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Networks, GEN2 2024 VCAA 15

An upgrade to the supermarket requires the completion of 11 activities, \(A\) to \(K\).

The directed network below shows these activities and their completion time, in weeks.

The minimum completion time for the project is 29 weeks.
 

 

  1. Write down the critical path.   (1 mark)

  2. Which activity can be delayed for the longest time without affecting the minimum completion time of the project?   (1 mark)

Use the following information to answer parts c-e.

A change is made to the order of activities.

The table below shows the activities and their new latest starting times in weeks.

\begin{array}{|c|c|}
\hline
\textbf{Activity} & \textbf{Latest Starting}\\
&\textbf{time} \text{(weeks)}\\
\hline A & 0 \\
\hline B & 2 \\
\hline C & 10 \\
\hline D & 9 \\
\hline E & 13 \\
\hline F & 14 \\
\hline G & 18 \\
\hline H & 17 \\
\hline I & 19 \\
\hline J & 25 \\
\hline K & 22 \\
\hline
\end{array}

A dummy activity is now required in the network.

  1. On the directed network below, draw a directed edge to represent the dummy activity. Include a label.  (1 mark)

     

  1. What is the new minimum completion time of the project?  (1 mark)
  2. The owners of the supermarket want the project completed earlier.
  3. They will pay to reduce the time of some of the activities.
  4. A reduction in completion time of an activity will incur an additional cost of $10 000 per week.
  5. Activities can be reduced by a maximum of two weeks.
  6. The minimum number of weeks an activity can be reduced to is seven weeks.
  7. What is the minimum amount the owners of the supermarket will have to pay to reduce the completion time of the project as much as possible?   (1 mark)

Show Answers Only

a.    \(A, C, H, J\)

b.    \(\text{Activity E}\)

c.    

d.    \(\text{30 weeks}\)

e.    \($50\,000\)

Show Worked Solution

a.    \(\text{Critical path:  }A, C, H, J\)
 

 
b.    \(\text{Activity with the largest float time can be delayed the longest.}\)

\(\text{Consider Activity E:}\)

\(\text{EST = 11, LST}= 18-4=14\rightarrow\ \text{Float time = 3 weeks}\)

\(\therefore\ \text{Activity E can be delayed the longest.}\)
 

♦♦ Mean mark (b) 34%.

c.    

♦♦♦ Mean mark (c) 10%.

d.    \(\text{New minimum completion time is 30 weeks.}\)
 

♦♦♦ Mean mark (d) 27%.

e.    \(\text{Activities that can be reduced:}\)

\(-A\ \text{can be reduced by 2 weeks}\)

\(-B, D\ \text{can each be reduced by 1 week each}\)

\(-H\ \text{can be reduced by 1 week}\)

\(\text{Total reduction = 5 weeks}\)

\(\Rightarrow \ \text{Minimum payment}=$50\,000\)

♦♦♦ Mean mark (e) 7%.

Networks, GEN1 2022 VCAA 7-8 MC

A project involves 11 activities, \(A\) to \(K\).

The table below shows the earliest start time and duration, in days, for each activity.

The immediate predecessor(s) of each activity is also shown.

\begin{array} {|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \textbf{Activity}\ \ & \textbf{Earliest} & \ \ \textbf{Duration}\ \ & \textbf{Immediate}\\
& \textbf{start time} \rule[-1ex]{0pt}{0pt} & &\textbf{predecessor}\\
\hline
\rule{0pt}{2.5ex} A \rule[-1ex]{0pt}{0pt} & \text{0} & \text{6} & \text{-}\\
\hline
\rule{0pt}{2.5ex} B \rule[-1ex]{0pt}{0pt} & \text{0} & \text{7} & \text{-}\\
\hline
\rule{0pt}{2.5ex} C \rule[-1ex]{0pt}{0pt} & \text{6} & \text{10} & A\\
\hline
\rule{0pt}{2.5ex} D \rule[-1ex]{0pt}{0pt} & \text{6} & \text{7} & A\\
\hline
\rule{0pt}{2.5ex} E \rule[-1ex]{0pt}{0pt} & \text{7} & \text{8} & B\\
\hline
\rule{0pt}{2.5ex} F \rule[-1ex]{0pt}{0pt} & \text{15} & \text{2} & D,\ E\\
\hline
\rule{0pt}{2.5ex} G \rule[-1ex]{0pt}{0pt} & \text{15} & \text{2} & E\\
\hline
\rule{0pt}{2.5ex} H \rule[-1ex]{0pt}{0pt} & \text{17} & \text{3} & G\\
\hline
\rule{0pt}{2.5ex} I \rule[-1ex]{0pt}{0pt} & \text{20} & \text{6} & C,\ F,\ H\\
\hline
\rule{0pt}{2.5ex} J \rule[-1ex]{0pt}{0pt} & \text{17} & \text{5} & G\\
\hline
\rule{0pt}{2.5ex} K \rule[-1ex]{0pt}{0pt} & \text{26} & \text{2} & I,\ J\\
\hline
\end{array}

