5. A carpet manufacturer has two warehouses, \(W _ { 1 }\) and \(W _ { 2 }\), which supply carpets for three sales outlets, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). At one point \(S _ { 1 }\) requires 40 rolls of carpet, \(S _ { 2 }\) requires 23 rolls of carpet and \(S _ { 3 }\) requires 37 rolls of carpet. At this point \(W _ { 1 }\) has 45 rolls in stock and \(W _ { 2 }\) has 40 rolls in stock. The following table shows the cost, in pounds, of transporting one roll from each warehouse to each sales outlet:
| \cline { 2 - 4 }
\multicolumn{1}{c|}{} | \(S _ { 1 }\) | \(S _ { 2 }\) | \(S _ { 3 }\) |
| \(W _ { 1 }\) | 8 | 7 | 11 |
| \(W _ { 2 }\) | 9 | 10 | 11 |
The company's manager wishes to supply the 85 rolls that are in stock such that transportation costs are kept to a minimum.
- Use the north-west corner rule to obtain an initial solution to the problem.
- Calculate improvement indices for the unused routes.
- Use the stepping-stone method to obtain an optimal solution.
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