Simplex Tableau – Basic Variables and the Solution

-5x - y + profit  = 0
1x  + 0y         <= 250
0x  + 1y         <= 200
1x  + 1y         <= 300

became

  x     y    s1   s2   s3   constraint
[1.0,  0.0,  1.0, 0.0, 0.0, 250.0]
[0.0,  1.0,  0.0, 1.0, 0.0, 200.0]
[1.0,  1.0,  0.0, 0.0, 1.0, 300.0]
[-5.0, -1.0, 0.0, 0.0, 0.0, 0.0]

The tableau is useful due to its two kinds of variables, basic and non-basic. In the tableau's current initial state, all the slack variables, s1, s2, and s3 are basic. x and y are non-basic.

In a simplex tableau, a variable is basic if there's only one nonzero value in its column and that value is exactly 1.

For any given simplex tableau, there's an associated solution to the system of equations: just set all the non-basic variables to zero and compute the values of the basic variables. For the tableau above the associated solution is:

x = 0
y = 0
s1 = 250
s2 = 200
s3 = 300

As you can probably tell, this isn't the optimal solution to our problem. This is just the initial state of the tableau; it represents the point (0, 0) on our graph, which is the starting point, not the final vertex.