7.05c Total float: calculation and interpretation

4 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 hours). \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-05_680_1125_424_244} (ii) Key: \begin{figure}[h] \end{figure} Minimum completion time = \(\_\_\_\_\) hours Critical activities: \(\_\_\_\_\) (iii) \(\_\_\_\_\) (iv) \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-11_513_1189_543_520} Number of workers required = \(\_\_\_\_\)

\(\_\_\_\_\)

	\(J\)	\(K\)	\(L\)	\(M\)	\(N\)	\(O\)
\(A\)	2	5	2	2	5	2
\(B\)	2	5	2	0	5	5
\(C\)	5	0	5	5	2	2
\(D\)
\(E\)
\(F\)

Answer part (iv) in your answer booklet.

6 Answer parts (i), (ii) and (iii) of this question on the insert provided. The activity network for a project is shown below. The durations are in minutes. The events are numbered 1, 2, 3, etc. for reference. \includegraphics[max width=\textwidth, alt={}, center]{406831f5-74a3-415e-8849-2c381bfe47f4-06_747_1249_482_447}

1. Complete the table in the insert to show the immediate predecessors for each activity.
2. Explain why the dummy activity is needed between event 2 and event 3, and why the dummy activity is needed between event 4 and event 5 .
3. Carry out a forward pass to find the early event times and a backward pass to find the late event times. Record your early event times and late event times in the table in the insert. Write down the minimum project completion time and the critical activities. Suppose that the duration of activity \(K\) changes to \(x\) minutes.
4. Find, in terms of \(x\), expressions for the early event time and the late event time for event 9 .
5. Find the maximum duration of activity \(K\) that will not affect the minimum project completion time found in part (iii). \section*{ADVANCED GCE
   MATHEMATICS} Decision Mathematics 2
   INSERT for Questions 5 and 6 (ii) Dummy activity is needed between event 2 and event 3 because \(\_\_\_\_\) Dummy activity is needed between event 4 and event 5 because \(\_\_\_\_\) (iii)

   Event	1	2	3	4	5	6	7	8	9	10
   Early event time
   Late event time

   Minimum project completion time = \(\_\_\_\_\) minutes Critical activities: \(\_\_\_\_\) \section*{Answer part (iv) and part (v) in your answer booklet.} OCR
   RECOGNISING ACHIEVEMENT

3. A project consists of 11 activities, some of which are dependent on others having been completed. The following precedence table summarises the relevant information.

1. Draw an activity network for the project.
2. Find the critical path and the minimum time in which the project can be completed. Activity \(F\) can be carried out more cheaply if it is allocated more time.
3. Find the maximum time that can be allocated to activity \(F\) without increasing the minimum time in which the project can be completed.

4.
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Construct an activity network to model the work involved in laying the foundations and putting in services for an industrial complex.

1. Execute a forward scan to find the minimum time in which the project can be completed.
2. Execute a backward scan to determine which activities lie on the critical path. The contractor is committed to completing the project in this minimum time and faces a penalty of \(\pounds 50000\) for each day that the project is late. Unfortunately, before any work has begun, flooding means that activity \(E\) will take 3 days longer than the 7 days allocated.
3. Activity \(K\) could be completed in 1 day at an extra cost of \(\pounds 90000\). Explain why doing this is not economical.
   (2 marks)
4. If the time taken to complete any one activity, other than \(E\), could be reduced by 2 days at an extra cost of \(\pounds 80000\), for which activities on their own would this be profitable. Explain your reasoning.
   (3 marks)
   11 marks

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.

1. A project involves six tasks, some of which cannot be started until others have been completed. This is shown in the table below.

1. Draw an activity network for this project.
2. By labelling your network, find the critical path and the minimum duration of the project. An extra condition is now imposed. Task \(A\) may not begin until task \(B\) has been underway for at least 10 minutes.
3. Draw a new network taking into account this restriction.
4. Find a revised value for the minimum duration of the project and state the new critical path.
   

1. A project consists of the activities listed in the table below. For each activity the table shows how long it will take, which other activites must be completed before it can be done and the number of workers needed to complete it.

