5. A car-hire firm has six branches in a region. Three of the branches, $A , B$ and $C$, have spare cars, whereas the other three, $D , E$ and $F$, require cars. The total number of cars required is equal to the number of cars available. The table below shows the cost in pounds of sending one car from each branch with spares to each branch needing more cars and the number of cars available or required by each branch.

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\backslashbox{Branches with spare cars}{Branches needing cars} & $D$ & $E$ & $F$ & Available \\
\hline
$A$ & 6 & 4 & 7 & 7 \\
\hline
B & 8 & 5 & 3 & 8 \\
\hline
C & 4 & 4 & 2 & 5 \\
\hline
Required & 5 & 9 & 6 &  \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the north-west corner method to obtain a possible pattern of moving cars and find its cost.

The firm wishes to minimise the cost of redistributing the cars.
\item Calculate shadow costs for the pattern found in part (a) and improvement indices for each unoccupied cell.
\item State, with a reason, whether or not the pattern found in part (a) is optimal.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D2  Q5 [10]}}