5. A standard transportation problem is described in the linear programming formulation below.

Let $X _ { i j }$ be the number of units transported from $i$ to $j$\\
where $i \in \{ \mathrm {~A} , \mathrm {~B} , \mathrm { C } , \mathrm { D } \}$

$$j \in \{ \mathrm { R } , \mathrm {~S} , \mathrm {~T} \} \text { and } x _ { i j } \geqslant 0$$

Minimise $P = 23 x _ { \mathrm { AR } } + 17 x _ { \mathrm { AS } } + 24 x _ { \mathrm { AT } } + 15 x _ { \mathrm { BR } } + 29 x _ { \mathrm { BS } } + 32 x _ { \mathrm { BT } }$

$$+ 25 x _ { \mathrm { CR } } + 25 x _ { \mathrm { CS } } + 27 x _ { \mathrm { CT } } + 19 x _ { \mathrm { DR } } + 20 x _ { \mathrm { DS } } + 25 x _ { \mathrm { DT } }$$

subject to

$$\begin{aligned}
& \sum x _ { \mathrm { A } j } \leqslant 34 \\
& \sum x _ { \mathrm { B } j } \leqslant 27 \\
& \sum x _ { \mathrm { C } j } \leqslant 41 \\
& \sum x _ { \mathrm { D } j } \leqslant 18 \\
& \sum x _ { i \mathrm { R } } \geqslant 44 \\
& \sum x _ { i \mathrm {~S} } \geqslant 37 \\
& \sum x _ { i \mathrm {~T} } \geqslant k
\end{aligned}$$

Given that the problem is balanced,
\begin{enumerate}[label=(\alph*)]
\item state the value of $k$.
\item Explain precisely what the constraint $\sum x _ { i \mathrm { R } } \geqslant 44$ means in the transportation problem.
\item Use the north-west corner method to obtain the cost of an initial solution to this transportation problem.
\item Perform one 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 2022 Q5 [9]}}