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this modeling tool allows to distinguish the roles played by the different parties
in the network inducing new type of distribution patterns, see Kalcsics et al.
(
2010a
,
b
). This type of formulation incorporates flexibility through rank dependent
compensation factors, and it allows one to model that the driving force in a
distribution problem is shared by its different parties.
The goal of the ordered median location problem is to minimize the
ordered weighted average of the distances or transportation costs, between the
clients/demand points and the server, once we have applied rank dependent
compensation factors on them. These rank dependent weights allow, for instance,
to compensate unfair situations. Indeed, if a solution places a set of facilities so that
the accessibility cost of a demand point at j is in the s-th position in the ordered
sequence of cost between each client and its corresponding server and the cost of
a demand point at j
0
is in the t-th position with s<t, the model tries to favor j
with respect to j
0
by assigning weights
s
t
. (Note that these weights do not
penalize site j
0
but instead they compensate site j because these lambdas reduce
the dispersion of the costs.) In order to incorporate this ordinal information in the
overall transportation cost, the objective function applies a correction factor to the
transportation cost for each demand point (to reach the facility) which is dependent
on the position of that cost relative to similar costs from other demand points. For
example, a different penalty might be applied if the transportation cost of a demand
point at j was the 5th-most expensive cost rather than the 2nd-most expensive, see
Boland et al. (
2006
), Marín et al. (
2009
), Nickel and Puerto (
2005
), Puerto and
Fernández (
2000
), Rodríguez-Chía et al. (
2000
). It is even possible to neglect some
costs by assigning a zero penalty. This adds a “sorting”-problem to the underlying
location problem, making formulation and solution more challenging.
This type of objective function has been extensively studied and successfully
applied in a variety of problems within the literature of Location Analysis. Puerto
and Fernández (
2000
) and Papini and Puerto (
2004
) characterize the structure of
optimal solutions sets. Rodríguez-Chía et al. (
2000
,
2010
), Blanco et al. (
2013
,
2014a
), Espejo et al. (
2009
), Nickel et al. (
2005
), Drezner (
2007
) and Drezner
and Nickel (
2009a
,
b
), among others, develop algorithms for different continuous
ordered median location problems. In addition, there are nowadays some successful
approaches available when the framework space is either discrete (see Boland et al.
2006
; Domínguez-Marín et al.
2005
;Espejoetal.
2009
; Marín et al.
2009
,
2010
;
Puerto et al.
2011
,
2013
,
2014
) or a network (see Berman et al.
2009
; Kalcsics et al.
2003
,
2002
; Nickel and Puerto
1999
; Puerto and Tamir
2005
; Puerto and Rodríguez-
Chía
2005
).
The aim of this chapter is to introduce the reader into the field of ordered median
location providing some modeling tools and properties. These elements will allow
to formulate and solve location problems in different solution spaces (continuous,
networks and discrete settings) using this unifying tool. To achieve this goal, in
the next section we formally introduce the family of ordered median functions
(
OMf
). Sections
10.3.2
,
10.4
and
10.5
are devoted to analyze the ordered median
location problem in three different frameworks: continuous, networks and discrete,
respectively. The chapter ends with some concluding remarks.
