3. The table below shows the stock held at each supply point and the stock required at each demand point in a standard transportation problem. The table also shows the cost, in pounds, of transporting the stock from each supply point to each demand point.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 2 - 5 }
\multicolumn{1}{c|}{} & Q & R & S & Supply \\
\hline
A & 23 & 18 & 12 & 45 \\
\hline
B & 8 & 10 & 14 & 27 \\
\hline
C & 11 & 14 & 21 & 34 \\
\hline
D & 19 & 15 & 11 & 50 \\
\hline
Demand & 75 & 37 & 44 &  \\
\hline
\end{tabular}
\end{center}

The problem is partially described by the linear programming formulation below.\\
Let $x _ { i j }$ be the number of units transported from i to j

$$\begin{aligned}
& \text { where } \quad i \in \{ A , B , C , D \} \\
& \quad j \in \{ Q , R , S \} \text { and } x _ { i j } \geqslant 0 \\
& \text { Minimise } P = 23 x _ { A Q } + 18 x _ { A R } + 12 x _ { A S } + 8 x _ { B Q } + 10 x _ { B R } + 14 x _ { B S } \\
& \quad + 11 x _ { C Q } + 14 x _ { C R } + 21 x _ { C S } + 19 x _ { D Q } + 15 x _ { D R } + 11 x _ { D S }
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Write down, as inequalities, the constraints of the linear program.
\item Use the north-west corner method to obtain an initial solution to this transportation problem.
\item Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
\item Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and the

\begin{itemize}
  \item shadow costs
  \item improvement indices
  \item entering cell and exiting cell
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{Edexcel FD2 2023 Q3 [9]}}