Mathematics Standard • Year 12 • Module 6 • Lesson 8
Multiple Critical Paths, Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on identifying, analysing and crashing networks with more than one critical path.
1. Short-answer questions
1.1 A project has activities: A(3,−), B(4,A), C(4,A), D(2,B), E(2,C), F(1,D,E). List all critical paths and state the project duration. 3 marks Band 3
1.2 A network has: A(2,−), B(5,A), C(4,A), D(2,B), E(3,C), F(1,D,E).
(a) Find all critical paths and the project duration.
(b) Identify any shared critical activities.
(c) If activity C's duration increases by 1 day, what happens to the number of critical paths? 4 marks Band 3-4
1.3 A project has two critical paths sharing no common activities, each 14 days long. Three crash options exist: A on Path 1 ($300/day), B on Path 2 ($500/day), and C on Path 2 ($400/day, max 2 days). The site manager wants to reduce the project to 12 days at minimum cost. Calculate the cheapest crash plan, showing all working. 4 marks Band 4
Stuck on 1.3? With no shared activities, every cut day needs ONE activity crashed on Path 1 AND ONE on Path 2.2. Extended response
2.1 A regional NSW council is upgrading a community centre. The activity network has:
Site demo (D): 3 days, no predecessor
North-wing slab (N): 4 days, predecessor D
South-wing slab (S): 4 days, predecessor D
North-wing frame (Nf): 3 days, predecessor N
South-wing frame (Sf): 3 days, predecessor S
Roofing across both wings (R): 2 days, predecessors Nf and Sf
(a) Carry out a forward scan and state the project duration.
(b) List all critical paths and identify the shared activities.
(c) The council can crash one activity by 1 day for cost: D = $1500, N = $400, S = $400, R = $1200. They want a 1-day cut at minimum cost. Recommend an action and show all working, including a clear conclusion sentence. 7 marks Band 5-6
Explicit marking criteria
Part (a), 2 marks
• 1 mark correct forward scan (ES and EF for all six activities).
• 1 mark correct project duration (12 days).
Part (b), 2 marks
• 1 mark both critical paths listed: D → N → Nf → R and D → S → Sf → R.
• 1 mark shared activities D and R identified.
Part (c), 3 marks
• 1 mark recognises that crashing N alone (or S alone) does NOT reduce duration because the other critical path stays at 12.
• 1 mark compares the shared-activity options D ($1500) vs R ($1200) against the combined non-shared plan (N + S = $400 + $400 = $800).
• 1 mark explicit conclusion sentence naming N + S = $800 as the cheapest plan, with clear justification it cuts both critical paths.
Your response:
Stuck on (c)? Compute the cost of every option that cuts BOTH paths by 1 day. The cheapest may be two simultaneous non-shared crashes.How did this worksheet feel?
What I'll revisit before next class:
1.1, Symmetric network (3 marks)
Sample response. Forward: A(0,3), B(3,7), C(3,7), D(7,9), E(7,9), F(9,10). Paths: A-B-D-F = 3+4+2+1 = 10. A-C-E-F = 3+4+2+1 = 10. Both critical. Project duration = 10 days. Shared: A, F.
Marking notes. 1 mark, forward scan. 1 mark, both paths listed (or numerically equal). 1 mark, duration stated with both paths declared critical.
1.2, Asymmetric network (4 marks)
(a) Forward: A(0,2), B(2,7), C(2,6), D(7,9), E(6,9), F(9,10). Paths: A-B-D-F = 2+5+2+1 = 10. A-C-E-F = 2+4+3+1 = 10. Both critical. Duration = 10 days.
(b) Shared critical activities: A and F.
(c) If C increases by 1: A-C-E-F = 2+5+3+1 = 11. A-B-D-F = 10 (unchanged). Now only A-C-E-F is critical; duration = 11. Number of critical paths drops from 2 to 1.
Marking notes. (a) 1, forward scan + paths. 1, both critical, duration 10. (b) 1, A, F. (c) 1, explanation of path becoming sole critical, duration 11.
1.3, Crash plan with disjoint critical paths (4 marks)
Sample response. No shared activities, so every day cut requires ONE activity-crash on each path.
Per-day cuts needed = 14 − 12 = 2 days.
Path 1: only A is available ($300/day). 2 days × $300 = $600.
Path 2: cheaper of B ($500) or C ($400), use C (up to 2 days). 2 days × $400 = $800.
Total = $600 + $800 = $1,400.
Marking notes. 1 mark, recognises disjoint paths require independent crashes on both. 1 mark, correctly identifies C as cheaper than B per day on Path 2. 1 mark, correct totals on each path. 1 mark, final sum $1,400 with units.
2.1, Community centre (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Forward scan and project duration.
D: ES=0, EF=3.
N: ES=3, EF=7. S: ES=3, EF=7.
Nf: ES=7, EF=10. Sf: ES=7, EF=10.
R: ES = max(10, 10) = 10, EF = 12. [forward scan, 1 mark]
Project duration = 12 days. [1 mark]
(b) Critical paths and shared activities.
Paths: D → N → Nf → R = 3+4+3+2 = 12. D → S → Sf → R = 3+4+3+2 = 12.
Both critical. Critical paths: D → N → Nf → R and D → S → Sf → R. [1 mark]
Shared critical activities: D and R. [1 mark]
(c) Cheapest 1-day cut.
Crashing N alone ($400) reduces D-N-Nf-R to 11 but leaves D-S-Sf-R at 12 ⇒ duration still 12. Crashing one non-shared activity alone does NOT cut the project. [1 mark]
Options that genuinely cut 1 day:
Crash D (shared) by 1: cost = $1,500. Both paths drop to 11.
Crash R (shared) by 1: cost = $1,200. Both paths drop to 11.
Crash N + S simultaneously, each by 1: cost = $400 + $400 = $800. Path 1 drops by 1, Path 2 drops by 1 ⇒ both now 11. [comparison of options, 1 mark]
Conclusion: the cheapest 1-day cut is to crash N and S simultaneously by 1 day each, at a total cost of $800. This is cheaper than crashing either shared activity (D = $1,500 or R = $1,200) and is the only option that reduces both critical paths within budget. [1 mark, explicit conclusion]
Total: 7/7.
Band descriptors for marker.
Band 3: Forward scan attempted; only one critical path identified; suggests crashing N or S alone. ≈ 3 marks.
Band 4: Both critical paths found; recognises duration is 12; either misses the shared-vs-disjoint cost comparison or doesn't compute N + S = $800. ≈ 5 marks.
Band 5: Comparison of shared crash vs paired non-shared crash done numerically, but conclusion sentence missing the explicit dollar amount or activity names. ≈ 6 marks.
Band 6: Complete forward scan; both critical paths and shared activities named; cost comparison of all options; explicit conclusion sentence "Crash N and S simultaneously at $800". 7/7.