Geoscience Reference
In-Depth Information
Fernández
2009
; Muyldermans et al.
2003
). However, this leaves considerable
freedom for assignments below the maximal distance and typically increases the
overlap. A slightly different approach is based on the maximal pairwise distance
and the weighted sum of pairwise distances
cmp
spw
.D
k
/
D
X
i;j2D
k
;i¤j
cmp
mpw
.D
k
/
D
max
i;j2D
k
;i¤j
d
ij
w
i
w
j
d
ij
(Ríos-Mercado and Salazar-Acosta
2011
and Blais et al.
2003
, respectively).
In case of measures based on the sum (maximum) of distances, the global
compactness of a districting plan is then usually also computed as the sum
(maximum) over all districts. But sometimes also a sum-max combination is used
or a convex combination of sum and max (Muyldermans et al.
2003
; Silva de Assis
et al.
2014
;Butschetal.
2014
).
23.4.4.3
Mathematical Modelling
The majority of districting models has compactness as an objective function to
be optimized. In addition, sometimes the maximal distance between a basic unit
and its district center or between two basic unit of the same district is restricted
(Benzarti et al.
2013
). The appeal of distance-based measures is that they easily
give rise to linear or, in case of pairwise distances, quadratic expressions. Therefore,
these measures are sometimes also used for polygonal basic units, even if geometric
measures could have been applied (Ríos-Mercado and Fernández
2009
).
23.4.5
District Center
Strictly speaking, determining district centers is in most cases not an optimization
criterion in itself. However, several measures for contiguity and compactness rely on
district centers. Thus, if no centers are predefined for the districts, seeking district
centers is part of the optimization process. Typically, a district center is the basic
unit of the district that minimizes the respective compactness measure. But also the
(weighted) center of gravity can be used to determine a district center. Note however
that this center usually does not coincide with a basic unit, which is problematic if
distance computations are based on road networks.
23.4.6
Other Criteria
There are a few other criteria for districting problems that are included from time to
time in districting models. For example, for re-districting problems the changes in
