4 The Simplex method is to be used to maximise $P = 3 x + 2 y + z$ subject to the constraints

$$\begin{aligned}
- x + y + z & \leqslant 4 \\
2 x + y + 4 z & \leqslant 10 \\
4 x + 2 y + 3 z & \leqslant 21
\end{aligned}$$

The initial Simplex tableau is given below.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
$\boldsymbol { P }$ & $\boldsymbol { x }$ & $\boldsymbol { y }$ & $\boldsymbol { z }$ & $s$ & $t$ & $\boldsymbol { u }$ & value \\
\hline
1 & -3 & -2 & -1 & 0 & 0 & 0 & 0 \\
\hline
0 & -1 & 1 & 1 & 1 & 0 & 0 & 4 \\
\hline
0 & 2 & 1 & 4 & 0 & 1 & 0 & 10 \\
\hline
0 & 4 & 2 & 3 & 0 & 0 & 1 & 21 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item The first pivot is to be chosen from the $x$-column. Identify the pivot and explain why this particular value is chosen.
\item Perform one iteration of the Simplex method and 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 write down the initial inequality that still has slack.

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{172c5c92-4254-4593-b741-1caa83a1e833-11_2486_1714_221_153}
\end{center}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA D2 2011 Q4 [15]}}