Moderate -0.8 This is a standard D1 critical path analysis question covering routine algorithmic procedures (forward/backward pass, float calculation, Gantt charts, scheduling). While multi-part with 16 marks total, each component follows textbook methods with no novel problem-solving required. The dummy activity explanation and lower bound calculation are direct recall of standard techniques.
\includegraphics{figure_2}
A project is modelled by the activity network shown in Figure 2. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the activity. The numbers in circles are the event numbers. Each activity requires one worker.
Explain the significance of the dummy activity
from event 5 to event 6
from event 7 to event 9.
\hfill [2]
Complete Diagram 3 in the answer book to show the early event times and the late event times. \hfill [4]
State the minimum project completion time. \hfill [1]
Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time. You must show your working. \hfill [2]
On Grid 1 in your answer book, draw a cascade (Gantt) chart for this project. \hfill [4]
On Grid 2 in your answer book, construct a scheduling diagram to show that this project can be completed with three workers in just one more hour than the minimum project completion time. \hfill [3]
(i) The dummy from event 5 to event 6 is needed to show that J depends on F but I depends on D, E and F
B1
(ii) The dummy from event 7 to event 9 is because activities G and H must be able to be described uniquely in terms of the events at each end
B1
(2)
6(b)
Answer
Marks
[Activity on Arrow network diagram with all activities and dummies correctly shown]
M1, A1
[Network diagram repeated with correct values]
M1, A1
(4)
All top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue. cao (top boxes). All bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue. cao (bottom boxes).
6(c)
Answer
Marks
21 (hours)
B1
(1)
6(d)
Answer
Marks
\(\frac{64}{21} \approx 3.048\) so at least 4 workers required
M1 A1
(2)
Attempt to find lower bound: (a value in the interval [55 – 73] / their finish time) or (sum of the activities / their finish time) or (as a minimum) an awrt 3.05 or 3.04 (truncated). cso – either a correct calculation seen or awrt 3.05 (or 3.04) then 4. An answer of 4 with no working scores M0A0.
6(e)
Answer
Marks
[Gantt chart showing resource scheduling]
M1, A1
[Gantt chart repeated]
M1, A1
(4)
At least 8 activities added including 5 floats. Scheduling diagram scores M0. Critical activities dealt with correctly and 4 non-critical activities dealt with correctly. All 11 activities including all 8 floats (on the correct non-critical activities). cso (all activities correct and present only once).
6(f)
Answer
Marks
[Gantt chart with 3 workers, 11 activities, no cascading, completion time 22 hours]
M1, A1, A1
(3)
Not a cascade chart, 3 workers used and at least 9 activities placed. The completion time must be no greater than one hour more than the minimum completion time stated in (c) or seen in (b). 3 workers, All 11 activities present (just once). Condone one error either precedence or activity length. The completion time must be one hour greater than the minimum completion time stated in (c) or seen in (b). 3 workers, All 11 activities present (just once). No errors. The completion time must be 22.
General Notes for Question 6
In (a), any use of the terms 'activity' and 'event' must be correct.
6(a) Notes:
- B1: cao dependency – all relevant activities must be referred to – activities I, J, F and either D or E must be mentioned.
- B1: cao uniqueness – please note that, for example, 'so that activities can be defined uniquely' is not sufficient to earn this mark. There must be some mention of describing activities in terms of the event at each end. However, give bod on statements that imply that an activity begins and ends at the same event.
6(b) Notes:
- M1: All top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue.
- A1: cao (top boxes).
- M1: All bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue.
- A1: cao (bottom boxes).
6(c) Notes:
- B1: cao (21)
6(d) Notes:
- M1: Attempt to find lower bound: (a value in the interval [55 – 73] / their finish time) or (sum of the activities / their finish time) or (as a minimum) an awrt 3.05 or 3.04 (truncated).
- A1: cso – either a correct calculation seen or awrt 3.05 (or 3.04) then 4. An answer of 4 with no working scores M0A0.
6(e) Notes:
- M1: At least 8 activities added including 5 floats. Scheduling diagram scores M0.
- A1: Critical activities dealt with correctly and 4 non-critical activities dealt with correctly.
- M1: All 11 activities including all 8 floats (on the correct non-critical activities).
- A1: cso (all activities correct and present only once).
6(f) Notes:
- M1: Not a cascade chart, 3 workers used and at least 9 activities placed. The completion time must be no greater than one hour more than the minimum completion time stated in (c) or seen in (b).
- A1: 3 workers, All 11 activities present (just once). Condone one error either precedence or activity length. The completion time must be one hour greater than the minimum completion time stated in (c) or seen in (b).
- A1: 3 workers, All 11 activities present (just once). No errors. The completion time must be 22.
