| Exam Board | OCR |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Find range for variable duration |
| Difficulty | Standard +0.3 This is a standard Critical Path Analysis question requiring routine application of forward/backward pass algorithms and finding how a variable duration affects completion time. Parts (i)-(iii) are textbook exercises, while parts (iv)-(v) involve straightforward algebraic manipulation of the critical path with a variable duration—slightly above pure recall but well within standard D2 technique. |
| 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 |
| Event | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Early event time | ||||||||||
| Late event time |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Predecessors correct for \(A\) to \(F\) (entries for \(A\) and \(B\) may be blank) | B1 | |
| Substantially correct attempt at predecessors for other activities (at most 2 errors) | M1 | |
| Predecessors all correct for \(G\) to \(N\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Activity | Duration | Predecessors |
| \(A\) | 6 | — |
| \(B\) | 5 | — |
| \(C\) | 3 | \(A, B\) |
| \(D\) | 9 | \(A\) |
| \(E\) | 4 | \(A, B\) |
| \(F\) | 2 | \(A, B\) |
| \(G\) | 2 | \(E, H\) |
| \(H\) | 3 | \(C, F\) |
| \(I\) | 5 | \(D, G\) |
| \(J\) | 6 | \(E, H\) |
| \(K\) | 10 | \(C, F\) |
| \(L\) | 4 | \(I\) |
| \(M\) | 12 | \(I\) |
| \(N\) | 6 | \(J, K, L\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Dummy needed between nodes 2 and 3 so that \(C\), \(E\) and \(F\) follow both \(A\) and \(B\) but \(D\) follows \(A\) only | B1 | \(D\) does not follow \(B\) (\(D\) follows \(A\) only) |
| Dummy needed between nodes 4 and 5 so that \(C\) and \(F\) do not share both a common start and a common finish | B1 | Identifying \(C\) and \(F\) appropriately |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Early event times correct, in table | B1 | |
| Substantially correct backwards pass (at most 2 errors in total) | M1 | |
| Late event times correct, in table | A1 | |
| Minimum project completion time \(= 32\) minutes | B1 | 32, cao |
| Critical activities: \(A, D, I\) and \(M\) and no others, cao | B1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Node | 1 | 2 |
| Early | 0 | 6 |
| Late | 0 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Early event time at node 9 becomes the larger of 24 and \(9 + x\) | M1 | \(9 + x\); larger of 24 and \(9 + x\) |
| Early event time at node 10 becomes the larger of 32 and \(15 + x\), which then also becomes the late event time at node 10 | M1 | Considering the event times at node 10 |
| Late event time at node 9 then becomes 26 or \(9 + x\) | A1 | Correct consideration of 26 and \(9 + x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = 17\) | B1 | 17 |
# Question 6:
## Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Predecessors correct for $A$ to $F$ (entries for $A$ and $B$ may be blank) | B1 | |
| Substantially correct attempt at predecessors for other activities (at most 2 errors) | M1 | |
| Predecessors all correct for $G$ to $N$ | A1 | [3] |
Full predecessor table:
| Activity | Duration | Predecessors |
|---|---|---|
| $A$ | 6 | — |
| $B$ | 5 | — |
| $C$ | 3 | $A, B$ |
| $D$ | 9 | $A$ |
| $E$ | 4 | $A, B$ |
| $F$ | 2 | $A, B$ |
| $G$ | 2 | $E, H$ |
| $H$ | 3 | $C, F$ |
| $I$ | 5 | $D, G$ |
| $J$ | 6 | $E, H$ |
| $K$ | 10 | $C, F$ |
| $L$ | 4 | $I$ |
| $M$ | 12 | $I$ |
| $N$ | 6 | $J, K, L$ |
## Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Dummy needed between nodes 2 and 3 so that $C$, $E$ and $F$ follow both $A$ and $B$ but $D$ follows $A$ only | B1 | $D$ does not follow $B$ ($D$ follows $A$ only) |
| Dummy needed between nodes 4 and 5 so that $C$ and $F$ do not share both a common start and a common finish | B1 | Identifying $C$ and $F$ appropriately | [2] |
## Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Early event times correct, in table | B1 | |
| Substantially correct backwards pass (at most 2 errors in total) | M1 | |
| Late event times correct, in table | A1 | |
| Minimum project completion time $= 32$ minutes | B1 | 32, cao |
| Critical activities: $A, D, I$ and $M$ and no others, cao | B1 | [5] |
Event times table:
| Node | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Early | 0 | 6 | 6 | 9 | 9 | 12 | 15 | 20 | 24 | 32 |
| Late | 0 | 6 | 7 | 10 | 10 | 13 | 15 | 20 | 26 | 32 |
## Part (iv)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Early event time at node 9 becomes the larger of 24 and $9 + x$ | M1 | $9 + x$; larger of 24 and $9 + x$ |
| Early event time at node 10 becomes the larger of 32 and $15 + x$, which then also becomes the late event time at node 10 | M1 | Considering the event times at node 10 |
| Late event time at node 9 then becomes 26 or $9 + x$ | A1 | Correct consideration of 26 and $9 + x$ | [4] |
## Part (v)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = 17$ | B1 | 17 | [1] |6 Answer parts (i), (ii) and (iii) of this question on the insert provided.
The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference.\\
\includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-06_747_1249_482_447}\\
(i) Complete the table in the insert to show the immediate predecessors for each activity.\\
(ii) Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .\\
(iii) Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities.
Suppose that the duration of activity $K$ changes to $x$ minutes.\\
(iv) Find, in terms of $x$, expressions for the early event time and the late event time for event 9 .\\
(v) Find the maximum duration of activity $K$ that will not affect the minimum project completion time found in part (iii).
\section*{ADVANCED GCE \\
MATHEMATICS}
Decision Mathematics 2\\
INSERT for Questions 5 and 6
(ii) Dummy activity is needed between event 2 and event 3 because $\_\_\_\_$\\
Dummy activity is needed between event 4 and event 5 because $\_\_\_\_$\\
(iii)
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Event & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Early event time & & & & & & & & & & \\
\hline
Late event time & & & & & & & & & & \\
\hline
\end{tabular}
\end{center}
Minimum project completion time = $\_\_\_\_$ minutes
Critical activities: $\_\_\_\_$
\section*{Answer part (iv) and part (v) in your answer booklet.}
OCR\\
RECOGNISING ACHIEVEMENT
\hfill \mbox{\textit{OCR D2 2010 Q6 [15]}}