1. Draw an activity network for the project.
2. Find the critical path and the minimum time in which the project can be completed.
3. Represent all of the activities on a Gantt diagram.
4. By drawing a resource histogram, find out the maximum number of workers required at any one time if each activity is begun as soon as possible.
5. Draw another resource histogram to show how the project can be completed in the minimum time possible using a maximum of 10 workers at any one time. Sheet for answering question 4 \section*{Please hand this sheet in for marking} \includegraphics[max width=\textwidth, alt={}, center]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-6_729_1227_482_338} \includegraphics[max width=\textwidth, alt={}, center]{b8eb80d5-5af5-4a8b-8335-6fae95f3aa73-6_723_1223_1466_338}

5 A group of friends want to prepare a meal. They start preparing the meal at 6:30 pm Activities to prepare the meal are shown in Figure 1 below. \begin{table}[h] \end{table} 5

1. 1. Use Figure 2 shown below to complete Figure 1 above. 5
      1. (ii) Complete Figure 2 showing the earliest start time and latest finishing time for each activity. \begin{figure}[h]
         \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-06_700_1650_1781_194}
         \end{figure} 5
   2. 1. State the activity which must be started first so that the meal is served in the shortest possible time. Fully justify your answer.
         5
   3. (ii) Determine the earliest possible time at which the preparation of the meal can be completed.
      Question 5 continues on the next page 5
   4. The group of friends want to cook spring rolls so that they are served at the same time as the rest of the meal. This requires the additional activities shown in Figure 3. \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3}

      Label	Activity	Duration	Immediate predecessors
      J	Switch on and heat oven		-
      K	Put spring rolls in oven and cook
      \(L\)	Transfer spring rolls to serving dish

      \end{table} It takes 15 seconds to switch on the oven. The oven must be allowed to heat up for 10 minutes before the spring rolls are put in the oven. It takes 15 seconds to put the spring rolls in the oven.
      The spring rolls must cook in the hot oven for 8 minutes.
      It takes 30 seconds to transfer the spring rolls to a serving dish.
      5
   5. 1. Complete Figure 3 above. 5
   6. (ii) Determine the latest time at which the oven can be switched on in order for the spring rolls to be served at the same time as the rest of the meal.
      [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{5a826f8b-4751-4589-ad0a-109fc5c821f2-09_2488_1716_219_153}

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 the precedence table in the answer book.
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. 1. State the minimum project completion time.
   2. List the critical activities.
4. Calculate the maximum number of hours by which activity H 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.
6. Draw a cascade chart for this project on Grid 1 in the answer book.
7. Using the answer to (f), explain why it is not possible to complete the project in the shortest possible time using the number of workers found in (e).

1. Bernie makes garden sheds. He can build up to four sheds each month.

If he builds more than two sheds in any one month, he must hire an additional worker at a cost of \(\pounds 250\) for that month. In any month in which sheds are made, the overhead costs are \(\pounds 35\) for each shed made that month. A maximum of three sheds can be held in storage at the end of any one month, at a cost of \(\pounds 80\) per shed per month. Sheds must be delivered at the end of the month.
The order schedule for sheds is There are no sheds in storage at the beginning of January and Bernie plans to have no sheds left in storage after the May delivery. Use dynamic programming to determine the production schedule that minimises the costs given above. Complete the working in the table provided in the answer book and state the minimum cost.

\includegraphics{figure_2} 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 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 5 to event 6
   2. from event 7 to event 9.
   \hfill [2]
2. Complete Diagram 3 in the answer book to show the early event times and the late event times. \hfill [4]
3. State the minimum project completion time. \hfill [1]
4. Calculate a lower bound for the minimum number of workers required to complete the project in the minimum time. You must show your working. \hfill [2]
5. On Grid 1 in your answer book, draw a cascade (Gantt) chart for this project. \hfill [4]
6. On Grid 2 in your answer book, construct a scheduling diagram to show that this project can be completed with three workers in just one more hour than the minimum project completion time. \hfill [3]

This question should be answered on the sheet provided in the answer booklet. \includegraphics{figure_2} 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.

1. 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]

\includegraphics{figure_4} A trainee at a building company is using critical path analysis to help plan a project. Figure 4 shows the trainee's activity network. Each activity is represented by an arc and the number in brackets on each arc is the duration of the activity, in hours.