## 6(a)
(i) The dummy from event 5 to event 6 is needed to show that J depends on F but I depends on D, E and F | B1 |
(ii) The dummy from event 7 to event 9 is because activities G and H must be able to be described uniquely in terms of the events at each end | B1 |
| | (2) |
## 6(b)
[Activity on Arrow network diagram with all activities and dummies correctly shown] | M1, A1 |
[Network diagram repeated with correct values] | M1, A1 |
| | (4) |
All top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue. cao (top boxes). All bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue. cao (bottom boxes).
## 6(c)
21 (hours) | B1 |
| | (1) |
## 6(d)
$\frac{64}{21} \approx 3.048$ so at least 4 workers required | M1 A1 |
| | (2) |
Attempt to find lower bound: (a value in the interval [55 – 73] / their finish time) or (sum of the activities / their finish time) or (as a minimum) an awrt 3.05 or 3.04 (truncated). cso – either a correct calculation seen or awrt 3.05 (or 3.04) then 4. An answer of 4 with no working scores M0A0.
## 6(e)
[Gantt chart showing resource scheduling] | M1, A1 |
[Gantt chart repeated] | M1, A1 |
| | (4) |
At least 8 activities added including 5 floats. Scheduling diagram scores M0. Critical activities dealt with correctly and 4 non-critical activities dealt with correctly. All 11 activities including all 8 floats (on the correct non-critical activities). cso (all activities correct and present only once).
## 6(f)
[Gantt chart with 3 workers, 11 activities, no cascading, completion time 22 hours] | M1, A1, A1 |
| | (3) |
Not a cascade chart, 3 workers used and at least 9 activities placed. The completion time must be no greater than one hour more than the minimum completion time stated in (c) or seen in (b). 3 workers, All 11 activities present (just once). Condone one error either precedence or activity length. The completion time must be one hour greater than the minimum completion time stated in (c) or seen in (b). 3 workers, All 11 activities present (just once). No errors. The completion time must be 22.
---
# General Notes for Question 6
In (a), any use of the terms 'activity' and 'event' must be correct.
**6(a) Notes:**
- B1: cao dependency – all relevant activities must be referred to – activities I, J, F and either D or E must be mentioned.
- B1: cao uniqueness – please note that, for example, 'so that activities can be defined uniquely' is not sufficient to earn this mark. There must be some mention of describing activities in terms of the event at each end. However, give bod on statements that imply that an activity begins and ends at the same event.
**6(b) Notes:**
- M1: All top boxes complete, values generally increasing in the direction of the arrows ('left to right'), condone one rogue.
- A1: cao (top boxes).
- M1: All bottom boxes complete, values generally decreasing in the opposite direction of the arrows ('right to left'), condone one rogue.
- A1: cao (bottom boxes).
**6(c) Notes:**
- B1: cao (21)
**6(d) Notes:**
- M1: Attempt to find lower bound: (a value in the interval [55 – 73] / their finish time) or (sum of the activities / their finish time) or (as a minimum) an awrt 3.05 or 3.04 (truncated).
- A1: cso – either a correct calculation seen or awrt 3.05 (or 3.04) then 4. An answer of 4 with no working scores M0A0.
**6(e) Notes:**
- M1: At least 8 activities added including 5 floats. Scheduling diagram scores M0.
- A1: Critical activities dealt with correctly and 4 non-critical activities dealt with correctly.
- M1: All 11 activities including all 8 floats (on the correct non-critical activities).
- A1: cso (all activities correct and present only once).
**6(f) Notes:**
- M1: Not a cascade chart, 3 workers used and at least 9 activities placed. The completion time must be no greater than one hour more than the minimum completion time stated in (c) or seen in (b).
- A1: 3 workers, All 11 activities present (just once). Condone one error either precedence or activity length. The completion time must be one hour greater than the minimum completion time stated in (c) or seen in (b).
- A1: 3 workers, All 11 activities present (just once). No errors. The completion time must be 22.
\includegraphics{figure_2}
A project is modelled by the activity network shown in Figure 2. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the activity. The numbers in circles are the event numbers. Each activity requires one worker.
\begin{enumerate}[label=(\alph*)]
\item Explain the significance of the dummy activity
\begin{enumerate}[label=(\roman*)]
\item from event 5 to event 6
\item from event 7 to event 9.
\end{enumerate}
\hfill [2]
\item Complete Diagram 3 in the answer book to show the early event times and the late event times. \hfill [4]
\item State the minimum project completion time. \hfill [1]
\item Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time. You must show your working. \hfill [2]
\item On Grid 1 in your answer book, draw a cascade (Gantt) chart for this project. \hfill [4]
\item On Grid 2 in your answer book, construct a scheduling diagram to show that this project can be completed with three workers in just one more hour than the minimum project completion time. \hfill [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2018 Q6 [16]}}