Draw cascade/Gantt chart

6 Joan and Keith have to clear and tidy their garden. The table shows the jobs that have to be completed, their durations and their precedences.

1. Draw an activity on arc network for these activities.
2. Mark on your diagram the early time and the late time for each event. Give the minimum completion time and the critical activities.
3. Each job is to be done by one person only. Joan and Keith are equally able to do all jobs. Draw a cascade chart indicating how to organise the jobs so that Joan and Keith can complete the project in the least time. Give that least time and explain why the minimum project completion time is shorter.

1 Figure 1 below shows an activity diagram for a cleaning project. The duration of each activity is given in days.

1. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 1.
2. Find the critical paths and state the minimum time for completion of the project.
3. Find the activity with the greatest float time and state the value of its float time.
4. On Figure 2 opposite, draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as late as possible.
   1. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-02_846_1488_1391_292}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-03_1295_1714_219_150} \captionsetup{labelformat=empty} \caption{Figure 2}
      \end{figure} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{(d)} \includegraphics[alt={},max width=\textwidth]{1aca4e91-d1b3-4a78-8880-e42a4fbf3ddb-03_1023_1584_1589_278}
      \end{figure}

3 The table lists the activities involved in preparing for a cycle ride, their expected durations and their immediate predecessors.

1. Draw an activity network to represent the information in the table. Show the activities on the arcs and indicate the direction of each activity and dummy activity. You are advised to make your network quite large.
2. Carry out a forward pass and a backward pass to determine the minimum completion time for preparing for the ride. List the critical activities.
3. Construct a cascade chart, showing each activity starting at its earliest possible time. Two people, John and Kerry, are intending to go on the cycle ride. Activities \(A , B , F\) and \(G\) will each be done by just one person (either John or Kerry), but both are needed (at the same time) for activities \(C , D\) and \(E\). Also, each of John and Kerry must carry out activity \(H\), although not necessarily at the same time. All timings and precedences in the original table still apply.
4. Draw up a schedule showing which activities are done by each person at which times in order to complete preparing for the ride in the shortest time possible. The schedule should have three columns, the first showing times in 4-minute intervals, the second showing which activities John does and the third showing which activities Kerry does.

4 A project is represented by the activity network below. The activity durations are given in hours. \includegraphics[max width=\textwidth, alt={}, center]{f20391b2-e3c1-4021-9a87-47fd4ea7c490-5_346_1033_351_244}

1. By carrying out a forward pass, determine the minimum project completion time.
2. By carrying out a backward pass, determine the (total) float for each activity.
3. For each non-critical activity, determine the independent float and the interfering float.
4. Construct a cascade chart showing all the critical activities on one row and each non-critical activity on a separate row, starting at its earliest start time, and using dashed lines to indicate (total) float. You may not need to use all the grid. Each activity requires exactly one worker.
5. Construct a schedule to show how exactly two workers can complete the project as quickly as possible. You may not need to use all the grid. Issues with deliveries delay the earliest possible start of activity D by 3 hours.
6. Construct a schedule to show how exactly two workers can complete the project with this delay as quickly as possible. You may not need to use all the grid.

2 Kirstie has bought a house that she is planning to renovate. She has broken the project into a list of activities and constructed an activity network, using activity on arc. \includegraphics[max width=\textwidth, alt={}, center]{0c9513fe-a471-427e-ba30-b18df11271e3-3_887_1751_1030_207}

1. Construct a cascade chart for the project, showing the float for each non-critical activity.
2. Calculate the float for remodelling the internal layout stating how much of this is independent float and how much is interfering float. Kirstie needs to supervise the project. This means that she cannot allow more than three activities to happen on any day.
3. Describe how Kirstie should organise the activities so that the project is completed in the minimum project completion time and no more than three activities happen on any day.

3. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 2. 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 Grid 1 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 time and activities. (You do not need to provide a schedule of the activities.)

7. \begin{figure}[h] \end{figure} A company models a project 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. Add early and late event times to Diagram 1 in the answer book.
2. State the critical path and its length.
3. On Diagram 2 in the answer book, construct a cascade (Gantt) chart.
4. By using your cascade chart, state which activities must be happening at
   1. time 7.5
   2. time 16.5 It is decided that the company may use up to 25 days to complete the project.
5. On Diagram 3 in the answer book, construct a scheduling diagram to show how this project can be completed within 25 days using as few workers as possible.

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.)

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}

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.

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.

