Question 6 - A-Level Maths

Edexcel D1 2021 January — Question 6 13 marks

Exam Board	Edexcel
Module	D1 (Decision Mathematics 1)
Year	2021
Session	January
Marks	13
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Critical Path Analysis
Type	Calculate early and late times
Difficulty	Moderate -0.3 This is a standard Critical Path Analysis question covering routine D1 techniques: completing an activity network, calculating early/late times, identifying critical activities, finding a lower bound for workers, and scheduling. While multi-part with several steps, each component is a textbook application of well-practiced algorithms with no novel problem-solving required. Slightly easier than average due to the mechanical nature of the procedures.
Spec	7.05a Critical path analysis: activity on arc networks 7.05b Forward and backward pass: earliest/latest times, critical activities

Activity	Duration (days)	Immediately preceding activities
A	4	-
B	7	-
C	6	-
D	10	A
E	5	A
F	7	C
G	6	B, C, E
H	6	B, C, E
I	7	B, C, E
J	9	D, H
K	8	B, C, E
L	4	F, G, K
M	6	F, G, K
N	7	F, G
P	5	M, N

The table above shows the activities required for the completion of a building project. For each activity the table shows the duration, in days, and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{48e785c0-7de5-450f-862c-4dd4d169adf9-08_668_1271_1658_397} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a partially completed activity network used to model the project. The activities are represented by the arcs and the numbers in brackets on the arcs are the times taken, in days, to complete each activity.

Complete the network in Diagram 1 in the answer book by adding activities \(\mathrm { G } , \mathrm { H }\) and I and the minimum number of dummies.
Add the early event times and the late event times to Diagram 1 in the answer book.
State the critical activities.
Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. You must show your working.
Schedule the activities on Grid 1 in the answer book, using the minimum number of workers, so that the project is completed in the minimum time.

Show mark scheme Show mark scheme source

Question 6:

Parts (a) and (b)

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Any two of the five arcs (G, H, I or the two dummies) drawn correctly	B1	Activities labelled with correct letter; dummies as dashed lines with no weight
Four of the five arcs drawn correctly	B1	Activities labelled correctly; dummies as dashed lines
All five arcs (G, H, I and two dummies) drawn correctly with no extras; correct arrows, correct weights	B1	CSO — all arrows must be present
All top boxes complete, values generally increasing in direction of arrows	M1	Condone lack of 0 for M mark; dependent on first mark in (a)
All values in top boxes correct	A1	CAO
All bottom boxes complete, values generally decreasing opposite to arrow direction	M1	Condone lack of 28/0; dependent on first mark in (a)
All values in bottom boxes correct	A1	CAO

Part (c)

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Critical activities: A, E, K, M, P	B1	CAO — these five activities only

Part (d)

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Lower bound \(= \dfrac{97}{28} = 3.46\ldots\) so 4 workers	M1 A1	M1 for attempt at lower bound using value in \([87-107]\) divided by finish time; A1 CSO — correct calculation seen or awrt 3.5 then 4

Part (e)

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
Not a cascade: Gantt chart with 5 workers used, at most, at least 11 activities placed	M1
4 workers; all 15 activities present (just once); condone at most two errors	A1
4 workers; all 15 activities present (just once); no errors	A1

# Question 6:

## Parts (a) and (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any two of the five arcs (G, H, I or the two dummies) drawn correctly | B1 | Activities labelled with correct letter; dummies as dashed lines with no weight |
| Four of the five arcs drawn correctly | B1 | Activities labelled correctly; dummies as dashed lines |
| All five arcs (G, H, I and two dummies) drawn correctly with no extras; correct arrows, correct weights | B1 | CSO — all arrows must be present |
| All top boxes complete, values generally increasing in direction of arrows | M1 | Condone lack of 0 for M mark; dependent on first mark in (a) |
| All values in top boxes correct | A1 | CAO |
| All bottom boxes complete, values generally decreasing opposite to arrow direction | M1 | Condone lack of 28/0; dependent on first mark in (a) |
| All values in bottom boxes correct | A1 | CAO |

## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Critical activities: A, E, K, M, P | B1 | CAO — these five activities only |

## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Lower bound $= \dfrac{97}{28} = 3.46\ldots$ so 4 workers | M1 A1 | M1 for attempt at lower bound using value in $[87-107]$ divided by finish time; A1 CSO — correct calculation seen or awrt 3.5 then 4 |

## Part (e)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Not a cascade: Gantt chart with 5 workers used, at most, at least 11 activities placed | M1 | |
| 4 workers; all 15 activities present (just once); condone at most two errors | A1 | |
| 4 workers; all 15 activities present (just once); no errors | A1 | |

---

Show LaTeX source

6.

\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Activity & Duration (days) & Immediately preceding activities \\
\hline
A & 4 & - \\
\hline
B & 7 & - \\
\hline
C & 6 & - \\
\hline
D & 10 & A \\
\hline
E & 5 & A \\
\hline
F & 7 & C \\
\hline
G & 6 & B, C, E \\
\hline
H & 6 & B, C, E \\
\hline
I & 7 & B, C, E \\
\hline
J & 9 & D, H \\
\hline
K & 8 & B, C, E \\
\hline
L & 4 & F, G, K \\
\hline
M & 6 & F, G, K \\
\hline
N & 7 & F, G \\
\hline
P & 5 & M, N \\
\hline
\end{tabular}
\end{center}

The table above shows the activities required for the completion of a building project. For each activity the table shows the duration, in days, and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{48e785c0-7de5-450f-862c-4dd4d169adf9-08_668_1271_1658_397}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

Figure 2 shows a partially completed activity network used to model the project. The activities are represented by the arcs and the numbers in brackets on the arcs are the times taken, in days, to complete each activity.
\begin{enumerate}[label=(\alph*)]
\item Complete the network in Diagram 1 in the answer book by adding activities $\mathrm { G } , \mathrm { H }$ and I and the minimum number of dummies.
\item Add the early event times and the late event times to Diagram 1 in the answer book.
\item State the critical activities.
\item Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. You must show your working.
\item Schedule the activities on Grid 1 in the answer book, using the minimum number of workers, so that the project is completed in the minimum time.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2021 Q6 [13]}}

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