Mathematics Standard • Year 12 • Module 6 • Lesson 6
Crashing and Resource Allocation — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on crashing and resource decisions: crash-cost calculations, choosing activities to crash, and one structured extended response.
1. Short-answer questions
1.1 Two critical activities are candidates for crashing. X: normal 10 days at $8,000; crash 7 days at $9,500. Y: normal 6 days at $5,000; crash 5 days at $5,700. Reduce the project duration by 3 days at minimum extra cost. Show your working. 3 marks Band 3
1.2 A non-critical activity has float 5 days and a crash cost of $200/day. The project needs to finish 2 days earlier. Decide whether crashing this activity is a sensible way to achieve the goal, and explain in one or two sentences. 3 marks Band 3-4
1.3 A project has three critical paths. Activity X is on all three critical paths, with crash cost $400/day. Activity Y is on only one critical path, with crash cost $250/day.
(a) State which activity (X or Y) should be crashed to reduce the project duration by 1 day, and explain in one sentence why.
(b) The site manager argues "Y is cheaper per day so we should crash Y." Briefly respond to this argument. 4 marks Band 4
2. Extended response
2.1 An events company has been contracted to build a temporary stage at the Sydney Royal Easter Show. The current critical path through the project is A → B → C → D, with project duration 25 days. The client wants the stage ready 3 days earlier — a total reduction of 3 days. The activity data is:
A (frame): normal 8 days $6,000; crash 6 days $7,200.
B (rigging): normal 6 days $4,000; crash 4 days $5,200.
C (decking): normal 7 days $3,500; crash 5 days $4,700.
D (test): normal 4 days $2,000; crash 3 days $2,500.
(a) Calculate the crash cost per day and the maximum days each activity can be crashed.
(b) Determine which activities to crash, and by how many days, to reduce the project by 3 days at minimum total extra cost. State the total extra cost.
(c) The client offers a $2,500 early-completion bonus if the project finishes 3 days early. Use your part (b) total to advise whether the contractor should accept the bonus. Give the net financial outcome. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — correct crash cost per day for all four activities (with units).
• 1 mark — correct maximum crash days for all four activities.
Part (b) — 3 marks
• 1 mark — ranks activities by $/day and identifies the cheapest critical activity.
• 1 mark — applies crash limits correctly when distributing the 3-day reduction.
• 1 mark — correct total extra cost with units.
Part (c) — 2 marks
• 1 mark — correctly subtracts the part (b) cost from the $2,500 bonus to get the net financial outcome.
• 1 mark — clear sentence advising whether to accept the bonus, with the net figure.
Your response:
Stuck on (c)? Bonus − total crash cost. If positive, accept. If negative, reject.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Reduce by 3 days at minimum cost (3 marks)
Sample response.
X $/day = ($9,500 − $8,000) ÷ (10 − 7) = $1,500 ÷ 3 = $500/day. Max crash = 3 days.
Y $/day = ($5,700 − $5,000) ÷ (6 − 5) = $700 ÷ 1 = $700/day. Max crash = 1 day.
X is cheaper. Crash X by 3 days (max) = 3 × $500 = $1,500. No need to crash Y.
Total extra cost = $1,500.
Marking notes. 1 mark — both $/day correct. 1 mark — correctly identifies X as cheaper and within crash limit. 1 mark — correct total $1,500 with units. A response that crashes Y first scores 1/3 (wrong choice).
1.2 — Crash a non-critical activity? (3 marks)
Sample response. No — do not crash this activity. Non-critical activities have float and do not control the project duration; crashing them costs money but does not bring the finish forward. To shorten the project by 2 days, the contractor must crash a critical activity instead.
Marking notes. 1 mark — clear "no" answer. 1 mark — reason: non-critical activities do not control duration. 1 mark — states what must be done instead (crash a critical activity). A "yes" answer scores 0/3.
1.3 — Shared critical activity vs single-path activity (4 marks)
(a) Sample response. Crash X. X lies on all three critical paths, so crashing it by 1 day shortens every critical path by 1 day, reducing the project by 1 day for a single $400 payment.
(b) Sample response. The argument is wrong. Although Y has a lower $/day ($250 vs $400), Y is only on one of the three critical paths. Shortening just that one path leaves the other two unchanged, so the project's longest path (and finish date) is unchanged — Y's $250 buys zero days of project reduction. X is the correct choice even though X is more expensive per day.
Marking notes. (a) 1 mark — picks X; 1 mark — reasons that X is on all three paths. (b) 1 mark — recognises Y crashes only one of three paths; 1 mark — explicit conclusion that Y's $250 produces no actual project reduction.
2.1 — Easter Show stage build (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Crash cost per day and max crash for each activity.
A: ($7,200 − $6,000) ÷ (8 − 6) = $1,200 ÷ 2 = $600/day; max crash = 2 days.
B: ($5,200 − $4,000) ÷ (6 − 4) = $1,200 ÷ 2 = $600/day; max crash = 2 days.
C: ($4,700 − $3,500) ÷ (7 − 5) = $1,200 ÷ 2 = $600/day; max crash = 2 days.
D: ($2,500 − $2,000) ÷ (4 − 3) = $500 ÷ 1 = $500/day; max crash = 1 day.
[1 mark — all $/day correct; 1 mark — all max-crash days correct.]
(b) Crashing plan for a 3-day reduction.
Rank by $/day: D ($500/day, max 1) is cheapest, then A, B, C tied at $600/day. [1 mark — ranking.]
Crash D by 1 day = 1 × $500 = $500. Need 2 more days; pick any one of A/B/C (they tie). Crash A by 2 days = 2 × $600 = $1,200. [1 mark — applies crash limits, uses D fully then a cheap $600 activity for the remaining 2 days.]
Total extra cost = $500 + $1,200 = $1,700. [1 mark — correct total with units.]
(c) Bonus decision.
Net = Bonus − crash cost = $2,500 − $1,700 = +$800. [1 mark — correct subtraction giving net figure.]
Conclusion: Yes — the contractor should accept the bonus. The minimum-cost crashing plan adds $1,700 of expense, and the $2,500 bonus more than covers it, leaving a net profit of $800 from the early-completion arrangement. [1 mark — clear sentence advising acceptance with the net figure.]
Total: 7/7.
Band descriptors for marker.
Band 3: Calculates $/day for some activities and chooses one to crash; total cost partially correct; no bonus comparison. ≈ 3 marks.
Band 4: All $/day correct, picks the cheapest activity correctly for most of the crash, total cost approximately right; bonus subtraction attempted but conclusion missing. ≈ 5 marks.
Band 5: Complete crash plan with correct total $1,700; bonus subtraction correct; conclusion lacks the net figure or doesn't explicitly say "accept the bonus". ≈ 6 marks.
Band 6: Complete — all $/day, max crashes, optimal plan with total $1,700, net = +$800, AND a clear conclusion sentence advising acceptance with the $800 net profit named. 7/7.