| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2020 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Find missing early/late times |
| Difficulty | Standard +0.3 This is a standard critical path analysis question requiring systematic application of forward/backward pass algorithms to find early/late times, identify critical activities, and consider resource constraints. While multi-part with several calculations, each step follows routine procedures taught in Decision Maths with no novel problem-solving insight required—slightly easier than average A-level questions. |
| 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 float7.05e Cascade charts: scheduling and effect of delays |
| Activity | A | B | C | D | E | F | G | H | I | J | K |
| Workers | 2 | 1 | 1 | 2 | \(n\) | 1 | 2 | 1 | 1 | 1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | For reference: | |
| 6 | (a) | (i) |
| Answer | Marks |
|---|---|
| 0 ≤ x ≤ 3 | M1 |
| Answer | Marks |
|---|---|
| [3] | Forward pass, early event times at every vertex, shown on |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | (ii) |
| Answer | Marks |
|---|---|
| A, F, H, K | M1 |
| Answer | Marks |
|---|---|
| [3] | Backward pass, late event times at every vertex |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (b) | Max value for n is 2 |
| Answer | Marks |
|---|---|
| so only 2 workers are needed | B1 |
| Answer | Marks |
|---|---|
| [4] | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Event | | |
| Early time | 0 | 2 |
| Event | | |
| Late time | 0 | 3 |
| Activity | A | B |
| Workers | 2 | 1 |
| (c) | 18 minutes |
| Answer | Marks |
|---|---|
| (n = 1) (x = 0) | B1 |
| [1] | 18 or 18 + x |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | 0 minutes | B1FT |
| [1] | Their answer to part c minus 18 |
| Answer | Marks | Guidance |
|---|---|---|
| (e) | 1 worker | B1 |
| [1] | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Time | 0 | 2 |
| Start | A, B | C |
| (n = 1) | (x = 0) |
Question 6:
6 | For reference:
6 | (a) | (i) | Event
Early time 0 2 4 4 7 10 10+x 15
0 ≤ x ≤ 3 | M1
A1
B1
[3] | Forward pass, early event times at every vertex, shown on
network or listed (increasing from 0)
All correct, apart possibly from
≤ 3
6 | (a) | (ii) | Event
Late time 0 3 4 4 7 10 13 15
A, F, H, K | M1
A1
B1
[3] | Backward pass, late event times at every vertex
(decreasing, with start and finish matching the forward pass)
All correct, may have max{12+x, 15}or 12+x, 15 at
A, F, H, K
Allow A, F, H, I, J, K (with both I and J) but no others
6 | (b) | Max value for n is 2
2 do A while another does B then C
4 workers do K while the other does I then J
D, (E), F, G and H must be completed between time 4 and time 10
G needs 2 workers
so 3 workers are available for D, (E), F and H
G takes 5 minutes
F followed by H uses 1 worker for 6 minutes
E can be fitted in before D
so only 2 workers are needed | B1
M1*
M1dep*
A1
[4] | 2
For reference:
Activity A B C D E F G H I J K
Workers 2 1 1 2 n 1 2 1 1 1 4
Reasoning may be seen on grids
D, F, G and H must be done between time 4 and time 10
3 workers available for D, F and H
(or implied from E followed by D or vice versa)
E done before D
Event | | | | | | | |
Early time | 0 | 2 | 4 | 4 | 7 | 10 | 10+x | 15
Event | | | | | | | |
Late time | 0 | 3 | 4 | 4 | 7 | 10 | 13 | 15
Activity | A | B | C | D | E | F | G | H | I | J | K
Workers | 2 | 1 | 1 | 2 | n | 1 | 2 | 1 | 1 | 1 | 4
(c) | 18 minutes
Time 0 2 4 6 8 11 16
Start A, B C E, F, D G H K I, J
(n = 1) (x = 0) | B1
[1] | 18 or 18 + x
Working is not necessary but may be seen on grids
(d) | 0 minutes | B1FT
[1] | Their answer to part c minus 18
Working is not necessary but may be seen on grids
(e) | 1 worker | B1
[1] | 1
Working is not necessary but may be seen on grids
Time | 0 | 2 | 4 | 6 | 8 | 11 | 16
Start | A, B | C | E, F, D | G | H | K | I, J
(n = 1) | (x = 0)
PMT
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recorded or monitored6 A project is represented by the activity on arc network below.\\
\includegraphics[max width=\textwidth, alt={}, center]{cc58fb7a-efb6-4548-a8e1-e40abe1eb722-7_410_1095_296_486}
The duration of each activity (in minutes) is shown in brackets, apart from activity I.
\begin{enumerate}[label=(\alph*)]
\item Suppose that the minimum completion time for the project is 15 minutes.
\begin{enumerate}[label=(\roman*)]
\item By calculating the early event times, determine the range of values for $x$.
\item By calculating the late event times, determine which activities must be critical.
The table shows the number of workers needed for each activity.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Activity & A & B & C & D & E & F & G & H & I & J & K \\
\hline
Workers & 2 & 1 & 1 & 2 & $n$ & 1 & 2 & 1 & 1 & 1 & 4 \\
\hline
\end{tabular}
\end{center}
\end{enumerate}\item Determine the maximum possible value for $n$ if 5 workers can complete the project in 15 minutes. Explain your reasoning.
The duration of activity F is reduced to 1.5 minutes, but only 4 workers are available. The minimum completion time is no longer 15 minutes.
\item Determine the minimum project completion time in this situation.
\item Find the maximum possible value for $x$ for this minimum project completion time.
\item Find the maximum possible value for $n$ for this minimum project completion time.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2020 Q6 [13]}}