| Exam Board | OCR |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Draw resource histogram |
| Difficulty | Moderate -0.3 This is a standard Decision Maths 2 critical path analysis question requiring routine application of well-practiced algorithms (forward/backward pass, drawing networks and resource histograms). While multi-part with several steps, each component follows textbook procedures with no novel problem-solving or insight required, making it slightly easier than average A-level difficulty. |
| 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 |
| Activity | Duration (hours) | Immediate predecessors | Number of workers |
| \(A\) | 3 | - | 1 |
| \(B\) | 2 | - | 1 |
| C | 2 | \(A\) | 2 |
| \(D\) | 3 | \(A\), \(B\) | 2 |
| E | 3 | \(C\) | 3 |
| \(F\) | 3 | C, D | 3 |
| \(G\) | 2 | D | 3 |
| \(H\) | 5 | \(E , F\) | 1 |
| I | 4 | \(F , G\) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Activity network with \(A(3), B(2), C(2), D(3), E(3), F(3), G(2), H(5), I(4)\) and exactly five directed dummies used correctly | M1, M1d, A1 | Correct structure even without directions; exactly five directed dummies used correctly; completely correct with all arcs directed |
| Answer | Marks | Guidance |
|---|---|---|
| Forward pass and backward pass giving early/late times; minimum project completion time \(= 14\) hours | M1, M1, A1ft, B1 | Substantially correct forward pass (up to 2 errors); substantially correct backward pass (up to 2 errors); both passes wholly correct; 14 cao |
| Critical activities \(A, D, F, H\) | B1 | \(ADFH\) cao |
| Answer | Marks | Guidance |
|---|---|---|
| Resource histogram with axes scaled, no holes or overhangs, axes labelled, histogram completely correct | M1, A1 | Axes scaled appropriately (or implied from lines) and plausible histogram; axes labelled and histogram completely correct, cao |
| Answer | Marks | Guidance |
|---|---|---|
| Delay \(G\) by 2 hours so it starts after \(E\) has finished (6 to 8 \(\to\) 8 to 10); delay \(I\) by 1 hour (9 to 13 \(\to\) 10 to 14) | M1, A1 | May be shown as diagram with activities marked so shift of \(G\) and \(I\) can be seen |
# Question 3:
## Part (i)
| Activity network with $A(3), B(2), C(2), D(3), E(3), F(3), G(2), H(5), I(4)$ and exactly five directed dummies used correctly | M1, M1d, A1 | Correct structure even without directions; exactly five directed dummies used correctly; completely correct with all arcs directed |
## Part (ii)
| Forward pass and backward pass giving early/late times; minimum project completion time $= 14$ hours | M1, M1, A1ft, B1 | Substantially correct forward pass (up to 2 errors); substantially correct backward pass (up to 2 errors); both passes wholly correct; 14 cao |
| Critical activities $A, D, F, H$ | B1 | $ADFH$ cao |
## Part (iii)
| Resource histogram with axes scaled, no holes or overhangs, axes labelled, histogram completely correct | M1, A1 | Axes scaled appropriately (or implied from lines) and plausible histogram; axes labelled and histogram completely correct, cao |
## Part (iv)
| Delay $G$ by 2 hours so it starts after $E$ has finished (6 to 8 $\to$ 8 to 10); delay $I$ by 1 hour (9 to 13 $\to$ 10 to 14) | M1, A1 | May be shown as diagram with activities marked so shift of $G$ and $I$ can be seen |
---3 The table lists the duration, immediate predecessors and number of workers required for each activity in a project.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Activity & Duration (hours) & Immediate predecessors & Number of workers \\
\hline
$A$ & 3 & - & 1 \\
\hline
$B$ & 2 & - & 1 \\
\hline
C & 2 & $A$ & 2 \\
\hline
$D$ & 3 & $A$, $B$ & 2 \\
\hline
E & 3 & $C$ & 3 \\
\hline
$F$ & 3 & C, D & 3 \\
\hline
$G$ & 2 & D & 3 \\
\hline
$H$ & 5 & $E , F$ & 1 \\
\hline
I & 4 & $F , G$ & 2 \\
\hline
\end{tabular}
\end{center}
(i) Represent the project by an activity network, using activity on arc. You should make your diagram quite large so that there is room for working.\\
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event times and late event times clearly at the vertices of your network.
State the minimum project completion time and list the critical activities.\\
(iii) Draw a resource histogram to show the number of workers required each hour when each activity begins at its earliest possible start time.\\
(iv) Show how it is possible for the project to be completed in the minimum project completion time when only six workers are available.
\hfill \mbox{\textit{OCR D2 2011 Q3 [12]}}