Mathematics Standard • Year 12 • Module 6 • Lesson 3
Drawing Activity Networks — Problem Set
Apply AOA network drawing to five Australian project scenarios: a community hall fit-out, a small business launch, a film production, an apartment renovation and a charity-event rollout.
Problem 1 — Community hall fit-out (regional NSW)
A small contractor is fitting out a community hall (days):
Site clear SC(2, —), Paint walls PW(3, SC), Lay floor LF(4, SC), Install lighting IL(2, PW), Place furniture PF(2, LF and IL).
Set up: What are we solving for?
(i) Sketch the AOA network, numbering every event. 2 marks
(ii) State the number of events in your diagram and whether any dummies are required. 1 mark
(iii) Identify which two activities run in parallel after SC, and which event is the merge point for PF. 2 marks
Stuck? Revisit lesson § AOA Rules — PW and LF both leave the event after SC, then converge at the event where PF starts.Problem 2 — Small-business launch (Newcastle)
A new gym is opening. Tasks (days):
Sign lease SL(1, —), Fit-out FO(7, SL), Order equipment OE(5, SL), Hire trainers HT(4, SL), Install equipment IE(3, FO and OE), Onboard trainers OT(2, HT), Opening day OD(1, IE and OT).
Set up: What are we solving for?
(i) Sketch the AOA network, numbering every event. 2 marks
(ii) Explain why FO and OE both finish at the same event in your diagram (i.e. why their end events are the merge point for IE). 2 marks
(iii) State whether any dummies are required and explain in one sentence why or why not. 2 marks
Stuck? Revisit lesson § Dummy Activities — dummies are only needed when two activities would otherwise share both start and end events, OR when a partial dependency must be separated.Problem 3 — Short film production (Sydney film school)
A student-film project (days):
Script SC(3, —), Location scout LS(2, SC), Cast audition CA(4, SC), Shoot SH(6, LS and CA), Edit picture EP(5, SH), Edit sound ES(3, SH), Final mix FM(2, EP and ES).
Set up: What are we solving for?
(i) Sketch the AOA network with events numbered. 2 marks
(ii) List every pair of activities that runs in parallel in your diagram. 2 marks
(iii) State how many events and how many dummies appear in your network. 1 mark
Stuck? Revisit lesson § Worked Example — LS and CA both leave SC's end event and both feed SH; EP and ES both leave SH's end event and both feed FM.Problem 4 — Apartment renovation (Melbourne)
A renovator lists tasks (days):
Demo DM(2, —), Electrical EL(3, DM), Plumbing PL(3, DM), Drywall DW(4, EL), Tiling TL(3, PL), Painting PT(2, DW and TL).
Set up: What are we solving for?
(i) Sketch the AOA network with events numbered. 2 marks
(ii) A student draws the diagram with both EL and PL ending at the same event, then connects DW from there. State which rule has been broken and how to fix the diagram. 2 marks
(iii) Suppose a new activity Cabinet install CI(3, EL only) is added, and PT now needs CI in addition to DW and TL. Re-draw or describe the modified network and state whether any extra dummy is needed. 2 marks
Stuck on (ii)? If both EL and PL share start AND end events, you cannot distinguish which arrow is which. A dummy after one of them gives separate end events.Problem 5 — Charity fun-run rollout (Brisbane)
A charity is organising a fun-run (weeks):
Permits PM(2, —), Route plan RP(3, PM), Marketing MK(4, PM), Volunteer recruit VR(3, MK), Course setup CS(2, RP), Event day ED(1, CS and VR).
Set up: What are we solving for?
(i) Sketch the AOA network with events numbered. 2 marks
(ii) Identify the two activities that can run in parallel after PM. 1 mark
(iii) An organiser proposes adding a Sponsor sign-off SS(2, RP and MK). Describe (in words) the change to the network and state whether a dummy is needed for SS. 2 marks
Stuck on (iii)? SS needs both RP and MK to finish first. Their end events are different, so you may need a dummy to bring them to a common event before SS starts.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Community hall fit-out
Set up. We are drawing the AOA diagram from SC to PF, then identifying parallel activities and merge point.
