6. \begin{figure}[h] \end{figure} 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.
3. Calculate the total float on activities \(D , E\) and \(F\).
4. Determine the critical activities.
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.
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. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 6} \includegraphics[alt={},max width=\textwidth]{6a0cf9b1-e6a0-4c38-a2d9-deb9c0c76015-08_1502_1655_285_196}
   \end{figure} The captain of the Malde Mare takes passengers on trips across the lake in her boat.
   The number of children is represented by \(x\) and the number of adults by \(y\).
   Two of the constraints limiting the number of people she can take on each trip are $$x < 10$$ and $$2 \leqslant y \leqslant 10$$ These are shown on the graph in Figure 6, where the rejected regions are shaded out.
7. Explain why the line \(x = 10\) is shown as a dotted line.
   (1)
8. Use the constraints to write down statements that describe the number of children and the number of adults that can be taken on each trip.
   (3) For each trip she charges \(\pounds 2\) per child and \(\pounds 3\) per adult. She must take at least \(\pounds 24\) per trip to cover costs. The number of children must not exceed twice the number of adults.
9. Use this information to write down two inequalities.
   (2)
10. Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R .
11. Use your graph to determine how many children and adults would be on the trip if the captain takes:
    1. the minimum number of passengers,
    2. the maximum number of passengers. \begin{figure}[h]
       \captionsetup{labelformat=empty} \caption{Figure 7} \includegraphics[alt={},max width=\textwidth]{6a0cf9b1-e6a0-4c38-a2d9-deb9c0c76015-10_1018_1600_285_221}
       \end{figure} In solving a network flow problem using the labelling procedure, the diagram in Figure 7 was created.
       The arrow on each arc indicates the direction of the flow along that arc.
       The arrows above and below each arc show the direction and value of the flow as indicated by the labelling procedure.
12. Add a supersource S , a supersink T and appropriate arcs to Diagram 2 in the answer book, and complete the labelling procedure for these arcs.
13. Write down the value of the initial flow shown in Figure 7.
14. Use Diagram 2, the initial flow and the labelling procedure to find the maximal flow of 124 through this network. List each flow-augmenting path you use, together with its flow.
15. Show your flow on Diagram 3 and state its value.
16. Prove that your flow is maximal.