7.05a Critical path analysis: activity on arc networks

1. The precedence table for activities involved in a small project is shown below

Draw an activity network, using activity on edge and without using dummies, to model this project.
(5)

6. \begin{figure}[h] \end{figure} Figure 5 is the activity network relating to a development project. 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 the precedence table in the answer book.
   (2)
2. Complete Diagram 1 in the answer book to show the early event times and late event times.
   (4)
3. Calculate the total float for activity E. You must make the numbers you use in your calculation clear.
   (2)
4. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
   (2)
5. Schedule the activities using the minimum number of workers so that the project is completed in the minimum time.

3. \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 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 late event times.
2. Calculate the total float for activity H. You must make the numbers you use in your calculation clear.
3. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. Show your calculation. Diagram 2 in the answer book shows a partly completed scheduling diagram for this project.
4. Complete the scheduling diagram, using the minimum number of workers, so that the project is completed in the minimum time.

6.

1. Draw the activity network described in the precedence table, using activity on arc and exactly two dummies.
   (5)
2. Explain why each of the two dummies is necessary.
   (2)

7. \begin{figure}[h] \end{figure} \section*{[The sum of the duration of all activities is 172 days]} 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 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. Calculate the total float for activity M. You must make the numbers you use in your calculation clear.
3. For each of the situations below, explain the effect that the delay would have on the project completion date.
   1. A 2 day delay on the early start of activity P.
   2. A 2 day delay on the early start of activity Q .
4. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. Diagram 2 in the answer book shows a partly completed cascade chart for this project.
5. Complete the cascade chart.
6. Use your cascade chart to determine a second lower bound on the number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities.
7. State which of the two lower bounds found in (d) and (f) is better. Give a reason for your answer.
   (Total 17 marks)

6.

1. Draw the activity network described in the precedence table below, using activity on arc and the minimum number of dummies.

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

2. Explain why each of your dummies is necessary.

7. \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 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. Calculate the total float for activity D. You must make the numbers you use 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. The project is to be completed in the minimum time using as few workers as possible.
4. Schedule the activities using Grid 1 in the answer book.

2.

1. Draw the activity network described in the precedence table below, using activity on arc and exactly two dummies.

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

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

7.

1. In the context of critical path analysis, define the term 'total float'. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{818ba207-5839-4698-aacb-75dab88b218f-08_1310_1563_340_251} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 is the activity network for a building project. 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.
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 maximum number of days by which activity G 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.
   

7. The table shows the activities required for the completion of a building project. For each activity the table shows the time taken, 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] \end{figure} Figure 6 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.

1. Add activities, E, F and I, and exactly one dummy to Diagram 1 in the answer book.
2. Complete Diagram 1 in the answer book to show the early event times and late event times.
3. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. You must show your working.
   (2)
4. Schedule the activities, using the minimum number of workers, so that the project is completed in the minimum time.
   (Total 13 marks)

2. Draw the activity network described in the precedence table below, using activity on arc and exactly three dummies.

7. \begin{figure}[h] \end{figure} The network in Figure 6 shows the activities that need to be undertaken 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 exactly one worker. The early event times and late event times are shown at each vertex. Given that the total float on activity D is 1 day,

1. find the values of \(\boldsymbol { w } , \boldsymbol { x } , \boldsymbol { y }\) and \(\boldsymbol { z }\).
2. On Diagram 1 in the answer book, draw a cascade (Gantt) chart for the project.
3. Use your cascade chart to determine a lower bound for the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities. It is decided that the company may use up to 36 days to complete the project.
4. On Diagram 2 in the answer book, construct a scheduling diagram to show how the project can be completed within 36 days using as few workers as possible.
   (3)

6. \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 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. Draw a Gantt chart for the project on the grid provided in the answer book.
3. State the activities that must be happening at time 18.5 An additional activity, P , is now included in the activity network shown in Figure 6. Activity P is immediately preceded only by activity D . No activity is dependent on the completion of activity P . Each activity still requires exactly one worker and the revised project is to be completed in the shortest possible time.
4. Explain, briefly, whether or not the revised project can be completed in the same time as the original project if the duration of activity P is
   1. 10 days
   2. 17 days

3.

