Draw resource histogram

4 Table 4.1 shows some of the activities involved in preparing for a meeting. \begin{table}[h] \end{table}

1. Draw an activity-on-arc network to represent the precedences.
2. Find the early event time and the late event time for each vertex of your network, and list the critical activities.
3. Assuming that each activity requires one person and that each activity starts at its earliest start time, draw a resource histogram.
4. In fact although activity A has duration 1 hour, it actually involves only 0.5 hours work, since 0.5 hours involves waiting for replies. Given this information, and the fact that there is only one person available to do the work, what is the shortest time needed to prepare for the meeting? Fig. 4.2 shows an activity network for the tasks which have to be completed after the meeting. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c429bfed-9241-409a-9cd5-9553bf16c9df-5_533_844_1688_294} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} P: Clean room
   Q: Prepare draft minutes
   R: Allocate action tasks
   S: Circulate draft minutes
   T: Approve task allocations
   U: Obtain budgets for tasks
   V: Post minutes
   W: Pay refreshments bill
5. Draw a precedence table for these activities.

1 [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 float for activity \(G\).
3. Find the critical paths and state the minimum time for completion.
4. 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	2	2	3	2	3	2	1	3	5	2

   Given that each activity starts as late 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 The table shows the activities involved in a project, their durations, precedences and the number of workers needed for each activity. The graph gives a schedule with each activity starting at its earliest possible time. \includegraphics[max width=\textwidth, alt={}, center]{3d8f3593-7923-40f7-b5c0-ac5c3bc21292-03_473_1591_964_278}

1. Using the graph, find the minimum completion time for the project and state which activities are critical.
2. Draw a resource histogram, using graph paper, assuming that there are no delays and that every activity starts at its earliest possible time. Assume that only four workers are available but that they are equally skilled at all tasks. Assume also that once an activity has been started it continues until it is finished.
3. The critical activities are to start at their earliest possible times. List the start times for the non-critical activities for completion of the project in the minimum possible time. What is this minimum completion time?

3 The table lists the duration (in hours), immediate predecessors and number of workers required for each activity in a project.

1. Draw an activity network, using activity on arc, to represent the project. 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 and late event times clearly at the vertices of your network. State the minimum project completion time and list the critical activities.
3. Using graph paper, draw a resource histogram to show the number of workers required each hour. Each activity begins at its earliest possible start time. Once an activity has started it runs for its duration without a break. A delay from the supplier means that the start of activity \(F\) is delayed.
4. By how much could the start of activity \(F\) be delayed without affecting the minimum project completion time? Suppose that only six workers are available after the first four hours of the project.
5. Explain carefully what delay this will cause on the completion of the project. What is the maximum possible delay on the start of activity \(F\), compared with its earliest possible start time in part (iii), without affecting the new minimum project completion time? Justify your answer.

3 The table lists the duration, immediate predecessors and number of workers required for each activity in a project.

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 clearly 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. Show how it is possible for the project to be completed in the minimum project completion time when only six workers are available.

2. \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, in hours, of the corresponding activity is shown in brackets.

1. 1. Complete Diagram 1 in the answer book to show the early event times and the late event times.
   2. State the minimum completion time of the project. The table below lists the number of workers required for each activity in the project.

      Activity	Number of workers
      A	2
      B	1
      C	2
      D	2
      E	3
      F	2
      G	1
      H	3

      Each worker is able to do any of the activities. Once an activity is started it must be completed without interruption. It is given that each activity begins at its earliest possible start time.
2. 1. On Grid 1 in the answer book, draw a resource histogram to show the number of workers required at each time.
   2. Hence state the time interval(s) when six workers are required.

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 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.

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 [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.

A company undertakes a project which consists of 12 activities, \(A\), \(B\), \(C\), \(\ldots\), \(L\) Each activity requires one worker. The resource histogram below shows the duration of each activity. Each activity begins at its earliest start time. The path \(ADGJL\) is critical. \includegraphics{figure_1} The company only has two workers available to work on the project. Which of the following could be a correctly levelled histogram? Tick \((\checkmark)\) one box. [1 mark] \includegraphics{figure_2} \includegraphics{figure_3} \includegraphics{figure_4} \includegraphics{figure_5}