Geoscience Reference
In-Depth Information
¼
v
0
x
coefficients and, hence,
q
as the total value added of an economy associated
with a vector of total outputs—the economy's gross domestic product, we can
rewrite the generalized input output formulation as a linear programming
(LP) problem to determine the minimum value of
q
that satisfies deliveries to
final demand or, equivalently, what values of the elements of
x
minimize
q
while
satisfying final demand?
11
0
Min
q
¼
v
x
subject to
: Gx x
Since the input-output model has a unique solution for a given level of final
demand, and those conditions are part of the constraint equations,
, then
either of two situations apply: (1) the additional constraints beyond the Leontief
conditions included in the constraint equations (e.g., energy, environmental and
employment equations) over-constrain the problem, i.e., present conflicting
constraints, so that there is no feasible region and, hence, no possible LP solution
or (2) the additional conditions are not binding constraints, i.e., they are fully
satisfied by the Leontief conditions. In our example so far the latter situation applies
and the LP solution is identical to the impact analysis solution determined earlier. In
the former case, however, when there is no feasible region, we will need to resort to
other approaches in order to find a solution, e.g., multiobjective programming
techniques.
12
Even the basic LP formulation to the generalized input-output planning problem
gives us the flexibility to include alternative or multiple objective functions in
approaching planning problems. For example, one might be interested in
minimizing the value added cost to meeting a target final demand, minimizing
pollution emissions and energy conservation all as goals. However, the problem of
conflicting constraints requires other methods of solution. As an illustration, linear
goal programming (GP) is one commonly applied extension to LP that
accommodates multiple objectives in a very straightforward manner and can be
used to extend the Leontief framework to deal with environmental issues involving
conflicting objectives and constraints.
Gx x
11
More extensive economic interpretations of the Leontief model as a linear programming
problem are included in Dorfman et al. (
1958
) and Intriligator (
1971
). An important advantage
of posing the input-output framework in this way is that we can include alternative objective
functions and/or additional constraints as part of a planning problem.
12
Decision making with multiple objectives is another well-developed area in operations research
with many approaches available. Surveys of such approaches are found in Cohen (
1978
), Cochrane
and Zeleny (
1973
), Nijkamp and Rietveld (
1976
), Trzaskalik and Michnik (
2002
), and Tanino
et al. (
2003
).
