| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Explain dummy activities |
| Difficulty | Moderate -0.8 This is a standard D1 critical path analysis question testing routine procedures: explaining dummy activities (textbook definitions), calculating early/late times (algorithmic), finding float (formula application), and drawing a Gantt chart. All parts are procedural recall with no problem-solving or novel insight required, making it easier than average. |
| 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 |
| I depends on B, E and F only; K depends on B, E, F and D | B1 DB1 | K, I, D and at least one of B, E, F referred to. Correct but maybe incomplete: bod |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| So that G and H will not share the same start and end events; so that G and H can be uniquely described in terms of their end events | B1 | Clear correct statement. No bod. Must refer to either events or activities. 'unique' alone not enough. 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| All top boxes complete, values generally increasing left to right | M1 A1 | Condone one rogue. CAO |
| All bottom boxes complete, values generally decreasing R to L | M1 A1 | Condone one rogue. CAO. 4 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct calculation of floats, all three numbers correct (ft) | M1 A1ft | One float \((\geq 0)\) correct |
| Both floats correct | B1 | Independent of working. 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Total float on \(D = 18 - 5 - 6 = 7\) | M1 A1cso | Attempt to calculate a lower bound. Accept awrt 2.81. 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Total float on \(G = 17 - 4 - 7 = 6\) | M1 A1 | At least 7 activities including at least 4 floats. Do not accept scheduling diagram. Critical activities dealt with correctly |
| Lower bound \(= \dfrac{59}{21} = 3\) workers | M1 A1 | All 11 activities including at least 8 floats. Non-critical activities dealt with correctly. 4 marks total |
# Question 7:
## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| I depends on B, E and F only; K depends on B, E, F **and** D | B1 DB1 | K, I, D and at least one of B, E, F referred to. Correct but maybe incomplete: bod |
## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| So that G and H will not share the same start and end events; so that G and H can be uniquely described in terms of their end events | B1 | Clear correct statement. No bod. Must refer to either events or activities. 'unique' alone not enough. **3 marks total** |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| All top boxes complete, values generally increasing left to right | M1 A1 | Condone one rogue. CAO |
| All bottom boxes complete, values generally decreasing R to L | M1 A1 | Condone one rogue. CAO. **4 marks total** |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct calculation of floats, all three numbers correct (ft) | M1 A1ft | One float $(\geq 0)$ correct |
| Both floats correct | B1 | Independent of working. **3 marks total** |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Total float on $D = 18 - 5 - 6 = 7$ | M1 A1cso | Attempt to calculate a lower bound. Accept awrt 2.81. **2 marks total** |
## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Total float on $G = 17 - 4 - 7 = 6$ | M1 A1 | At least 7 activities including at least 4 floats. Do not accept scheduling diagram. Critical activities dealt with correctly |
| Lower bound $= \dfrac{59}{21} = 3$ workers | M1 A1 | All 11 activities including at least 8 floats. Non-critical activities dealt with correctly. **4 marks total** |
**Total: 16 marks**7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-9_1042_1426_267_315}
\captionsetup{labelformat=empty}
\caption{Figure 7}
\end{center}
\end{figure}
A project is modelled by the activity network shown in Figure 7. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the activity. The numbers in circles are the event numbers. Each activity requires one worker.
\begin{enumerate}[label=(\alph*)]
\item Explain the significance of the dummy activity
\begin{enumerate}[label=(\roman*)]
\item from event 4 to event 6 ,
\item from event 5 to event 7\\
(3)
\end{enumerate}\item Calculate the early time and the late time for each event. Write these in the boxes in the answer book.
\item Calculate the total float on each of activities D and G. You must make the numbers you use in your calculations clear.
\item Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time.
\item On the grid in your answer book, draw a cascade (Gantt) chart for this project.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2012 Q7 [16]}}