7.05d Latest start and earliest finish: independent and interfering float

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.

5. The precedence table shows the eleven activities required to complete a project.

1. Draw the activity network for the project, using activity on arc and the minimum number of dummies.
   (5) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{27296f39-bd03-47ff-9a5e-c2212d0c68ed-07_314_1385_1464_347} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a schedule for the project. Each of the activities shown in the precedence table requires one worker. The time taken to complete each activity is in hours and the project is to be completed in the minimum possible time.
2. 1. State the minimum completion time for the project.
   2. State the critical activities.
   3. State the total float on activity G and the total float on activity K .
      (4)

6.

1. Draw the activity network for the project described in the precedence table, using activity on arc and the minimum number of dummies. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ba9337bf-7a3c-49aa-b395-dd7818cf1d13-10_880_1154_1464_452} \captionsetup{labelformat=empty} \caption{Grid 1}
   \end{figure} A cascade chart for all the activities of the project, except activity \(\mathbf { L }\), is shown on Grid 1. The time taken to complete each activity is given in hours and each activity requires one worker. The project is to be completed in the minimum time using as few workers as possible.
2. State the critical activities of the project.
3. Use the 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.) The duration of activity L is \(x\) hours. Given that the total float of activity L is at most 7 hours,
4. determine the range of possible values for \(\chi\).

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.
   

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.

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.

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 that activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Calculate the early time and the late time for each event, using Diagram 1 in the answer book.
2. On Grid 1 in the answer book, complete the cascade (Gantt) chart for this project.
3. On Grid 2 in the answer book, draw a resource histogram to show the number of workers required each day when each activity begins at its earliest time. The supervisor of the project states that only three workers are required to complete the project in the minimum time.
4. Use Grid 2 to determine if the project can be completed in the minimum time by only three workers. Give reasons for your answer.

5 Answer this question on the insert provided. The diagram shows an activity network for a project. The table lists the durations of the activities (in days). \includegraphics[max width=\textwidth, alt={}, center]{9c9b1a42-8d16-446a-85a1-4c08e5e368be-4_652_867_429_393}

1. Explain why each of the dummy activities is needed.
2. Complete the blank column of the table in the insert to show the immediate predecessors for each activity.
3. Carry out a forward pass to find the early start times for the events. Record these at the eight vertices on the copy of the network on the insert. Also calculate the late start times for the events and record these at the vertices. Find the minimum completion time for the project and list the critical activities.
4. By how much would the duration of activity \(C\) need to increase for \(C\) to become a critical activity? Assume that each activity requires one worker and that each worker is able to do any of the activities. The activities may not be split. The duration of \(C\) is 4 days.
5. Draw a resource histogram, assuming that each activity starts at its earliest possible time. How many workers are needed with this schedule?
6. Describe how, by delaying the start of activity \(E\) (and other activities, to be determined), the project can be completed in the minimum time by just three workers.

5 Answer this question on the insert provided. The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days. \includegraphics[max width=\textwidth, alt={}, center]{95fbb09b-0301-4fc1-b694-838b8d0b64a6-06_956_921_495_612}

1. Complete the table in the insert to show the precedences for the activities.
2. Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Find the minimum project duration and list the critical activities. The number of people required for each activity is shown in the table below. The workers are all equally skilled at all of the activities.

   Activity	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
   Number of workers	4	1	2	2	3	2	3	3	1	2

3. On graph paper, draw a resource histogram for the project with each activity starting at its earliest possible time.
4. Describe how the project can be completed in 21 days using just six workers.

2 Answer this question on the insert provided. The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days. \includegraphics[max width=\textwidth, alt={}, center]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-3_497_1230_493_459}

1. Complete the table in the insert to show the precedences for the activities.
2. Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Show that the minimum project completion time is 28 days and list the critical activities. The resource histogram below shows the number of workers required each day when the activities each begin at their earliest possible start time. Once an activity has been started it runs for its duration without a break. \includegraphics[max width=\textwidth, alt={}, center]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-3_457_1543_1503_299}
3. By considering which activities are happening each day, complete the table in the insert to show the number of workers required for each activity. You are advised to start at day 28 and work back through the days towards day 1 . Only five workers are actually available, but they are all equally skilled at each of the activities. The project can still be completed in 28 days by delaying the start of activity \(E\).
4. Find the minimum possible delay and the maximum possible delay on activity \(E\) in this case.

4 Answer this question on the insert provided. The table shows activities involved in a "perm" in a hair salon, their durations and immediate predecessors. \begin{table}[h] \end{table}

1. Complete the activity-on-arc network in the insert to represent the precedences.
2. Perform a forward pass and a backward pass to find early and late event times. Give the critical activities and the time needed to complete the perm.
3. Give the total float time for the activity \(G\). Activities \(\mathrm { D } , \mathrm { F } , \mathrm { H } , \mathrm { K }\) and N require a stylist.
   Activities \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { E } , \mathrm { G } , \mathrm { J }\) and M are done by a trainee.
   Activities \(I\) and \(L\) require no-one in attendance.
   A stylist and a trainee are to give a perm to a customer.
4. Use the chart in the insert to show a schedule for the activities, assuming that all activities are started as early as possible.
5. Which activity would be better started at its latest start time?

4 A project is represented by the activity network below. The times are in days. \includegraphics[max width=\textwidth, alt={}, center]{22571082-016b-409b-bfeb-e7ebf48ccac7-4_384_935_1110_566}

1. Explain the reason for each dummy activity.
2. Calculate the early and late event times.
3. Identify the critical activities.
4. Calculate the independent float and interfering float on activity A .
5. (a) Draw a cascade chart to represent the project, using the grid in the Printed Answer Booklet.
   (b) Describe the effect on
   The number of workers needed for each activity is shown below.

   Activity	A	B	C	D	E	F	G	H
   Workers	2	1	1	2	1	1	1	1

   The project needs to be completed in at most 3 weeks ( 21 days).
   The duration of activity D is 9 days.
6. Find the minimum number of workers needed. You should explain your reasoning carefully.

5. This question should be answered on the sheet provided in the answer booklet. \begin{figure}[h] \end{figure} Figure 2 shows the activity network used to model a small building project. The activities are represented by the edges and the number in brackets on each edge represents the time, in hours, taken to complete that activity.

1. Calculate the early time and the late time for each event. Write your answers in the boxes on the answer sheet.
   (6 marks)
2. Hence determine the critical activities and the length of the critical path.
   (2 marks)
   Each activity requires one worker. The project is to be completed in the minimum time.
3. Schedule the activities for the minimum number of workers using the time line on the answer sheet. Ensure that you make clear the order in which each worker undertakes his activities.
   (5 marks)

3 [Figures 1 and 2, printed on the insert, are provided for use in this question.] A building 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 and state the minimum time for completion.
5. State the float time for each non-critical activity.
6. Given that each activity starts as early as possible, draw a resource histogram for the project on Figure 2.
7. There are only 3 workers available at any time. Use resource levelling to explain why the project will overrun and state the minimum extra time required.

1 [Figure 1, printed on the insert, is provided for use in this question.]
A building project is to be undertaken. The table shows the activities involved.

1. Complete an 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. State the minimum completion time for the building project and identify the critical paths.