Explain dummy activities

5.

1. Draw the activity network described in the precedence table, using activity on arc. Your activity network must contain only the minimum number of dummies.
   (5)
2. Explain why, in general, dummies may be required in an activity network.

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 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
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. State the minimum project completion time.
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.
5. On Grid 1 in your answer book, draw a cascade (Gantt) chart for this project.
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.
   (3)

6. (a) Draw the activity network described in this precedence table, using activity on arc and dummies only where necessary. (b) Explain the possible reasons dummies may be needed in activity networks.

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 required, in hours, to complete the corresponding activity. The numbers in circles are the event numbers.

1. Explain the significance of the dummy activity
   1. from event 2 to event 3
   2. from event 6 to event 7
2. Complete Diagram 1 in the answer book to show the early event times and the late event times.
3. State the minimum project completion time and list the critical activities. The duration of activity H changes to \(x\) hours.
4. Find, in terms of \(x\) where necessary,
   1. the possible new early event time for event 7
   2. the possible new late event time for event 7 Given that the duration of activity H is such that the minimum project completion time is four hours greater than the time found in (c),
5. determine the value of \(x\).

5. (a) Draw the activity network described in this precedence table, using activity on arc and exactly two dummies.
(4) (b) Explain why each of the two dummies is necessary.
(3)

3. (a) Draw the activity network described in this precedence table, using activity on arc and exactly two dummies.
(5) (b) Explain why each of the two dummies is necessary.

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.

1. Draw the activity network described in the precedence table, using activity on arc and exactly two dummies.
   (5)
2. Explain why each of the two dummies is necessary.
   (2)

6. (i) Draw the activity network described in the precedence table below, using activity on arc and the minimum number of dummies. (ii) Explain why each of your dummies is necessary.

2. (a) Draw the activity network described in the precedence table below, using activity on arc and exactly two dummies. (b) Explain why each of the two dummies is necessary.

5. The precedence table shows the activities involved in planning an opening ceremony. An activity on arc network is to be drawn to model this planning process.

1. Draw the activity network using exactly two dummies.
2. Explain why each of the two dummies is necessary.

2. The precedence table for activities involved in manufacturing a toy is shown below.

1. Draw an activity network, using activity on arc, and exactly one dummy, to model the manufacturing process.
2. Explain briefly why it is necessary to use a dummy in this case.

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 required, in hours, to complete the corresponding activity. The numbers in circles are the event numbers. Each activity requires one worker, and the project is to be completed in the shortest possible time.

1. Explain the significance of the dummy activity from event 3 to event 4
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 a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.
5. Draw a Gantt chart for this project on Grid 1 in the answer book.

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

1. Explain why each of the dummy activities is required.
2. Complete the table in the answer book to show the immediately preceding activities for each activity.
3. 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 for the project.
   3. State the critical activities. Each activity requires one worker. Each worker is able to do any of the activities. Once an activity is started it must be completed without interruption.
4. On Grid 1 in the answer book, draw a resource histogram to show the number of workers required at each time when each activity begins at its earliest possible start time.
5. Determine whether or not the project can be completed in the minimum possible time using fewer workers than the number indicated by the resource histogram in (d). You must justify your answer with reference to the resource histogram and the completed Diagram 1.

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.

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.

\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]

\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)

The precedence table shows the activities involved in a project.

1. Draw the activity network for this project, using activity on arc and using two dummies. [4]
2. Explain why each of the two dummies is necessary. [3]

(Total 7 marks)