7.05a Critical path analysis: activity on arc networks

6. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires exactly one worker. The project is to be completed in the shortest possible time.

1. Complete Diagram 1 in the answer book to show the early event times and late event times.
2. State the critical activities.
3. Calculate the maximum number of days by which activity E could be delayed without lengthening the completion time of the project. You must make the numbers used in your calculation clear.
4. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
5. Draw a cascade (Gantt) chart for this project on the grid provided in the answer book.

4. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 3. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the corresponding activity. Each activity requires exactly one worker. The project is to be completed in the shortest possible time.

1. Complete Diagram 1 in the answer book to show the early event times and late event times.
2. Determine the critical activities and the length of the critical path.
3. Calculate the total float for activity D. You must make the numbers you use in your calculation clear.
4. Draw a cascade (Gantt) chart for this project on Grid 1 in the answer book.
5. Use your cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities.

7. Draw the activity network described in this precedence table, using activity on arc and dummies only where necessary.

5. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 6. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Complete the precedence table in the answer book.
2. Complete Diagram 1 in the answer book to show the early event times and late event times.
3. State the minimum project completion time and list the critical activities.
4. Calculate the maximum number of hours by which activity E could be delayed without affecting the shortest possible completion time of the project. You must make the numbers used in your calculation clear.
5. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working. The project is to be completed in the minimum time using as few workers as possible.
6. Schedule the activities using Grid 1 in the answer book.
   (3) Before the project begins it becomes apparent that activity E will require an additional 6 hours to complete. The project is still to be completed in the shortest possible time and the time to complete all other activities is unchanged.
7. State the new minimum project completion time and list the new critical activities.

4. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 3. The activities are represented by the arcs. The number in brackets on each arc gives the time required, in hours, to complete the corresponding activity. The numbers in circles are the event numbers.

1. Explain the significance of the dummy activity
   1. from event 2 to event 3
   2. from event 6 to event 7
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. State the minimum project completion time and list the critical activities. The duration of activity H changes to \(x\) hours.
4. Find, in terms of \(x\) where necessary,
   1. the possible new early event time for event 7
   2. the possible new late event time for event 7 Given that the duration of activity H is such that the minimum project completion time is four hours greater than the time found in (c),
5. determine the value of \(x\).

4.

1. Draw the activity network described by the precedence table below, using activity on arc. Use dummies only where necessary.
   (5)

   Activity	Immediately preceding activities
   A	-
   B	-
   C	A
   D	A, B
   E	C, D
   F	D
   G	C
   H	G
   I	G
   J	E, F, I
   K	F

   Given that K is a critical activity,
2. state which other activities must also be critical.
   (1) Given instead that all activities shown in the precedence table have the same duration and K is not necessarily critical,
3. state the critical path for the network.
   (1)
   

5. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the corresponding activity. Each activity requires exactly one worker. The project is to be completed in the shortest possible time.

1. Complete Diagram 1 in the answer book to show the early event times and the late event times.
2. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
3. 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. Additional resources become available, which can shorten the duration of one of activities D, G or P by one day.
4. Determine which of these three activities should be shortened to allow the project to be completed in the minimum time. You must give reasons for your answer.

2. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the corresponding activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Complete Diagram 1 in the answer book to show the early event times and the late event times.
2. Draw a cascade chart for this project on Grid 1 in the answer book.
3. Use your cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to time and activities. (You do not need to provide a schedule of the activities.)

6.

1. Draw the activity network for the project described in the precedence table above, using activity on arc and the minimum number of dummies.
   (5)
2. State which activity is guaranteed to be critical, giving a reason for your answer.
   (2) It is given that each activity in the table takes two hours to complete.
3. State the minimum completion time and write down the critical path for the project.
   (2)

2. \begin{figure}[h] \end{figure} The network in Figure 1 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration of the activity, in days, is shown in brackets. The early event times and late event times are to be shown at each vertex and some have been completed. Given that

• CHN is the critical path for the project
• the total float on activity B is twice the duration of the total float on activity I
  1. find the value of \(x\) and show that the value of \(y\) is 7
  2. Calculate the missing early event times and late event times and hence complete Diagram 1 in your answer book.

