5. (a) Describe a practical problem that could be solved using the transportation algorithm.
A problem is to be solved using the transportation problem. The costs are shown in the table. The supply is from \(A , B\) and \(C\) and the demand is at \(d\) and \(e\).
| \cline { 2 - 4 }
\multicolumn{1}{c|}{} | \(d\) | \(e\) | Supply |
| \(A\) | 5 | 3 | 45 |
| \(B\) | 4 | 6 | 35 |
| \(C\) | 2 | 4 | 40 |
| Demand | 50 | 60 | |
(b) Explain why it is necessary to add a third demand \(f\).
(c) Use the north-west corner rule to obtain a possible pattern of distribution and find its cost.
| \cline { 2 - 5 }
\multicolumn{1}{c|}{} | \(d\) | \(e\) | \(f\) | Supply |
| \(A\) | 5 | 3 | | 45 |
| \(B\) | 4 | 6 | | 35 |
| \(C\) | 2 | 4 | | 40 |
| Demand | 50 | 60 | | |
(d) Calculate shadow costs and improvement indices for this pattern.
(e) Use the stepping-stone method once to obtain an improved solution and its cost.
(Total 16 marks)