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.

We choose one basic variable per constraint row and track its column index in self.basis. A basic variable has 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.