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

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

   Given that D is a critical activity,
2. state which other activities must also be critical.
   (2)
   

6. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 4. 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 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. State the critical activities.
3. Draw a cascade (Gantt) chart for this project on the grid in the answer book.
4. 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. (You do not need to provide a schedule of the activities.)

5.

1. Draw the activity network described in the precedence table above, using activity on arc and exactly 4 dummies.
2. Explain why one of the activities I or J must be critical. It is given that activity C is a critical activity.
3. State the activities that are therefore guaranteed to be critical.

5. The precedence table shows the activities involved in planning an opening ceremony. An activity on arc network is to be drawn to model this planning process.

1. Draw the activity network using exactly two dummies.
2. Explain why each of the two dummies is necessary.

9. \begin{figure}[h] \end{figure} Figure 7 shows an activity network. Each activity is represented by an arc and the number in brackets on each arc is the duration of the activity in days.

1. Complete Figure 7 in the answer book showing the early and late event times.
2. List the critical path for this network. The sum of all the activity times is 95 days and each activity requires just one worker. The project must be completed in the minimum time.
3. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must make your method clear.
4. On the grid in your answer book, draw a cascade (Gantt) chart for this network.

2. The precedence table for activities involved in manufacturing a toy is shown below.

1. Draw an activity network, using activity on arc, and exactly one dummy, to model the manufacturing process.
2. Explain briefly why it is necessary to use a dummy in this case.

4. (a) Draw an activity network described in this precedence table, using as few dummies as possible.

1. A different project is represented by the activity network shown in Fig. 3. The duration of each activity is shown in brackets. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-05_710_1580_1509_239}
   \end{figure} Find the range of values of \(x\) that will make \(D\) a critical activity.
   (2)

8. \begin{figure}[h] \end{figure} The network in Figure 5 shows activities that need to be undertaken in order to complete a project. Each activity is represented by an arc. The number in brackets is the duration of the activity in hours. The early and late event times are shown at each node. The project can be completed in 24 hours.

1. Find the values of \(x , y\) and \(z\).
2. Explain the use of the dummy activity in Figure 5.
3. List the critical activities.
4. Explain what effect a delay of one hour to activity \(B\) would have on the time taken to complete the whole project. The company which is to undertake this project has only two full time workers available. The project must be completed in 24 hours and in order to achieve this, the company is prepared to hire additional workers at a cost of \(\pounds 28\) per hour. The company wishes to minimise the money spent on additional workers. Any worker can undertake any task and each task requires only one worker.
5. Explain why the company will have to hire additional workers in order to complete the project in 24 hours.
6. Schedule the tasks to workers so that the project is completed in 24 hours and at minimum cost to the company.
7. State the minimum extra cost to the company.

3 Deva Construction Ltd undertakes a small building project. The activity network for this project is shown below in Figure 1, where each activity's duration is given in hours. \begin{figure}[h] \end{figure} 3

1. Complete the activity network for the building project. 3
2. Deva Construction Ltd is able to reduce the duration of a single activity to 1 hour by using specialist equipment. State, with a reason, which activity should have its duration reduced to 1 hour in order to minimise the completion time for the building project.
   3
3. State one limitation in the building project used by Deva Construction Ltd. Explain how this limitation affects the project.
   [0pt] [2 marks]

3. The table above shows the activities required for the completion of a building project. For each activity, the table shows the time it takes, 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] \end{figure} Figure 2 shows a partially completed activity network used to model the project. The activities are represented by the arcs and the number in brackets on each arc is the time taken, in days, to complete the corresponding activity.

1. Add the missing activities and necessary dummies to Diagram 1 in the answer book.
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. State the critical activities. At the beginning of the project it is decided that activity G is no longer required.
4. Explain what effect, if any, this will have on
   1. the shortest completion time of the project if activity G is no longer required,
   2. the timing of the remaining activities.

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. The exact duration, \(x\), of activity N is unknown, but it is given that \(5 < x < 10\) 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 the late event times.
3. List the critical activities. It is given that activity J can be delayed by up to 4 hours without affecting the shortest possible completion time of the project.
4. Determine the value of \(x\). You must make the numbers used in your calculation clear.
5. Draw a cascade chart for this project on Grid 1 in the answer book.

2.

1. Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies only.
2. Explain why it is necessary to draw a dummy from the end of activity A . Every activity shown in the precedence table has the same duration.
3. State which activity cannot be critical, justifying your answer.