functions g
(see Chap.
10
)
are possible.
If the new facility X is required to be a point, or a set of points, we are in the
situation of classical continuous facility location, see Drezner et al. (
2001
). In this
chapter, however, we assume that X is not a point but a dimensional structure such
as a line, a circle, a hyperplane, a hypersphere, a polygonal line, etc. This, in turn,
means that the distance d.X;v/ in (
7.1
) is the distance between a set X (which
