Geoscience Reference
In-Depth Information
a
b
Fig. 23.4
Districting
plans
for
two
center-based
compactness
measures
with
contiguity.
(
a
) Districts for
cmp
ud
./.(
b
) Districts for
cmp
wd
2
./
overlap and travel distances are between the other two measures, with the former
being slightly better.
The situation is different if we try to enforce contiguity. Assume that an approxi-
mate neighborhood graph has been computed using the approach in Haugland et al.
(
2007
). Using the contiguity constraints of Shirabe (
2009
), the resulting districting
plans for
cmp
ud
.
/ and
cmp
wd
2
.
/ are shown in Fig.
23.4
. The separation between
the districts for
cmp
ud
.
/ is clearer than before. However, even if the total length of
the TSP tours reduces considerably (from 92.78 to 81.15), the districts consisting
of the diamond, pentagon, and square shaped basic units are still distorted and will
receive little approval from planners. (The square shaped district is connected since
there exists an edge along the top of the point set.) For
cmp
wd
2
.
/ the overlap is not
much different from the previous plan, and the total travel distance even slightly
decreased to 72.97. The main difference is that the centers are now all included in
their districts, if only at the boundary.
This example illustrates the considerable differences between districting plans
for different compactness measures and the influence of contiguity constraints.
However, this is just a single example, and the observations cannot be generalized
without further testing. Also, the length of a TSP tour is just an indicator for travel
distances, as a sales person may not visit all customers on a single day.
The fact that squared distances produce compact but non-contiguous districts
for fixed centers has been observed several times in the past (Hojati
1996
;
Schröder
2001
). An important factor influencing the shape of districts is the spatial
distribution of the district centers. If they are spread evenly, the differences between
the measures in terms of district overlap will decrease, see Example
23.3
.However,
this uneven distribution is not unusual as sales force residences often concentrate
in certain areas, e.g., larger cities, and sometimes even have the same address. Also
the threshold for the allowed balance deviation has an impact on the compactness of
solutions. The smaller the threshold value is, the larger the overlap between districts
will get.
Instead of taking the sum, one could also take the maximum for each of
the center-based measures (cf. Elizondo-Amaya et al.
2014
; Ríos-Mercado and
