Anti-covering Problems - Location Science - page 115

The basic form of ' is an all-or-nothing function, already suggested in Church
and ReVelle ( 1974 ), see also e.g. Drezner and Wesolowsky ( 1994 ),
c.a;x/ D '. k x a k / D 1; if k x a k R
0; otherwise;

(6.2)

where the threshold value R is called the range (Christaller 1966 )or coverage
standard . For an attractive facility, R represents the highest distance a user is willing
to overcome to utilize a facility, whereas for undesirable facilities, R represents the
distance of the boundary of the zone within which the facility would have a negative
impact (Farhan and Murray 2006 ). Extensions of ( 6.2 ) abound in the literature,
leading to so-called gradual covering models (Berman et al. 2009c , 2003 ; Drezner
et al. 2004 ). For instance the all-or-nothing function above is replaced by a piecewise
constant function modeling different levels of coverage in Berman and Krass ( 2002 ),
by a piecewise linear function in Berman et al. ( 2003 ), Berman and Wang ( 2011 ),
Drezner et al. ( 2004 ), or by more general nonlinear functions, such as the logistic
model

1
1 C exp .Ǜ a C LJ a k x a k / ;

c.a;x/ D '. k x a k / D

(6.3)

in Fernández et al. ( 2000 ), see also Berman et al. ( 2010 , 2003 ), Karasakal and
Karasakal ( 2004 ). Observe that in some of the papers cited above the coverage
functions c are introduced for attractive facilities, and thus maximization, instead
of minimization, is pursued. However, the models for c are the very same.
Expressions above for c,as( 6.2 ), are adequate just for the single-facility case.
When several facilities are to be located, the covering model ( 6.1 ) can be extended
in several ways, by first defining, for each facility i D 1;2;:::;p; the function
' i converting distances into coverage. In the simplest and most popular model in
the literature, for a p-tuple of facility locations x D .x 1 ;:::;x p /; covering c of an
individual location a by x is given by

c.a; x / D max
1ip

c i .a;x i /: (6.4)

In the particular form of individual covering c i givenby( 6.2 )using' i instead of '
and R i instead of R; one considers the individual location a to be covered by the
p-tuple of facility locations x D .x 1 ;:::;x p / if it is covered by at least one of the
p facilities, i.e., if at least one facility i is at a distance smaller than its threshold
value R i :
Multifacility covering functions other than ( 6.4 ) can be found in the literature,
see Berman et al. ( 2010 ) for an updated review. One may consider fuzzy operators