| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Draw resource histogram |
| Difficulty | Moderate -0.3 This is a comprehensive but standard Critical Path Analysis question covering all routine techniques (network drawing, forward/backward pass, critical path, float, resource histogram, and basic resource levelling). While lengthy with multiple parts, each step follows textbook procedures with no novel problem-solving required, making it slightly easier than average for A-level. |
| 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 |
| Activity | Immediate Predecessors | Duration (days) | Number of Workers Required |
| A | - | 2 | 3 |
| B | A | 4 | 2 |
| C | A | 6 | 1 |
| D | \(B , C\) | 8 | 3 |
| E | C | 3 | 2 |
| F | D | 2 | 2 |
| G | D, E | 4 | 2 |
| H | D, E | 6 | 1 |
| I | \(F , G , H\) | 2 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Activity network SCA | M1 | |
| Almost correct (up to 2 slips) | A1 | |
| All correct | A1 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Forward pass for earliest times | M1 | |
| Correct | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Backward pass | M1 | |
| Correct | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Critical path is \(ACDHI\) | B1 | |
| Minimum completion 24 days | B1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Non-critical activities \(B,\ E,\ F,\ G\) with floats \(2,\ 5,\ 4,\ 2\) respectively | M1 | At least 3 activities and float in one activity correct |
| Correct earliest and latest times shown | \(A1\sqrt{}\) | 2 – \(\sqrt{}\) their earliest and latest times |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Resource histogram \(\leq 11\) | M1 | |
| Correct | A1 | |
| Rest as histogram – generally start activities ok | M1 | |
| All correct | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Problems with \(D\ \&\ E\) solved by \(E\) coming after \(D\) | M1 | |
| Problem at 16–18 days with \(F\) can be solved by moving \(F\) to 20–22 | A1 | |
| Must overrun by equivalent to duration of \(E\) (3 days) | B1 | 3 |
## Question 3:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Activity network SCA | M1 | |
| Almost correct (up to 2 slips) | A1 | |
| All correct | A1 | **3** |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Forward pass for earliest times | M1 | |
| Correct | A1 | **2** |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Backward pass | M1 | |
| Correct | A1 | **2** |
### Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Critical path is $ACDHI$ | B1 | |
| Minimum completion 24 days | B1 | **2** |
### Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Non-critical activities $B,\ E,\ F,\ G$ with floats $2,\ 5,\ 4,\ 2$ respectively | M1 | At least 3 activities and float in one activity correct |
| Correct earliest and latest times shown | $A1\sqrt{}$ | **2** – $\sqrt{}$ their earliest and latest times |
### Part (f)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Resource histogram $\leq 11$ | M1 | |
| Correct | A1 | |
| Rest as histogram – generally start activities ok | M1 | |
| All correct | A1 | **4** |
### Part (g)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Problems with $D\ \&\ E$ solved by $E$ coming after $D$ | M1 | |
| Problem at 16–18 days with $F$ can be solved by moving $F$ to 20–22 | A1 | |
| Must overrun by equivalent to duration of $E$ (3 days) | B1 | **3** |
---3 [Figures 1 and 2, printed on the insert, are provided for use in this question.] A building project is to be undertaken. The table shows the activities involved.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Activity & Immediate Predecessors & Duration (days) & Number of Workers Required \\
\hline
A & - & 2 & 3 \\
\hline
B & A & 4 & 2 \\
\hline
C & A & 6 & 1 \\
\hline
D & $B , C$ & 8 & 3 \\
\hline
E & C & 3 & 2 \\
\hline
F & D & 2 & 2 \\
\hline
G & D, E & 4 & 2 \\
\hline
H & D, E & 6 & 1 \\
\hline
I & $F , G , H$ & 2 & 3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the activity network for the project on Figure 1.
\item Find the earliest start time for each activity.
\item Find the latest finish time for each activity.
\item Find the critical path and state the minimum time for completion.
\item State the float time for each non-critical activity.
\item Given that each activity starts as early as possible, draw a resource histogram for the project on Figure 2.
\item There are only 3 workers available at any time. Use resource levelling to explain why the project will overrun and state the minimum extra time required.
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2006 Q3 [18]}}