1. Find the values of \(x\), \(y\) and \(z\). [3]
2. State the total length of the project and list the critical activities. [2]
3. Calculate the total float time on
   1. activity \(N\),
   2. activity \(H\). [3]

The trainee's activity network is checked by the supervisor who finds a number of errors and omissions in the diagram. The project should be represented by the following precedence table.

1. By adding activities and dummies amend the diagram in the answer book so that it represents the precedence table. (The durations of activities \(A\), \(B\), ..., \(N\) are all correctly given on the diagram in the answer book.) [4]
2. Find the total time needed to complete this project. [2]

\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 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. The numbers in circles are the event numbers. Each activity requires one worker.

1. Explain the purpose of the dotted line from event 6 to event 8. (1)
2. Calculate the early time and late time for each event. Write these in the boxes in the answer book. (4)
3. Calculate the total float on activities \(D\), \(E\) and \(F\). (3)
4. Determine the critical activities. (2)
5. Given that the sum of all the times of the activities is 95 hours, calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working. (2)
6. Given that workers may not share an activity, schedule the activities so that the process is completed in the shortest time using the minimum number of workers. (4)

(Total 16 marks)

\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} 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_4} An engineering 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 activity. Each activity requires one worker. The project is to be completed in the shortest time.

1. Calculate the early time and late time for each event. Write these in boxes in Diagram 1 in the answer book. [4]
2. State the critical activities. [1]
3. Find the total float on activities D and F. You must show your working. [3]
4. On the grid in the answer book, draw a cascade (Gantt) chart for this project. [4]

The chief engineer visits the project on day 15 and day 25 to check the progress of the work. Given that the project is on schedule,

1. which activities must be happening on each of these two days? [3]

\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)

\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

Cassi is managing the building of a house. The table shows the major activities that are involved, their durations and their precedences.

1. Draw an activity-on-arc network to represent this information. [5]
2. Find the early time and the late time for each event. Give the project duration and list the critical activities. [6]
3. Calculate total and independent floats for each non-critical activity. [2]

Cassi's clients wish to take delivery in 42 days. Some durations can be reduced, at extra cost, to achieve this.

• The tiler will finish activity I in 9 days for an extra £250, or in 8 days for an extra £500.
• The bricklayer will cut his total of 7 days on activity B by up to 3 days at an extra cost of £350 per day.
• The electrician could be paid £300 more to cut a day off activity D, or £600 more to cut two days.

1. What is the cheapest way in which Cassi can get the house built in 42 days? [3]

Bill Durrh Ltd undertake a construction project. The activity network for the project is shown below. The duration of each activity is given in weeks. \includegraphics{figure_7}

1. 1. Find the earliest start time and the latest finish time for each activity and show these values on the activity network above. [3 marks]
   2. Identify all of the critical activities. [1 mark]
2. The manager of Bill Durrh Ltd recruits some additional temporary workers in order to reduce the duration of one activity by 2 weeks. The manager wants to reduce the minimum completion time of the project by the largest amount. State, with a reason, which activity the manager should choose. [2 marks]

A project is undertaken by Higton Engineering Ltd. The project is broken down into 11 separate activities \(A\), \(B\), \(\ldots\), \(K\) Figure 3 below shows a completed activity network for the project, along with the earliest start time, duration, latest finish time and the number of workers required for each activity. All times and durations are given in days. \includegraphics{figure_3}

1. Write down the critical path. [1 mark]
2. Using Figure 4 below, draw a resource histogram for the project to show how the project can be completed in the minimum possible time. Assume that each activity is to start as early as possible. [3 marks] \includegraphics{figure_4}
3. Higton Engineering Ltd only has four workers available to work on the project. Find the minimum completion time for the project. Use Figure 5 below in your answer. [3 marks] \includegraphics{figure_5} Minimum completion time _____________________________________

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]

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{figure_1}

1. Construct a cascade chart for the project, showing the float for each non-critical activity. [7]
2. Calculate the float for remodelling the internal layout stating how much of this is independent float and how much is interfering float. [3]

Kirstie needs to supervise the project. This means that she cannot allow more than three activities to happen on any day.

1. 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]