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Networks, STD2 N3 EQ-Bank 38

The precedence table below shows the 12 activities required to complete a project. The duration in days and immediate predecessors are shown.

\begin{array}{|c|c|c|}
\hline \rule{0pt}{2.5ex}\ \ \textbf{Activity} \ \ & \ \ \textbf{Duration} \ \ & \textbf{Immediate predecessors} \\
\hline \rule{0pt}{2.5ex}A \rule[-1ex]{0pt}{0pt}& 4 & - \\
\hline \rule{0pt}{2.5ex}B \rule[-1ex]{0pt}{0pt}& 6 & A \\
\hline \rule{0pt}{2.5ex}C \rule[-1ex]{0pt}{0pt}& 8 & A \\
\hline \rule{0pt}{2.5ex}D \rule[-1ex]{0pt}{0pt}& 3 & A \\
\hline \rule{0pt}{2.5ex}E \rule[-1ex]{0pt}{0pt}& 9 & B\\
\hline \rule{0pt}{2.5ex}F \rule[-1ex]{0pt}{0pt}& 6 & C \\
\hline \rule{0pt}{2.5ex}G \rule[-1ex]{0pt}{0pt}& 7 & B, D, F \\
\hline \rule{0pt}{2.5ex}H \rule[-1ex]{0pt}{0pt}& 12 & C \\
\hline \rule{0pt}{2.5ex}I \rule[-1ex]{0pt}{0pt}& 6 & G, H \\
\hline \rule{0pt}{2.5ex}J \rule[-1ex]{0pt}{0pt}& 4 & E, I \\
\hline \rule{0pt}{2.5ex}K \rule[-1ex]{0pt}{0pt}& 3 & G, H \\
\hline \rule{0pt}{2.5ex}L \rule[-1ex]{0pt}{0pt}& 9 & J \\
\hline
\end{array}

The project is to be completed in minimum time.

  1. Sketch the network, identifying each activity and its duration, including any dummy activities.   (3 marks)
  2. Determine the critical path of the network.   (1 mark)
Show Answers Only

a.    \(\text{Sketch the network:}\) 
 

b.    \(\text {Critical Path:}\ ACFGIJL\)

Show Worked Solution

a.    \(\text{Sketch the network:}\) 
 

 
b.   \(\text{Critical path only requires the forward scanning in the above network.}\)

\(\text {Critical Path:}\ ACFGIJL\)

Networks, STD2 N3 2024 HSC 39

A project involving nine activities is shown in the network diagram.

The duration of each activity is not yet known.
 

The following table gives the earliest start time (EST) and latest start time (LST) for three of the activities. All times are in hours.

\begin{array} {|c|c|c|}
\hline
\rule{0pt}{2.5ex} \textit{Activity} \rule[-1ex]{0pt}{0pt} & EST & LST \\
\hline
\rule{0pt}{2.5ex} A \rule[-1ex]{0pt}{0pt} & \ \ \ \ \ \ 0\ \ \ \ \ \  & \ \ \ \ \ \ 2\ \ \ \ \ \  \\
\hline
\rule{0pt}{2.5ex} C \rule[-1ex]{0pt}{0pt} & 0 & 1 \\
\hline
\rule{0pt}{2.5ex} I \rule[-1ex]{0pt}{0pt} & 12 & 12 \\
\hline
\end{array}

  1. What is the critical path?   (1 mark)
  2. The minimum time required for this project to be completed is 19 hours.
  3. What is the duration of activity \(I\)?   (1 mark)

  4. The duration of activity \(C\) is 3 hours.
  5. What is the maximum amount of time that could occur between the start of activity \(F\) and the end of activity \(H\)?   (1 mark)

Show Answers Only

a.   \(\text{Critical Path:}\ BEGI\)

b.   \(\text{Duration of}\ I =7\ \text{hours}\)

c.  \(\text{Max time}\ =8\ \text{hours}\)

Show Worked Solution

a.   \(\text{Activity}\ A\ \text{and}\ C: \ LST \gt EST\)

\(\Rightarrow\ \text{Activity}\ A\ \text{and}\ C\ \text{not on critical path.}\)

\(\text{Critical Path:}\ BEGI\)
 

♦ Mean mark (a) 43%.

b.   \(\text{Duration of}\ I = 19-12=7\ \text{hours}\)
 

c.   \(\text{Since}\ C + F + H + I\ \text{is not a critical path:}\)

\(C + F + H + I = 18\ \text{or less (C.P. = 19 hours)}\)

\(3+F+H+7 = 18\ \text{or less}\)

\(\Rightarrow\ F+H = 8\ \text{or less}\)

\(\therefore\ \text{Max time from start of}\ F\ \text{to end of}\ H = 8\ \text{hours}\)

♦♦♦ Mean mark (c) 10%.

