I have the following main program to be solved by simplex method.
$$Min \quad P=0.2x_1 + 0.5x_2 + 0.6x_3 + 0.1x_4 + 0.2x_5 + 0.3x_6 \\ \text{subject to: } 0.2x_1 + 0.3x_2 + 0.25x_3 \le 0.8 \\ \quad\quad\quad\quad 0.1x_4 + 0.4x_5 + 0.2x_6 \le 0.8 \\ x_1 + x_4 = 1 \\ x_2 + x_5 = 1 \\ x_3 + x_6 = 1 $$
whose standard form is:
$$Max \quad P = -0.2x_1 - 0.5x_2 - 0.6x_3 - 0.1x_4 - 0.2x_5 - 0.3x_6 \\ \text{subject to: } 0.2x_1 + 0.3x_2 + 0.25x_3 + u = 0.8 \\ \quad\quad\quad\quad 0.1x_4 + 0.4x_5 + 0.2x_6 + v = 0.8 \\ x_1 + x_4 = 1 \quad (1)\\ x_2 + x_5 = 1 \quad (2)\\ x_3 + x_6 = 1 \quad (3) $$
According to some advice, I used the last three equations in the standard problem to eliminate three variables from other equations. I chose variables $x_1$, $x_2$ and $x_3$ to be eliminated in all other equations.
Therefore, I have the following program to be solved by simplex method:
$$Max \quad P = 0.1x_4 + 0.3x_5 + 0.3x_6 -1.3\\ \text{subject to: } -0.2x_4 - 0.3x_5 - 0.25x_6 + u = 0.05 \\ \quad\quad\quad\quad 0.1x_4 + 0.4x_5 + 0.2x_6 + v = 0.8 \\ $$
After solving above problem with simplex method, I get these basic variables: $u=1.05$ and $x_6=4$. All other variables of above program are non-basic, hence equal zero.
The values of eliminated variables according to those equations are: $x_1=1$, $x_2=1$ and $x_3=-3$. The values of $x_3$ is negative which is not correct according to the general constraint of non-negative values of input variables of linear programs.
I expected to see $0 \le x_1,x_2, x_3, x_4, x_5, x_6 \le 1$ as I have the three equations (1), (2), (3).
Is there something wrong with my math or assumptions? All computations are done by a computer program I wrote.
