4 A linear programming problem involving the variables $x , y$ and $z$ is to be solved. The objective function to be maximised is $P = 2 x + 3 y + 5 z$. The initial Simplex tableau is given below.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$\boldsymbol { P }$ & $\boldsymbol { x }$ & $\boldsymbol { y }$ & $\boldsymbol { z }$ & $s$ & $\boldsymbol { t }$ & $\boldsymbol { u }$ & value \\
\hline
1 & -2 & -3 & -5 & 0 & 0 & 0 & 0 \\
\hline
0 & 1 & 0 & 1 & 1 & 0 & 0 & 9 \\
\hline
0 & 2 & 1 & 4 & 0 & 1 & 0 & 40 \\
\hline
0 & 4 & 2 & 3 & 0 & 0 & 1 & 33 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item In addition to $x \geqslant 0 , y \geqslant 0 , z \geqslant 0$, write down three inequalities involving $x , y$ and $z$ for this problem.
\item \begin{enumerate}[label=(\roman*)]
\item By choosing the first pivot from the $z$-column, perform one iteration of the Simplex method.
\item Explain how you know that the optimal value has not been reached.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Perform one further iteration.
\item Interpret the final tableau and state the values of the slack variables.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA D2 2008 Q4 [14]}}