6 A linear programming problem is\\
Maximise $\mathrm { P } = 2 \mathrm { x } - \mathrm { y }$\\
subject to

$$\begin{aligned}
3 x + y - 4 z & \leqslant 24 \\
5 x - 3 z & \leqslant 60 \\
- x + 2 y + 3 z & \leqslant 12
\end{aligned}$$

and $x \geqslant 0 , y \geqslant 0 , z \geqslant 0$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Represent this problem as an initial simplex tableau.
\item Carry out one iteration of the simplex algorithm.

After two iterations the resulting tableau is

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$P$ & $x$ & $y$ & $z$ & $s$ & $t$ & $u$ & RHS \\
\hline
1 & 0 & $\frac { 5 } { 11 }$ & 0 & $- \frac { 6 } { 11 }$ & $\frac { 8 } { 11 }$ & 0 & $30 \frac { 6 } { 11 }$ \\
\hline
0 & 1 & $- \frac { 3 } { 11 }$ & 0 & $- \frac { 3 } { 11 }$ & $\frac { 4 } { 11 }$ & 0 & $15 \frac { 3 } { 11 }$ \\
\hline
0 & 0 & $- \frac { 5 } { 11 }$ & 1 & $- \frac { 5 } { 11 }$ & $\frac { 3 } { 11 }$ & 0 & $5 \frac { 5 } { 11 }$ \\
\hline
0 & 0 & $\frac { 34 } { 11 }$ & 0 & $\frac { 12 } { 11 }$ & $- \frac { 5 } { 11 }$ & 1 & $10 \frac { 10 } { 11 }$ \\
\hline
\end{tabular}
\end{center}
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Write down the basic variables after two iterations.
\item Write down the exact values of the basic feasible solution for $x , y$ and $z$ after two iterations.
\item State what you can deduce about the optimal value of the objective for the original problem.

You are now given that, in addition to the constraints above, $\mathrm { x } + \mathrm { y } + \mathrm { z } = 9$.
\end{enumerate}\item Use the additional constraint to rewrite the original constraints in terms of $x$ and $y$ but not $z$.
\item Explain why the simplex algorithm cannot be applied to this new problem without some modification.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Discrete 2022 Q6 [15]}}