| Exam Board | OCR |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Draw activity network from table |
| Difficulty | Moderate -0.8 This is a standard Critical Path Analysis question covering routine techniques (drawing activity network, forward/backward pass, identifying critical path and float). The scheduling constraint in parts (iv)-(v) adds mild problem-solving, but the question explicitly guides students through the approach. Typical D2 exam question requiring methodical application of learned algorithms rather than novel insight. |
| 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 |
| Activity | Duration (hours) | Immediate predecessors | |
| A | Sort through cupboard and throw out rubbish | 4 | - |
| B | Get packing boxes | 1 | - |
| C | Sort out items from desk and throw out rubbish | 3 | - |
| D | Pack remaining items from cupboard in boxes | 2 | \(A\), \(B\) |
| E | Put personal items from desk into briefcase | 0.5 | C |
| \(F\) | Pack remaining items from desk in boxes | 1.5 | \(B , C\) |
| G | Take certificates down and put into briefcase | 1 | - |
| H | Label boxes to be stored | 0.5 | D, F |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Activity network drawn with correct precedences | B1 | Nodes for events, arcs for activities |
| \(A\), \(B\), \(C\) start from source node; \(D\) follows \(A,B\); \(E\) follows \(C\); \(F\) follows \(B,C\); \(G\) from source; \(H\) follows \(D,F\) | B1 | Correct immediate predecessors |
| Dummy activity used correctly if needed | B1 | |
| All durations labelled correctly | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Forward pass completed correctly | M1 | |
| Early event times correct | A1 | |
| Backward pass completed correctly | M1 | |
| Late event times correct | A1 | |
| Minimum project completion time \(= \mathbf{6}\) hours | A1 | |
| Critical activities: \(B, D, H\) (or \(A, D, H\) depending on network structure) | B1 | ft from network |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Float on activity \(C = \) late start of successor \(-\) early start of \(C\) \(-\) duration of \(C\) | M1 | |
| \(C\) could take \(\mathbf{0.5}\) hours longer (total 3.5 hours) without affecting completion | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| If Khalid does \(A\) first (duration 4): then \(C\) can start at \(t=4\), \(E\) at \(t=7\), \(G\) at \(t=0\)... total for Khalid \(= A+C+E+G = 4+3+0.5+1=8.5\) but must be sequential so finishes at \(4+3+0.5+1=8.5\), but project requires \(H\) at \(t\geq6\)... | M1 | Correct argument for one ordering |
| In both orderings Khalid cannot finish his activities within 8.5 hours while respecting precedences, so project takes more than 8.5 hours | A1 | Both cases demonstrated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Valid schedule drawn for Khalid and Mia completing project in 9 hours | B1 | Khalid does \(A,C,E,G\); Mia does \(B,D,F,H\) with correct sequencing |
| Schedule consistent with precedence constraints and completes by \(t=9\) | B1 |
# Question 5:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Activity network drawn with correct precedences | B1 | Nodes for events, arcs for activities |
| $A$, $B$, $C$ start from source node; $D$ follows $A,B$; $E$ follows $C$; $F$ follows $B,C$; $G$ from source; $H$ follows $D,F$ | B1 | Correct immediate predecessors |
| Dummy activity used correctly if needed | B1 | |
| All durations labelled correctly | B1 | |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Forward pass completed correctly | M1 | |
| Early event times correct | A1 | |
| Backward pass completed correctly | M1 | |
| Late event times correct | A1 | |
| Minimum project completion time $= \mathbf{6}$ hours | A1 | |
| Critical activities: $B, D, H$ (or $A, D, H$ depending on network structure) | B1 | ft from network |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Float on activity $C = $ late start of successor $-$ early start of $C$ $-$ duration of $C$ | M1 | |
| $C$ could take $\mathbf{0.5}$ hours longer (total 3.5 hours) without affecting completion | A1 | |
## Part (iv)
| Answer | Mark | Guidance |
|--------|------|----------|
| If Khalid does $A$ first (duration 4): then $C$ can start at $t=4$, $E$ at $t=7$, $G$ at $t=0$... total for Khalid $= A+C+E+G = 4+3+0.5+1=8.5$ but must be sequential so finishes at $4+3+0.5+1=8.5$, but project requires $H$ at $t\geq6$... | M1 | Correct argument for one ordering |
| In both orderings Khalid cannot finish his activities within 8.5 hours while respecting precedences, so project takes more than 8.5 hours | A1 | Both cases demonstrated |
## Part (v)
| Answer | Mark | Guidance |
|--------|------|----------|
| Valid schedule drawn for Khalid and Mia completing project in 9 hours | B1 | Khalid does $A,C,E,G$; Mia does $B,D,F,H$ with correct sequencing |
| Schedule consistent with precedence constraints and completes by $t=9$ | B1 | |5 Following a promotion at work, Khalid needs to clear out his office to move to a different building. The activities involved, their durations (in hours) and immediate predecessors are listed in the table below. You may assume that some of Khalid's friends will help him and that once an activity is started it will be continued until it is completed.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{Activity} & Duration (hours) & Immediate predecessors \\
\hline
A & Sort through cupboard and throw out rubbish & 4 & - \\
\hline
B & Get packing boxes & 1 & - \\
\hline
C & Sort out items from desk and throw out rubbish & 3 & - \\
\hline
D & Pack remaining items from cupboard in boxes & 2 & $A$, $B$ \\
\hline
E & Put personal items from desk into briefcase & 0.5 & C \\
\hline
$F$ & Pack remaining items from desk in boxes & 1.5 & $B , C$ \\
\hline
G & Take certificates down and put into briefcase & 1 & - \\
\hline
H & Label boxes to be stored & 0.5 & D, F \\
\hline
\end{tabular}
\end{center}
(i) Represent this project using an activity network.\\
(ii) Carry out a forward pass and a backward pass through the activity network, showing the early event time and late event time at each vertex of your network. State the minimum project completion time and list the critical activities.\\
(iii) How much longer could be spent on sorting the items from the desk and throwing out the rubbish (activity $C$ ) without it affecting the overall completion time?
Khalid says that he needs to do activities $A , C , E$ and $G$ himself. These activities take a total of 8.5 hours.\\
(iv) By considering what happens if Khalid does $A$ first, and what happens if he does $C$ first, show that the project will take more than 8.5 hours.\\
(v) Draw up a schedule to show how just two people, Khalid and his friend Mia, can complete the project in 9 hours. Khalid must do $A , C , E$ and $G$ and activities cannot be shared between Khalid and Mia. [2]
\hfill \mbox{\textit{OCR D2 2014 Q5 [14]}}