5. The table below shows the cost of transporting one unit of stock from each of four supply points, 1 , 2,3 and 4, to each of three demand points, A, B and C. It also shows the stock held at each supply point and the stock required at each demand point. A minimal cost solution is required.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
 & A & B & C & Supply \\
\hline
1 & 18 & 23 & 20 & 15 \\
\hline
2 & 22 & 17 & 25 & 36 \\
\hline
3 & 24 & 21 & 19 & 28 \\
\hline
4 & 21 & 22 & 17 & 20 \\
\hline
Demand & 40 & 20 & 25 &  \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain why it is necessary to add a dummy demand point.
\item Add a dummy demand point and appropriate values to Table 1 in the answer book.
\item Use the north-west corner method to obtain a possible solution.

After one iteration of the stepping-stone method the table becomes

\begin{center}
\begin{tabular}{ | r | r | r | r | r | }
\hline
 & A & B & C & D \\
\hline
1 & 15 &  &  &  \\
\hline
2 & 19 & 17 &  &  \\
\hline
3 &  & 3 & 25 &  \\
\hline
4 & 6 &  &  & 14 \\
\hline
\end{tabular}
\end{center}
\item Taking D3 as the entering cell, use the stepping-stone method twice to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, routes, entering cells and exiting cells.
\item Determine whether your solution from (d) is optimal. Justify your answer.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D2 2016 Q5 [12]}}