Geoscience Reference
In-Depth Information
7.4.5.2
Locating Euclidean Concentric Circles
In a recent paper, Drezner and Brimberg (
2014
) introduce the following interesting
extension of the circle location problem: They look for p concentric circles with
different radii r
1
;:::;r
p
which minimize the distances to a given set of points. In
their paper they assume a partition of V into sets V
1
;:::;V
p
and require that each
point in V
i
is served by the circle with radius r
i
.Thismeansthevariablestobe
determined are the center point x
2
2
and the radii r
1
;:::;r
p
of the p circles. The
model is considered for the least squares objective function, the minsum, and the
minmax objective function. Using that
R
d.S
x;r
j
;v
j
/
Dj
d.x;v
j
/
r
j
the objective functions which are considered are given as
p
X
X
w
j
d.x;v
j
/
r
2
f
2
.x;r
1
;:::;r
p
/
D
qD1
v
j
2V
q
p
X
X
f
1
.x;r
1
;:::;r
p
/
D
w
j
j
d.x;v
j
/
r
j
qD1
v
j
2V
q
f
max
.x;r
1
;:::;r
p
/
D
max
qD1;:::;p
max
v
j
2V
q
w
j
j
d.x;v
j
/
r
j
:
Drezner and Brimberg (
2014
) solve the problem by global optimization methods,
using a reformulation of the circle location problem as an ordered median point
location problem (see the location of a Euclidean minsum circle in Sect.
7.4.3
)and
applying the Big-Triangle-Small-Triangle method (Drezner and Suzuki
2004
).
7.4.5.3
Location of a Circle with Fixed Radius
The location of a circle with fixed radius is considered in Brimberg et al. (
2009a
). In
this case, it can be shown that considering every triple of points separately yields an
optimal solution, i.e., a finite dominating set can be derived by solving
3
smaller
optimization problems.
7.4.5.4
Generalized Circle Location: Locating the Unit Ball of One Norm
Measuring Distances with Respect to Another Norm
The circle location problem treated so far is to translate and scale a circle S
Df
x
2
R
2
Wk
x
k
1
g
(derived from norm
kk
) in such a way that the distances to the set V
are minimized, where the distances are measured with respect to the same norm
kk
.