Geoscience Reference
In-Depth Information
with
2
.0;1/. The convex combination alleviates some of the weaknesses of
bal
sum
and
bal
max
. The latter does not take into account the balance of all districts
and sometimes yields rather poor solutions on average whereas the former allows a
few highly unbalanced districts to be compensated by some well-balanced districts.
A different global approach computes the range of district sizes (Tavares-Pereira
et al.
2007
)
bal
rng
.
D
/
D
max
kD1;:::;p
w
.D
k
/
min
kD1;:::;p
w
.D
k
/:
23.4.2.1
Mathematical Modelling
In districting models, there is no clear trend on whether to treat balance as a hard
constraint (Hess et al.
1965
; Fleischmann and Paraschis
1988
; Zoltners and Sinha
2005
) or to include it in the objective function (Blais et al.
2003
; Ricca and Simeone
2008
; Silva de Assis et al.
2014
). In the former case, the size of each district is
required to lie between a given lower and upper bound. Some authors even do both
(Bergey et al.
2003
; Salazar-Aguilar et al.
2013b
). All of the above measures easily
give rise to linear expressions.
23.4.3
Contiguity
Almost all districting approaches require districts to be contiguous. In political
districting, this criterion should prevent gerrymandering. For the other types of
applications, contiguous districts reduce the day-to-day travel distances for sales
persons, delivery vans, snow ploughs, mailmen, etc. Unfortunately, a rigid and con-
cise mathematical formulation of contiguity is difficult for basic units representing
points.
23.4.3.1
Graph-Based Measures
If basic units are lines or polygons, it is easy to derive explicit neighborhood
information. For example, two zip-code areas are neighboring if they share a
common border, or two streets if they meet in a crossroad. In the former case,
sometimes an additional requirement is the existence of a direct road connection
between the two basic units. In general, two basic units are called
neighboring
,if
their geometric representations have a nonempty intersection. This information is
stored in the neighborhood graph G
D
.V;E/, and a district is contiguous if the
basic units of the district induce a connected subgraph in G.
If basic units are represented by points, e.g., customer addresses, it is not
clear how to assess contiguity. Over the years, different surrogate definitions for
