Calculate lower bound for workers

7. \begin{figure}[h] \end{figure} The network in Figure 4 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 for you. Given that activity F is a critical activity and that the total float on activity G is 2 days,

1. write down the value of \(x\) and the value of \(y\),
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.
3. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time.
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. (You do not need to provide a schedule of the activities.)

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 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. Explain what is meant by a critical path.
3. List the critical path for this network.
4. For each of the situations below, state the effect that the delay would have on the project completion date.
   1. A 4-day delay during activity J.
   2. A 4-day delay during activity M . The delays mentioned in (d) do not occur.
5. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
6. Schedule the activities using the minimum number of workers so that the project is completed in the minimum time.

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 hours, to complete the corresponding activity. Each activity requires one worker. The project is to be completed in the shortest possible time using as few workers as possible.

1. Complete Diagram 1 in the answer book to show the early event times and the late event times.
2. Calculate the total float for activity D. You must make the numbers used in your calculation clear.
3. Calculate a lower bound for the minimum number of workers required to complete the project in the shortest possible time. You must show your working.
4. Draw a cascade 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 time and activities. (You do not need to provide a schedule of the activities.)

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.

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.

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.

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.

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)

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.

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. (a) In the context of critical path analysis, define the term 'total float'. \begin{figure}[h] \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.
(b) Complete Diagram 1 in the answer book to show the early event times and the late event times.
(c) State the critical activities.
(d) 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.
(e) 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.
(f) Schedule the activities using Grid 1 in the answer book.

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.

\includegraphics{figure_3} The network in Fig. 3 shows the activities involved in the process of producing a perfume. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, taken to complete the activity.

1. Calculate the early time and the late time for each event, showing them on Diagram 1 in the answer booklet. [4]
2. Hence determine the critical activities. [2]
3. Calculate the total float time for \(D\). [2]

Each activity requires only one person.

1. Find a lower bound for the number of workers needed to complete the process in the minimum time. [2]

Given that there are only three workers available, and that workers may not share an activity,

1. schedule the activities so that the process is completed in the shortest time. Use the time line in the answer booklet. State the new shortest time. [5]

\includegraphics{figure_5} The network in Figure 5 shows the activities that need to be undertaken to complete a 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 to be shown at each vertex and some have been completed for you.

1. Calculate the missing early and late times and hence complete Diagram 2 in your answer book. [3]
2. List the two critical paths for this network. [2]
3. Explain what is meant by a critical path. [2]

The sum of all the activity times is 110 days and each activity requires just one worker. The project must be completed in the minimum time.

1. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working. [2]
2. List the activities that must be happening on day 20. [2]
3. Comment on your answer to part (e) with regard to the lower bound you found in part (d). [1]
4. Schedule the activities, using the minimum number of workers, so that the project is completed in 30 days. [3]

(Total 15 marks)