P= 40x1 + 60x2 + 50x3
Subject to the constraints
2x1 + 2x2 + x3 < = 8
x1 - 4x2 +3x3 < = 12
x1 > = 0 x2 > = 0 x3 > = 0
We introduce non-negative, "slack" variables x_4 and x_5 and rewrite the
problem as follows:
Maximize P = 40 x_1 + 60 x_2 + 50 x_3
s.t. 2 x_1 + 2 x_2 + x_3 + x_4 = 0
x_1 - 4 x_2 + 3 x_3 + x_5 = 12
We note that we have an initial feasible solution (0,0,0,8,12).
We call (x_4,x_5) a basis for this solution and x_4 and x_5 "basic" variables".
We call x_1, x_2, and x_3 non-basic variables for this solution.
We look to see if there are any non-basic variables that will increase the
solution from P=0. This is done by choosing the variable with the largest
positive coefficient in our expression of P above (in terms of non-basics).
We choose to put x_2 into the basis because it would increase P most quickly
(per unit change in x_2).
We next turn to how much we can increase x_2 and still remain feasible.
In general, if "e" denotes the subscript of the entering variable. a_(i,e) is
the coefficient of this variable for ...
This shows how to use Simplex method to maximize a function with given constraints.