Calculate early and late times

5 The table shows some of the activities involved in building a block of flats. The table gives their durations and their immediate predecessors.

1. Draw an activity on arc network for these activities.
2. Mark on your diagram the early and late times for each event. Give the minimum completion time and the critical activities. Each of the tasks E, F, H and J can be speeded up at extra cost. The maximum number of weeks by which each task can be shortened, and the extra cost for each week that is saved, are shown in the table below.

   Task	E	F	H	J
   Maximum number of weeks by which task may be shortened	3	3	1	3
   Cost per week of shortening task (in thousands of pounds)	30	15	6	20

3. Find the new shortest time for the flats to be completed.
4. List the activities which will need to be speeded up to achieve the shortest time found in part (iii), and the times by which each must be shortened.
5. Find the total extra cost needed to achieve the new shortest time.

5 The tasks involved in turning around an "AirGB" aircraft for its return flight are listed in the table. The table gives the durations of the tasks and their immediate predecessors.

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. Because of delays on the outbound flight the aircraft has to be turned around within 50 minutes, so as not to lose its air traffic slot for the return journey. There are four tasks on which time can be saved. These, together with associated costs, are listed below.

   Task	A	B	D	E
   New time (mins)	20	20	15	15
   Extra cost	250	50	50	100

3. List the activities which need to be speeded up in order to turn the aircraft around within 50 minutes at minimum extra cost. Give the extra cost and the new set of critical activities.

1 The table shows the activities involved in a project, 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 critical activities.

4 The table shows the tasks involved in preparing breakfast, and their durations.

1. Show the immediate predecessors for each of these tasks.
2. Draw an activity on arc network modelling your precedences.
3. Perform a forward pass and a backward pass to find the early time and the late time for each event.
4. Give the critical activities, the project duration, and the total float for each activity.
5. Given that only one person is available to do these tasks, and noting that tasks B, D and F do not require that person's attention, produce a cascade chart showing how breakfast can be prepared in the least possible time.

6 The table shows the tasks involved in making a salad, their durations and their precedences.

1. Draw an activity on arc network for these activities.
2. Mark on your diagram the early and late times for each event. Give the minimum completion time and the critical activities.
3. Given that each task can only be done by one person, how many people are needed to prepare the salad in the minimum time? What is the minimum time required to prepare the salad if only one person is available?
4. Show how two people can prepare the salad as quickly as possible.

5 The activity network and table together show the tasks involved in constructing a house extension, their durations and precedences. \includegraphics[max width=\textwidth, alt={}, center]{2e03f6fb-69db-438a-a79e-3e04fab0d08a-5_231_985_338_539}

1. Show the immediate predecessors for each activity.
2. Perform a forward pass and a backward pass to find the early time and the late time for each event.
3. Give the critical activities, the project duration, and the total float for each activity.
4. The activity network includes one dummy activity. Explain why this dummy activity is needed. Whilst the foundations are being dug the customer negotiates the installation of a decorative corbel. This will take one day. It must be done after the walls have been built, and before the roof is constructed. The windows and doors cannot be installed until it is completed. It will not have any effect on the construction of the floor.
5. Redraw the activity network incorporating this extra activity.
6. Find the revised critical activities and the revised project duration.

6 The table shows the tasks involved in making a batch of buns, the time in minutes required for each task, 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. Preparing the batch for baking consists of tasks A to G ; each of these tasks can only be done by one person. Baking, task H, requires no people.
3. How many people are required to prepare the batch for baking in the minimum time?
4. What is the minimum time required to prepare the batch for baking if only one person is available? Jim is preparing and baking three batches of buns. He has one oven available for baking. For the rest of the question you should consider 'preparing the batch for baking' as one activity.
5. Assuming that the oven can bake only one batch at a time, draw an activity on arc diagram for this situation and give the minimum time in which the three batches of buns can be prepared and baked.
6. Assuming that the oven is big enough to bake all three batches of buns at the same time, give the minimum time in which the three batches of buns can be prepared and baked.

5 The table lists activities which are involved in framing a picture. The table also lists their durations and their immediate predecessors. Except for activities C and H, each activity is undertaken by one person. Activities C and H require no people.

