2 A manufacturing company is planning to build three new machines, \(A , B\) and \(C\), at the rate of one per month. The order in which they are built is a matter of choice, but the profits will vary according to the number of workers available and the suppliers' costs. The expected profits in thousands of pounds are given in the table.
| \multirow[b]{2}{*}{Month} | \multirow[b]{2}{*}{Already built} | Profit (in units of £1000) |
| | \(\boldsymbol { A }\) | \(\boldsymbol { B }\) | \(\boldsymbol { C }\) |
| 1 | - | 52 | 47 | 48 |
| \multirow[t]{3}{*}{2} | A | - | 58 | 54 |
| B | 70 | - | 54 |
| \(\boldsymbol { C }\) | 68 | 63 | - |
| \multirow[t]{3}{*}{3} | \(\boldsymbol { A }\) and \(\boldsymbol { B }\) | - | - | 64 |
| \(\boldsymbol { A }\) and \(\boldsymbol { C }\) | - | 67 | - |
| \(\boldsymbol { B }\) and \(\boldsymbol { C }\) | 69 | - | - |
- Draw a labelled network such that the most profitable order of manufacture corresponds to the longest path within that network.
- Use dynamic programming to determine the order of manufacture that maximises the total profit, and state this maximum profit.