Optimum Design
155
VI. LINEAR OPTIMIZATION WITH FUNCTIONAL
CONSTRAINTS
A criterion function is
C = C(xl... xn)
(3.25)
with linear functional constraints
R~ < Fl(X~ ... x,~) 5 R’~
:
:
(3.26)
Rm < Fm(x~ "" x,) < R~m
When Eqs. (3.25) and (3.26) are linear, it means sums of single power
variables. This condition is called linear programming. It should also be
realized that a regional constraint is
x~, x2 .... , xn >_ 0
(3.27)
It has been found that the optimum solution is found at the corners defined
by the regional constraints. The constraints in two variables can be easily
handled by plotting on graph paper, however, for three or more variables
a simplex method is used. Most of the linear programming problems involve
mixing, production scheduling, and transportation. These problems tend to
be industrial process, chemical, or civil engineering in nature. A few comments are in order about the simplex method.
A. Simplex method [3.14]
The simplex method makes use of the fact an optimum solution is
obtained in the corners of the region defined by the regional constraints.
The following process is followed for two or three variables.
1. Select a corner of the region as a starting point. The farthest the
criterion’function is translated from the origin, will yield a minimum or maximum.
2. Choose an edge through this corner such that C increases in value
along the edge.
3. Proceed along the edge of the next corner.
4. Repeat steps (2) and (3) until an optimum solution is reached.
5. If a function constraint is parallel to the criterion function any
point on the function constraint line yields the same value for
the criterion function.