Find missing early/late times

1 The diagram shows the activity network and the duration, in days, of each activity for a particular project. Some of the earliest start times and latest finish times are shown on the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-02_830_1447_678_301}

1. Find the values of the constants \(x , y\) and \(z\).
2. Find the critical paths.
3. Find the activity with the largest float and state the value of this float.
4. The number of workers required for each activity is shown in the table.

   Activity	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
   Number of workers required	4	2	3	4	2	4	3	3	5	6

   Given that each activity starts as early as possible and assuming that there is no limit to the number of workers available, draw a resource histogram for the project on Figure 1 below, indicating clearly which activities are taking place at any given time.
5. It is later discovered that there are only 9 workers available at any time. Use resource levelling to find the new earliest start time for activity \(J\) so that the project can be completed with the minimum extra time. State the minimum extra time required. (d) Number of workers \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{b23828c8-01ee-4b5a-b6d2-41b7e27190d6-03_803_1330_1224_468}
   \end{figure}

6 A project is represented by the activity on arc network below. \includegraphics[max width=\textwidth, alt={}, center]{cc58fb7a-efb6-4548-a8e1-e40abe1eb722-7_410_1095_296_486} The duration of each activity (in minutes) is shown in brackets, apart from activity I.

1. Suppose that the minimum completion time for the project is 15 minutes.
   1. By calculating the early event times, determine the range of values for \(x\).
   2. By calculating the late event times, determine which activities must be critical. The table shows the number of workers needed for each activity.

      Activity	A	B	C	D	E	F	G	H	I	J	K
      Workers	2	1	1	2	\(n\)	1	2	1	1	1	4

2. Determine the maximum possible value for \(n\) if 5 workers can complete the project in 15 minutes. Explain your reasoning. The duration of activity F is reduced to 1.5 minutes, but only 4 workers are available. The minimum completion time is no longer 15 minutes.
3. Determine the minimum project completion time in this situation.
4. Find the maximum possible value for \(x\) for this minimum project completion time.
5. Find the maximum possible value for \(n\) for this minimum project completion time.

2. \begin{figure}[h] \end{figure} The network in Figure 3 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 of the activity, 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 B is 2 days and the total float on activity F is also 2 days,

1. find the values of \(w , x , y\) and \(z\).
2. Draw a cascade (Gantt) 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.)

3. \begin{figure}[h] \end{figure} The network in Figure 2 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 one worker. The early event times and late event times are shown at each vertex. The total float on activity D is twice the total float on activity E .

1. Find the values of \(x , y\) and \(z\).
2. Draw a cascade chart for this project on Grid 1 in the answer book.
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 time and activities. (You do not need to provide a schedule of the activities.)

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.

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} 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.

4. \begin{figure}[h] \end{figure} The network in Figure 3 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 one late event time has been completed for you. The total float of activity H is 7 days.

1. Explain, with detailed reasoning, why \(x = 11\)
2. Determine 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 using as few workers as possible.
3. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time.
4. Schedule the activities using Grid 1 in the answer book.

5. \begin{figure}[h] \end{figure} The network in Figure 2 shows the activities that need to be completed for 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 are shown in Figure 2.

1. Complete Table 1 in the answer book to show the immediately preceding activities for each activity. It is given that \(4 < x \leqslant m\)
2. State the largest possible integer value of \(m\).
3. 1. Complete Diagram 1 in the answer book to show the late event times.
   2. State the activities that must be critical.
4. Calculate the total float for activity G. The resource histogram in Figure 3 shows the number of workers required when each activity starts at its earliest possible time. The histogram also shows which activities happen at each time. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{27586973-89f4-45e1-9cc4-04c4044cd3db-09_682_1612_356_230} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}
5. Complete Table 2 in the answer book to show the number of workers required for each activity of the project.
6. Draw a Gantt chart on Grid 1 in the answer book to represent the activity network.

8 An activity diagram for a project is shown below. The duration of each activity is given in weeks. The earliest start time and the latest finish time for each activity are shown on the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-22_640_1626_475_209}

1. Find the values of \(x , y\) and \(z\).
2. State the critical path.
3. Some of the activities can be speeded up at an additional cost. The following table lists the activities that can be speeded up together with the minimum possible duration of these activities. The table also shows the additional cost of reducing the duration of each of these activities by one week.

2 Part of an activity network is shown in the diagram below. \(A B C\) is part of the critical path of the activity network. \includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-04_264_908_447_566} The duration of activity \(B\) is \(d\).
Which of the following statements about \(d\) is correct? Circle your answer. $$0 < d < 10 \quad d = 10 \quad 10 < d < 20 \quad d = 20$$

\includegraphics{figure_3} A project is modelled by the activity network in Fig. 3. The activities are represented by the arcs. One worker is required for each activity. The number in brackets on each arc gives the time, in hours, to complete the activity. The earliest event time and the latest event time are given by the numbers in the left box and right box respectively.

1. State the value of \(x\) and the value of \(y\). [2]
2. List the critical activities. [2]
3. Explain why at least 3 workers will be needed to complete this project in 38 hours. [2]
4. Schedule the activities so that the project is completed in 38 hours using just 3 workers. You must make clear the start time and finish time of each activity. [4]