| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2015 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Calculate lower bound for workers |
| Difficulty | Moderate -0.5 This is a standard Critical Path Analysis question covering routine D1 techniques: finding activity durations using critical activity and float information, calculating early/late event times, finding lower bound (sum of durations ÷ project duration), and drawing a cascade chart. While multi-part with several marks, each step follows textbook procedures without requiring novel insight or complex problem-solving. |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 12\) \(y = 3\) | B1 B1 (2) | |
| Network diagram with all values correct | M1 A1 A1 (3) | |
| Lower bound = \(\frac{99}{37} = 2.675\ldots\) so 3 workers | B1 (1) | |
| At least nine activities including at least five floats. | M1 A1 A1 A1 (4) | |
| Lower bound is 5 workers – e.g. activities H, I, J, K and L together with \(27 < \text{time} < 28\) | M1 A1 (2) | |
| 12 marks |
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 12$ $y = 3$ | B1 B1 (2) | |
| Network diagram with all values correct | M1 A1 A1 (3) | |
| Lower bound = $\frac{99}{37} = 2.675\ldots$ so 3 workers | B1 (1) | |
| At least nine activities including at least five floats. | M1 A1 A1 A1 (4) | |
| Lower bound is 5 workers – e.g. activities H, I, J, K and L together with $27 < \text{time} < 28$ | M1 A1 (2) | |
| | 12 marks | |
**Notes for Question 7:**
- a1B1: Correct value (12) for $x$.
- a2B1: Correct value (3) for $y$.
- b1M1: All (but one) boxes complete and any three values correct.
- b1A1: Any five values correct.
- b2A1: CAO (all seven values correct).
- c1B1: CSO – no incorrect working – if 3 workers with no working then give on the bod.
- d1M1: At least nine activities including at least five floats. **Scheduling diagram scores M0.**
- d1A1: The correct critical activities (B, F, H and M) dealt with correctly.
- d2A1: All correct non-critical activities present with floats with five non-critical activities correct.
- d3A1: All nine non-critical activities correct.
- e1M1 A1: A statement with the correct number of workers (5) **and** the correct activities (H, I, J, K and L) with some mention of time.
- e1A1: A completely correct statement with details of both time **and** activities. Candidates only need to give a time within the correct interval. Please note the strict inequalities for the time interval. Allow for example, 'on day 28' as equivalent to $27 < \text{time} < 28$.
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-8_980_1577_229_268}
\captionsetup{labelformat=empty}
\caption{Figure 4\\[0pt]
[The sum of all the activity durations is 99 days]}
\end{center}
\end{figure}
The network in Figure 4 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration of the activity, in days, is shown in brackets. The early event times and late event times are to be shown at each vertex and some have been completed for you.
Given that activity F is a critical activity and that the total float on activity G is 2 days,
\begin{enumerate}[label=(\alph*)]
\item write down the value of $x$ and the value of $y$,
\item calculate the missing early event times and late event times and hence complete Diagram 1 in your answer book.
Each activity requires one worker and the project must be completed in the shortest possible time.
\item Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time.
\item Draw a cascade (Gantt) chart for this project on Grid 1 in the answer book.
\item Use your cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities. (You do not need to provide a schedule of the activities.)
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2015 Q7 [12]}}