Networks, STD2 N3 2021 FUR1 6

The directed graph below shows the sequence of activities required to complete a project.

The time taken to complete each activity, in hours, is also shown.
 

The minimum completion time for this project is 18 hours.

The time taken to complete activity `E` is labelled  `x`.

What is the maximum value of  `x`?   (2 marks)

Show Answers Only

`text(3 hours)`

Show Worked Solution

`text{Consider all possible network paths:}`

`ADHJ \ -> \ text{18 hours}`

`BEJ \ -> \ (12 + x) \ text{hours}`

`CGFEJ \ -> \  (15 + x) \ text{hours}`

`CGIJ \ -> \ text{18 hours}`
 

`text{Minimum completion time = 18 hours}`

`=>\ text{No path can be longer than 18 hours}`

`:. \ x_text{max} = 3 \ text{hours}`

Networks, STD2 N3 2021 HSC 36

A project requires completion of 11 tasks  `A, B, C, . . . , K.`

A network diagram for the project giving the completion time for each task, in minutes, is shown.
 

  1. Find the minimum time to complete the project.  (1 mark)

  2. State the critical path for this project.  (1 mark)

  3. A new task, `X`, is to be added to the project. The earliest starting time for `X` is 17 minutes, the latest starting time for `X` is 18 minutes and `X` has a completion time of 12 minutes.
  4. Add task `X` to the given network diagram above AND state the float time for this task.  (3 marks)

Show Answers Only
  1. `40 \ text{minutes}`
  2. `A – C – D – E – G – J – K`
  3. `text{Float time = 1minute}`
     
Show Worked Solution
a.    `text{Scanning forwards:}`

♦ Mean mark part (a) 37%.

 


 
`text{Minimum completion time} = 40 \ text{minutes}`
♦ Mean mark part (b) 44%.
 
 
b.   `text{Critical Path:} \ A C D E G J K`
 
 
♦♦ Mean mark part (c) 32%.
c.   
`text{Float time}\ (X)` `= 18 – 17`  
  `= 1 \ text{minute}`  

Networks, STD2 N3 FUR1 2020 6

The activity network below shows the sequence of activities required to complete a project.

The number next to each activity in the network is the time it takes to complete that activity, in days.

What is the critical path and minimum completion time for this project.   (2 marks)

Show Answers Only

`BEIJM = 22\ text(days)`

Show Worked Solution

`text(Scanning forward:)`
 

`text(Critical path)` `= BEIJM`
  `= 7 + 4 + 3 + 3 + 5`
  `= 22\ text(days)`

Networks, STD2 N3 SM-Bank 33

A project requires nine activities (A–I) to be completed. The duration, in hours, and the immediate predecessor(s) of each activity are shown in the table below.
 


 

  1. Sketch the network, identifying each activity and its duration.  (2 marks)

  2. Identify the critical path and the minimum completion time of the project.  (2 marks)

Show Answers Only
  1.   
  2. `text(Critical path is)\ ACFGI.`
    `text(20 hours)`
Show Worked Solution

i.  `text(Sketch the network:)`

 

 
ii.   `text(Completion time of possible paths:)`

`ABEGI = 4 + 3 + 5 + 4 + 3 = 19\ text(hours)`

`ACFGI = 4 + 7 + 2 + 4 + 3 = 20\ text(hours)`

`ADHI = 4 + 2 + 5 + 3 = 14\ text(hours)`
 

`:.\ text(Critical path is)\ ACFGI.`

`:.\ text{Minimum completion time  = 20 hours}`

Networks, STD2 N3 2018 FUR1 5 MC

The directed network below shows the sequence of 11 activities that are needed to complete a project.

