Demand Point Aggregation for Some Basic Location Models - Location Science - page 476

Table 18.1 (a) shows how some DRC performance measures changed with various r values for
the idealized stage 1; (b) shows similar results for the actual stage 2 results
(a) Idealized (b) Actual

Travel limit r (miles) 10 15 20 10 15 20

Maximum travel distance (miles) 10.9 15.1 20.3 14.0 25.8 26.94

Average travel distance (miles) 4.9 9.1 7.6 4.86 6.76 7.36

% Parcels covered 99.78 99.96 99.92 97.7 89.8 97.4

Average covering violation (miles) 0.184 0.84 0.184 1.05 2.80 2.55

Note that max f d ( v i , v i 0 ): all unchosen DPs v i g b when the algorithm terminates,
so keeping b small guarantees a small aggregation error. Aggregation error is
discussed in the next section.
Once the DPs were aggregated, and the potential DRC sites were also similarly
aggregated, the covering location problem could be solved. We call the covering
location problem the idealized problem, while we call the one that considers all
eleven criteria the actual problem. The team solved the idealized problem first,
and then sought good solutions to the actual problem that were “close” to those
of the idealized problem. This approach greatly simplified the problem and worked
acceptably.
Because of initial uncertainty about an appropriate value of r , the greatest
distance any resident should need to travel to a closest DRC, it was decided to treat
r as a parameter of the study, try various r values, and then evaluate the resultant
solutions. The team eventually chose r values of 10, 15 and 20 miles (16.1, 24.2 and
32.3 km respectively). By solving the idealized covering model with these three r
values, solutions were found requiring 8, 4 and 3 DRC's respectively; see Fig. 18.2
for the case of 3 DRCs; note Fig. 18.2 illustrates three B ( s ,20) regions. The team
then proceeded to solve the actual problem by finding potential DRC locations near
the idealized solutions which would meet the other evaluation requirements. To aid
in this effort, they and the sponsor developed a score card, much like a grade card,
on which they could score each potential location considered; most of the buildings
considered were schools, churches, recreation centers, or government buildings.
Tab le 18.1a illustrates some DRC performance measures for the solutions to the
idealized problem. Discrepancies between Table 18.1a performance measures and
the three different radius measures are due to aggregation effects, and can be seen
to be quite small. Table 18.1b shows performance measures for the actual problem.
There are some bigger discrepancies than in Table 18.1a , but these locations scored
well on all the other criteria. Also it was recognized that the proper choice of a
radius value r was somewhat subjective.
A number of modeling insights were gained in the course of this study, including
the following. (1) Sponsors may not have a principal objective. (2) The choice of
a model may be somewhat subjective. (3) Getting and working with all the data
can be most of the work in an aggregation/location study. (4) Data aggregation can
be essential and helpful. (5) The covering location model solutions were easy to