Geoscience Reference
In-Depth Information
principal objective of the study. The team spent a substantial effort discussing with
their Alachua County sponsor possible principal objectives for the study; eventually
they agreed upon the following idealized one: minimize the total number of DRCs
needed, subject to each county resident being within a specified distance r (called
the “radius”) of a closest DRC. Thus if B ( s , r ) denotes the set of all points in the
plane whose distance from a given point s is at most r , a requirement meaning that
each county resident location must be in at least one B ( s , r ) for some DRC location
s ; that is, each resident point in the county must be “covered” by at least one B ( s , r )
for some DRC location s . Hence the location requirement specifies a “covering”
problem (see Chap. 5 ) . It was the belief of the team (eventually confirmed) that if
they could solve this idealized problem meeting the location requirement, then they
could find nearby locations that would meet all the other requirements.
A natural and important question was how to measure distances between points.
Ideally, shortest path distances on the existing road network would have been used,
but these were unavailable due to the very limited study budget. Since the county
had a largely rectilinear/right-angle road network, the team, with the agreement
of its sponsor, settled on the use of rectilinear distances: for any planar points
s D (s 1 , s 2 ), t D (t 1 , t 2 ), d(s , t ) Dj s 1 t 1 jCj s 2 t 2 j defines the rectilinear distance
between s and t.
We refer to resident locations as “demand points”, abbreviated as DPs. For any
real aggregation location problem, obtaining and dealing with DP data will very
probably be a major part of the problem-solving effort. Interaction with the county
property appraiser's office elicited the information that principal DP data sources
could be obtained from GIS data available in a library, and from the county property
appraiser's office. The county DP data was arranged by “parcels” of land. There
were about 6,600 parcels, and for each parcel the following information was known:
x and y coordinates of the parcel center, the total heated square footage of the parcel
buildings, and whether parcel buildings were residential or commercial. The parcel
locations were used as residential location/DPs, and as possible DRC sites. As many
as 3,900 of the parcels seemed possibly usable for DRC sites, as they had public or
commercial buildings whose total usable footage exceeded 2,000 ft 2 . Figure 18.1
shows a plot of all the DPs, as well as the aggregated DPs (yet to be discussed).
Covering models are discussed in Chap. 5 ; they provide a way to compute, for
a specified covering radius r , a minimal set of locations, say S Df s 1 , :::, s k g ,so
that each DP is contained in at least one B ( s i , r ). To formulate the covering problem
using all the available data as an integer program model would give a constraint
matrix with about 6,600 rows and 3,900 columns. The size of this model was
beyond the resources available to the team to deal with. The covering algorithm
readily available to the team was one in Excel, which could deal with at most 200
variables/columns. The need to somehow aggregate the DP data and the potential
site data thus became quite evident.
In a later section we discuss a useful error bound for covering location problems,
max n d v j ;v 0 j W j D 1;:::;n o ,where v j is the location of DP j ,and v j is the ADP
(aggregate DP) that replaces v j ;the v j are distinct while the v j
are not. Choosing the
Search WWH ::




Custom Search