The time, in weeks, that it takes to complete each activity is also shown.
 


 

How many of these activities could be delayed without affecting the minimum completion time of the project?

  1. 3
  2. 4
  3. 5
  4. 6
Show Answers Only

`B`

Show Worked Solution

`text(Activities not on critical path can be delayed.)`

`text{Scanning forward: two critical paths exist (15 weeks)}`

♦ Mean mark 43%.

`ADHK\ text(and)\ BFJK`

`:.\ text(Activity)\ C, E, G\ text(and)\ I\ text(could be delayed)`

`=> B`

Networks, STD2 N3 SM-Bank 49

A project requires nine activities (A–I) to be completed. The duration, in hours, and the immediate predecessor(s) of each activity are shown in the table below.


 

  1. Sketch the network diagram. (3 marks)

  2. Find the minimum completion time for this project, in hours.   (1 mark)

Show Answers Only
  2. 20 `text(hours)`
Show Worked Solution

i.   `text(Sketch the network:)`
 

 
ii.  `text(Completion time of paths:)`

`ABEGI = 4 + 3 + 5 + 4 + 3 = 19\ text(hours)`

`ACFGI = 4 + 7 + 2 + 4 + 3 = 20\ text(hours)`

`ADHI = 4 + 2 + 5 + 3 = 14\ text(hours)`
 

`:.\ text{Minimum completion time (critical path) = 20 hours}`

Networks, STD2 N3 2012 FUR1 8 MC

networks-fur1-2012-vcaa-8-mc 
 

Eight activities, `A, B, C, D, E, F, G`  and  `H`, must be completed for a project.

The network above shows these activities and their usual duration in hours.

The duration of each activity can be reduced by one hour.

To complete this project in 16 hours, the minimum number of activities that must be reduced by one hour each is

A.   `1`

B.   `2`

C.   `3`

D.   `4`

Show Answers Only

`C`

Show Worked Solution

`text(Two critical paths)\ AFH\ text(and)\ BCFH => 18\ text(hours)`

♦♦ Mean mark 30%.

`text(Paths)\ AEG\ text(and)\ BCEG => 17\ text(hours)`

`text(Reducing)\ A\ text(and)\ B\ text(by 1 hour each reduces each)`

`text(path above by 1 hour.)`

 

`text(Need to reduce)\ AFH\ text(and)\ BCFH\ text(by 1 hour to 16 hours.)`

`=>\ text(Reducing either)\ F\ text(or)\ H\ text(by 1 hour brings the)`

`text(critical path down to 16 hours.)`

`rArr C`

Networks, STD2 N3 2010 FUR1 8 MC

A project has 12 activities. The network below gives the time (in hours) that it takes to complete each activity.
 

 
The critical path for this project is

A.   `ADGK`

B.   `ADGIL`

C.   `BHJL`

D.   `CEGIL`

Show Answers Only

`D`

Show Worked Solution

`text(Scanning forward:)`

♦ Mean mark 41%.
 

 
`text(Critical path is)\ \ CEGIL`

`=>  D`

Networks, STD2 N3 2006 FUR1 9 MC

The network below shows the activities and their completion times (in hours) that are needed to complete a project.
 


 

The project is to be optimised by reducing the completion time of one activity only.

This will reduce the completion time of the project by a maximum of

A.   1 hour 

B.   3 hours

C.   4 hours

D.   5 hours

Show Answers Only

`C`

Show Worked Solution

`text(Scanning forward:)`
 


 

`text(Critical path:)`

♦♦♦ Mean mark 17%.
MARKER’S COMMENT: When choosing an activity to crash, take care that a new critical path is not created.

`=> BDCEHJ\ text{(19 hours)}`
 

`text(Other routes not through)\ B,`

`ACEHJ\ text{(15 hours),}\ AFJ\ text{(14 hours)}`
 

`:.\ text(Activity)\ B\ text(could be reduced by 4 hours without)`

`text(a new critical path emerging.)`
 

`rArr C`

Networks, STD2 N3 SM-Bank 49

The directed graph below shows the sequence of activities required to complete a project.

The time to complete each activity, in hours, is also shown.
 


