7.05b Forward and backward pass: earliest/latest times, critical activities

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)

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.

6. \begin{figure}[h] \end{figure} A building project is modelled by the activity network shown in Fig. 4. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, taken to complete the activity. The left box entry at each vertex is the earliest event time and the right box entry is the latest event time.

1. Determine the critical activities and state the length of the critical path.
2. State the total float for each non-critical activity.
3. On the grid in the answer booklet, draw a cascade (Gantt) chart for the project. Given that each activity requires one worker,
4. draw up a schedule to determine the minimum number of workers required to complete the project in the critical time. State the minimum number of workers.
   (3)

7. \begin{figure}[h] \end{figure} The network in Figure 6 shows the activities that need to be undertaken to complete a building project. Each activity is represented by an arc. The number in brackets is the duration of the activity in days. The early and late event times are shown at each vertex.

1. Find the values of \(v , w , x , y\) and \(z\).
2. List the critical activities.
3. Calculate the total float on each of activities H and J .
4. Draw a cascade (Gantt) chart for the project. The engineer in charge of the project visits the site at midday on day 8 and sees that activity E has not yet been started.
5. Determine if the project can still be completed on time. You must explain your answer. Given that each activity requires one worker and that the project must be completed in 35 days,
6. use your cascade chart to determine a lower bound for the number of workers needed. You must justify your answer.

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.

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)

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.

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)

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

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.)

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.

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.