D2 June 2013 (R) Q8
8. A factory can process up to five units of carrots each month.
Each unit can be sold fresh or frozen or canned.
The profits, in £100s, for the number of units sold, are shown in the table.
The total monthly profit is to be maximised.
| Number of units | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Fresh | 0 | 45 | 85 | 120 | 150 | 175 |
| Frozen | 0 | 45 | 70 | 100 | 120 | 130 |
| Canned | 0 | 35 | 75 | 125 | 155 | 195 |
Use dynamic programming to determine how many of the five units should be sold fresh, frozen and canned in order to maximise the monthly profit. State the maximum monthly profit. (12)
| Stage | State | Action | Dest. | Value | Marks |
|---|---|---|---|---|---|
| Fresh | 0 | 0 | 0 | 0 | |
| 1 | 1 | 0 | 45 | ||
| 2 | 2 | 0 | 85 | 1M1 1A1 (stage 1) | |
| 3 | 3 | 0 | 120 | ||
| 4 | 4 | 0 | 150 | ||
| 5 | 5 | 0 | 175 | ||
| Frozen | 0 | 0 | 0 | 0 | |
| 1 | 1 | 0 | 45 + 0 = 45* | ||
| 0 | 1 | 0 + 45 = 45* | |||
| 2 | 2 | 0 | 70 + 0 = 70 | ||
| 1 | 1 | 45 + 45 = 90* | 2M1 2A1 (1st 4 states of stage 2) | ||
| 0 | 2 | 0 + 85 = 85 | |||
| 3 | 3 | 0 | 100 + 0 = 100 | ||
| 2 | 1 | 70 + 45 = 115 | |||
| 1 | 2 | 45 + 85 = 130* | 3A1 (state 3) | ||
| 0 | 3 | 0 + 120 = 120 | |||
| 4 | 4 | 0 | 120 + 0 = 120 | ||
| 3 | 1 | 100 + 45 = 145 | |||
| 2 | 2 | 70 + 85 = 155 | 3M1 4A1 | ||
| 1 | 3 | 45 + 120 = 165* | (Last 2 states of stage 2) | ||
| 0 | 4 | 0 + 150 = 150 | |||
| 5 | 5 | 0 | 130 + 0 = 130 | ||
| 4 | 1 | 120 + 45 = 165 | |||
| 3 | 2 | 100 + 85 = 185 | |||
| 2 | 3 | 70 + 120 = 190 | 5A1 | ||
| 1 | 4 | 45 + 150 = 195* | (state 5) | ||
| 0 | 5 | 0 + 175 = 175 | |||
| Canned | 5 | 5 | 0 | 195 + 0 = 195 | |
| 4 | 1 | 155 + 45 = 200 | |||
| 3 | 2 | 125 + 90 = 215* | 4M1 6A1ft | ||
| 2 | 3 | 75 + 130 = 205 | (3rd stage) | ||
| 1 | 4 | 35 + 165 = 200 | |||
| 0 | 5 | 0 + 195 = 195 |
| Scheme | Marks |
|---|---|
| Fresh = 1, Frozen = 1, Canned = 3 | 5M1 |
| Monthly income = £21 500 | 7A1ft |
| 12 marks |
Notes
- ALL M marks – Must bring earlier optimal results into calculations. Must have necessary right ‘ingredients’ (profit values from the table) at least once per stage.
- Penalise inconsistency/errors with the state/destination columns with the first two A marks earned only.
- Penalise empty/errors in stage column with first A mark earned only.
1M1: First stage completed (bod something in each cell). Condone missing state 0 for this mark. Must have columns for stage, state, value and one of either action or destination.
1A1: CAO for first stage – condone missing * in this stage.
2M1: Second stage – states 0, 1 and 2 complete (bod something in each cell). Condone missing state 0 for this mark.
2A1: States 0, 1 and 2 correct for second stage. Penalise * errors only twice in the question on the first occurrences.
3A1: State 3 correct for second stage. Penalise * errors only twice in the question.
3M1: Second stage – states 3 and 4 complete (bod something in each cell).
4A1: State 4 correct for second stage. Penalise * errors only twice in the question.
5A1: State 5 correct for second stage. Penalise * errors only twice in the question.
4M1: Third stage complete (bod something in each cell).
6A1ft: Third stage correct (ft from previous stage). Penalise * errors only twice in the question.
5M1: For Fresh = 1, Frozen = 1, Canned = 3 – dependent on all previous M marks having been awarded.
7A1ft: Income correct for their table. Must have earned the first 4 M marks only (not dependent on 5M1).