Geoscience Reference
In-Depth Information
Chapter 7
Location of Dimensional Facilities
in a Continuous Space
Anita Schöbel
Abstract
In many cases, the facilities to be located cannot be represented by
isolated points, but may be modeled as dimensional structures. Examples for one-
dimensional facilities are straight lines, line segments, or circles while boxes, strips,
or balls are two-dimensional facilities. The goal of this chapter is to review the
location of lines and circles in the plane and the location of hyperplanes and
hyperspheres in higher dimensional spaces. We also discuss the location of some
other dimensional facilities. We formulate the resulting location problems, point
out some of their important properties and review the basic solution techniques and
algorithmic approaches. Our focus lies on presenting a unified understanding of the
common characteristics these problems have, and on reviewing the new findings
obtained in this field within the last 10 years.
Keywords
Circle location Hyperplane location Line location
7.1
Introduction
Within the locational context, the problem of locating a dimensional facility was
first posed in Wesolowsky (
1972
,
1975
) where the location of a line minimizing the
sum of rectangular or Euclidean distances to a set of existing points was introduced.
Since this time, the subject of locating lines and hyperplanes, circles, spheres,
and other dimensional facilities has been intensively studied. Surveys are given in
Martini and Schöbel (
1998
), Díaz-Bánez et al. (
2004
), an extensive list of papers
dealing with the location of dimensional structures (most of them before 2000) is
also given in Blanquero et al. (
2009
).
Within the last 10 years, many new results have been found and published. In
this chapter, one goal is to review these new findings. More importantly, another
goal is to present a unified understanding of the subject which is now possible since
the field has become more mature. We hence not only present a list of problems
