Usage example
Consider the following linear programming problem (in standard form):
$$
\begin{cases}\begin{aligned}\min &&&& -x_1 &&-x_2\\ \text{s.t.} &&&& 3x_1 &&+2x_2 &&+x_3 && && = 4\\ &&&& && x_2 && &&+x_4 && = 3\\ \\ &&&& x_1,&&x_2,&&x_3,&&x_4 &&\ge 0\\ \end{aligned}\end{cases}
$$
To solve this problem from the Python console, I would write
>>> import numpy as np
>>> from simplex import simplex
>>> A = np.matrix([[3, 2, 1, 0], [0, 1, 0, 1]])
>>> b = np.array([4, 3])
>>> c = np.array([-1, -1, 0, 0])
>>> simplex(A, b, c)
The output would be:
Executing phase I...
Iteration no. 1: q = 1 rq = -3.00 B[p] = 1 theta* = 1.3333 z = 3.00
Iteration no. 2: q = 2 rq = -1.00 B[p] = 2 theta* = 2.0000 z = 1.00
Iteration no. 3: q = 4 rq = -1.00 B[p] = 4 theta* = 1.0000 z = 0.00
Iteration no. 4: found optimum
Phase I terminated.
Found initial BFS at x =
[0. 2. 0. 1.].
Executing phase II...
Iteration no. 1: found optimum
Phase II terminated.
---------------------------------
| Found optimal solution at x = |
| [0. 2. 0. 1.]. |
| |
| Basic indices: {1, 3} |
| Nonbasic indices: {0, 2} |
| |
| Optimal cost: -2.0. |
---------------------------------
4 iterations in phase I, 1 iterations in phase II (5 total).
(0, array([0., 2., 0., 1.]), -2.0, None)