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. A picture is to be framed as quickly as possible. Two people are available to do the job.
3. Produce a schedule to show how two people can complete the picture framing in the minimum time. To reduce the completion time an instant glue is to be used. This will reduce the time for activities C and H to 0 minutes.
4. Produce a schedule for two people to complete the framing in the new minimum completion time, and give that time.

3. This question should be answered on the sheet provided. \begin{figure}[h] \end{figure} Figure 2 shows an activity network. The nodes represent events and the arcs represent the activities. The number in each bracket gives the time, in days, needed to complete the activity.

1. Calculate the early and late times for each event using appropriate forward and backward scanning.
   (5 marks)
2. Hence, determine the activities which lie on the critical path.
3. State the minimum number of days needed to complete the entire project.

7. This question should be answered on the sheet provided. \begin{figure}[h] \end{figure} The activity network in Figure 5 models the work involved in laying the foundations and putting in services for an industrial complex. The activities are represented by the arcs and the numbers in brackets give the time, in days, to complete each activity. Activity \(C\) is a dummy.

1. Execute a forward scan to calculate the early times and a backward scan to calculate the late times, for each event.
2. Determine which activities lie on the critical path and list them in order.
3. State the minimum length of time needed to complete the project. 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 \(F\) will take 3 days longer than the 7 days allocated.
4. Activity \(N\) could be completed in 1 day at an extra cost of \(\pounds 90000\). Explain why doing this is not economical.
   (3 marks)
5. If the time taken to complete any one activity, other than \(F\), 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) END \section*{Please hand this sheet in for marking}

   	A	B	C	D	E	\(F\)
   A	-	130	190	155	140	125
   B	130	-	215	200	190	170
   C	190	215	-	110	180	100
   D	155	200	110	-	70	45
   E	140	190	180	70	-	75
   \(F\)	125	170	100	45	75	-

   \section*{Please hand this sheet in for marking}

   1. \(n\)	\(x _ { n }\)	\(a\)	Any more data?	\(x _ { n + 1 }\)	\(b\)	\(( b - a ) > 0\) ?	\(a\)
      1	8	8	Yes	2	2	No	2
      2	-	-

      Final output
   2. \(\_\_\_\_\) Sheet for answering question 3
      NAME \section*{Please hand this sheet in for marking}
   4. 1. \includegraphics[max width=\textwidth, alt={}, center]{acc09687-11a3-4392-af17-3d4d331d5ab4-11_716_1218_502_331}
      2. \includegraphics[max width=\textwidth, alt={}, center]{acc09687-11a3-4392-af17-3d4d331d5ab4-11_709_1214_1498_333} Maximum flow =
   5. 1. \(\_\_\_\_\)
      2. \(\_\_\_\_\) \section*{Please hand this sheet in for marking}
   6. \includegraphics[max width=\textwidth, alt={}, center]{acc09687-11a3-4392-af17-3d4d331d5ab4-12_764_1612_402_255}
   7. \(\_\_\_\_\)
   8. \(\_\_\_\_\)
   9. \(\_\_\_\_\)
   10. \(\_\_\_\_\)

1
A group of workers is involved in a decorating project. The table shows the activities involved. Each worker can perform any of the given activities. The activity network for the project is given in Figure 1 below.

1. Find the earliest start time and the latest finish time for each activity, inserting their values on Figure 1.
2. Hence find:
   1. the critical path;
   2. the float time for activity \(D\).
      1. \begin{figure}[h]
         \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-02_647_1657_1640_180}
         \end{figure}
      2. 1. The critical path is \(\_\_\_\_\)
         2. The float time for activity \(D\) is \(\_\_\_\_\)
   3. 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 below, indicating clearly which activities are taking place at any given time.
   4. It is later discovered that there are only 8 workers available at any time. Use resource levelling to construct a new resource histogram on Figure 3 below, showing how the project can be completed with the minimum extra time. State the minimum extra time required.
   5. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_586_1708_922_150}
      \end{figure}
   6. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{172c5c92-4254-4593-b741-1caa83a1e833-03_496_1705_1672_153}
      \end{figure} The minimum extra time required is \(\_\_\_\_\)

4 Jamil is building a summerhouse in his garden. The activities involved, the duration, immediate predecessors and number of workers required for each activity are listed in the table.

