5. The table shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to each of three sales points, \(\mathrm { P } , \mathrm { Q }\) and R . It also shows the stock held at each supply point and the amount required at each sales point. A minimum cost solution is required. The north-west corner method gives the following initial solution.

1. Taking AQ as the entering cell, use the stepping stone method to find an improved solution. Make your route clear.
2. Perform one further iteration of the stepping stone method to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.
3. Determine whether your current solution is optimal. Justify your answer.
4. State the cost of the solution you found in (b).
5. Formulate this problem as a linear programming problem. You must define your decision variables and make the objective function and constraints clear.