| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Dynamic programming order sequencing |
| Difficulty | Moderate -0.8 This is a standard textbook dynamic programming problem with a pre-structured table to fill in. Students simply work backwards through the table using given cost values, performing basic arithmetic to find minimum costs at each stage. The algorithm is mechanical and requires no problem-solving insight beyond following the standard DP procedure taught in D2. |
| 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 float |
| Year | Already renovated | House renovated | Calculation | Value |
| 3 | \(A\) and \(B\) | C | ||
| \(A\) and \(C\) | B | |||
| \(B\) and \(C\) | A | |||
| 2 | A | B | ||
| C | ||||
| B | A | |||
| C | ||||
| C | A | |||
| B | ||||
| 1 | ||||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Least annual cost for order \(BAC\) is 55 (thousand pounds) | B1 | From table: Year 2, already renovated B, cost of A = 55 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Least annual cost for \(ABC\): min of 60, 75, 75 = 60 | B1 | Reading minimum cost across years for order ABC |
| Order \(ABC\) is better than \(BAC\) | M1 | Comparing 60 > 55 — no, 60 > 55 so ABC is better |
| Because least annual cost of \(ABC\) (60) \(>\) least annual cost of \(BAC\) (55) | A1 | Must give reason with correct values |
| Answer | Marks | Guidance |
|---|---|---|
| Year | Already renovated | House renovated |
| 3 | \(A\) and \(B\) | \(C\) |
| 3 | \(A\) and \(C\) | \(B\) |
| 3 | \(B\) and \(C\) | \(A\) |
| 2 | \(A\) | \(B\) |
| 2 | \(A\) | \(C\) |
| 2 | \(B\) | \(A\) |
| 2 | \(B\) | \(C\) |
| 2 | \(C\) | \(A\) |
| 2 | \(C\) | \(B\) |
| 1 | — | \(A\) |
| 1 | — | \(B\) |
| 1 | — | \(C\) |
| Answer | Mark | Guidance |
| Optimum order is \(\mathbf{BAC}\) with minimum = 115 (i.e. maximin = 115) | A1 | Follow through from working; maximising the minimum annual cost gives order BAC |
# Question 5:
## Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Least annual cost for order $BAC$ is **55** (thousand pounds) | B1 | From table: Year 2, already renovated B, cost of A = 55 |
## Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Least annual cost for $ABC$: min of 60, 75, 75 = **60** | B1 | Reading minimum cost across years for order ABC |
| Order $ABC$ is **better** than $BAC$ | M1 | Comparing 60 > 55 — no, 60 > 55 so ABC is better |
| Because least annual cost of $ABC$ (60) $>$ least annual cost of $BAC$ (55) | A1 | Must give reason with correct values |
## Part (b):
| Year | Already renovated | House renovated | Calculation | Value | Mark |
|------|------------------|-----------------|-------------|-------|------|
| 3 | $A$ and $B$ | $C$ | 75 | **75** | B1 |
| 3 | $A$ and $C$ | $B$ | 80 | **80** | B1 |
| 3 | $B$ and $C$ | $A$ | 60 | **60** | B1 |
| 2 | $A$ | $B$ | $75 + 80 = 155$ | **155** | M1A1 |
| 2 | $A$ | $C$ | $70 + 75 = 145$ | **145** | A1 |
| 2 | $B$ | $A$ | $55 + 60 = 115$ | **115** | A1 |
| 2 | $B$ | $C$ | $60 + 75 = 135$ | **135** | A1 |
| 2 | $C$ | $A$ | $65 + 60 = 125$ | **125** | A1 |
| 2 | $C$ | $B$ | $80 + 80 = 160$ | **160** | A1 |
| 1 | — | $A$ | $60 + 115 = 175$ | **175** | M1 |
| 1 | — | $B$ | $70 + 125 = 195$ | **195** | A1 |
| 1 | — | $C$ | $65 + 145 = 210$ | **210** | A1 |
| Answer | Mark | Guidance |
|--------|------|----------|
| Optimum order is $\mathbf{BAC}$ with minimum = **115** (i.e. maximin = **115**) | A1 | Follow through from working; maximising the minimum annual cost gives order BAC |
---5 Dave plans to renovate three houses, $A , B$ and $C$, at the rate of one per year. The order in which they are renovated is a matter of choice, but some costs vary over the three years. The expected costs, in thousands of pounds, are given in the table below.
(b)
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Year & Already renovated & House renovated & Calculation & Value \\
\hline
3 & $A$ and $B$ & C & & \\
\hline
& $A$ and $C$ & B & & \\
\hline
& $B$ and $C$ & A & & \\
\hline
& & & & \\
\hline
2 & A & B & & \\
\hline
& & C & & \\
\hline
& & & & \\
\hline
& B & A & & \\
\hline
& & C & & \\
\hline
& & & & \\
\hline
& C & A & & \\
\hline
& & B & & \\
\hline
& & & & \\
\hline
1 & & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
& & & & \\
\hline
\end{tabular}
\end{center}
Optimum order $\_\_\_\_$\\
\hfill \mbox{\textit{AQA D2 2012 Q5 [10]}}