Geoscience Reference
In-Depth Information
Linear Goal Programming
13
13.6
In a traditional linear programming problem the optimal solution must lie within the
feasible region defined by the constraint equations. LP maximize or minimizes the
objective function subject to having already satisfied what can be considered a
preliminary objective of considering solutions that are within the feasible region. In
goal programming the implicit objective of any solution lying within the feasible
region in the linear programming approach is considered instead as an explicit
rather than implicit objective. A traditional method of accomplishing this is to
introduce so-called
deviational
variables
d
k
, for
k
m
, for
m
constraint
equations. The values of
d
k
measure “deviation” from the right-hand side values of
the constraint equations or deviation from an established goal for the explicit
objective function,
q
. In all cases
d
k
¼1, 2,
...
0.
We can think of the deviational variables as either exhibiting
overachievement
or
underachievement
of the established goal. For overachievement of goal
k
we call
the level of overachievement a
positive deviation
, which is indicated by a non-zero
value for a
positive deviational variable, d
k
+
(sometimes referred to as a
surplus
variable)
.
Likewise, for underachievement of the established goal, we call the level
of underachievement a
negative deviation
from the goal indicated by a non-zero
value for a
negative deviational variable
,
d
k
(sometimes referred to as a
slack
variable). In GP if
d
k
+
0 then
d
k
¼
0 and vice versa, i.e., at most one of the paired
positive and negative deviational variables,
d
k
and
d
k
+
, can be greater than 0; if
both are 0 then, of course, the goal is achieved exactly.
In GP, to account for the relative order of priority on the objectives, we assign
the objectives to
pre-emptive priority
classes, each of which is denoted by
P
l
, for
l
>
,
L
where
L
is the total number of priority classes. The GP equivalent of an
LP would be to two pre-emptive priority classes,
P
1
and
P
2
, associated with first
satisfying all LP constraint equations as the highest priority (the implicit objective
of any solution falling within the feasible region) and then maximizing
q
as
the second priority
.
Continuing the relationship with the LP case, in general,
minimizing positive deviational variables is equivalent to satisfying a
constraint
or minimizing an objective function and minimizing negative deviational variables
is equivalent to satisfying a
¼
1,
...
constraint or maximizing an objective function.
Hence, seeking to drive
d
k
+
for the constraint equations to zero is equivalent to
satisfying the constraints originally specified in the LP problem and seeking to drive
d
k
to zero for
q
is equivalent to maximizing the objective function specified in the
LP problem. The key difference is that in GP we no longer really distinguish
between what in the LP formulation were objective and constraint equations
14
13
Goal programming was first suggested by Charnes and Cooper (
1961
). Useful characterizations
of the approach and further refinements are included in Lane (
1970
), Lee (
1971
,
1972
,
1973
),
and Ignizio (
1976
).
14
Constraints in LP are generally specified as inequalities, but by introduction of slack or surplus
variables, they can be specified as equations.
