| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Schedule with limited workers - determine minimum time |
| Difficulty | Moderate -0.5 This is a standard Critical Path Analysis question covering routine D2 techniques (earliest/latest times, critical path, float, resource histograms, and scheduling with limited workers). While multi-part with several marks, each component follows textbook procedures without requiring novel insight or complex problem-solving—slightly easier than average due to its procedural nature. |
| 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 |
| Activity | Immediate predecessor(s) |
| A | |
| B | |
| C | |
| D | |
| E | |
| \(F\) | |
| G | |
| H | |
| I | |
| J | |
| \(K\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Earliest start times: \(A=0, B=0, C=8, D=8, E=8, F=5, G=14, H=14, I=18, J=20, K=30\) | M1 | Method for forward pass |
| Latest finish times: \(K=31, I=30, J=30, G=18, H=26, D=14, E=15, F=15, C=11, A=8, B=8\) | A1 | Correct earliest start times |
| A1 | Correct latest finish times (ft from EST) | |
| A1 | All values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(A\): none; \(B\): none; \(C\): \(A\); \(D\): \(A, C\); \(E\): \(B, C\); \(F\): \(B\); \(G\): \(D, E\); \(H\): \(E, F\); \(I\): \(G\); \(J\): \(G, H\); \(K\): \(I, J\) | B2 | B1 for 4 or 5 correct rows |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(A, D, G, I, K\) | B1 | Must be stated as a path |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Float of \(E = 1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct histogram with all activities starting as early as possible | B3 | B1 for correct blocks 0–8; B1 for correct blocks 8–14; B1 for remainder correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Minimum completion time \(= 31\) hours | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Total time \(= 8+5+3+6+1+7+4+6+12+9+1 = 62\) hours | B1 |
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Earliest start times: $A=0, B=0, C=8, D=8, E=8, F=5, G=14, H=14, I=18, J=20, K=30$ | M1 | Method for forward pass |
| Latest finish times: $K=31, I=30, J=30, G=18, H=26, D=14, E=15, F=15, C=11, A=8, B=8$ | A1 | Correct earliest start times |
| | A1 | Correct latest finish times (ft from EST) |
| | A1 | All values correct |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $A$: none; $B$: none; $C$: $A$; $D$: $A, C$; $E$: $B, C$; $F$: $B$; $G$: $D, E$; $H$: $E, F$; $I$: $G$; $J$: $G, H$; $K$: $I, J$ | B2 | B1 for 4 or 5 correct rows |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $A, D, G, I, K$ | B1 | Must be stated as a path |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| Float of $E = 1$ | B1 | |
## Part (e)
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct histogram with all activities starting as early as possible | B3 | B1 for correct blocks 0–8; B1 for correct blocks 8–14; B1 for remainder correct |
## Part (f)
| Answer | Mark | Guidance |
|--------|------|----------|
| Minimum completion time $= 31$ hours | B1 | |
## Part (g)
| Answer | Mark | Guidance |
|--------|------|----------|
| Total time $= 8+5+3+6+1+7+4+6+12+9+1 = 62$ hours | B1 | |
---
1\\
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.
\begin{enumerate}[label=(\alph*)]
\item Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
\item On Figure 2 opposite, complete the precedence table.
\item Find the critical path.
\item Find the float time of activity $E$.
\item Using Figure 3 on page 5, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
\item Given that there are two workers available for the project, find the minimum completion time for the project.
\item Given that there is only one worker available for the project, find the minimum completion time for the project.
Figure 1
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(a)}
\includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-02_629_1550_1818_292}
\end{center}
\end{figure}
(b)
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\begin{tabular}{|l|l|}
\hline
Activity & Immediate predecessor(s) \\
\hline
A & \\
\hline
B & \\
\hline
C & \\
\hline
D & \\
\hline
E & \\
\hline
$F$ & \\
\hline
G & \\
\hline
H & \\
\hline
I & \\
\hline
J & \\
\hline
$K$ & \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-05_2486_1717_221_150}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2013 Q1 [13]}}