I have a few MCQ questions regarding the tableau Simplex Method. I am having a hard time wrapping my head around the examples with missing data points. Below are some of the examples that I find weird, and I’m not entirely sure how to solve:
a) For the first example, I have a final Phase $2$ tableau for some problem (P) given by:
$$ \begin{array}{c|c|c|c|c|c|c} 1 & \color{red}\alpha & 0 & 0 & 0 & 0 & 0 \\ x_1&x_2& y_1& y_2&y_3&z_1&z_2&RHS \\ \hline 1 & 0 & -1 & 0 & 0 & 1 & 0 & 2 \\ 0 & 1 & 1 & 0 & 1 & -1 & 0 & 4\\ 0 & 0 & 1 & 1 & 1 & -1 & -1 & 3\\ \hline \end{array}$$
The question asks for which $\alpha$ the corresponding basic solution remains optimal?
b) I have both the initial Phase $1$ tableau:
$$ \begin{array}{c|c|c|c|c|c|c} 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ x_1&x_2& y_1& y_2&y_3&z_1&z_2&RHS \\ \hline 1 & 0 & -1 & 0 & 0 & 1 & 0 & 2 \\ 0 & 1 & 0 & -1 & 0 & 0 & 1 & 1\\ 1 & 1 & 0 & 0 & 1 & 0 & 0 & \color{red} \beta\\ \hline -1 & -1 & 1 & 1 & 0 & 0 &0 & -3 \\ \end{array}$$
and final Phase $2$ tableau for problem (P):
$$ \begin{array}{c|c|c|c|c|c|c} 1 & 4 & 0 & 0 & 0 & 0 & 0 \\ x_1&x_2& y_1& y_2&y_3&z_1&z_2&RHS \\ \hline 1 & 0 & -1 & 0 & 0 & 1 & 0 & \\ 0 & 1 & 1 & 0 & 1 & -1 & 0 & \\ 0 & 0 & 1 & 1 & 1 & -1 & -1 & \\ \hline 0 & 0 & 3 & 0 & 4 & -3 &0 & \\ \end{array}$$
The question is for which value $\beta$ is the basis in the final tableau degenerate?