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Chapter 3
The five step outline is the same as
C = ClXl d- c2x2 d- e3x3 +.. ¯ -+- CnXn
(3.28)
and the functional constraints which bound the solution of the criterion
function are
alXl d-- a2x2 q-- ... q- anXn = b~
anlxl ~- an2X2 -1-"" q- annXn = bn
(3.29)
with regional constraints for positive values
Xn > O n= 1,2 .... n
(3.30)
There are other conditions which arise in an actual programming of the
above equations.
A simple example of the simplex method in two dimensions follows.
EXAMPLE 3.5 [3.14]. Optimize a two variable criterion function
F = x - 2y + 4
(3.31)
with the functional constraints
x+y<4
x + 2y >_ -2
x - y _> -2
(3.32)
x_<3
Plot the functional constraints in Fig. 3.2 and follow the five step simplex
method steps outlined. Start at point B where F=-I (step 1) and move
along the two edges (step 2) to corner A where F= 2 and C where F=
Now with steps 2 and 3 travel along the edges from A to D and C to D
where at D, F= 12. The five steps can be used in another fashion.
(a) Plot the criterion function through the origin where F=4 then
(b) Take perpendicular distances dl and d2 where the largest translation of the criterion function is a maximum or minimum. In
this case F=- 1 at B which happens to be a minimum and at
D, F= 12 the desired maximum.
(c) Also note statement 5 where a functional constraint is parallel to
the function constraint. As seen from Fig. 3.2 along any or
the parametric lines for F the value of F is constant.
As can be seen corners A and C need not be evaluated and in fact since
de >dl corner D is the only corner to be evaluated but one does not know
where the regional minimum is located.