| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2016 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Explain dummy activities |
| Difficulty | Moderate -0.3 Part (a) tests conceptual understanding of dummy activities in activity networks—a standard D1 topic requiring explanation rather than calculation. While students must understand precedence relationships, this is routine bookwork with no problem-solving or novel insight required. The remaining parts involve standard critical path analysis procedures (early/late times, lower bound calculation, scheduling diagrams) that are methodical rather than conceptually challenging. |
| 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 |
|---|---|---|
| (a)(i) | The dummy from event 5 to event 6 is needed to show that J depends on F but I depends on D, E and F | B1 |
| (b) | Network diagram shown with activities and durations labeled. Event boxes with early and late times. | M1 A1 M1 A1 |
| (c) | \(\frac{21}{64} \approx 3.048\) so at least 4 workers required | B1 M1 A1 |
| (d) | Gantt chart shown with activities B, F, J, A, C, D, E, G, H, I, K with correct timing and resource allocation | M1 A1 M1 A1 |
| (e) | Gantt chart shown: Activities B, A, C, D, E, G, H, I, K with correct timing and appropriate shading | M1 A1 M1 A1 |
| Answer | Marks |
|---|---|
| a1B1 | Cao – all relevant activities must be referred to - so activities I, J, F and either D or E must be mentioned. |
| a2B1 | Cao – mention of describing activities uniquely in terms of the events at each end. |
| b1M1 | All top boxes complete, values generally increasing from left to right, condone one 'rogue' |
| b1A1 | Cao on top boxes |
| b2M1 | All bottom boxes complete, values generally decreasing right to left, condone one 'rogue' |
| b2A1 | Cao on bottom boxes |
| c1B1 | Cao |
| d1M1 | Attempt to find lower bound: \([55 - 73 / \text{their finish time}]\) or \([\text{sum of the activities} / \text{their finish time}]\) |
| d1A1 | Cao – correct calculation seen then 4. |
| e1M1 | At least 8 activities added including 5 floats. Scheduling diagram scores M0. Critical activites dealt with correctly and four other non-critical activities dealt with correctly. |
| e1A1 | All 11 activities including all 8 floats |
| e2M1 | Not a cascade chart. 3 workers used and at least 9 activities placed. |
| f1A1 | 3 workers. All 11 activities present (just once). Condone one error either precedence or activity length. |
| f2A1 | 3 workers. All 11 activities present (just once). No errors. |
(a)(i) | The dummy from event 5 to event 6 is needed to show that J depends on F but I depends on D, E and F | B1 | (ii) | The dummy from event 7 to event 9 is because activities G and H must be able to be described uniquely in terms of the events at each end | B1 | (2) |
(b) | Network diagram shown with activities and durations labeled. Event boxes with early and late times. | M1 A1 M1 A1 | (4) |
(c) | $\frac{21}{64} \approx 3.048$ so at least 4 workers required | B1 M1 A1 | (3) |
(d) | Gantt chart shown with activities B, F, J, A, C, D, E, G, H, I, K with correct timing and resource allocation | M1 A1 M1 A1 | (4) |
(e) | Gantt chart shown: Activities B, A, C, D, E, G, H, I, K with correct timing and appropriate shading | M1 A1 M1 A1 | (4) |
**Notes:**
| a1B1 | Cao – all relevant activities must be referred to - so activities I, J, F and either D or E must be mentioned. |
| a2B1 | Cao – mention of describing activities uniquely in terms of the events at each end. |
| b1M1 | All top boxes complete, values generally increasing from left to right, condone one 'rogue' |
| b1A1 | Cao on top boxes |
| b2M1 | All bottom boxes complete, values generally decreasing right to left, condone one 'rogue' |
| b2A1 | Cao on bottom boxes |
| c1B1 | Cao |
| d1M1 | Attempt to find lower bound: $[55 - 73 / \text{their finish time}]$ or $[\text{sum of the activities} / \text{their finish time}]$ |
| d1A1 | Cao – correct calculation seen then 4. |
| e1M1 | At least 8 activities added including 5 floats. Scheduling diagram scores M0. Critical activites dealt with correctly and four other non-critical activities dealt with correctly. |
| e1A1 | All 11 activities including all 8 floats |
| e2M1 | Not a cascade chart. 3 workers used and at least 9 activities placed. |
| f1A1 | 3 workers. All 11 activities present (just once). Condone one error either precedence or activity length. |
| f2A1 | 3 workers. All 11 activities present (just once). No errors. |
**Total: 16 marks**6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a3ca2743-2311-4225-8b78-dcd5eb592704-7_664_1520_239_276}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
A project is modelled by the activity network shown in Figure 4. 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 5 to event 6
\item from event 7 to event 9
\end{enumerate}\item Complete Diagram 1 in the answer book to show the early event times and the late event times.
\item State the minimum project completion time.
\item Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time. You must show your working.
\item On Grid 1 in your answer book, draw a cascade (Gantt) chart for this project.
\item On Grid 2 in your answer book, construct a scheduling diagram to show that this project can be completed with three workers in just one more hour than the minimum project completion time.\\
(3)
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2016 Q6 [16]}}