7. A steel manufacturer has 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) which can produce 35,25 and 15 kilotonnes of steel per year, respectively. Three businesses \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) have annual requirements of 20,25 and 30 kilotonnes respectively. The table below shows the cost \(C _ { i j }\) in appropriate units, of transporting one kilotonne of steel from factory \(F _ { i }\) to business \(B _ { j }\).
| \cline { 3 - 5 }
\multicolumn{2}{c|}{} | Business |
| \cline { 3 - 5 }
\multicolumn{2}{c|}{} | \(B _ { 1 }\) | \(B _ { 2 }\) | \(B _ { 3 }\) |
| \multirow{3}{*}{Factory} | \(F _ { 1 }\) | 10 | 4 | 11 |
| \cline { 2 - 5 } | \(F _ { 2 }\) | 12 | 5 | 8 |
| \cline { 2 - 5 } | \(F _ { 3 }\) | 9 | 6 | 7 |
The manufacturer wishes to transport the steel to the businesses at minimum total cost.
- Write down the transportation pattern obtained by using the North-West corner rule.
- Calculate all of the improvement indices \(I _ { i j }\), and hence show that this pattern is not optimal.
- Use the stepping-stone method to obtain an improved solution.
- Show that the transportation pattern obtained in part (c) is optimal and find its cost.