Question

While solving a standard form linear programming problem using the simplex method, we arrive at the following tableau: \begin{tabular}{|c|c|c|c|c|} \hline $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ \\ \hline 0 & 0 & $\bar{c}_3$ & 0 & $\bar{c}_5$ & $-\zeta$ \\ 0 & 1 & -1 & 0 & $\beta$ & 1 \\ 0 & 0 & 2 & 1 & $\gamma$ & 2 \\ 1 & 0 & 4 & 0 & $\epsilon$ & 3 \\ \hline \end{tabular} Suppose also that the last three columns of the matrix A form an identity matrix. (c) Suppose that $\gamma > 0$. Show that there exists an optimal basic feasible solution regardless of the values of $\bar{c}_3$ and $\bar{c}_5$.

          While solving a standard form linear programming problem using the simplex method, we arrive at the following tableau:
\begin{tabular}{|c|c|c|c|c|}
\hline
$x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ \\
\hline
0 & 0 & $\bar{c}_3$ & 0 & $\bar{c}_5$ & $-\zeta$ \\
0 & 1 & -1 & 0 & $\beta$ & 1 \\
0 & 0 & 2 & 1 & $\gamma$ & 2 \\
1 & 0 & 4 & 0 & $\epsilon$ & 3 \\
\hline
\end{tabular}
Suppose also that the last three columns of the matrix A form an identity matrix.
(c) Suppose that $\gamma > 0$. Show that there exists an optimal basic feasible solution regardless of the values of $\bar{c}_3$ and $\bar{c}_5$.

While solving a standard form linear programming problem using the simplex method, we arrive at the following tableau:

x1 x2 x3 x4 x5

0 0 c̅3 0 c̅5 -ζ

0 1 -1 0 β 1

0 0 2 1 γ 2

1 0 4 0 ϵ 3

Suppose also that the last three columns of the matrix A form an identity matrix.
(c) Suppose that γ > 0. Show that there exists an optimal basic feasible solution regardless of the values of c̅3 and c̅5.

Added by Angela F.

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