 
Question 7

A directed network for this project will require a dummy activity.

The dummy activity will be drawn from the end of

  1. activity \(A\) to the start of activity \(D\).
  2. activity \(E\) to the start of activity \(F\).
  3. activity \(F\) to the start of activity \(I\).
  4. activity \(G\) to the start of activity \(H\).
  5. activity \(I\) to the start of activity \(J\).

 
Question 8

When this project is completed in the minimum time, the sum of all the float times, in days, will be

  1. 0
  2. 16
  3. 18
  4. 20
  5. 28
Show Answers Only

\(\text{Question 7:}\ B\)

\(\text{Question 8:}\ D\)

Show Worked Solution

\(\text{Question 7}\)

Draw network from table:
 

 

→ \(F\) starts after completion of both \(D\) and \(E\).

→ \(G\) starts after activity \(E\) only.

\(\Rightarrow B\)


♦ Mean mark (Q7) 49%.

 
\(\text{Question 8}\)

Scanning forwards and backwards on network diagram:

→ The critical path is \(B E G H I K\)

→ Float times occur at all points not on the critical path.

\(\text{Total Float Times}\) \(= A + C + D + F + J\)  
  \(= 4 + 4 + 5 + 3 + 4\)  
  \(= 20\)  

 
\(\Rightarrow D\)


♦♦ Mean mark (Q8) 34%.

Networks, GEN1 2023 VCAA 38 MC

A particular building project has ten activities that must be completed.

These activities and their immediate predecessor(s) are shown in the table below.

\begin{array} {|c|c|}
\hline
\rule{0pt}{2.5ex} \textbf{Activity} \rule[-1ex]{0pt}{0pt} & \textbf{Immediate predecessor(s)} \\
\hline
A  & - \\
\hline
B  & - \\
\hline
C  & A \\
\hline
 D  & A \\
\hline
E  & B \\
\hline
 F  & D, E \\
\hline
 G  & C, F \\
\hline
 H  & F \\
\hline
 I  & D, E \\
\hline
 J  & H, I \\
\hline
\end{array}

A directed graph that could represent this project is
 

Show Answers Only

\(D\)

Show Worked Solution

\(\text{Activity}\ H\ \text{has only one immediate predecessor,}\ F. \)

\(\text{Eliminate}\ A, B, C\ \text{and}\ E. \)

\(\Rightarrow D\)

NETWORKS, FUR2 2020 VCAA 5

The Sunny Coast cricket clubroom is undergoing a major works project.

This project involves nine activities: `A` to `I`.

The table below shows the earliest start time (EST) and duration, in months, for each activity.

The immediate predecessor(s) is also shown.

The duration for activity `C` is missing.
 

   

The information in the table above can be used to complete a directed network.

This network will require a dummy activity.

  1. Complete the following sentence by filling in the boxes provided.    (1 mark)

    This dummy activity could be drawn as a directed edge from the end of activity to the start of activity  
  1. What is the duration, in months, of activity  `C`?   (1 mark)

  2. Name the four activities that have a float time.   (1 mark)

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

     

    What is the minimum time, in months, that the project can be completed in?    (1 mark)

Show Answers Only
  1. `B\ text{to the start of activity}\ C.`
  2. `text{2 months}`
  3. `A, E, F, H`
  4. `text{17 months}`
Show Worked Solution

a.   `B\ text{to the start of activity}\ C.`

♦♦ Mean mark part (a) 26%.
   

b.   `text{Sketch network diagram.}`

♦ Mean mark part (b) 49%.

