Critical Path Analysis — practice questions

A project has activities: P (4 days, no predecessor), Q (3, after P), R (6, after P), S (2, after Q), T (5, after R and S). Describe the network structure, state how many nodes it needs, and number them.

A project has activities: A (2, no predecessor), B (5, no predecessor), C (4, after A), D (3, after A and B), E (6, after C and D). Explain why a dummy activity is required, where it must go, and what would go wrong without it.

Explain why a project network must contain no cycles, and devise a systematic test that detects whether a given precedence table can be drawn as a valid network. Apply it to the table: A (—), B (A), C (B, E), D (C), E (D).

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.

A project takes 30 days with critical path P → Q → R. The manager shortens activity S (which has 4 days of float) by 3 days at a cost of $1,500. What is the new project duration, and was the money well spent?

A project has paths: A→B→C = 20 days, A→D→C = 17 days, A→E→C = 14 days. Activity B (8 days) can be shortened by up to 5 days at $300 per day, and A (4 days) by up to 2 days at $800 per day. Find the shortest achievable duration and its cost, explaining how the critical path changes.

A project's earliest-start schedule requires 2 workers on days 1–4, 5 on days 5–8 and 1 on days 9–12, but only 3 workers are available. Activities on days 5–8 comprise a critical activity needing 2 workers and two non-critical activities needing 2 and 1 workers, with 3 and 6 days of total float respectively. Devise a feasible schedule, and analyse what the resource limit costs the project.