Anti-covering Problems - Location Science - page 113

For such undesirable facilities, a push model, pushing facilities away from the sites
affected by facilities nearness, is more suitable: the location for the facilities is
then sought maximizing a certain non-increasing function of the distances from the
individuals to the facilities. For both desirable and undesirable facilities, interactions
may be measured as a function of the individual-facility distance (or time), or, as
studied here, via coverage ; see e.g. Kolen and Tamir ( 1990 ), Li et al. ( 2011 ), Murray
et al. ( 2009 ), Schilling et al. ( 1993 ) for extensive reviews on covering models and
solution approaches. It is important to stress here that, independently of the nature of
the facility, either attractive or repulsive, the very same models for covering function
apply (Farhan and Murray 2006 ), the difference being algorithmic: such covering is
to be maximized for desirable facilities and minimized for undesirable facilities.
On top of individual-facility interactions, facility-facility interactions are also
likely to be relevant. Such interactions may be critical when facilities are obnoxious,
and risk or damage to population scales nonlinearly (e.g., with hazardous materials
deposits or dangerous plants which may suffer chain reactions) and thus negative
impacts are to be dispersed. Facility-facility interactions are also important in
models for locating facilities which, although they are perceived as attractive by
the users, they are perceived as repelling by other facilities competing for the very
same market. In these models, locating the facilities far away from each other
avoids cannibalization and optimizes competitive market advantage (Christaller
1966 ; Curtin and Church 2006 ; Lei and Church 2013 ).
Although the models described are general, the algorithmic approach presented
here is restricted to the planar case (Drezner and Wesolowsky 1994 ;Plastria 2002 ;
Plastria and Carrizosa 1999 ): facilities are identified with points in the plane, and
interact with the remaining facilities and with individuals, also identified with points
in the plane. Interactions are measured via distances in the plane. See Plastria ( 1992 )
for an excellent review of planar distances and planar location models and e.g.
Berman et al. ( 1996 ), Berman and Huang ( 2008 ), Berman and Wang ( 2011 )for
covering models for which interactions are not measured via planar distances, but
network distances instead, typically shortest-path distances.
Contrary to most papers in the literature, affected individuals are not assumed
here to be concentrated at a finite number of points, and, instead, an arbitrary
distribution (in particular, a continuous distribution) on their location is given. This
way we can directly address models in which affected individuals are densely spread
on a region, but we also address models in which uncertainties exist about the exact
location of the individuals, due to their mobility (Carrizosa et al. 1998b ).
Regional models are not so common in the location literature, since, even when
individuals are assumed to be continuously distributed, a discretization process is
usually done, and such continuous distribution is replaced by a discrete one, by e.g.
replacing all points in each district by its centroid, or other central point, see e.g.
Francis and Lowe ( 2011 ), Francis et al. ( 2008 , 2000 , 2002 ), Murray and O'Kelly
( 2002 ), Plastria ( 2001 ), Tong and Church ( 2012 ). Nevertheless, discretization is well
known not to perform well in applications, this issue being especially relevant in
covering models, since significant discrepancies may exist between what is modeled
as covered and what is actually covered, see e.g. Current and Schilling ( 1990 ),