 

`text{Duration of Activity C = 2 months}`
  

c.   `text{Critical path:}\   BCDGI`

♦♦ Mean mark part (c) 23%.

`text{Activities with a float time are activities}`

`text{not on critical path.}`

`:. \ text{Four activities are:}\ \ A, E, F, H`
  

d.   `text{Completion time of}\ BCDGI = 20\ text{months}`

♦♦♦ Mean mark part (d) 12%.

`text{Reduce the completion of}\ B\ text{by 3 months to create}`

`text{a new minimum completion time of 17 months.}`

NETWORKS, FUR1 2019 VCAA 7 MC

A project involves nine activities, `A` to `I`.

The immediate predecessor(s) of each activity is shown in the table below.
 

             Activity            Immediate
     predecessor(s)     
  `A` `-`
  `B` `A`
  `C` `A`
  `D` `B`
  `E` `B, C`
  `F` `D`
  `G` `D`
  `H` `E, F`
  `I` `G, H`

 
A directed network for this project will require a dummy activity.

The dummy activity will be drawn from the end of

  1. activity `B` to the start of activity `C`.
  2. activity `B` to the start of activity `E`.
  3. activity `D` to the start of activity `E`.
  4. activity `E` to the start of activity `H`.
  5. activity `E` to the start of activity `F`.
Show Answers Only

`B`

Show Worked Solution

`text(Sketch network diagram:)`
 


 

`text(The dummy activity needs to be drawn)`

`text(from the end of activity)\ B\ text(to the start)`

`text(of activity)\ E.`

`=>  B`

NETWORKS, FUR1 2006 VCAA 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 crashed 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.   2 hours

C.   3 hours

D.   4 hours

E.   5 hours

Show Answers Only

`D`

Show Worked Solution

`text(The critical path is)\ BDCEHJ\ text{(19 hours)}`

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

 
`text(Also,)\ ACEHJ\ text{(15 hours)}`

`:.\ text(Activity)\ B\ text(could be crashed by 4 hours)`

`text(without a new critical path emerging.)`

`rArr D`

NETWORKS, FUR1 2007 VCAA 5-6 MC

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


 

Part 1

Which one of the following statements regarding this project is false?

A.   Activities `A, B` and `C` all have the same earliest start time.

B.   There is only one critical path for this project.

C.   Activity `J` may start later than activity `H.`

D.   The shortest path gives the minimum time for project completion.

E.   Activity `L` must be on the critical path.

 

Part 2

The earliest start time for activity `L`, in hours, is

A.   11

B.   12

C.   14

D.   15

E.   16

Show Answers Only

`text (Part 1:)\ D`

`text (Part 2:)\ E`

Show Worked Solution

`text (Part 1)`

♦ Mean mark 43%.

`A, B, C,\ text(and)\ E\ text(can be shown to be true.)`

`=>  D`

 

`text (Part 2)`

`text(Critical path is)\ CDFKL`

`:.\ text(EST for)\ L` `= 5 + 0 + 4 + 7`
  `= 16\ text(hours)`

  
`=>  E`

NETWORKS, FUR1 2011 VCAA 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.`

E.   `G and J.`

Show Answers Only

`D`

Show Worked Solution

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

`=>  D`

NETWORKS, FUR2 2015 VCAA 3

Nine activities are needed to prepare a daily delivery of groceries from the factory to the towns.

The duration, in minutes, earliest starting time (EST) and immediate predecessors for these activities are shown in the table below.
 

   Networks, FUR2 2015 VCAA 31
 

The directed network that shows these activities is shown below.
 

 Networks, FUR2 2015 VCAA 32
 

All nine of these activities can be completed in a minimum time of 26 minutes.

  1. What is the EST of activity `D`?   (1 mark)

  2. What is the latest starting time (LST) of activity `D`?   (1 mark)

  3. Given that the EST of activity `I` is 22 minutes, what is the duration of activity `H`?   (1 mark)

  4. Write down, in order, the activities on the critical path.   (1 mark)

  5. Activities `C` and `D` can only be completed by either Cassie or Donna.

     

    One Monday, Donna is sick and both activities `C` and `D` must be completed by Cassie. Cassie must complete one of these activities before starting the other.