 

  1. Find the earliest starting time, in hours, for activity `N`.   (2 marks)

To complete the project in minimum time, some activities cannot be delayed.

  1. Calculate the number of activities that cannot be delayed.  (1 mark)
Show Answers Only
  1. `12\ text(hours)`
  2. `4`
Show Worked Solution

i.   `text{Scanning forward:}`
 


 

`:.\ text(EST for)\ N` `= CGJ`
  `= 4 +3+5`
  `= 12`

 

ii.   `text(Critical path is:)\ CFHM`

`:. 4\ text(activities can’t be delayed.)`

Networks, STD2 N3 SM-Bank 47

The directed graph below shows the sequence of activities required to complete a project.

All times are in hours.
 


 

  1.  Find the number of activities that have exactly two immediate predecessors. (1 mark)

  2. Identify the critical path for this project. (3 marks)
  3. If Activity E is reduced by one hour, identify the two new critical paths. (1 mark)

Show Answers Only
  1. `2`
  2. `BEIL`
  3. `ADIL and CHKL`
Show Worked Solution

i.    `I\ text(and)\ J`

`:.\ text(2 activities have exactly two immediate predecessors.)`

 

ii.   `text(Scanning forwards:)`
 


 

`:.\ text(Initial critical path)\ BEIL`

 

iii.   `text(If)\ E\ text(reduced by 1 hour, critical path)\ BEIL`

`text(reduces to 19 hours.)`

`:.\ text(Other critical paths of 19 hours are:)`

`ADIL = 5 + 2 + 4 + 8 = 19`

`CHKL = 2 + 6 + 3 + 8 = 19`

Networks, STD2 N3 SM-Bank 48

The network shows the activities that are needed to complete a particular project.
 

networks-fur1-2009-vcaa-5-6-mc
 

  1. Find the total number of activities that need to be completed before activity L can begin.   (1 mark)

The duration of every activity is initially 5 hours. 

  1. If the completion times of both activity F and activity K are reduced to 3 hours each, calculate the effect on the completion time for the project.   (2 marks)

Show Answers Only
  1. `7`
  2. `text(Completion time is unchanged.)`
Show Worked Solution

i.   `A, B, C, D, E, H\ text(and)\ I\ text(must be completed before)\ L.`

`:.\ text(7 activities need to be completed.)`

 

ii.   `text{Scanning forward (all activities take 5 hours):}`
 


 

`text(Activity)\ F\ text(and)\ K\ text(are not on any critical path.)`

`:.\ text(Reducing either will not change the completion time)`

`text(for the project.)`

Networks, STD2 N3 2008 FUR1 8-9 MC

The network below shows the activities that are needed to finish a particular project and their completion times (in days).
 

networks-fur1-2008-vcaa-8-mc

 
Part 1

The earliest start time for Activity K, in days, is

A.     `7`

B.   `15`

C.   `16`

D.   `19`

 

Part 2

This project currently has one critical path.

A second critical path, in addition to the first, would be created by

A.   increasing the completion time of D by 7 days.

B.   increasing the completion time of G by 1 day.

C.   increasing the completion time of I by 2 days.

D.   decreasing the completion time of C by 1 day.

Show Answers Only

`text(Part 1:)\ C`

`text(Part 2:)\ A`

Show Worked Solution

`text(Part 1)`

`text(Scanning forwards:)`
 

 
`text(EST for Activity)\ K`

`=\ text(Duration)\ ACFI`

`= 2 + 5 + 6 + 3`

`= 16`

`=> C`

 

`text(Part 2)`

♦♦ Mean mark of Part 2 was 35%.

`text(Original critical path:)\ ACFHJL\ text{(22 days)}`
 

`text(Consider option)\ A,`

`text(New critical path:)\ ABDJL\ text{(22 days)}`

`=> A`

Networks, STD2 N3 2011 FUR1 8 MC

The diagram shows the tasks that must be completed in a project.

Also shown are the completion times, in minutes, for each task.
 

 
The critical path for this project includes activities

A.   `B and I.`

B.   `C and H.`

C.   `D and E.`

D.   `F and K.`

Show Answers Only

`D`

Show Worked Solution

`text(Scanning forward:)`
 

`:.\ text(The critical path is)\ \ ACFIK.`

`=>  D`