| Exam Board | AQA |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Effect of activity delay/change |
| Difficulty | Moderate -0.8 This is a standard Critical Path Analysis question testing routine algorithmic procedures (forward/backward pass, identifying critical paths, drawing Gantt charts, and analyzing simple delays). While multi-part, each component follows textbook methods with no novel problem-solving required, making it easier than average A-level material. |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Forward pass: EST for node after A = 4, after B = 3, after C = 2 | B1 | |
| EST for D = 4, E = max(4,3) = 4 (not 3), F = max(3,2) = 3 | M1 | Method for finding ESTs correctly |
| ESTs: G = 7, H = 8, I = 8; J = 15, K = 14, L = 17; Project = 19 | A1 | All ESTs correct |
| Latest finish times found by backward pass; LFT = 19 throughout critical path | A1 | All LFTs correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Critical path: \(B \to E \to H \to J \to L\) | B1 | |
| Critical path: \(B \to E \to H \to K \to L\) | B1 | |
| Minimum completion time = 19 days | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Activities labelled correctly on Gantt chart | B1 | |
| All activities starting at correct earliest start times | B1 | |
| All durations correct | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Activity \(I\) now takes 9 days, so earliest finish of \(I\) = \(8 + 9 = 17\) | M1 | |
| Earliest start of \(K\) = 17, earliest start of \(L\) = max(17+5, 15+2) = 22 | A1 | Both correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Project delayed by 3 days (completion now 22 days) | B1 |
# Question 1:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Forward pass: EST for node after A = 4, after B = 3, after C = 2 | B1 | |
| EST for D = 4, E = max(4,3) = 4 (not 3), F = max(3,2) = 3 | M1 | Method for finding ESTs correctly |
| ESTs: G = 7, H = 8, I = 8; J = 15, K = 14, L = 17; Project = 19 | A1 | All ESTs correct |
| Latest finish times found by backward pass; LFT = 19 throughout critical path | A1 | All LFTs correct |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Critical path: $B \to E \to H \to J \to L$ | B1 | |
| Critical path: $B \to E \to H \to K \to L$ | B1 | |
| Minimum completion time = 19 days | B1 | |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Activities labelled correctly on Gantt chart | B1 | |
| All activities starting at correct earliest start times | B1 | |
| All durations correct | B1 | |
## Part (d)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Activity $I$ now takes 9 days, so earliest finish of $I$ = $8 + 9 = 17$ | M1 | |
| Earliest start of $K$ = 17, earliest start of $L$ = max(17+5, 15+2) = 22 | A1 | Both correct |
## Part (d)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Project delayed by 3 days (completion now 22 days) | B1 | |
---1 Figure 1 below shows an activity diagram for a construction project. The time needed for each activity is given in days.
\begin{enumerate}[label=(\alph*)]
\item Find the earliest start time and latest finish time for each activity and insert their values on Figure 1.
\item Find the critical paths and state the minimum time for completion of the project.
\item On Figure 2 opposite, draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as early as possible.
\item A delay in supplies means that Activity $I$ takes 9 days instead of 2 .
\begin{enumerate}[label=(\roman*)]
\item Determine the effect on the earliest possible starting times for activities $K$ and $L$.
\item State the number of days by which the completion of the project is now delayed.\\
(1 mark)
\section*{Figure 1}
(a)\\
\includegraphics[max width=\textwidth, alt={}, center]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-02_815_1337_1573_395}\\
(b) Critical paths are $\_\_\_\_$\\
Minimum completion time is $\_\_\_\_$ days.
QUESTION PART REFERENCE\\
(c)
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{c4dc61a7-47ee-4d5c-bf6d-30a5da2ee8dd-03_978_1207_354_461}
\end{center}
\end{figure}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D2 2010 Q1 [13]}}