Demand Point Aggregation for Some Basic Location Models - Location Science - page 483

and/or error bounds, together with ADP-DP distances. The bound in the inequality

LJ LJ LJ

1-f.S W V/=f.S 0 W V/ LJ LJ LJ

2 eb =f .S 0 W V/seems particularly promising.
An alternative to using some low computational order approach to aggregate
the original demand point set, and then solving the resulting aggregated location
model to optimality, is to use some low computational order metaheuristic approach
(Pardalos and Resende 2002 ;Reeves 1993 ; Resende and de Sousa 2004 ) to approxi-
mately minimize the original, unaggregated location model. The first approach gives
bounds on optimality to the original model. The second approach introduces an
additional source of error, since a heuristic is used, but may possibly result in a
better solution. Given the current state of the art, which approach is best is not
known. Indeed, “best” may not even be well-defined, since there is no generally
accepted measure of aggregation error.

18.5 Error Bounds

We have argued that an upper bound on the absolute error is among the best
representations and measures of the error associated with an aggregation. We have
used the symbol eb to represent this upper bound so that with f ( S , V ) a general
location model, j f.S W V 0 / -f.S W V/ j eb .
Consider now obtaining error bounds for the PMM and PCM, say eb pmm and
eb pcm , with these two models defined in Examples 1 and 2 respectively. Both error
bounds are direct consequences of the triangle inequality for shortest distances,
which holds for all j 2 J and all S ǝ:
d v 0 j ;v j D S;v 0 j -D S;v j d v 0 j ;v j
() LJ LJ LJ D S;v 0 j -D S;v j LJ LJ LJ d v 0 j ;v j : (18.1)

The p -median and the p -center models have the following error bounds respec-
tively:
eb pmm D X n w j d v 0 j ;v j W j 2 J o ;eb pcm D max n w j d v 0 j ;v j W j 2 J o :

The error bounds themselves can be viewed as location models; if v j is the closest

ADP to v j (which is reasonable), then we have
eb pmm D X ˚ w j D ;v j W j 2 J ;eb pcm D max ˚ w j D ;v j W j 2 J :

Since it is of interest to have small error bounds when doing aggregation, we can
view each of the latter two error bound expressions as a location model, and use