7. A tailor makes two types of garment, $A$ and $B$. He has available $70 \mathrm {~m} ^ { 2 }$ of cotton fabric and $90 \mathrm {~m} ^ { 2 }$ of woollen fabric. Garment $A$ requires $1 \mathrm {~m} ^ { 2 }$ of cotton fabric and $3 \mathrm {~m} ^ { 2 }$ of woollen fabric. Garment $B$ requires $2 \mathrm {~m} ^ { 2 }$ of each fabric.

The tailor makes $x$ garments of type $A$ and $y$ garments of type $B$.
\begin{enumerate}[label=(\alph*)]
\item Explain why this can be modelled by the inequalities

$$\begin{aligned}
& x + 2 y \leq 70 \\
& 3 x + 2 y \leq 90 \\
& x \geq 0 , y \geq 0
\end{aligned}$$

The tailor sells type $A$ for $\pounds 30$ and type $B$ for $\pounds 40$. All garments made are sold. The tailor wishes to maximise his total income.
\item Set up an initial Simplex tableau for this problem.
\item Solve the problem using the Simplex algorithm.

Figure 4 shows a graphical representation of the feasible region for this problem.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-008_686_1277_1319_453}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\item Obtain the coordinates of the points A, $C$ and $D$.
\item Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4.

6689 Decision Mathematics 1 (New Syllabus)

Order of selecting edges\\

Final tree\\
(b) Minimum total length of cable\\

Question 4 to be answered on this page\\
(a) $A$

\begin{itemize}
  \item Monday (M)\\
$B$ ◯
  \item Tuesday (Tu)\\
$C \odot$
  \item Wednesday (W)\\
$D$ ◯
  \item Thursday (Th)\\
$E$ -
  \item Friday (F)\\
(b)\\
(c)
\end{itemize}

Question 5 to be answered on this page\\
Key\\
(a)

Early\\
Time\\
Late\\
Time\\
\includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_433_227_534_201}\\
$F ( 3 )$\\
\includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_117_222_1016_992}\\
H(4)

K(6)\\
(b) Critical activities\\

Length of critical path $\_\_\_\_$\\

(c)\\
\includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_492_1604_1925_266}

Question 6 to be answered on pages 4 and 5\\
(a) (i) SAET $\_\_\_\_$\\
(ii) SBDT $\_\_\_\_$\\
(iii) SCFT $\_\_\_\_$\\
(b)

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-012_691_1307_893_384}
\captionsetup{labelformat=empty}
\caption{Diagram 1}
\end{center}
\end{figure}

(c)

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_699_1314_167_382}
\captionsetup{labelformat=empty}
\caption{Diagram 2}
\end{center}
\end{figure}

Flow augmenting routes\\

(d)

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_693_1314_1368_382}
\captionsetup{labelformat=empty}
\caption{Diagram 3}
\end{center}
\end{figure}

(e) $\_\_\_\_$
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1  Q7}}