8. Polly has a bird food stall at the local market. Each week she makes and sells three types of packs $A , B$ and C.

Pack $A$ contains 4 kg of bird seed, 2 suet blocks and 1 kg of peanuts.\\
Pack $B$ contains 5 kg of bird seed, 1 suet block and 2 kg of peanuts.\\
Pack $C$ contains 10 kg of bird seed, 4 suet blocks and 3 kg of peanuts.\\
Each week Polly has 140 kg of bird seed, 60 suet blocks and 60 kg of peanuts available for the packs.\\
The profit made on each pack of $A , B$ and $C$ sold is $\pounds 3.50 , \pounds 3.50$ and $\pounds 6.50$ respectively. Polly sells every pack on her stall and wishes to maximise her profit, $P$ pence.

Let $x , y$ and $z$ be the numbers of packs $A , B$ and $C$ sold each week.\\
An initial Simplex tableau for the above situation is

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 4 & 5 & 1 & 1 & 0 & 0 & 140 \\
$s$ & 2 & 1 & 4 & 0 & 1 & 0 & 60 \\
$t$ & 1 & 2 & 3 & 0 & 0 & 1 & 60 \\
\hline
$P$ & - 350 & - 350 & - 650 & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain the meaning of the variables $r , s$ and $t$ in the context of this question.
\item Perform one complete iteration of the Simplex algorithm to form a new tableau $T$. Take the most negative number in the profit row to indicate the pivotal column.
\item State the value of every variable as given by tableau $T$.
\item Write down the profit equation given by tableau $T$.
\item Use your profit equation to explain why tableau $T$ is not optimal.

Taking the most negative number in the profit row to indicate the pivotal column,
\item identify clearly the location of the next pivotal element.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D2 2005 Q8 [15]}}