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Fig. 6.2 Intersection of
covered areas from ext.X i /
giving the region which is
covered by all points in the
box. The integral is computed
over the inscribed circle of
this region, I i .X i /
I (X
)
i
i
)
I (X
*
i
i
X i
In what follows, the so obtained bound will be denoted by LB . X /,
Z
Z
p
p
X
X
LB . X / D
d.a/
d.a/:
I i .X i / T I j .X j /
I i .X i /
iD1
i;jD1
i<j
Notice, that the integral could be computed directly as R A f.a/min x 2 X c.a; x / da ,
but that is not practical for the all-or-nothing covering function. Numerical inte-
grators take many sample points around discontinuities, that are introduced with
c.a; x /, therefore taking a very long time for a single integration.
6.4
Numerical Examples
The branch and bound method outlines above was implemented in Fortran 90
(Intel©Fortran Compiler XE 12.0), using the integration tools of the IMSL Fortran
Numerical Library. Executions were carried out on an Intel Core i7 computer with
8.00 Gb of RAM memory at 2.8 GHz, running Windows 7.
Two types of experiments were performed. First, a series of problems with
randomly generated demand functions were solved for p D 1 and p D 2: The
demand function was generated as a mixture of r bivariate gaussian distribution
functions ( 6.16 ) with centers and weights uniformly generated in Œ0;10 2 and
Œ0:1;0:1 C 1=.10r/, respectively. We set the covariance matrix to w i E,thatis
the identity matrix scaled by the knot weight. The location of the facilities were
sought in the square Œ2;8 2 . Three parameters were considered, leading to different
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