| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2009 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Dynamic programming resource allocation |
| Difficulty | Moderate -0.3 This is a standard textbook dynamic programming problem with a small state space (6 possible allocations across 3 schemes). The table structure is provided, requiring only systematic enumeration of states and straightforward maximization at each stage—no novel insight or complex optimization needed, making it slightly easier than average. |
| Spec | 7.05a Critical path analysis: activity on arc networks |
| \(\mathbf { \pounds 0 }\) | \(\mathbf { \pounds 5 0 0 0 0 }\) | \(\mathbf { \pounds 1 0 0 0 0 0 }\) | \(\mathbf { \pounds 1 5 0 0 0 0 }\) | \(\mathbf { \pounds 2 0 0 0 0 0 }\) | \(\mathbf { \pounds 2 5 0 0 0 0 }\) | |
| Scheme1 | 0 | 60 | 120 | 180 | 240 | 300 |
| Scheme 2 | 0 | 65 | 125 | 190 | 235 | 280 |
| Scheme 3 | 0 | 55 | 110 | 170 | 230 | 300 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | Complete table with all values filled in correctly showing optimal strategy of investing scheme 1 £100,000, scheme 2 £150,000, scheme 3 £0, with Maximum income £310,000 | 1M1 A1 (10) |
| Scheme | 1 | 2 |
| Invest (in £1000s) | 100 | 150 |
| (b) | Stage: Scheme being considered; State: Money available to invest; Action: Amount chosen to invest | B1 B1 B1 (3) |
(a) | Complete table with all values filled in correctly showing optimal strategy of investing scheme 1 £100,000, scheme 2 £150,000, scheme 3 £0, with Maximum income £310,000 | 1M1 A1 (10) | All values shown with asterisks marking optimal decisions |
| Scheme | 1 | 2 | 3 |
| Invest (in £1000s) | 100 | 150 | 0 | B1 B1 | |
(b) | Stage: Scheme being considered; State: Money available to invest; Action: Amount chosen to invest | B1 B1 B1 (3) | |
**Total [13]**
---7. Minty has $\pounds 250000$ to allocate to three investment schemes. She will allocate the money to these schemes in units of $\pounds 50000$. The net income generated by each scheme, in $\pounds 1000$ s, is given in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
& $\mathbf { \pounds 0 }$ & $\mathbf { \pounds 5 0 0 0 0 }$ & $\mathbf { \pounds 1 0 0 0 0 0 }$ & $\mathbf { \pounds 1 5 0 0 0 0 }$ & $\mathbf { \pounds 2 0 0 0 0 0 }$ & $\mathbf { \pounds 2 5 0 0 0 0 }$ \\
\hline
Scheme1 & 0 & 60 & 120 & 180 & 240 & 300 \\
\hline
Scheme 2 & 0 & 65 & 125 & 190 & 235 & 280 \\
\hline
Scheme 3 & 0 & 55 & 110 & 170 & 230 & 300 \\
\hline
\end{tabular}
\end{center}
Minty wishes to maximise the net income. She decides to use dynamic programming to determine the optimal allocation, and starts the table shown in your answer book.
\begin{enumerate}[label=(\alph*)]
\item Complete the table in the answer book to determine the amount Minty should allocate to each scheme in order to maximise the income. State the maximum income and the amount that should be allocated to each scheme.
\item For this problem give the meaning of the table headings
\begin{enumerate}[label=(\roman*)]
\item Stage,
\item State,
\item Action.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel D2 2009 Q7 [13]}}