7. \begin{figure}[h] \end{figure} \section*{[The sum of the duration of all activities is 172 days]} 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 days, to complete the activity. Each activity requires 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. Calculate the total float for activity M. You must make the numbers you use in your calculation clear.
3. For each of the situations below, explain the effect that the delay would have on the project completion date.
   1. A 2 day delay on the early start of activity P.
   2. A 2 day delay on the early start of activity Q .
4. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time. Diagram 2 in the answer book shows a partly completed cascade chart for this project.
5. Complete the cascade chart.
6. Use your cascade chart to determine a second lower bound on the number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities.
7. State which of the two lower bounds found in (d) and (f) is better. Give a reason for your answer.
   (Total 17 marks)