(i) 1 → SC → 2; 2 → PW → 3; 2 → LF → 4; 3 → IL → 4; 4 → PF → 5. (Alternative: keep IL ending at event 5 with a dummy, but the canonical drawing merges IL and LF at event 4 because PF needs both.)
(ii) Events = 5. Dummies = 0 (no two activities share both start and end events). Note: some students draw IL and LF with separate end events and then add a dummy to converge them — also valid, just with 1 dummy and 6 events.
(iii) Parallel after SC: PW and LF. PF starts at the merge event where LF and IL end (event 4 in the canonical drawing).
Problem 2 — Small-business launch
Set up. We are drawing the AOA for a gym launch and explaining why FO and OE share an end event.
(i) 1 → SL → 2; 2 → FO → 3; 2 → OE → 4; 2 → HT → 5; dummy 3 → 4 (to merge FO and OE for IE); 4 → IE → 6; 5 → OT → 6; 6 → OD → 7.
(ii) FO and OE both feed IE. To draw IE leaving a single event, FO and OE must converge there. In AOA we make this happen by giving them a common end event (often via a dummy) so that IE's start event is the unique merge point for both.
(iii) One dummy is needed (from event 3 to event 4) to satisfy the unique-event rule when FO and OE both leave event 2 — without the dummy, they would share start AND end events, which is not allowed.
Problem 3 — Short film production
Set up. We are drawing the AOA for the film and naming parallel pairs.
(i) 1 → SC → 2; 2 → LS → 3; 2 → CA → 4; dummy 3 → 4; 4 → SH → 5; 5 → EP → 6; 5 → ES → 7; dummy 6 → 7; 7 → FM → 8.
(ii) Parallel pairs: LS and CA (both start after SC and both feed SH); EP and ES (both start after SH and both feed FM).
(iii) Events = 8. Dummies = 2 (one to merge LS and CA for SH; one to merge EP and ES for FM). Some students draw without the dummies by placing LS/CA at the same end event — that violates the unique-event rule, so the canonical answer keeps the dummies.
Problem 4 — Apartment renovation
Set up. We are drawing the AOA, fixing a student's drawing error, and adding a new activity CI.
(i) 1 → DM → 2; 2 → EL → 3; 2 → PL → 4; 3 → DW → 5; 4 → TL → 5; 5 → PT → 6. (DW and TL converge at event 5 because PT needs both.) Events = 6, dummies = 0.
(ii) If EL and PL share both start (event 2) and end events, the unique-event rule is broken — two activities cannot be drawn between the same pair of events. Fix: insert a dummy on EL or PL so they have different end events.
(iii) Add CI starting from event 3 (after EL): 3 → CI → 7. Then PT now needs DW, TL and CI — so DW, TL, CI must all end at PT's start event. If they don't already, add one or more dummies to converge them. With the canonical drawing above, the simplest extension is: 3 → CI → 5 (so CI ends at event 5 with DW and TL); then PT 5 → 6 as before. No extra dummy needed if CI's end event is event 5. (Alternative: CI ends at event 7 and a dummy 7 → 5 brings it back — but then 5 is later than 7 which breaks numbering, so prefer the first approach.)
Problem 5 — Charity fun-run
Set up. We are drawing the AOA for the fun-run and adding a sponsor sign-off.
(i) 1 → PM → 2; 2 → RP → 3; 2 → MK → 4; 3 → CS → 5; 4 → VR → 5; 5 → ED → 6. Events = 6, dummies = 0.
(ii) Parallel after PM: RP and MK (both have PM as the only predecessor).
(iii) SS needs both RP and MK. In the diagram above, RP ends at event 3 and MK ends at event 4. SS therefore needs to start where both have arrived — we can add a dummy from event 3 to event 4 so SS starts from event 4: 4 → SS → 7. Yes — one dummy is needed so that SS can wait for both RP and MK to finish without changing CS's start event (which only needs RP).