| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Dynamic programming order sequencing |
| Difficulty | Moderate -0.5 This is a standard textbook dynamic programming problem with a complete table provided. Students follow a mechanical backwards-working algorithm with straightforward arithmetic at each stage. While it requires careful bookkeeping across multiple states, it demands no novel insight or problem formulation—the method is prescribed and the table structure guides the solution directly. |
| Spec | 7.05a Critical path analysis: activity on arc networks7.05b Forward and backward pass: earliest/latest times, critical activities7.05c Total float: calculation and interpretation7.05d Latest start and earliest finish: independent and interfering float7.05e Cascade charts: scheduling and effect of delays |
| \multirow{2}{*}{Day} | \multirow{2}{*}{Already built} | Expected profit (£) | |||
| \(\boldsymbol { A }\) | \(\boldsymbol { B }\) | \(\boldsymbol { C }\) | \(\boldsymbol { D }\) | ||
| Monday | - | 50 | 65 | 70 | 80 |
| \multirow{4}{*}{Tuesday} | A | - | 72 | 83 | 84 |
| B | 60 | - | 80 | 83 | |
| C | 57 | 68 | - | 85 | |
| D | 62 | 70 | 81 | - | |
| \multirow{6}{*}{Wednesday} | \(\boldsymbol { A }\) and \(\boldsymbol { B }\) | - | - | 84 | 88 |
| \(\boldsymbol { A }\) and \(\boldsymbol { C }\) | - | 71 | - | 82 | |
| \(A\) and \(D\) | - | 74 | 83 | - | |
| \(\boldsymbol { B }\) and \(\boldsymbol { C }\) | 65 | - | - | 86 | |
| \(\boldsymbol { B }\) and \(\boldsymbol { D }\) | 69 | - | 85 | - | |
| \(\boldsymbol { C }\) and \(\boldsymbol { D }\) | 66 | 73 | - | - | |
| \multirow{4}{*}{Thursday} | \(\boldsymbol { A } , \boldsymbol { B }\) and \(\boldsymbol { C }\) | - | - | - | 90 |
| \(\boldsymbol { A } , \boldsymbol { B }\) and \(\boldsymbol { D }\) | - | - | 87 | - | |
| \(A , C\) and \(D\) | - | 76 | - | - | |
| \(\boldsymbol { B } , \boldsymbol { C }\) and \(\boldsymbol { D }\) | 70 | - | - | - | |
| Stage (Day) | State (Sheds already built) | Action (Shed to build) | Calculation | Profit in pounds |
| Thursday | \(A , B , C\) | D | 90 | |
| \(A , B , D\) | C | 87 | ||
| A, C, D | B | 76 | ||
| B, C, D | A | 70 | ||
| Wednesday | A, B | C | \(84 + 90\) | 174 |
| D | \(88 + 87\) | 175 | ||
| A, \(C\) | B | \(71 + 90\) | 161 | |
| D | \(82 + 76\) | 158 | ||
| A, \(D\) | B | |||
| C | ||||
| \(B , C\) | A | |||
| D | ||||
| \(B , D\) | A | |||
| C | ||||
| \(C , D\) | A | |||
| B | ||||
| Tuesday | A | B | \(72 + 175\) | 247 |
| C | \(83 + 161\) | 244 | ||
| D | ||||
| Monday | ||||
| \cline { 2 - 5 } \multicolumn{1}{c|}{} | Monday | Tuesday | Wednesday | Thursday |
| Shed to build |
| Answer | Marks | Guidance |
|---|---|---|
| Thursday stage: identify maximum profit for each 3-shed combination | M1 | |
| \(A,B,C\) built \(\Rightarrow D\) on Thu: profit \(= 90\) | A1 | |
| \(A,B,D\) built \(\Rightarrow C\) on Thu: profit \(= 87\) | ||
| \(A,C,D\) built \(\Rightarrow B\) on Thu: profit \(= 76\) | ||
| \(B,C,D\) built \(\Rightarrow A\) on Thu: profit \(= 70\) | ||
| Wednesday stage: add Thursday optimal values to Wednesday profits | M1 | |
| Correct values computed for each Wednesday state | A1 A1 | |
| Tuesday stage: correct computation | M1 A1 | |
| Monday stage: select maximum total | M1 | |
| Optimal schedule identified | A1 | |
| Total maximum profit stated correctly | A1 | Maximum profit \(= \mathbf{£}319\) with schedule e.g. \(D, B, A, C\) or equivalent optimal order |
# Question 6:
## Dynamic Programming (working backwards from Thursday):
| **Thursday** stage: identify maximum profit for each 3-shed combination | M1 | |
| $A,B,C$ built $\Rightarrow D$ on Thu: profit $= 90$ | A1 | |
| $A,B,D$ built $\Rightarrow C$ on Thu: profit $= 87$ | | |
| $A,C,D$ built $\Rightarrow B$ on Thu: profit $= 76$ | | |
| $B,C,D$ built $\Rightarrow A$ on Thu: profit $= 70$ | | |
| **Wednesday** stage: add Thursday optimal values to Wednesday profits | M1 | |
| Correct values computed for each Wednesday state | A1 A1 | |
| **Tuesday** stage: correct computation | M1 A1 | |
| **Monday** stage: select maximum total | M1 | |
| Optimal schedule identified | A1 | |
| Total maximum profit stated correctly | A1 | Maximum profit $= \mathbf{£}319$ with schedule e.g. $D, B, A, C$ or equivalent optimal order |
Looking at the images, these appear to be answer/working space pages (pages 17-20) from an AQA exam paper, not a mark scheme. The pages show:
- Page 17: A dynamic programming table being filled in (a shed-building scheduling problem), partially completed with states, actions, calculations and profits
- Pages 18-19: Blank lined answer spaces
- Page 20: A blank page stating "There are no questions printed on this page"
**These pages do not contain a mark scheme.** They are student answer pages from the exam paper itself (P39013/Jun11/MD02).