Each activity requires one worker, and the project must be completed in the shortest possible time.

Draw a cascade chart for this project on Grid 1 in your answer book, and use it to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to time and activities.

5. The precedence table shows the eleven activities required to complete a project.

1. Draw the activity network for the project, using activity on arc and the minimum number of dummies.
   (5) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{27296f39-bd03-47ff-9a5e-c2212d0c68ed-07_314_1385_1464_347} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a schedule for the project. Each of the activities shown in the precedence table requires one worker. The time taken to complete each activity is in hours and the project is to be completed in the minimum possible time.
2. 1. State the minimum completion time for the project.
   2. State the critical activities.
   3. State the total float on activity G and the total float on activity K .
      (4)

1. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 1. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires exactly one worker. The project is to be completed in the shortest possible time.

1. Complete Diagram 1 in the answer book to show the early event times and the late event times.
2. Calculate the maximum number of days by which activity H could be delayed without lengthening the completion time of the project. You must make the numbers used in your calculation clear.
3. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
4. 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.

5.

1. Draw the activity network described in the precedence table above, using activity on arc and the minimum number of dummies. A project is modelled by the activity network drawn in (a). Each activity requires exactly one worker. The project is to be completed in the shortest possible time. The table below gives the time, in hours, to complete three of the activities.

   Activity	Duration (in hours)
   A	10
   E	7
   F	8

   The length of the critical path AEFK is 33 hours.
2. Determine the range of possible values for the duration of activity J. You must make your method and working clear.

2. \begin{figure}[h] \end{figure} [The sum of the durations of all the activities is 59 days.]
The network in Figure 1 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration, in days, of the corresponding activity is shown in brackets. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. 1. Complete Diagram 1 in the answer book to show the early event times and the late event times.
   2. State the minimum completion time of the project.
2. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
3. Schedule the activities using the minimum number of workers so that the project is completed in the minimum time.

6.

1. Draw the activity network for the project described in the precedence table, using activity on arc and the minimum number of dummies. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ba9337bf-7a3c-49aa-b395-dd7818cf1d13-10_880_1154_1464_452} \captionsetup{labelformat=empty} \caption{Grid 1}
   \end{figure} A cascade chart for all the activities of the project, except activity \(\mathbf { L }\), is shown on Grid 1. The time taken to complete each activity is given in hours and each activity requires one worker. The project is to be completed in the minimum time using as few workers as possible.
2. State the critical activities of the project.
3. Use the cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to time and activities. (You do not need to provide a schedule of the activities.) The duration of activity L is \(x\) hours. Given that the total float of activity L is at most 7 hours,
4. determine the range of possible values for \(\chi\).

4. \begin{figure}[h] \end{figure} The network in Figure 2 shows the activities that need to be carried out by a company to complete a project. Each activity is represented by an arc, and the duration, in days, is shown in brackets. Each activity requires one worker. The early event times and the late event times are shown at each vertex.

1. Complete the precedence table in the answer book.
   (2) A cascade chart for this project is shown on Grid 1. \includegraphics[max width=\textwidth, alt={}, center]{d409aaae-811d-4eca-b118-efc927885f97-07_885_1358_276_356} \section*{Grid 1}
2. Use Figure 2 and Grid 1 to find the values of \(v , w , x , y\) and \(z\). The project is to be completed in the minimum time using as few workers as possible.
3. Calculate a lower bound for the minimum number of workers required. You must show your working.
4. On Grid 2 in your answer book, construct a scheduling diagram for this project. Before the project begins it is found that activity F will require an additional 5 hours to complete. The durations of all other activities are unchanged. The project is still to be completed in the shortest possible time using as few workers as possible.
5. State the new minimum project completion time and state the new critical path.

8. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 7. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Complete Diagram 2 in the answer book to show the early and late event times.
2. State the critical activities.
3. On Grid 1 in the answer book, draw a cascade (Gantt) chart for this project.
4. Use your cascade chart to determine a lower bound for the number of workers needed. You must justify your answer. \section*{END}

4. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the activity. Some of the early and late times for each event are shown.

1. Calculate the missing early and late times and hence complete Diagram 1 in your answer book.
2. Calculate the total float on activities D, G and I. You must make your calculations clear.
3. List the critical activities. Each activity requires one worker.
4. Calculate a lower bound for the number of workers needed to complete the project in the minimum time.
   (2)

5.

1. Draw the activity network described in this precedence table, using activity on arc and exactly two dummies.
   (4)

   Activity	Immediately preceding activities
   A	-
   B	-
   C	A
   D	B
   E	B, C
   F	B, C

2. Explain why each of the two dummies is necessary.
   (3)
   

3.

1. Draw the activity network described in this precedence table, using activity on arc and exactly two dummies.
   (5)

   Activity	Immediately preceding activities
   A	-
   B	-
   C	-
   D	B
   E	B, C
   F	B, C
   G	F
   H	F
   I	G, H
   J	I

2. Explain why each of the two dummies is necessary.
   

8. \begin{figure}[h] \end{figure} The network in Figure 5 shows the activities involved in a process. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, taken to complete the activity.

1. Calculate the early time and the late time for each event, showing them on the diagram in the answer book.
2. Determine the critical activities and the length of the critical path.
3. Calculate the total float on activities F and G . You must make the numbers you used in your calculation clear.
4. On the grid in the answer book, draw a cascade (Gantt) chart for the process. Given that each task requires just one worker,
5. use your cascade chart to determine the minimum number of workers required to complete the process in the minimum time. Explain your reasoning clearly.

6. \begin{figure}[h] \end{figure} Figure 5 is the activity network relating to a building project. The number in brackets on each arc gives the time taken, in days, to complete the activity.

1. Explain the significance of the dotted line from event (2) to event (3).
2. Complete the precedence table in the answer booklet.
3. Calculate the early time and the late time for each event, showing them on the diagram in the answer booklet.
4. Determine the critical activities and the length of the critical path.
5. On the grid in the answer booklet, draw a cascade (Gantt) chart for the project.

7. \begin{figure}[h] \end{figure} The network in Figure 7 shows the activities that need to be undertaken to complete a maintenance project. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. The numbers in circles are the events. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Complete the precedence table for this network in the answer book.
2. Explain why each of the following is necessary.
   1. The dummy from event 6 to event 7 .
   2. The dummy from event 8 to event 9 .
3. Complete Diagram 2 in the answer book to show the early and the late event times.
4. State the critical activities.
5. Calculate the total float on activity K . You must make the numbers used in your calculation clear.
6. Calculate a lower bound for the number of workers needed to complete the project in the minimum time.

7. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 7. 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.

1. Explain the significance of the dummy activity
   1. from event 4 to event 6 ,
   2. from event 5 to event 7
      (3)
2. Calculate the early time and the late time for each event. Write these in the boxes in the answer book.
3. Calculate the total float on each of activities D and G. You must make the numbers you use in your calculations clear.
4. Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time.
5. On the grid in your answer book, draw a cascade (Gantt) chart for this project.

7. \begin{figure}[h] \end{figure} Figure 7 is the activity network relating to a building project. The activities are represented by the arcs. The number in brackets on each arc gives the time to complete the activity. Each activity requires one worker. The project must be completed in the shortest possible time.

1. Explain the reason for the dotted line from event 4 to event 6 as shown in Figure 7.
   (2)
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. State the critical activities.
4. Calculate the total float for activity G. You must make the numbers you use in your calculation clear.
5. Draw a Gantt chart for this project on the grid provided in the answer book.
6. State the activities that must be happening at time 5.5
7. Use your Gantt chart to determine the minimum number of workers needed to complete the project in the minimum time. You must justify your answer.