initial simplex tableau

Example: Product Planning

This is to continue the example introducing slack and surplus variables. The reformulated model was written into the initial simplex tableau:

The initial basic solution consists of the slack variables  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.