     

    What is the least effect of this on the usual minimum preparation time for the delivery of groceries from the factory to the five towns?   (1 mark)

  6. Every Friday, a special delivery to the five towns includes fresh seafood. This causes a slight change to activity `G`, which then cannot start until activity `F` has been completed.
      
    i.     Michael was the best player in 2014 and he considered purchasing cricket equipment that was valued at $750.
    On the directed graph below, show this change without duplicating any activity?   (1 mark)


    Networks, FUR2 2015 VCAA 32
     
  7. ii. What effect does the inclusion of seafood on Fridays have on the usual minimum preparation time for deliveries from the factory to the five towns?   (1 mark)

Show Answers Only
  1. `text(3 minutes)`
  2. `text(4 minutes)`
  3. `text(3 minutes)`
  4. `A-C-F-H-I`
  5. `text(The critical path is increased by)`
    `text(7 minutes to 33 minutes.)`
  6. i. 
    Networks, FUR2 2015 VCAA 3 Answer
    ii. `text(The critical path is increased by)`
         `text(2 minutes to 28 minutes.)`

Show Worked Solution

a.   `text(3 minutes)`
 

b.   `text(4 minutes)`
 

c.   `text(3 minutes)`
 

d.   `A-C-F-H-I`
 

e.   `text(The critical path is increased by 7 minutes)`

`text(to 33 minutes.)`

 

f.i.    Networks, FUR2 2015 VCAA 3 Answer

 

f.ii.   `text(The critical path is increased by 2 minutes)`

`text(to 28 minutes.)`

NETWORKS, FUR2 2012 VCAA 2

Thirteen activities must be completed before the produce grown on a farm can be harvested. 

The directed network below shows these activities and their completion times in days.

 

NETWORKS, FUR2 2012 VCAA 2
  

  1. Determine the earliest starting time, in days, for activity `E`.   (1 mark)

  2. A dummy activity starts at the end of activity `B`.

     

    Explain why this dummy activity is used on the network diagram.   (1 mark)

  3. Determine the earliest starting time, in days, for activity `H`.   (1 mark)

  4. In order, list the activities on the critical path.   (1 mark)

  5. Determine the latest starting time, in days, for activity `J`.   (1 mark)

Show Answers Only
  1. `12\ text(days)`
  2. `F\ text(has)\ B\ text(as a predecessor while)\ G\ text(and)\ H`
    `text(have)\ B\ text(and)\ C\ text(as predecessors.)`
    `text(S)text(ince there cannot be 2 activities called)\ B,`
    `text{a dummy activity is drawn as an extension of}`
    `B\ text(to show that it is also a predecessor of)\ G\ text(and)`
    `H\ text{(with zero time).}`
  3. `15\ text(days)`
  4. `A-B-H-I-L-M`
  5. `25\ text(days)`
Show Worked Solution
a.    `text(EST of)\ E` `= 10 + 2`
    `= 12\ text(days)`
♦ Mean mark of all parts (combined) 47%.

 

b.   `F\ text(has)\ B\ text(as a predecessor while)\ G\ text(and)\ H`

`text(have)\ B\ text(and)\ C\ text(as predecessors.)`

`text(S)text(ince there cannot be 2 activities called)\ B,`

`text{a dummy activity is drawn as an extension of}`

`B\ text(to show that it is also a predecessor of)\ G\ text(and)`

`H\ text{(with zero time).}`

 

♦♦ Exact data unavailable but “few students” were able to correctly deal with the dummy activity in this question.
c.    `text(EST of)\ H` `= 10 + 5`
    `= 15\ text(days)`

 

d.   `text(The critical path is)`

`A-B-H-I-L-M`

 

e.   `text(The shortest time to complete all the activities)`

MARKER’S COMMENT: A correct calculation based on an incorrect critical path in part (d) gained a consequential mark here. Show your working!

`= 10 + 5 + 4 + 3  + 4 + 2`

`= 28\ text(days)`

 

`:.\ text(LST of)\ J` `= 28-3`
  `= 25\ text(days)`