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.)

6. The precedence table below shows the twelve activities required to complete a project.

1. Draw the activity network described in the precedence table, using activity on arc. Your activity network must contain the minimum number of dummies only.
   (5) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6ccce35f-4e62-4b6b-acf6-f9b3e18d4b52-11_654_1358_153_356} \captionsetup{labelformat=empty} \caption{Figure 6}
   \end{figure} Figure 6 shows a partially completed cascade chart for the project. The non-critical activities F, J and K are not shown in Figure 6. The time taken to complete each activity is given in hours and the project is to be completed in the minimum possible time.
2. State the critical activities. Given that the total float of activity F is 2 hours,
3. state the duration of activity F . The duration of activity J is \(x\) hours, and the duration of activity K is \(y\) hours, where \(x > 0\) and \(y > 0\)
4. 1. State, in terms of \(y\), the maximum possible total float for activity K.
   2. State, in terms of \(x\) and \(y\), the total float for activity J .

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.] A construction project is to be undertaken. The table shows the activities involved.

1. Complete the activity network for the project on Figure 1.
2. Find the earliest start time for each activity.
3. Find the latest finish time for each activity.
4. Find the critical path.
5. State the float time for each non-critical activity.
6. On Figure 2, draw a cascade diagram (Gantt chart) for the project, assuming each activity starts as late as possible.

4. Construct an activity network to show the tasks involved in widening a bridge over the B451.

1. Find those tasks which lie on the critical path and list them in order.
2. State the minimum length of time needed to widen the bridge.
3. Represent the tasks on a Gantt diagram. Tasks \(F\) and \(J\) each require 3 workers, tasks \(B\), \(D\) and \(I\) each require 2 workers and the remaining tasks each require one worker.
4. Draw a resource histogram showing how it is possible for a team of 4 workers to complete the project in the minimum possible time.

7

1. 1. Find the earliest start time and the latest finish time for each activity and write these values on the activity network in Figure 1 7
      1. (ii) Write down the critical path. 7
   2. On Figure 2 below, draw a cascade diagram (Gantt chart) for the planned building project, assuming that each activity starts as early as possible. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-11_1127_1641_539_201}
      \end{figure} 7
   3. During further planning of the building project, Nova Merit Construction find that activity \(F\) is not necessary and they remove it from the project. Explain the effect removing activity \(F\) has on the minimum completion time of the project.

\includegraphics{figure_5} 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 late time for each event, showing them on the diagram in the answer book. [4]
2. Determine the critical activities and the length of the critical path. [2]
3. On the grid in the answer book, draw a cascade (Gantt) chart for the process. [4]

Each activity requires only one worker, and workers may not share an activity.

1. Use your cascade chart to determine the minimum numbers of workers required to complete the process in the minimum time. Explain your reasoning clearly. [2]
2. Schedule the activities, using the number of workers you found in part \((d)\), so that the process is completed in the shortest time. [3]

\includegraphics{figure_5} A project is modelled by the activity network shown in Fig. 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the activity. The numbers in circles give the event numbers. Each activity requires one worker.

1. Explain the purpose of the dotted line from event 4 to event 5. [1]
2. Calculate the early time and the late time for each event. Write these in the boxes in the answer book. [4]
3. Determine the critical activities. [1]
4. Obtain the total float for each of the non-critical activities. [3]
5. On the grid in the answer book, draw a cascade (Gantt) chart, showing the answers to parts (c) and (d). [4]
6. Determine the minimum number of workers needed to complete the project in the minimum time. Make your reasoning clear. [2]

\includegraphics{figure_7} 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. [4]
2. State the critical activities. [1]
3. On Grid 1 in the answer book, draw a cascade (Gantt) chart for this project. [4]
4. Use your cascade chart to determine a lower bound for the number of workers needed. You must justify your answer. [2]

(Total 11 marks) TOTAL FOR PAPER: 75 MARKS END

The activities involved in a project, their durations, immediate predecessors and the number of workers required for each activity are shown in the table.

Activity	Duration (hours)	Immediate predecessors	Number of workers
A	6	-	2
B	4	-	1
C	4	-	1
D	2	A	2
E	3	A, B	1
F	4	C	1
G	3	D	1
H	3	E, F	2

1. Model the project using an activity network.
2. Draw a cascade chart for the project, showing each activity starting at its earliest possible start time. [3]
3. Construct a schedule to show how three workers can complete the project in the minimum possible time. [4]