Geoscience Reference
In-Depth Information
23.3.1
Basic Units
A districting problem comprises a set J Df 1;:::;n g of basic units , sometimes
called sales coverage units, basic areas, or geographical units. Each basic unit
represents a geometric object in the plane: a point, e.g., a geo-coded address, a
line segment, e.g., a street, or a polygonal area, e.g., a zip code area, county, or
predefined company trading area. The distance between two basic units i;j 2 J is
denoted as d ij D d.i;j/. Typical examples for d ij are Euclidean (cf. Fleischmann
and Paraschis 1988 ) or road distances (cf. RĂ­os-Mercado and Salazar-Acosta 2011 ).
The latter have the advantage that they can properly reflect obstacles like rivers
or mountain ranges. For non-point objects, distances are either computed between
representative points, e.g., the midpoint of a street or the centroid of a polygon, or
as the surface-to-surface distance.
Moreover, one or more quantifiable attributes, called activity measures ,are
associated with each basic unit. Typical examples are service times, estimated sales
potential, or number of voters. Sometimes, they also include an estimate of the travel
time for visiting the basic unit (Jarrah and Bard 2012 ). The activity measures are all
assumed to be deterministic. Let w j denote the q-th activity measure of basic unit
j 2 J, 1 q Q,whereQ is the number of different attributes to be considered.
If Q D 1, the superscript is usually omitted.
If explicit neighborhood information is given for the basic units, then G D
.V;E/ denotes the neighborhood or contiguity graph where v j 2 V corresponds
to j 2 J and f v i ;v j g2 E iff basic units i and j are neighboring. The length of
edge f v i ;v j g is d ij . Finally, N.j/ V denotes the set of basic units adjacent to
v j 2 V .
23.3.2
Districts
A district D k , 1 k p, is a subset of basic units, where p is the total number
of districts. The number of districts can either be fixed in advance, e.g., the number
of political districts to create or the number of available nurses for elderly care, or
be subject to planning, e.g., the minimal number of salespersons required to service
all customers or the minimal number of patrol cars to ensure a certain response
time. The q-th activity measure of a district is the sum of the activity measures of
its basic units, i.e., w q .D k / D P j2D k w j .ForQ D 1, w 1 .D k / is simply called
the size of the district. Note that sometimes the size also includes an estimate of
the (expected) travel time. However, as travel times are represented through the
compactness criterion, we refrain from including them and just mention when this
may change things.
In some applications the location c k of a facility is associated with each district
D k . This may be some predefined site, e.g., a hospital providing preventive medical
care, or be an outcome of the districting process, e.g., the optimal location of a sales
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