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and, rather, consider them all goal equations that are optimized in a specified
preemptive order of priority.
15
It is important to note that GP provides considerable flexibility in handling
multiple, even conflicting objectives. In LP we are required to begin with a feasible
region in order to find our way to an optimal solution. In GP the feasible region is
essentially unspecified since any point can be completely specified in terms of the
deviational variables and it is the priority order of objectives that determines
the solution, because by specifying a priority order a solution can be found even if
the objectives are conflicting. This places much more of the planning burden on
determining the priority order, which in a policy context is often quite complex,
especiallywith sometimes competing policy objectives, such as economic growth and
environmental quality. Blair (
1979
) employs an approach called
analytic hierarchies
(Saaty
1980
) for this purpose. Others use a wide range of multiobjective decision-
making approaches, such as Nijkamp and van Delft (
1977
) and Cohen (
1978
).
Another commonly cited limitation of linear goal programming (and linear
programming generally for that matter) is that tightly constrained problems are
insensitive to how close one is to a given target because solutions, at least as we
have developed the methodology so far, are developed by satisfying goals
completely in pre-emptive order (ordinal) or lexicographic order. This means that
one has to fully satisfy a higher order goal equation before moving on to the next.
This can lead to illogical solutions, especially in tightly constrained problems. For
example, if an employment goal has a higher priority than, say, a pollution goal,
then the last unit of employment achieved could be at the expense of an enormous
amount of pollution. The literature includes many approaches to address this
problem, such as Lane (
1970
).
As with LP, when the number of variables and equations increases beyond two,
solution procedures become much more complex. However, there are a variety of
solution approaches to GP problems. In GP, as we have just seen with an example,
through sequential imposition of constraints we arrive at a solution—GP is some-
times referred to as “weighting within constraints.” Linear GP problems can be
solved via a basic simplex algorithm similar to that commonly used in LP.
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13.7 Applications to the Generalized Input-Output Planning
Problem
Let us return to our generalized input-output planning example noted earlier to
illustrate the GP solution in this context. Recall the constraint equations, which in
the GP context are no longer called constraints, but rather goal equations. We refer
15
This applies to the linear version of GP; the features of alternative GP formulations, such as
developed in Lane (
1970
) or Cohen (
1978
), address some of the limitations of the linear approach
that will be apparent in what follows, e.g., when problems are very tightly constrained.
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As in Blair (
1979
) and Lee (
1971
,
1972
,
1973
). Other approaches are explored in Ijiri (
1965
),
Cohen (
1978
), and Ignizio (
1976
).
