Geoscience Reference
In-Depth Information
eb/f
(
S
:
V
)and
eb/f
(
S
:
V
0
), in which case the model ratios would be nearly one and
we would have a good aggregation. By contrast, if the model is on a city/town scale
and the aggregation is also on a city/town scale, we might have a poor aggregation.
We need the aggregation scale to be substantially smaller than the model scale in
order to have a good aggregation
. This is one reason that aggregation may be of
more interest for problems of city/town/regional scope than those of national or
international scope.
There is another way to view the use of an aggregation error bound. The error
bound allows us to draw conclusions about a family of original models, instead of
just one. If the actual location model is
F
(
S
:
V
) instead of
f
(
S
:
V
), but the error bound
applies to both, that is
j
f.S
W
V
0
/ -F.S
W
V/
j
eb
and
j
f.S
W
V
0
/ -f.S
W
V/
j
eb
for all
S
, then whatever conclusions we draw about the function
f
using the error
bound inequality also apply to the function
F
. While we lose accuracy when we
aggregate, we gain the ability to draw approximate conclusions about a family of
original functions. As a general example of the function
F
, suppose instead of the
DP set
f
v
j
:
j
2
J
g
we have a different DP set, say
f
b
j
:
j
2
J
g
, defining
F
, while all
other model data is the same as for
f
(
S
:
V
). If each DP
b
j
is aggregated into
v
j
,then
each of the functions
F
and
f
will be aggregated into the same approximating model,
denoted by
f
0
. Further, if also d
v
j
;v
0
j
D
d
b
j
;v
0
j
for
j
2
J
, then the methods
we present later would provide both
F
and
f
0
,and
f
and
f
0
, with the same error
bound. The data for
F
and
f
differ, but are sufficiently similar that the aggregation
does not detect the differences.
Denote (globally) minimizing solutions to any original and approximating loca-
tion models
f
(
S
:
V
)and
f
(
S
:
V
0
)by
S*
and
S
0
respectively. While we usually cannot
expect to find
S*
if we must aggregate DPs, we can still obtain some information
about
S*
if we know an error bound
eb
and
S
0
. Geoffrion (
1977
) proves that if
j
f.S
0
W
V
0
/-f.S
W
V/
j
eb
,then
j
f.S
0
W
V/-f.S
W
V/
j
2
eb
. Supposing
f
(
S
0
:
V
) > 0, we thus have
LJ
LJ
LJ
1-f.S
W
V/=f.S
0
W
V/
LJ
LJ
LJ
2
eb
=f .S
0
W
V/. Hence, if
2eb
is small relative to
f
(
S
0
:
V
), we may reasonably accept
S
0
as a good substitute
for
S*
. We assume henceforth that we can compute
S
0
but not
S*
. Note that if we
wish to use
S
0
to approximate
S*
, then it makes no sense to allow
p
q
,forthenwe
can place a new facility at every ADP and may achieve a minimal approximating
function value of
f
(
S
0
:
V
0
)
D
0. Certainly it is desirable to have
p
q
.
Various authors, cited in Francis et al. (
2009
), have proposed different types of
optimality errors which we list in Table
18.3
. The first error can be computed, and
indicates how well the approximating function estimates the original function at
S
0
.
For large models, the second two errors cannot be computed without knowing
S*
.
They can be computed for smaller models where
S*
can be found without the need
to aggregate, or for larger models if one
assumes
the algorithm used to solve the
original problem provides
S*
. Unless one can be certain that
S*
is known, or that
some properties of
S*
are known, the latter two measures do not seem useful.
Although it is reasonable to use some measure of the difference between
f
(
S
:
V
0
)
and
f
(
S
:
V
) to represent aggregation error, doing so results in what may well be called
the
paradox of aggregation
(Francis and Lowe
1992
). Often our principal reason to
