| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Calculate early and late times |
| Difficulty | Moderate -0.5 This is a standard Critical Path Analysis question requiring systematic application of the forward and backward pass algorithms to find early/late times, identify the critical path, calculate float, and construct resource histograms. While it involves multiple parts and careful bookkeeping, it follows routine D2 procedures without requiring novel problem-solving insights or complex reasoning beyond the standard algorithm. |
| 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 | A | \(B\) | C | D | \(E\) | \(F\) | G | H | I | \(J\) | \(K\) | \(L\) |
| Duration (days) | 2 | 5 | 6 | 7 | 9 | 4 | 3 | 2 | 3 | 2 | 3 | 1 |
| Number of workers required | 6 | 3 | 5 | 2 | 5 | 2 | 4 | 4 | 5 | 3 | 2 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Earliest start times: \(A=0, B=2, C=2, D=2, E=7, F=7, G=11, H=14, I=14, J=16, K=17, L=19\) | B1 | Forward pass correct |
| Latest finish times: \(L=20, J=19, K=20, H=17, I=20, G=14, E=16, F=18, B=7, C=8, D=18, A=2\) | B1 | Backward pass correct |
| Critical path length = 20 days | B1 | |
| All values correctly placed on diagram | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(A \to C \to E \to G \to H \to J \to L\) | B2 | B1 for partially correct path |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Float \(= \) latest finish time \(-\) duration \(-\) earliest start time | M1 | Correct method shown |
| \(= 18 - 7 - 2 = 9\) days | A1 |
# Question 1:
## Part (a) - Earliest Start Times and Latest Finish Times
| Answer | Mark | Guidance |
|--------|------|----------|
| Earliest start times: $A=0, B=2, C=2, D=2, E=7, F=7, G=11, H=14, I=14, J=16, K=17, L=19$ | B1 | Forward pass correct |
| Latest finish times: $L=20, J=19, K=20, H=17, I=20, G=14, E=16, F=18, B=7, C=8, D=18, A=2$ | B1 | Backward pass correct |
| Critical path length = 20 days | B1 | |
| All values correctly placed on diagram | B1 | |
## Part (b)(i) - Critical Path
| Answer | Mark | Guidance |
|--------|------|----------|
| $A \to C \to E \to G \to H \to J \to L$ | B2 | B1 for partially correct path |
## Part (b)(ii) - Float time for activity D
| Answer | Mark | Guidance |
|--------|------|----------|
| Float $= $ latest finish time $-$ duration $-$ earliest start time | M1 | Correct method shown |
| $= 18 - 7 - 2 = 9$ days | A1 | |
---1\\
A group of workers is involved in a decorating project. The table shows the activities involved. Each worker can perform any of the given activities.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Activity & A & $B$ & C & D & $E$ & $F$ & G & H & I & $J$ & $K$ & $L$ \\
\hline
Duration (days) & 2 & 5 & 6 & 7 & 9 & 4 & 3 & 2 & 3 & 2 & 3 & 1 \\
\hline
Number of workers required & 6 & 3 & 5 & 2 & 5 & 2 & 4 & 4 & 5 & 3 & 2 & 4 \\
\hline
\end{tabular}
\end{center}
The activity network for the project is given in Figure 1 below.
\begin{enumerate}[label=(\alph*)]
\item Find the earliest start time and the latest finish time for each activity, inserting their values on Figure 1.
\item Hence find:
\begin{enumerate}[label=(\roman*)]
\item the critical path;
\item the float time for activity $D$.\\
(a)
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-02_647_1657_1640_180}
\end{center}
\end{figure}
(b) (i) The critical path is $\_\_\_\_$\\
(ii) The float time for activity $D$ is $\_\_\_\_$
\end{enumerate}\item Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on Figure 2 below, indicating clearly which activities are taking place at any given time.
\item It is later discovered that there are only 8 workers available at any time. Use resource levelling to construct a new resource histogram on Figure 3 below, showing how the project can be completed with the minimum extra time. State the minimum extra time required.\\
(c)
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_586_1708_922_150}
\end{center}
\end{figure}
(d)
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_496_1705_1672_153}
\end{center}
\end{figure}
The minimum extra time required is $\_\_\_\_$
\end{enumerate}
\hfill \mbox{\textit{AQA D2 2011 Q1 [14]}}