| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Dynamic programming resource allocation |
| Difficulty | Standard +0.3 This is a standard textbook dynamic programming problem with clear structure: 3 choices, 5 units capacity, and a straightforward table to fill using Bellman's principle. While it requires systematic working through states and decisions (hence 12 marks), the method is algorithmic with no novel insight needed—slightly easier than average A-level maths questions which typically require more problem-solving creativity. |
| Spec | 7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective7.07d Simplex terminology: basic feasible solution, basic/non-basic variable7.07e Graphical interpretation: iterations as edges of convex polygon7.07f Algebraic interpretation: explain simplex calculations |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Stage 1 (Fresh) | ||
| 0 → 0: value 0 | ||
| 1 → 0: value 45 | ||
| 2 → 0: value 85 | 1M1, 1A1: 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. | |
| 3 → 0: value 120 | ||
| 4 → 0: value 150 | ||
| 5 → 0: value 175 | ||
| Stage 2 (Frozen) | ||
| State 0: value 0 | ||
| State 1: 1 → 0 giving 45 + 0 = 45*; or 0 → 1 giving 0 + 45 = 45* | 2M1, 2A1 | 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. | ||
| State 2: 2 → 0 giving 70 + 0 = 70; or 1 → 1 giving 45 + 45 = 90*; or 0 → 2 giving 0 + 85 = 85 | ||
| State 3: 3 → 0 giving 100 + 0 = 100; 2 → 1 giving 70 + 45 = 115; 1 → 2 giving 45 + 85 = 130*; 0 → 3 giving 0 + 120 = 120 | 3A1 | 3A1: State 3 correct for second stage. Penalise * errors only twice in the question. |
| State 4: 4 → 0 giving 120 + 0 = 120; 3 → 1 giving 100 + 45 = 145; 2 → 2 giving 70 + 85 = 155; 1 → 3 giving 45 + 120 = 165*; 0 → 4 giving 0 + 150 = 150 | 3M1, 4A1 | 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. |
| State 5: 5 → 0 giving 130 + 0 = 130; 4 → 1 giving 120 + 45 = 165; 3 → 2 giving 100 + 85 = 185; 2 → 3 giving 70 + 120 = 190; 1 → 4 giving 45 + 150 = 195*; 0 → 5 giving 0 + 175 = 175 | 5A1 | 5A1: State 5 correct for second stage. Penalise * errors only twice in the question. |
| Stage 3 (Canned) | ||
| State 5: 5 → 0 giving 195 + 0 = 195; 4 → 1 giving 155 + 45 = 200; 3 → 2 giving 125 + 90 = 215*; 2 → 3 giving 75 + 130 = 205; 1 → 4 giving 35 + 165 = 200; 0 → 5 giving 0 + 175 = 175 | 4M1, 6A1ft | 4M1: Third stage complete (bod something in each cell). 6A1ft: Third stage correct (ft from previous stage). Penalise * errors only twice in the question. |
| Final answer | ||
| Fresh = 1, Frozen = 1, Canned = 3 | 5M1 | 5M1: For Fresh = 1, Frozen = 1, Canned = 3 – dependent on all previous M marks having been awarded. |
| Monthly income = £21 500 | 7A1ft | 7A1ft: Income correct for their table. Must have earned the first 4 M marks only (not dependent on 5M1). |
| Total | 12 marks |
| **Stage 1 (Fresh)** | | | |
|---|---|---|---|
| | 0 → 0: value 0 | | |
| | 1 → 0: value 45 | | |
| | 2 → 0: value 85 | | 1M1, 1A1: 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. |
| | 3 → 0: value 120 | | |
| | 4 → 0: value 150 | | |
| | 5 → 0: value 175 | | |
| **Stage 2 (Frozen)** | | | |
| | State 0: value 0 | | |
| | State 1: 1 → 0 giving 45 + 0 = 45*; or 0 → 1 giving 0 + 45 = 45* | 2M1, 2A1 | 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. |
| | State 2: 2 → 0 giving 70 + 0 = 70; or 1 → 1 giving 45 + 45 = 90*; or 0 → 2 giving 0 + 85 = 85 | | |
| | State 3: 3 → 0 giving 100 + 0 = 100; 2 → 1 giving 70 + 45 = 115; 1 → 2 giving 45 + 85 = 130*; 0 → 3 giving 0 + 120 = 120 | 3A1 | 3A1: State 3 correct for second stage. Penalise * errors only twice in the question. |
| | State 4: 4 → 0 giving 120 + 0 = 120; 3 → 1 giving 100 + 45 = 145; 2 → 2 giving 70 + 85 = 155; 1 → 3 giving 45 + 120 = 165*; 0 → 4 giving 0 + 150 = 150 | 3M1, 4A1 | 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. |
| | State 5: 5 → 0 giving 130 + 0 = 130; 4 → 1 giving 120 + 45 = 165; 3 → 2 giving 100 + 85 = 185; 2 → 3 giving 70 + 120 = 190; 1 → 4 giving 45 + 150 = 195*; 0 → 5 giving 0 + 175 = 175 | 5A1 | 5A1: State 5 correct for second stage. Penalise * errors only twice in the question. |
| **Stage 3 (Canned)** | | | |
| | State 5: 5 → 0 giving 195 + 0 = 195; 4 → 1 giving 155 + 45 = 200; 3 → 2 giving 125 + 90 = 215*; 2 → 3 giving 75 + 130 = 205; 1 → 4 giving 35 + 165 = 200; 0 → 5 giving 0 + 175 = 175 | 4M1, 6A1ft | 4M1: Third stage complete (bod something in each cell). 6A1ft: Third stage correct (ft from previous stage). Penalise * errors only twice in the question. |
| **Final answer** | | | |
| | Fresh = 1, Frozen = 1, Canned = 3 | 5M1 | 5M1: For Fresh = 1, Frozen = 1, Canned = 3 – dependent on all previous M marks having been awarded. |
| | Monthly income = £21 500 | 7A1ft | 7A1ft: Income correct for their table. Must have earned the first 4 M marks only (not dependent on 5M1). |
| | **Total** | **12 marks** | |
---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 $\pounds 100$ s, for the number of units sold, are shown in the table.\\
The total monthly profit is to be maximised.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Number of units & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
Fresh & 0 & 45 & 85 & 120 & 150 & 175 \\
\hline
Frozen & 0 & 45 & 70 & 100 & 120 & 130 \\
\hline
Canned & 0 & 35 & 75 & 125 & 155 & 195 \\
\hline
\end{tabular}
\end{center}
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.\\
(Total 12 marks)\\
\hfill \mbox{\textit{Edexcel D2 2013 Q8 [12]}}