Transportation LP formulation

5. Three warehouses \(W , X\) and \(Y\) supply televisions to three supermarkets \(J , K\) and \(L\). The table gives the cost, in pounds, of transporting a television from each warehouse to each supermarket. The warehouses have stocks of 34,57 and 25 televisions respectively, and the supermarkets require 20, 56 and 40 televisions respectively. The total cost of transporting the televisions is to be minimised. Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints.

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. 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 $$\begin{aligned} & \text { where } \quad i \in \{ A , B , C , D \} \\ & \quad j \in \{ Q , R , S \} \text { and } x _ { i j } \geqslant 0 \\ & \text { Minimise } P = 23 x _ { A Q } + 18 x _ { A R } + 12 x _ { A S } + 8 x _ { B Q } + 10 x _ { B R } + 14 x _ { B S } \\ & \quad + 11 x _ { C Q } + 14 x _ { C R } + 21 x _ { C S } + 19 x _ { D Q } + 15 x _ { D R } + 11 x _ { D S } \end{aligned}$$

1. Write down, as inequalities, the constraints of the linear program.
2. Use the north-west corner method to obtain an initial solution to this transportation problem.
3. Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
4. 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

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