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,

1. state the value of \(k\).
2. Explain precisely what the constraint \(\sum x _ { i \mathrm { R } } \geqslant 44\) means in the transportation problem.
3. Use the north-west corner method to obtain the cost of an initial solution to this transportation problem.
4. 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
   • shadow costs
   • improvement indices
   • entering cell and exiting cell.