| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Dynamic programming resource allocation |
| Difficulty | Moderate -0.5 This is a standard tabular dynamic programming problem with clear structure: 4 workers, 3 toy types, and a given table of values. Students follow a mechanical algorithm filling in a DP table by considering all allocations at each stage. While it requires careful bookkeeping and systematic working through 12 marks worth of calculations, it demands no novel insight—just application of the taught D2 method. Slightly easier than average A-level due to its algorithmic nature. |
| Spec | 7.05e Cascade charts: scheduling and effect of delays |
| \cline { 3 - 7 } \multicolumn{2}{c|}{} | Number of workers | |||||||||
| \cline { 3 - 7 } \multicolumn{2}{c|}{} | 0 | 1 | 2 | 3 | 4 | |||||
\multirow{2}{*}{
| Bicycle | 0 | 80 | 170 | 260 | 350 | ||||
| \cline { 2 - 7 } | Dolls House | 0 | 95 | 165 | 245 | 335 | ||||
| \cline { 2 - 7 } | Train Set | 0 | 100 | 180 | 260 | 340 | ||||
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(1M1 1A1\) | (2) |
| \(2M1 2A1 3A1\) | (3) | Second stage (Dolls house) completed for at least states 4 and 3. Bod something in each cell. States 4 + 3 |
| \(3M1 4A1 5A1\) | (3) | States 2 + 1 |
| \(4M1 6A1ft\) | (3) | Any three rows of third stage correct. Ft on * values only. No missing rows. Penalise * errors only once in the question |
| \(7A1\) | — | CAO for the third stage. No missing rows. Penalise * errors only once in the question |
| (b) | \(1B1\) | (1) |
| \(1B1\) | (1) | CAO. Must have attempted algorithm, getting all M marks in (a) |
| 13 marks |
**(a)** | $1M1 1A1$ | (2) | First stage (Bicycle) completed; bod something in each cell. Must have columns for stage, state, value and one of either action or destination |
| $2M1 2A1 3A1$ | (3) | Second stage (Dolls house) completed for at least states 4 and 3. Bod something in each cell. **States 4 + 3** |
| $3M1 4A1 5A1$ | (3) | **States 2 + 1** |
| $4M1 6A1ft$ | (3) | Any three rows of third stage correct. Ft on * values only. No missing rows. Penalise * errors only once in the question |
| $7A1$ | — | CAO for the third stage. No missing rows. Penalise * errors only once in the question |
**(b)** | $1B1$ | (1) | CAO. Must have attempted algorithm, getting all previous M marks |
| $1B1$ | (1) | CAO. Must have attempted algorithm, getting all M marks in (a) |
| | 13 marks | |
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# Notes for Questions:
**Notes for Question 7 (continued):**
- **ALL M marks:** Must bring earlier optimal results into calculations. Ignore extra rows. Must have right 'ingredients' (# number of workers) 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.7. Susie has hired a team of four workers who can make three types of toy. The total number of toys the team can produce will depend on which toys they make, and on how many workers are assigned to make each type of toy.
The table shows how many of each toy would be made if different numbers of workers were assigned to make them. Each worker is to be assigned to make just one type of toy and all four workers are to be assigned. Susie wishes to maximise the total number of toys produced.
\begin{center}
\begin{tabular}{ | c | l | c | c | c | c | c | }
\cline { 3 - 7 }
\multicolumn{2}{c|}{} & \multicolumn{5}{|c|}{Number of workers} \\
\cline { 3 - 7 }
\multicolumn{2}{c|}{} & 0 & 1 & 2 & 3 & 4 \\
\hline
\multirow{2}{*}{\begin{tabular}{ c }
T \\
O \\
Y \\
S \\
\end{tabular}} & Bicycle & 0 & 80 & 170 & 260 & 350 \\
\cline { 2 - 7 }
& Dolls House & 0 & 95 & 165 & 245 & 335 \\
\cline { 2 - 7 }
& Train Set & 0 & 100 & 180 & 260 & 340 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use dynamic programming to determine the allocation of workers which maximises the total number of toys made. You should show your working in the table provided in the answer book.\\
(12)
\item State the maximum total number of toys produced by this team.\\
(1)\\
(Total 13 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2014 Q7 [13]}}