Geoscience Reference
In-Depth Information
The variables defined in the system process functions are then separated into
clusters
, over which different policy makers (or others relevant to the planning
problem) have differing degrees of relative influence. The methodology of analytic
hierarchies is applied once again, this time to define weights of relative influence
of policy makers over clusters of decision variables. A
composite scenario
is then
assembled by bringing together all clusters of variables, with the values derived
from the corresponding clusters of variable values drawn from preference scenarios
weighted by the relative influence of policy makers over those clusters. This
composite scenario, however, will not likely be consistent (by the earlier definition)
with the set of system process functions since the relative weights over clusters are
derived independently and the composite scenario is not a linear combination of
consistent scenarios.
To show this analytically,
recall
that
the
Impact Analysis Form
is
* *
éù éù
x
D
specified
as
and
the
Planning Form
is
specified
x
=
=
Hf
=
f
êú êú
êú êú
x
L
ëû ëû
*
D
éù é
x
ù
as
x
%
=
=
Gx
=
x
. We define three new final demand vectors
f
1
,
f
2
and
êú
ê
ú
f
(
IA
-
)
êú
ë
û
ëû
f
3
that correspond to three different possible
future
scenario
s.
For each final demand
vector, we can compute the generalized impact of each as
x
i
¼
Hf
i
for
i
¼
1, 2 and
3. For convenience, if we define
f
1
⋮
f
2
⋮
f
3
] as
a
matrix, the columns of which
are the final demand vectors, then it is easy to define
F
¼
[
as the matrix of
corresponding generalized impact vectors, which can be expressed as
X
¼
x
1
⋮
x
2
⋮
x
3
½
é
*
*
*
ù
xx
x
XHFxxx
xxx
[
]
1
2
3
==
ê
ú
1
2
3
ê
ú
ë
û
1
2
3
These scenarios (each of which is specified in a column of
X
) are
consistent
as
x
*
¼
DLf
i
.
We can define a
composite scenario
as a linear combination of these future
scenarios. For example, consider the composite scenario
defined earlier, since, in each case,
f
c
defined as a simple
X
3
i
¼1
β
i
f
i
, where
average of the future scenarios, i.e.,
f
c
¼
β
i
¼
1/3 for
i
¼
1, 2 and
3 (note that
X
3
i
¼1
β
i
¼
x
*
1). It is easy to show that
¼
DLf
c
, which confirms that the
composite scenario is consistent as defined above. Blair (
1979
) shows that this is
true for any linear combination of consistent future scenarios (not just a simple
av
er
age as above). In the example, we can define the consistent composite scenario
as
x
c
¼
Hf
c
¼
H
β
1
f
1
þ
β
2
f
2
þ
β
3
f
3
½
.
*
éù
=
ê
ê
ëû
x
éù
=
ê
ê
ëû
x
*
c
and if
x
*
Since
x
, we write
x
¼
Dx
c
, then the composite scenario is
c
x
x
c
consistent as well as defined above; this will be true for any linear combination of
