7. This question is to be answered on the sheet provided in the answer booklet.

A chemical company makes 3 products $X , Y$ and $Z$. It wishes to maximise its profit $\pounds P$. The manager considers the limitations on the raw materials available and models the situation with the following Linear Programming problem.

Maximise

$$\begin{gathered}
P = 3 x + 6 y + 4 z \\
x \quad + \quad z \leq 4 \\
x + 4 y + 2 z \leq 6 \\
x + y + 2 z \leq 12 \\
x \geq 0 , \quad y \geq 0 , \quad z \geq 0
\end{gathered}$$

subject to\\
where $x , y$ and $z$ are the weights, in kg , of products $X , Y$ and $Z$ respectively.\\
A possible initial tableau is

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 1 & 0 & 1 & 1 & 0 & 0 & 4 \\
\hline
$s$ & 1 & 4 & 2 & 0 & 1 & 0 & 6 \\
\hline
$t$ & 1 & 1 & 2 & 0 & 0 & 1 & 12 \\
\hline
$P$ & - 3 & - 6 & - 4 & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain
\begin{enumerate}[label=(\roman*)]
\item the purpose of the variables $r , s$ and $t$,
\item the final row of the tableau.
\end{enumerate}\item Solve this Linear Programming problem by using the Simplex alogorithm. Increase $y$ for your first iteration and than increase $x$ for your second iteration.
\item Interpret your solution.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2001 Q7 [17]}}