Demand Point Aggregation for Some Basic Location Models - Location Science - page 470

the location of branch banks (Chelst et al. 1988 ), tax offices (Domich et al. 1991 ),
network traffic flow (Sheffi 1985 ), and vehicle exhaust emission inspection stations
(Francis and Lowe 1992 ) a popular aggregation approach is used: to suppose every
DP in each postal code area or zone of the larger urban area is at the centroid of
the postal code area or zone, and to compute distances accordingly. The result is a
smaller model to deal with, but one with an intrinsic error. If the modeler wishes
to aggregate to have a small number of aggregate demand points (abbreviated as
ADPs), and also desires a small error, then aggregation becomes a nontrivial matter.
It is tempting to ask the following question: How many ADPs are enough?
There are no general answers to this question. This is because there are important
tradeoffs in doing aggregation. Aggregation often decreases: (1) data collection cost,
(2) modeling cost, (3) computing cost, (4) confidentiality concerns and (5) data
statistical uncertainty. The first four items seem self-explanatory; item (5) occurs
because aggregation leads to pooled data, which provides larger samples and thus
smaller sample standard deviations. The price paid for aggregation is increased
model error: instead of working with the actual location model we work with some
approximate location model. How to trade off the benefits and costs of aggregation
is still an open question. The question is open in part because there is no general
agreement on how to measure the aggregation error, and also because there is
no accepted way to attach a cost to model aggregation error. To the best of our
knowledge, professional judgment is generally used to do the tradeoffs. Francis et al.
( 2009 ) provide a survey of various demand point aggregation error measures, and
an extensive literature discussion. In fact, much of the early material in this chapter,
and Table 18.4 , is from that paper.
One can categorize location models as strategic , tactical ,or operational in scope.
As pointed out by Bender et al. ( 2001 ), planar distances are often used for strategic-
level location models, and network distances for tactical-level location models. Such
models are often converted to equivalent mixed integer programming (MIP) models
for solution purposes, using some finite dominating set principle to reduce the set
of possible locations of interest to a finite set (Hooker et al. 1991 ). Thus results to
follow for these planar and network models also apply to their MIP representations,
including those for the p -median, p -center, and covering location models. These
models are the subject matter of Chaps. 2 , 4 , and 5 respectively. Operational-level
location models are not too common (mobile servers are one example), but for
such models no aggregation may be best. Note that the scope of the location model
may well indicate the degree of aggregation; a more detailed scope requires a more
detailed aggregation.

18.2 Terminology and Examples

We suppose that servers and DPs are all either points in the plane, or on some travel
network. In either case, there is some well-defined set of server points and DPs,
say ǝ, and a distance d ( x , y ) between any two points x , y in ǝ.Ifǝ is a travel