The Critical Path — practice questions

For a network with $1 \xrightarrow{P,4} 2$, $2 \xrightarrow{Q,3} 3$, $2 \xrightarrow{R,6} 4$, $3 \xrightarrow{S,2} 4$, $4 \xrightarrow{T,5} 5$, complete both passes and state the project duration.

A community event has activities: A book venue (2, —), B hire equipment (4, A), C recruit volunteers (7, A), D publicity (5, A), E set up (3, B and C), F run event (1, D and E). Complete both passes and explain what each junction rule means for the organisers.

Prove that the latest event time at the start node always equals zero when both passes are carried out correctly, and explain why this makes a useful check but not a complete one.

An activity M runs from node 3 to node 6 and takes 4 days. The event times are $E_3 = 9$, $L_3 = 11$, $E_6 = 18$, $L_6 = 19$. Find M's total float and free float, and state whether M is critical.

A project's activity table gives total floats: A 0, B 4, C 0, D 4, E 0, F 2, G 0, where B is followed by D. Explain what the manager can and cannot do with the floats on B, D and F, and identify the critical path.

Prove that free float never exceeds total float, and construct a project in which an activity joining two critical events is nevertheless not critical. Explain what this means for identifying the critical path.