Example: Product Planning
This is to continue the example introducing slack and surplus variables. The reformulated model was written into the initial simplex tableau:
and 
as
solution variables, and variables, 
and
as outside
variables. This is not a very profitable solution, with profit P = 0. It
calls for no production of either product A or B. The terms
and are
entered in the simplex table under the "Solution variables" column, and
their per-unit profits (the cj) are entered in the first line.
The solution variables can be identified as those having columns under
them containing one element with +1, and the remainder of the elements
0. Also, the 1 must be in a different row than the 1 of any other column.
Thus, has
a 1 in the first row, and has
a 1 in the second row. Note that this is equivalent to saying that we have
solved the constraint equations in terms of and :
The variables omitted from the solution are currently, and ;
they are equal to 0. Thus, for this first trial solution, =
24 and
= 16.
Note that we have a diagonal of 1s, and the remainder
of the numbers in the n columns are zeros.