| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 14 |
| 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 (precedence tables, earliest/latest times, critical paths, float, Gantt charts, and scheduling with limited workers). While multi-part with 7 sub-questions, each part follows textbook procedures with no novel problem-solving required. The limited worker scheduling (parts f-g) is slightly more challenging than basic CPA but remains algorithmic, making this slightly easier than average overall. |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| A – none; B – none; C – none; D – A, B; E – B, C; F – B, C; G – D, E; H – E, F; I – G, H; J – H | B1 | All correct for 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Earliest start times: A=0, B=0, C=0, D=7, E=5, F=5, G=13, H=13, I=18, J=19 | M1 A1 | M1 for attempt at forward pass; A1 all correct |
| Latest finish times: A=7, B=7, C=6, D=13, E=13, F=13, G=18, H=19, I=27, J=28 | M1 A1 | M1 for attempt at backward pass; A1 all correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| A – D – G – I – J | B1 | |
| B – E – H – J | B1 | Both paths required for 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Float of \(E\) = \(13 - 5 - 7 = 1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Gantt chart with critical activities correctly placed | B1 | Critical path activities correct |
| Non-critical activities placed at earliest start times | B1 | |
| All activities shown within correct time bounds | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(7+5+4+6+7+8+5+6+8+9 = 65\) hours | B1 | Sum of all activity durations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Minimum completion time = 33 hours, with a valid allocation of activities to two workers | M1 A1 | M1 for attempting to schedule; A1 for correct time with valid allocation shown |
# Question 1:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| A – none; B – none; C – none; D – A, B; E – B, C; F – B, C; G – D, E; H – E, F; I – G, H; J – H | B1 | All correct for 1 mark |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Earliest start times: A=0, B=0, C=0, D=7, E=5, F=5, G=13, H=13, I=18, J=19 | M1 A1 | M1 for attempt at forward pass; A1 all correct |
| Latest finish times: A=7, B=7, C=6, D=13, E=13, F=13, G=18, H=19, I=27, J=28 | M1 A1 | M1 for attempt at backward pass; A1 all correct |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| A – D – G – I – J | B1 | |
| B – E – H – J | B1 | Both paths required for 2 marks |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| Float of $E$ = $13 - 5 - 7 = 1$ | B1 | |
## Part (e)
| Answer | Mark | Guidance |
|--------|------|----------|
| Gantt chart with critical activities correctly placed | B1 | Critical path activities correct |
| Non-critical activities placed at earliest start times | B1 | |
| All activities shown within correct time bounds | B1 | |
## Part (f)
| Answer | Mark | Guidance |
|--------|------|----------|
| $7+5+4+6+7+8+5+6+8+9 = 65$ hours | B1 | Sum of all activity durations |
## Part (g)
| Answer | Mark | Guidance |
|--------|------|----------|
| Minimum completion time = 33 hours, with a valid allocation of activities to two workers | M1 A1 | M1 for attempting to schedule; A1 for correct time with valid allocation shown |
---1 Figure 2, on the page opposite, 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 On Figure 1 below, complete the precedence table.
\item Find the earliest start time and the latest finish time for each activity and insert their values on Figure 2.
\item List the critical paths.
\item Find the float time of activity $E$.
\item Using Figure 3 opposite, draw a Gantt diagram 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 is only one worker available for the project, find the minimum completion time for the project.
\item Given that there are two workers available for the project, find the minimum completion time for the project. Show a suitable allocation of tasks to the two workers.\\[0pt]
[2 marks]
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\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
\end{tabular}
\end{center}
\end{table}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_1071_1561_376_278}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_801_1301_1644_420}
\end{center}
\end{figure}
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2015 Q1 [14]}}