I'd like to analyse the LPP related to parameteres $\mu_i$ at the R.H.S. of the constraints. Specifically, I'd like to determine how the solution to the problem change when the parameters change and what is the range of the parameters where the problem has any solution. Here is the problem:
$$f = 3x_1+x_2 \rightarrow \max \\ 3x_1+2x_2 \le 18 + \mu_1 \\ -3x_1+x_2 \le 6+\mu_2 \\ 4x_1-3x_2 \le 12 + \mu_3 \\ x_1 \ge 0, x_2 \ge 0$$
First, I solve the linear problem disregarding the $\mu_i$ parameters:
\begin{array}{rr:rrrrr:r} T^{\textit{final}}& f & x_1 & x_2 & x_3 & x_4 & x_5 & b \\ \hline x_2 & 0 & 0 & 1 & \frac{4}{17}& 0 & -\frac{3}{17} & \frac{36}{17}\\ x_4 & 0 & 0 & 0 & \frac{5}{17}& 1 & \frac{9}{17} & \frac{300}{17}\\ x_1 & 0 & 1 & 0 & \frac{3}{17}& 0 & \frac{2}{17} & \frac{78}{17}\\ \hdashline f & 1 & 0 & 0 & \frac{13}{17}& 0 & \frac{3}{17} & \frac{270}{17}\\ \end{array}
Now the $f^{\mu}_{\textit{max}} = \frac{270}{17}+\frac{13}{17}\mu_1+\frac{3}{17}\mu_3$.
From there I write the new constraint system where the variables $x_i$ are replaced with the parameters $\mu_i$:
$$\frac{78}{17} + \frac{3}{17}\mu_1+\frac{2}{17}\mu_3 \ge 0 \\ \frac{36}{17}+ \frac{4}{17}\mu_1 -\frac{3}{17}\mu_3 \ge 0 \\ \frac{300}{17}+\frac{5}{17}\mu_1 + \frac{9}{17}\mu_3 + \mu_2 \ge 0$$
If I choose $\mu_2=\mu_3 = 0$ then
$$\mu_1 \ge -26 \\ \mu_1 \ge -9 \\ \mu_1\ge -60$$ From there it follows that $\mu_1 \in [-9;+\infty)$. But if I choose $\mu_1 = -12$ the LPP still has a feasible solution. So I gather, that there is a mistake somewhere in my computations. Can someone point me out the mistake?
