Complete precedence table from network

4 Colin is setting off for a day's sailing. The table and the activity network show the major activities that are involved, their durations and their precedences. \includegraphics[max width=\textwidth, alt={}, center]{21ab732d-435e-4f0b-bc88-21ddc2a398c9-3_480_912_555_925}

1. Complete the table in your answer book showing the immediate predecessors for each activity.
2. Find the early time and the late time for each event. Give the project duration and list the critical activities. When he sails on his own Colin can only do one thing at a time with the exception of activity G, motoring out of the harbour.
3. Use the activity network to determine which activities Colin can perform whilst motoring out of the harbour.
4. Find the minimum time to complete the activities when Colin sails on his own, and give a schedule for him to achieve this.
5. Find the minimum time to complete the activities when Colin sails with one other crew member, and give a schedule for them to achieve this.

2 A project is represented by the activity network and cascade chart below. The table showing the number of workers needed for each activity is incomplete. Each activity needs at least 1 worker. \includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_202_565_1605_201} \includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_328_560_1548_820}

1. Complete the table in the Printed Answer Booklet to show the immediate predecessors for each activity.
2. Calculate the latest start time for each non-critical activity. The minimum number of workers needed is 5 .
3. What type of problem (existence, construction, enumeration or optimisation) is the allocation of a number of workers to the activities? There are 8 workers available who can do activities A and B .
4. 1. Find the number of ways that the workers for activity A can be chosen.
   2. When the workers have been chosen for activity A , find the total number of ways of choosing the workers for activity B for all the different possible values of x , where \(\mathrm { x } \geqslant 1\).

4. \begin{figure}[h] \end{figure} \section*{Question 4 continued} \begin{figure}[h] \end{figure}

6. \begin{figure}[h] \end{figure} [The sum of the durations of all the activities is 142 days]
A project is modelled by the activity network shown in Figure 6. 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 the precedence table in the answer book.
2. Complete Diagram 1 in the answer book to show the early event times and late event times.
3. Calculate the total float for activity D. You must make the numbers you use in your calculation clear.
4. Calculate a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working. Diagram 2 in the answer book shows a partly completed scheduling diagram for this project.
5. Complete the scheduling diagram, using the minimum number of workers, so that the project is completed in the minimum time.

5. \begin{figure}[h] \end{figure} A project is modelled by the activity network shown in Figure 6. 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 late event times.
3. State the minimum project completion time and list the critical activities.
4. Calculate the maximum number of hours by which activity E 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. The project is to be completed in the minimum time using as few workers as possible.
6. Schedule the activities using Grid 1 in the answer book.
   (3) Before the project begins it becomes apparent that activity E will require an additional 6 hours to complete. The project is still to be completed in the shortest possible time and the time to complete all other activities is unchanged.
7. State the new minimum project completion time and list the new critical activities.

6. \begin{figure}[h] \end{figure} Figure 5 is the activity network relating to a building project. The number in brackets on each arc gives the time taken, in days, to complete the activity.

1. Explain the significance of the dotted line from event (2) to event (3).
2. Complete the precedence table in the answer booklet.
3. Calculate the early time and the late time for each event, showing them on the diagram in the answer booklet.
4. Determine the critical activities and the length of the critical path.
5. On the grid in the answer booklet, draw a cascade (Gantt) chart for the project.

7. \begin{figure}[h] \end{figure} The network in Figure 7 shows the activities that need to be undertaken to complete a maintenance project. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. The numbers in circles are the events. Each activity requires one worker. The project is to be completed in the shortest possible time.

1. Complete the precedence table for this network in the answer book.
2. Explain why each of the following is necessary.
   1. The dummy from event 6 to event 7 .
   2. The dummy from event 8 to event 9 .
3. Complete Diagram 2 in the answer book to show the early and the late event times.
4. State the critical activities.
5. Calculate the total float on activity K . You must make the numbers used in your calculation clear.
6. Calculate a lower bound for the number of workers needed to complete the project in the minimum time.

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.

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).