1. Represent the project by an activity network, using activity on arc. You should make your diagram quite large so that there is room for working.
2. Carry out a forward pass and a backward pass through the activity network, showing the early event times and late event times at the vertices of your network. State the minimum project completion time and list the critical activities.
3. Draw a resource histogram to show the number of workers required each hour when each activity begins at its earliest possible start time.
4. Describe how it is possible for the project to be completed in the minimum project completion time when only four workers are available.
5. Describe how two workers can complete the project as quickly as possible. Find the minimum time in which two workers can complete the project.

6 Tariq wants to advertise his gardening services. The activities involved, their durations (in hours) and immediate predecessors are listed in the table.

1. Draw an activity network, using activity on arc, to represent the project.
2. Carry out a forward pass and a backward pass through the activity network, showing the early event time and the late event time at each vertex of your network. State the minimum project completion time and list the critical activities. Tariq does not have time to complete all the activities on his own, so he gets some help from his friend Sally.
   Sally can help Tariq with any of the activities apart from \(C , H\) and \(J\). If Tariq and Sally share an activity, the time it takes is reduced by 1 hour. Sally can also do any of \(F , G\) and \(I\) on her own.
3. Describe how Tariq and Sally should share the work so that activity \(D\) can start 5 hours after the start of the project.
4. Show that, if Sally does as much of the work as she can, she will be busy for 18 hours. In this case, for how many hours will Tariq be busy?
5. Explain why, if Sally is busy for 18 hours, she will not be able to finish until more than 18 hours from the start. How soon after the start can Sally finish when she is busy for 18 hours?
6. Describe how Tariq and Sally can complete the project together in 18 hours or less.

2 The diagram below shows an activity network for a project. The figures in brackets show the durations of the activities, in hours. \includegraphics[max width=\textwidth, alt={}, center]{b3a3d522-2ec9-46ec-bd99-a8c698e3d1c0-3_371_1429_367_319}

1. Complete the table in your answer book to show the immediate predecessors for each activity.
2. Carry out a forward pass and a backward pass on the copy of the network in your answer book, showing the early event times and late event times. State the minimum project completion time, in hours, and list the critical activities.
3. How much longer could be spent on activity \(F\) without it affecting the overall completion time? Suppose that each activity requires one worker. Once an activity has been started it must continue until it is finished. Activities cannot be shared between workers.
4. (a) State how many workers are needed at the busiest point in the project if each activity starts at its earliest possible start time.
   (b) Suppose that there are fewer workers available than given in your answer to part (iv)(a). Explain why the project cannot now be completed in the minimum project completion time from part (ii). Suppose that activity \(C\) is delayed so that it starts 2 hours after its earliest possible start time, but there is no restriction on the number of workers available.
5. Describe what effect this will have on the critical activities and the minimum project completion time.

6 Sheona and Tim are making a short film. The activities involved, their durations and immediate predecessors are given in the table below.

1. By using an activity network, find:
   • the minimum project completion time
   • the critical activities
   • the float on each non-critical activity.
   • Give two reasons why the filming may take longer than the minimum project completion time.
   Each activity will involve either Sheona or Tim or both.
   • The activities that Sheona will do are ticked in the S column.
   • The activities that Tim will do are ticked in the T column.
   • They will do the planning and finishing together.
   • Some of the activities involve other people as well.
   An additional restriction is that Sheona and Tim can each only do one activity at a time.
2. Explain why the minimum project completion is longer than in part (i) when this additional restriction is taken into account.
3. The project must be completed in 14 days. Find:
   1. the longest break that either Sheona or Tim can take,
   2. the longest break that Sheona and Tim can take together,
   3. the float on each activity.

5 Hiro has been asked to organise a quiz.
The table below shows the activities involved, together with the immediate predecessors and the duration of each activity in hours.

