4 A linear programming problem consists of maximising an objective function $P$ involving three variables $x , y$ and $z$. Slack variables $s , t , u$ and $v$ are introduced and the Simplex method is used to solve the problem. Several iterations of the method lead to the following tableau.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
$\boldsymbol { P }$ & $x$ & $y$ & $\boldsymbol { Z }$ & $\boldsymbol { s }$ & $\boldsymbol { t }$ & $\boldsymbol { u }$ & $v$ & value \\
\hline
1 & 0 & -12 & 0 & 5 & -3 & 0 & 0 & 37 \\
\hline
0 & 1 & -8 & 0 & 1 & 2 & 0 & 0 & 16 \\
\hline
0 & 0 & 4 & 0 & 0 & 3 & 0 & 1 & 20 \\
\hline
0 & 0 & 2 & 0 & -3 & 2 & 1 & 0 & 14 \\
\hline
0 & 0 & 1 & 1 & 2 & 5 & 0 & 0 & 8 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item The pivot for the next iteration is chosen from the $\boldsymbol { y }$-column. State which value should be chosen and explain the reason for your choice.
\item Perform the next iteration of the Simplex method.
\end{enumerate}\item Explain why your new tableau solves the original problem.
\item State the maximum value of $P$ and the values of $x , y$ and $z$ that produce this maximum value.
\item State the values of the slack variables at the optimum point. Hence determine how many of the original inequalities still have some slack when the optimum is reached.
\end{enumerate}

\hfill \mbox{\textit{AQA D2 2008 Q4 [11]}}