AQA D2 2013 January — Question 1 13 marks

Exam BoardAQA
ModuleD2 (Decision Mathematics 2)
Year2013
SessionJanuary
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCritical Path Analysis
TypeSchedule with limited workers - determine minimum time
DifficultyModerate -0.5 This is a standard Critical Path Analysis question covering routine D2 techniques (earliest/latest times, critical path, float, resource histograms, and scheduling with limited workers). While multi-part with several marks, each component follows textbook procedures without requiring novel insight or complex problem-solving—slightly easier than average due to its procedural nature.
Spec7.05a 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

1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.
  1. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
  2. On Figure 2 opposite, complete the precedence table.
  3. Find the critical path.
  4. Find the float time of activity \(E\).
  5. Using Figure 3 on page 5, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
  6. Given that there are two workers available for the project, find the minimum completion time for the project.
  7. Given that there is only one worker available for the project, find the minimum completion time for the project. Figure 1 \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{(a)} \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-02_629_1550_1818_292}
    \end{figure} (b) \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2}
    ActivityImmediate predecessor(s)
    A
    B
    C
    D
    E
    \(F\)
    G
    H
    I
    J
    \(K\)
    \end{table}
    \includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-05_2486_1717_221_150}

Question 1:
Part (a)
AnswerMarks Guidance
AnswerMark Guidance
Earliest start times: \(A=0, B=0, C=8, D=8, E=8, F=5, G=14, H=14, I=18, J=20, K=30\)M1 Method for forward pass
Latest finish times: \(K=31, I=30, J=30, G=18, H=26, D=14, E=15, F=15, C=11, A=8, B=8\)A1 Correct earliest start times
A1Correct latest finish times (ft from EST)
A1All values correct
Part (b)
AnswerMarks Guidance
AnswerMark Guidance
\(A\): none; \(B\): none; \(C\): \(A\); \(D\): \(A, C\); \(E\): \(B, C\); \(F\): \(B\); \(G\): \(D, E\); \(H\): \(E, F\); \(I\): \(G\); \(J\): \(G, H\); \(K\): \(I, J\)B2 B1 for 4 or 5 correct rows
Part (c)
AnswerMarks Guidance
AnswerMark Guidance
\(A, D, G, I, K\)B1 Must be stated as a path
Part (d)
AnswerMarks Guidance
AnswerMark Guidance
Float of \(E = 1\)B1
Part (e)
AnswerMarks Guidance
AnswerMark Guidance
Correct histogram with all activities starting as early as possibleB3 B1 for correct blocks 0–8; B1 for correct blocks 8–14; B1 for remainder correct
Part (f)
AnswerMarks Guidance
AnswerMark Guidance
Minimum completion time \(= 31\) hoursB1
Part (g)
AnswerMarks Guidance
AnswerMark Guidance
Total time \(= 8+5+3+6+1+7+4+6+12+9+1 = 62\) hoursB1
# Question 1:

## Part (a)

| Answer | Mark | Guidance |
|--------|------|----------|
| Earliest start times: $A=0, B=0, C=8, D=8, E=8, F=5, G=14, H=14, I=18, J=20, K=30$ | M1 | Method for forward pass |
| Latest finish times: $K=31, I=30, J=30, G=18, H=26, D=14, E=15, F=15, C=11, A=8, B=8$ | A1 | Correct earliest start times |
| | A1 | Correct latest finish times (ft from EST) |
| | A1 | All values correct |

## Part (b)

| Answer | Mark | Guidance |
|--------|------|----------|
| $A$: none; $B$: none; $C$: $A$; $D$: $A, C$; $E$: $B, C$; $F$: $B$; $G$: $D, E$; $H$: $E, F$; $I$: $G$; $J$: $G, H$; $K$: $I, J$ | B2 | B1 for 4 or 5 correct rows |

## Part (c)

| Answer | Mark | Guidance |
|--------|------|----------|
| $A, D, G, I, K$ | B1 | Must be stated as a path |

## Part (d)

| Answer | Mark | Guidance |
|--------|------|----------|
| Float of $E = 1$ | B1 | |

## Part (e)

| Answer | Mark | Guidance |
|--------|------|----------|
| Correct histogram with all activities starting as early as possible | B3 | B1 for correct blocks 0–8; B1 for correct blocks 8–14; B1 for remainder correct |

## Part (f)

| Answer | Mark | Guidance |
|--------|------|----------|
| Minimum completion time $= 31$ hours | B1 | |

## Part (g)

| Answer | Mark | Guidance |
|--------|------|----------|
| Total time $= 8+5+3+6+1+7+4+6+12+9+1 = 62$ hours | B1 | |

---
1\\
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.
\begin{enumerate}[label=(\alph*)]
\item Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
\item On Figure 2 opposite, complete the precedence table.
\item Find the critical path.
\item Find the float time of activity $E$.
\item Using Figure 3 on page 5, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
\item Given that there are two workers available for the project, find the minimum completion time for the project.
\item Given that there is only one worker available for the project, find the minimum completion time for the project.

Figure 1

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{(a)}
  \includegraphics[alt={},max width=\textwidth]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-02_629_1550_1818_292}
\end{center}
\end{figure}

(b)

\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\begin{tabular}{|l|l|}
\hline
Activity & Immediate predecessor(s) \\
\hline
A &  \\
\hline
B &  \\
\hline
C &  \\
\hline
D &  \\
\hline
E &  \\
\hline
$F$ &  \\
\hline
G &  \\
\hline
H &  \\
\hline
I &  \\
\hline
J &  \\
\hline
$K$ &  \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{3ba973a1-6a45-4381-b634-e9c4673ef1fb-05_2486_1717_221_150}
\end{center}
\end{enumerate}

\hfill \mbox{\textit{AQA D2 2013 Q1 [13]}}