| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Explain dummy activities |
| Difficulty | Moderate -0.3 This is a standard critical path analysis question requiring explanation of dummy activities (routine concept), forward/backward pass calculations, and straightforward algebraic manipulation when a duration changes. While multi-part, each component is textbook-standard 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 |
|---|---|---|
| Answer | Marks | Guidance |
| The dummy from event 2 to event 3 is required because activity F (or G) relies on activity A and B but activity D (or E) relies on activity A only | B1 | a(i)1B1: CAO dependency - all relevant activities must be referred to - activities A and B and one of D or E and one of F or G (so four activities) must be mentioned |
| The dummy from event 6 to event 7 is required as otherwise activities J and K (which both begin at event 4) would end at the same event | B1 | a(ii)1B1: CAO uniqueness – please note that, for example, 'so that activities can be defined uniquely' is not sufficient to earn this mark. There must be some mention of describing activities in terms of the event at the end of activity. However, give bod on statements that imply that a activity begins and ends at the same event (for this mark candidates do not need to explicitly mention activities J and K) |
| All top boxes complete, values in the top boxes generally increasing in the direction of the arrows ('left to right'), condone one 'rogue' value (if values do not increase in the direction of the arrows then if one value is ignored and then the values do increase in the direction of the arrows then this is considered to be only one rogue value) | M1 | b1M1: All top boxes complete, values in the top boxes generally increasing in the direction of the arrows ('left to right'), condone one 'rogue' value (if values do not increase in the direction of the arrows then if one value is ignored and then the values do increase in the direction of the arrows then this is considered to be only one rogue value) |
| A1 | b1A1: CAO for the top boxes | |
| 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 26 (at the end event) for the M only | M1 | b2M1: 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 26 (at the end event) for the M only |
| A1 | b2A1: CAO for the bottom boxes | |
| Minimum completion time: 26 (hours); Critical activities: A, D, I and M | B1 B1 | c1B1: CAO (26). c2B1: CAO (A, D, I and M only) |
| The early event time at event 7 is (the larger of) 12 or \(9 + x\). The late event time at event 7 would then be either 15 or \(9 + x\) | M1 A1 | d1M1: One of 12 or \(9 + x\) as the early event time for event 7. d1A1: Both correct answers of 12, \(9 + x\) (A0 if 'linked' in some way e.g. \(12 > 9 + x\) but bod for the M mark) |
| \(x = 10\) | B1 | e1B1: CAO (10) |
| Total: 12 marks |
| Answer | Marks | Guidance |
|--------|-------|----------|
| The dummy from event 2 to event 3 is required because activity F (or G) relies on activity A and B but activity D (or E) relies on activity A only | B1 | a(i)1B1: CAO dependency - all relevant activities must be referred to - activities A and B and one of D or E and one of F or G (so four activities) must be mentioned |
| The dummy from event 6 to event 7 is required as otherwise activities J and K (which both begin at event 4) would end at the same event | B1 | a(ii)1B1: CAO uniqueness – please note that, for example, 'so that activities can be defined uniquely' is not sufficient to earn this mark. There must be some mention of describing activities in terms of the event at the end of activity. However, give bod on statements that imply that a activity begins and ends at the same event (for this mark candidates do not need to explicitly mention activities J and K) |
| All top boxes complete, values in the top boxes generally increasing in the direction of the arrows ('left to right'), condone one 'rogue' value (if values do not increase in the direction of the arrows then if one value is ignored and then the values do increase in the direction of the arrows then this is considered to be only one rogue value) | M1 | b1M1: All top boxes complete, values in the top boxes generally increasing in the direction of the arrows ('left to right'), condone one 'rogue' value (if values do not increase in the direction of the arrows then if one value is ignored and then the values do increase in the direction of the arrows then this is considered to be only one rogue value) |
| | A1 | b1A1: CAO for the top boxes |
| 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 26 (at the end event) for the M only | M1 | b2M1: 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 26 (at the end event) for the M only |
| | A1 | b2A1: CAO for the bottom boxes |
| Minimum completion time: 26 (hours); Critical activities: A, D, I and M | B1 B1 | c1B1: CAO (26). c2B1: CAO (A, D, I and M only) |
| The early event time at event 7 is (the larger of) 12 or $9 + x$. The late event time at event 7 would then be either 15 or $9 + x$ | M1 A1 | d1M1: One of 12 or $9 + x$ as the early event time for event 7. d1A1: Both correct answers of 12, $9 + x$ (A0 if 'linked' in some way e.g. $12 > 9 + x$ but bod for the M mark) |
| $x = 10$ | B1 | e1B1: CAO (10) |
| **Total: 12 marks** | | |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{aef6a6dd-76ec-47f7-b8c9-449006da29d3-06_677_1774_246_148}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
A project is modelled by the activity network shown in Figure 3. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the corresponding activity. The numbers in circles are the event numbers.
\begin{enumerate}[label=(\alph*)]
\item Explain the significance of the dummy activity
\begin{enumerate}[label=(\roman*)]
\item from event 2 to event 3
\item from event 6 to event 7
\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 and list the critical activities.
The duration of activity H changes to $x$ hours.
\item Find, in terms of $x$ where necessary,
\begin{enumerate}[label=(\roman*)]
\item the possible new early event time for event 7
\item the possible new late event time for event 7
Given that the duration of activity H is such that the minimum project completion time is four hours greater than the time found in (c),
\end{enumerate}\item determine the value of $x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2019 Q4 [12]}}