1. A sketch of the activity network is provided in the Printed Answer Booklet. Apply a forward pass to determine the minimum project completion time.
2. Use a backward pass to determine the critical activities. You can show your working on the activity network from part (a).
3. Give the total float for each non-critical activity. Hiro decides that there should be a final check of the answers which he will include as activity \(L\). Activity L needs to be done after checking the answers for rounds 1 and 2 and also after getting permission to use the pictures and music but before producing the answer sheets.
4. 1. Complete the activity network provided in the Printed Answer Booklet to show the new precedences, with the final check of the answers included as activity \(L\).
   2. As a result of including L , the minimum project completion time found in part (a) increases by 2.5 hours. Determine the duration of L .

2 The table below shows the activities involved in a project together with the immediate predecessors and the duration of each activity.

1. Model the project using an activity network.
2. Determine the minimum project completion time.
3. Calculate the total float for each non-critical activity.

6. The table above shows the activities required for the completion of a building project. For each activity the table shows the duration, in days, and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time. \begin{figure}[h] \end{figure} Figure 2 shows a partially completed activity network used to model the project. The activities are represented by the arcs and the numbers in brackets on the arcs are the times taken, in days, to complete each activity.

1. Complete the network in Diagram 1 in the answer book by adding activities \(\mathrm { G } , \mathrm { H }\) and I and the minimum number of dummies.
2. Add the early event times and the late event times to Diagram 1 in the answer book.
3. State the critical activities.
4. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. You must show your working.
5. Schedule the activities on Grid 1 in the answer book, using the minimum number of workers, so that the project is completed in the minimum time.

4. \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 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 late event times.
2. Determine the critical activities and the length of the critical path.
3. Calculate the total float for activity D. You must make the numbers you use in your calculation clear.
4. Draw a cascade (Gantt) chart for this project on Grid 1 in the answer book.
5. 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.

3. The table above shows the activities required for the completion of a building project. For each activity, the table shows the time taken in days to complete the activity and the immediately preceding activities. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Draw the activity network described in the table, using activity on arc. Your activity network must contain the minimum number of dummies only.
2. 1. Show that the project can be completed in 21 days, showing your working.
   2. Identify the critical activities.

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 = \(\_\_\_\_\) (ii) \(\_\_\_\_\) (iii) Answer part (iv) in your answer booklet.

5 A restoration project is divided into a number of activities. The duration and predecessor(s) of each activity are shown in the table below. 5

1. On the opposite page, construct an activity network for the project and fill in the earliest start time and latest finish time for each activity.
   [0pt] [4 marks] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-09_533_289_2124_1548} \captionsetup{labelformat=empty} \caption{Turn over -}
   \end{figure} 5
2. Due to a change of materials during the project, the duration of activity \(C\) is extended by 3 weeks. Determine the new minimum completion time of the project. \includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-11_2488_1716_219_153}

4 A community project consists of 10 activities \(A , B , \ldots , J\), as shown in the activity network below. \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-06_899_1083_367_466} The duration of each activity is shown in days. 4

1. 1. Complete the activity network in the diagram above, showing the earliest start time and latest finish time for each activity. 4
      1. (ii) State the minimum completion time for the community project.
         4
   2. Write down the critical activities of the network.
      4
   3. Glyn claims that a project's activity network can be used to determine its minimum completion time by adding together the durations of all the project's critical activities. 4
   4. 1. Show that Glyn's claim is false for this community project's activity network.
         4
   5. (ii) Describe a situation in which Glyn's claim would be true.

8 A motor racing team is undertaking a project to build next season's racing car. The project is broken down into 12 separate activities \(A , B , \ldots , L\), as shown in the precedence table below. Each activity requires one member of the racing team. 8

1. 1. Complete the activity network for the project on Figure 3. 8
      1. (ii) Find the earliest start time and the latest finish time for each activity and show these values on Figure 3. 8
   2. Write down the critical path(s).
      \section*{Figure 3} Figure 3 \includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-15_469_1360_356_338} 8
   3. 1. Using Figure 4, draw a resource histogram for the project to show how the project can be completed in the shortest possible time. Assume that each activity is to start as early as possible. \begin{figure}[h]
         \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_698_1534_541_251}
         \end{figure} 8
   4. (ii) The racing team's boss assigns two members of the racing team to work on the project. Explain the effect this has on the minimum completion time for the project.
      You may use Figure 5 in your answer. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{22f11ce2-8d07-4f51-9326-b578d1e454f9-16_704_1539_1695_248}
      \end{figure}

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]