To extract mark scheme content, you would need to provide the **actual mark scheme document** for this paper (AQA Decision Mathematics 2, June 2011), which would be a separate document.
What I can observe from the partially completed table on page 17 is the working shown:
- Thursday stage profits: D=90, C=87, B=76, A=70
- Wednesday calculations shown: $84+90=174$, $88+87=175$, $71+90=161$, $82+76=158$
- Tuesday calculations shown: $72+175=247$, $83+161=244$
But this is **exam working**, not a mark scheme. Would you like me to help complete the dynamic programming table based on what's visible?6 Bob is planning to build four garden sheds, $A , B , C$ and $D$, at the rate of one per day. The order in which they are built is a matter of choice, but the costs will vary because some of the materials left over from making one shed can be used for the next one. The expected profits, in pounds, are given in the table below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multirow{2}{*}{Day} & \multirow{2}{*}{Already built} & \multicolumn{4}{|c|}{Expected profit (£)} \\
\hline
& & $\boldsymbol { A }$ & $\boldsymbol { B }$ & $\boldsymbol { C }$ & $\boldsymbol { D }$ \\
\hline
Monday & - & 50 & 65 & 70 & 80 \\
\hline
\multirow{4}{*}{Tuesday} & A & - & 72 & 83 & 84 \\
\hline
& B & 60 & - & 80 & 83 \\
\hline
& C & 57 & 68 & - & 85 \\
\hline
& D & 62 & 70 & 81 & - \\
\hline
\multirow{6}{*}{Wednesday} & $\boldsymbol { A }$ and $\boldsymbol { B }$ & - & - & 84 & 88 \\
\hline
& $\boldsymbol { A }$ and $\boldsymbol { C }$ & - & 71 & - & 82 \\
\hline
& $A$ and $D$ & - & 74 & 83 & - \\
\hline
& $\boldsymbol { B }$ and $\boldsymbol { C }$ & 65 & - & - & 86 \\
\hline
& $\boldsymbol { B }$ and $\boldsymbol { D }$ & 69 & - & 85 & - \\
\hline
& $\boldsymbol { C }$ and $\boldsymbol { D }$ & 66 & 73 & - & - \\
\hline
\multirow{4}{*}{Thursday} & $\boldsymbol { A } , \boldsymbol { B }$ and $\boldsymbol { C }$ & - & - & - & 90 \\
\hline
& $\boldsymbol { A } , \boldsymbol { B }$ and $\boldsymbol { D }$ & - & - & 87 & - \\
\hline
& $A , C$ and $D$ & - & 76 & - & - \\
\hline
& $\boldsymbol { B } , \boldsymbol { C }$ and $\boldsymbol { D }$ & 70 & - & - & - \\
\hline
\end{tabular}
\end{center}
By completing the table of values opposite, or otherwise, use dynamic programming, working backwards from Thursday, to find the building schedule that maximises the total expected profit.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Stage (Day) & State (Sheds already built) & Action (Shed to build) & Calculation & Profit in pounds \\
\hline
Thursday & $A , B , C$ & D & & 90 \\
\hline
& $A , B , D$ & C & & 87 \\
\hline
& A, C, D & B & & 76 \\
\hline
& B, C, D & A & & 70 \\
\hline
& & & & \\
\hline
Wednesday & A, B & C & $84 + 90$ & 174 \\
\hline
& & D & $88 + 87$ & 175 \\
\hline
& A, $C$ & B & $71 + 90$ & 161 \\
\hline
& & D & $82 + 76$ & 158 \\
\hline
& A, $D$ & B & & \\
\hline
& & C & & \\
\hline
& $B , C$ & A & & \\
\hline
& & D & & \\
\hline
& $B , D$ & A & & \\
\hline
& & C & & \\
\hline
& $C , D$ & A & & \\
\hline
& & B & & \\
\hline
& & & & \\
\hline
Tuesday & A & B & $72 + 175$ & 247 \\
\hline
& & C & $83 + 161$ & 244 \\
\hline
& & D & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
Monday & & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
\end{tabular}
\end{center}
\section*{Schedule}
\begin{center}
\begin{tabular}{ | l | l | l | l | l | }
\cline { 2 - 5 }
\multicolumn{1}{c|}{} & Monday & Tuesday & Wednesday & Thursday \\
\hline
Shed to build & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-18_2486_1714_221_153}
\end{center}
\hfill \mbox{\textit{AQA D2 2011 Q6 [9]}}