| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Calculate lower bound for workers |
| Difficulty | Standard +0.3 This is a standard Critical Path Analysis question requiring calculation of early/late times, identification of critical path, and application of the lower bound formula (sum of activity times รท project duration). While it involves multiple steps, these are routine D1 techniques with no novel problem-solving required, making it slightly 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 interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| 2(a)(i) | Network diagram with all top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue; all bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue. Condone missing 0 and/or their 20 (at the end event) for the M mark only | M1, A1, M1, A1 |
| 2(a)(ii) | Minimum completion time is 20 (days) | A1 (ft) (5) |
| 2(b) | Lower bound \(= \frac{59}{20} = 2.95 = 3\) workers | B1 (1) |
| 2(c) | Gantt chart with: 3 or 4 workers. All 14 activities present (just once). Condone at most three errors. An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA; 3 or 4 workers. All 14 activities present (just once). Condone one error either precedence or time interval or activity length; An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA; 3 workers. All 14 activities present (just once). No errors. | M1, A1, A1, A1 (4) |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| 2(a)(i) | Network diagram with all top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue; all bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue. Condone missing 0 and/or their 20 (at the end event) for the M mark only | M1, A1, M1, A1 | See Notes for Question 2 |
| 2(a)(ii) | Minimum completion time is 20 (days) | A1 (ft) (5) | |
| 2(b) | Lower bound $= \frac{59}{20} = 2.95 = 3$ workers | B1 (1) | |
| 2(c) | Gantt chart with: 3 or 4 workers. All 14 activities present (just once). Condone at most three errors. An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA; 3 or 4 workers. All 14 activities present (just once). Condone one error either precedence or time interval or activity length; An activity can give rise to at most three errors; one on duration, one on time interval and only one on IPA; 3 workers. All 14 activities present (just once). No errors. | M1, A1, A1, A1 (4) | See Notes for Question 2 for reference table |
**Total: 10 marks**
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\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ba9337bf-7a3c-49aa-b395-dd7818cf1d13-03_942_1587_242_239}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
[The sum of the durations of all the activities is 59 days.]\\
The network in Figure 1 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration, in days, of the corresponding activity is shown in brackets. Each activity requires one worker. The project is to be completed in the shortest possible time.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Complete Diagram 1 in the answer book to show the early event times and the late event times.
\item State the minimum completion time of the project.
\end{enumerate}\item Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
\item Schedule the activities using the minimum number of workers so that the project is completed in the minimum time.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2024 Q2 [10]}}