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 .
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