5 [Figure 3, printed on the insert, is provided for use in this question.]
A maker of exclusive furniture is planning to build three cabinets \(A , B\) and \(C\) at the rate of one per month. The order in which they are built is a matter of choice, but the costs will vary because of the materials available and suppliers' costs. The expected costs, in pounds, are given in the table.
| \multirow[t]{2}{*}{Month} | \multirow[t]{2}{*}{Already built} | Cost |
| | \(\boldsymbol { A }\) | B | \(\boldsymbol { C }\) |
| 1 | - | 500 | 440 | 475 |
| 2 | A | - | 440 | 490 |
| B | 510 | - | 500 |
| \(\boldsymbol { C }\) | 520 | 490 | - |
| 3 | \(\boldsymbol { A }\) and \(\boldsymbol { B }\) | - | - | 520 |
| \(\boldsymbol { A }\) and \(\boldsymbol { C }\) | - | 500 | - |
| \(\boldsymbol { B }\) and \(\boldsymbol { C }\) | 510 | - | - |
- Use dynamic programming, working backwards from month 3, to determine the order of manufacture that minimises the total cost. You may wish to use Figure 3 for your working.
- It is discovered that the figures given were actually the profits, not the costs, for each item. Modify your solution to find the order of manufacture that maximises the total profit. You may wish to use the final column of Figure 3 for your working.