7.05c Total float: calculation and interpretation

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.

2. \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 hours, to complete the corresponding activity.

1. Complete Diagram 1 in the answer book to show the early event times and the late event times. Each activity requires one worker and the project must be completed in the shortest possible time using as few workers as possible.
2. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. You must show your working.
3. 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.

6. The precedence table below shows the 12 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) Each of the activities shown in the precedence table requires one worker. The project is to be completed in the minimum possible time. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{7f7546eb-0c1a-40da-bdf0-31e0574a9867-11_303_1547_296_260} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a schedule for the project using three workers.
2. 1. State the critical path for the network.
   2. State the minimum completion time for the project.
   3. Calculate the total float on activity B.
   4. Calculate the total float on activity G. Immediately after the start of the project, it is found that the duration of activity I, as shown in Figure 3, is incorrect. In fact, activity I will take 8 hours.
      The durations of all the other activities remain as shown in Figure 3.
3. Determine whether the project can still be completed in the minimum completion time using only three workers when the duration of activity I is 8 hours. Your answer must make specific reference to workers, times and activities.

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 Deva is having some work done on his house. The table shows the activities involved, their durations and their immediate predecessors.

1. Model this information as an activity network.
2. Find the minimum time in which the work can be completed.
3. Describe the effect on the minimum project completion time of each of the following happening individually.
   1. The duration of activity A is increased to 3.5 hours.
   2. The duration of activity D is increased to 4 hours.
   3. The duration of activity F is decreased to 2 hours. The decorators working on activity H cannot work for 3 hours without a break.
   4. How would you adapt your model to incorporate the break?

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.
2. Hence determine the critical activities and the length of the critical path. 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)

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.

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
A group of workers is involved in a building project. The table shows the activities involved. Each worker can perform any of the given activities.

1. Complete the activity network for the project on Figure 1.
2. Find the earliest start time and the latest finish time for each activity, inserting their values on Figure 1.
3. Find the critical path and state the minimum time for completion.
4. The number of workers required for each activity is given in the table above. Given that each activity starts as early as possible and assuming there is no limit to the number of workers available, draw a resource histogram for the project on Figure 2, indicating clearly which activities take place at any given time.
5. It is later discovered that there are only 7 workers available at any time. Use resource levelling to explain why the project will overrun and indicate which activities need to be delayed so that the project can be completed with the minimum extra time. State the minimum extra time required.

2 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
Figure 1 shows the activity network and the duration in days of each activity for a particular project.

1. On Figure 1:
   1. find the earliest start time for each activity;
   2. find the latest finish time for each activity.
2. Find the critical paths and state the minimum time for completion.
3. 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	3	3	4	2	3	4	1	2	2	5

   1. 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 2, indicating clearly which activities take place at any given time.
   2. It is later discovered that there are only 6 workers available at any time. Explain why the project will overrun, and use resource levelling to indicate which activities need to be delayed so that the project can be completed with the minimum extra time. State the minimum extra time required.

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.

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
The following diagram shows an activity diagram for a building project. The time needed for each activity is given in days. \includegraphics[max width=\textwidth, alt={}, center]{0c40b693-72d3-459c-bbb7-b9584a108b8e-02_698_1321_767_354}

1. Complete the precedence table for the project on Figure 1.
2. Find the earliest start times and latest finish times for each activity and insert their values on Figure 2.
3. Find the critical path and state the minimum time for completion of the project.
4. Find the activity with the greatest float time and state the value of its float time.

1 [Figures 1 and 2, printed on the insert, are provided for use in this question.]
The following diagram shows an activity network for a project. The time needed for each activity is given in days. \includegraphics[max width=\textwidth, alt={}, center]{f98d4434-458a-4118-92ed-309510d7975a-02_940_1698_721_164}

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.
3. On Figure 2, draw a cascade diagram (Gantt chart) for the project, assuming each activity starts as early as possible.
4. Activity \(C\) takes 5 days longer than first expected. Determine the effect on the earliest start time for other activities and the minimum completion time for the project.
   (2 marks)

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

1. Complete an activity network for the project on Figure 1.
2. On Figure 1, indicate:
   1. the earliest start time for each activity;
   2. the latest finish time for each activity.
3. State the minimum completion time for the decorating project and identify the critical path.
4. Activity \(F\) takes 4 days longer than first expected.
   1. Determine the new earliest start time for activities \(H\) and \(I\).
   2. State the minimum delay in completing the project.

1
Figure 1 below shows an activity diagram for a construction project. The time needed for 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. On Figure 2 opposite, draw a cascade diagram (Gantt chart) for the project, assuming that each activity starts as early as possible.
4. Activity \(J\) takes longer than expected so that its duration is \(x\) days, where \(x \geqslant 3\). Given that the minimum time for completion of the project is unchanged, find a further inequality relating to the maximum value of \(x\).
   1. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-02_910_1355_1414_411}
      \end{figure}
   2. Critical paths are \(\_\_\_\_\) Minimum completion time is \(\_\_\_\_\) days. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d0902228-7041-4449-9ccb-770352ce6bef-03_940_1160_390_520}
      \end{figure}
   3. \(\_\_\_\_\)

5 Dave plans to renovate three houses, \(A , B\) and \(C\), at the rate of one per year. The order in which they are renovated is a matter of choice, but some costs vary over the three years. The expected costs, in thousands of pounds, are given in the table below. (b) Optimum order \(\_\_\_\_\)

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.

1 Figure 2, on the page opposite, shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

1. On Figure 1 below, complete the precedence table.
2. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 2.
3. List the critical paths.
4. Find the float time of activity \(E\).
5. Using Figure 3 opposite, draw a Gantt diagram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
6. Given that there is only one worker available for the project, find the minimum completion time for the project.
7. Given that there are two workers available for the project, find the minimum completion time for the project. Show a suitable allocation of tasks to the two workers.
   [0pt] [2 marks] \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1}

   Activity	Immediate predecessor(s)
   A
   B
   C
   D
   E
   \(F\)
   G
   \(H\)
   I
   J

   \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_1071_1561_376_278}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_801_1301_1644_420}
   \end{figure}

1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.

1. Find the earliest start time and the latest finish time for each activity and insert these values on Figure 1.
2. 1. Find the critical path.
   2. Find the float time of activity \(F\).
3. Using Figure 2 on page 3, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
4. 1. Given that there are two workers available for the project, find the minimum completion time for the project.
   2. Write down an allocation of tasks to the two workers that corresponds to your answer in part (d)(i). \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-02_687_1655_1941_189}
      \end{figure} \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1115_1575_434_283}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1024_1593_1683_267}