Geoscience Reference
In-Depth Information
there exists a location for X such that all existing points are covered, this location
is clearly an optimal solution with objective value zero both for the minsum and for
the minmax problem. If it is not possible to cover all points, the minsum and the
minmax problem usually have different solutions.
A paper dealing with the location of a two-dimensional facility is Brimberg and
We s o l ow s ky (
2000
) where the rectangular distance is considered and special cases
could be transformed to classical point location problems. In the context of facility
layout the location of a rectangular office with minsum and minmax objective
function has been studied in Savas et al. (
2002
), Kelachankuttu et al. (
2007
)and
Sarkar et al. (
2007
). In these papers, already existing offices are treated as barriers.
Various problem variations for the location of an axis-parallel rectangle (with fixed
circumference, with fixed area, with fixed aspect ratio, or with fixed shape and size)
have been considered in Brimberg et al. (
2011b
). For most cases, a finite dominating
set could be derived.
The location of a two-dimensional ball
2
W
d.x;y/
r
g
B
x
Df
y
2
R
with given and fixed radius r has been considered in Brimberg et al. (
2013a
) both
for the minsum and the minmax objective function. Note that the distance between
B
x
and v
d.B
x
;v/
D
min
y2B
x
d.y;v/
is measured as the closest distance to any point in B, and not only to points on its
circumference S
x;r
. This means that
d.B
x
;v/
D
0 if v
2
B
x
d.S
x;r
;v/ otherwise.
Hence, Lemma
7.5
yields that d.B
x
;v/is a convex function and consequently, the
resulting optimization problems are much easier to solve than the circle location
problems of Sects.
7.4.3
and
7.4.4
. We remark that the location of a full-dimensional
ball has the following interesting interpretation as a point location problem with
partial coverage
:
Assume that we are looking for a new facility x
2
2
for which we know that
little or no service cost (or inconvenience) is associated with existing points that are
within an acceptable travel distance r from x. Thus, costs will be associated only
to those existing points that are further away from the facility than this threshold
distance r. If we assume that these costs are proportional to the distance in excess
of r, the resulting problem is equivalent to the location of a ball with radius r, and its
center point is the optimal location x we are looking for. This has been pointed out
in Brimberg et al. (
2013a
) where the behavior of the optimal solution with respect
to the threshold distance r is studied.
R
