4. A company produces $x _ { 1 }$ finished articles at the end of January, $x _ { 2 }$ finished articles at the end of February, $x _ { 3 }$ finished articles at the end of March, $x _ { 4 }$ finished articles at the end of April.

Other details for each month are as follows:

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Month & January & February & March & April \\
\hline
Demand at end of month & 200 & 350 & 250 & 200 \\
\hline
Production costs per article & £1000 & £1800 & £1600 & £1900 \\
\hline
\end{tabular}
\end{center}

The cost of storing each finished but unsold article is $\pounds 500$ per month. Thus, for example, any article unsold at the end of January would incur a $\pounds 500$ charge if it is stored until the end of February or a $\pounds 1000$ charge if it is stored until the end of March.

There must be no unsold stock at the end of April.\\
The selling price of each article is $\pounds 4000$ and the total profit ( $\pounds P$ ) must be maximised.
\begin{enumerate}[label=(\alph*)]
\item Rewrite $x _ { 4 }$ in terms of the other 3 variables.
\item Show that the total cost incurred $( \pounds C )$ is given by:

$$C = 600 x _ { 1 } + 900 x _ { 2 } + 200 x _ { 3 } + 1125000$$
\item Hence, show that $P = { } ^ { - } 600 x _ { 1 } - 900 x _ { 2 } - 200 x _ { 3 } + 2875000$.
\item Three of the constraints operating can be expressed as $x _ { 1 } \geq 200$, $x _ { 2 } \geq 0$ and $x _ { 3 } \geq 0$. Write down inequalities representing two further constraints.\\
(2 marks)
\item Explain why it is not appropriate to use a graphical method to solve this problem.
\item An employee of the company wishes to use the Simplex algorithm to solve the problem. He tries to generate an initial tableau with $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ as the non-basic variables.

Explain why this is not appropriate and explain what he should do instead. You are not required to generate an initial tableau or to solve the problem.\\
(2 marks)
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1  Q4 [12]}}