Mathematics Standard • Year 12 • Module 6 • Lesson 4
Forward and Backward Scan — Problem Set
Apply the forward and backward scan to realistic Australian projects: a backyard pool build, a cafe refurbishment, an outdoor concert, a hospital ward fit-out and a road resurfacing.
Problem 1 — Backyard pool installation (Western Sydney)
A pool company schedules a backyard install (days):
Site mark SM(1, —), Excavate EX(4, SM), Plumbing PL(3, EX), Steel reinforce SR(2, EX), Concrete CO(2, PL and SR), Cure CU(7, CO).
Set up: What are we solving for?
(i) Perform the forward scan and state the project duration in days. 2 marks
(ii) Perform the backward scan and report LS and LF for every activity. 2 marks
(iii) The site supervisor asks: "How much can the SR (steel reinforce) activity be delayed without delaying the project?" Use your ES and LS for SR to answer. 1 mark
Stuck? Revisit lesson § Backward Scan — slack on an activity is LS − ES.Problem 2 — Cafe refurbishment (Melbourne)
A cafe is being refurbished (days):
Close cafe CL(1, —), Strip out SO(2, CL), Plumbing rework PR(3, SO), Electrical rework ER(2, SO), Floor renew FR(4, PR and ER), Paint PT(2, ER), Reopen RO(1, FR and PT).
Set up: What are we solving for?
(i) Perform the forward scan and state the project duration. 2 marks
(ii) Perform the backward scan and produce a six-column table (Activity | Duration | ES | EF | LS | LF). 2 marks
(iii) The owner asks whether PT can be delayed without affecting the reopen date. Use your scan results to give a yes/no answer with one short reason. 2 marks
Stuck? Revisit lesson § Backward Scan — compare PT's LS to its ES.Problem 3 — Outdoor concert load-in (Brisbane Riverstage)
An events crew schedules a concert load-in (hours):
Site prep SP(2, —), Stage build SB(6, SP), Sound system SS(4, SB), Lighting LT(3, SB), Backdrop BD(2, SB), Cue check CC(1, SS and LT and BD), Doors open DO(1, CC).
Set up: What are we solving for?
(i) Perform the forward scan and report the project duration in hours. 2 marks
(ii) Perform the backward scan and identify, by LS, which two of SS, LT and BD have slack to start later than their ES. 2 marks
(iii) Doors are advertised to open at hour 14 from the start of site prep. Use your forward scan to decide whether the schedule meets this with the activities as written. 2 marks
Stuck? Revisit lesson § Forward Scan — project duration = EF of the finish activity = MAX over all paths.Problem 4 — Hospital ward fit-out (regional hospital)
A small NSW hospital refits a ward (days):
Approval AP(2, —), Demolish DM(3, AP), Rewire RW(4, DM), Replumb RP(3, DM), Install medical gas MG(2, RW), Drywall DW(5, RW and RP), Final paint FP(2, DW and MG), Sign-off SG(1, FP).
Set up: What are we solving for?
(i) Perform the forward scan and state the project duration. 2 marks
(ii) Perform the backward scan and produce LS, LF for every activity. 2 marks
(iii) The medical-gas team requests a 1-day delay to start MG. Use your scan results to decide whether this delays the ward sign-off. Justify in one short sentence. 2 marks
Stuck? Revisit lesson § Backward Scan — slack on MG = LS(MG) − ES(MG). If a 1-day delay is ≤ slack, no project impact.Problem 5 — Highway resurfacing (Pacific Highway, NSW)
A roadworks crew schedules an overnight resurfacing block (hours):
Close lanes CL(1, —), Mill old surface MO(3, CL), Sweep SW(1, MO), Tack coat TC(1, SW), Lay asphalt LA(4, TC), Roller RL(2, LA), Linemarking LM(2, RL), Reopen RP(1, LM).
Set up: What are we solving for?
(i) Perform the forward scan and state the project duration in hours. 2 marks
(ii) Perform the backward scan. Comment briefly on why the LS for every activity equals its ES. 2 marks
(iii) The night-works window is 14 hours. Decide whether the crew can finish in the window with the activities as written, and explain in one sentence using your project duration. 2 marks
Stuck? Revisit lesson § Forward Scan — in a strict chain (no parallel paths) every activity has zero slack.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Backyard pool installation
Set up. We are doing both scans to find duration and SR slack.
(i) Forward: SM(0,1), EX(1,5), PL(5,8), SR(5,7), CO(max(8,7)=8, 10), CU(10, 17). Project duration = 17 days.
(ii) Backward: CU LF=17, LS=10. CO LF=10, LS=8. PL LF=8, LS=5. SR LF=8, LS=6. EX LF=min(5,6)=5, LS=1. SM LF=1, LS=0.
(iii) SR slack = LS − ES = 6 − 5 = 1 day. SR can be delayed by up to 1 day without delaying the project.
Problem 2 — Cafe refurbishment
Set up. We are scanning the cafe job and checking PT's slack.
(i) Forward: CL(0,1), SO(1,3), PR(3,6), ER(3,5), FR(max(6,5)=6, 10), PT(5, 7), RO(max(10,7)=10, 11). Project duration = 11 days.
(ii) Backward (RO LF=11): RO LS=10. FR LF=10, LS=6. PT LF=10, LS=8. PR LF=6, LS=3. ER LF=min(6,8)=6, LS=4. SO LF=min(3,4)=3, LS=1. CL LF=1, LS=0.
Table:
CL | 1 | 0 | 1 | 0 | 1
SO | 2 | 1 | 3 | 1 | 3
PR | 3 | 3 | 6 | 3 | 6
ER | 2 | 3 | 5 | 4 | 6
FR | 4 | 6 | 10 | 6 | 10
PT | 2 | 5 | 7 | 8 | 10
RO | 1 | 10 | 11 | 10 | 11
(iii) Yes — PT can be delayed by up to 3 days (LS − ES = 8 − 5 = 3) without affecting RO. (Slip: confusing PT with FR — FR has zero slack.)
Problem 3 — Outdoor concert load-in
Set up. Forward + backward, then check the 14-hour doors-open target.
(i) Forward: SP(0,2), SB(2,8), SS(8,12), LT(8,11), BD(8,10), CC(max(12,11,10)=12, 13), DO(13, 14). Project duration = 14 hours.
(ii) Backward: DO LF=14, LS=13. CC LF=13, LS=12. SS LF=12, LS=8. LT LF=12, LS=9. BD LF=12, LS=10. SB LF=8, LS=2. SP LF=2, LS=0.
Slack: SS = 0 (critical); LT = 1; BD = 2. So LT and BD can each start later than their ES (LT by 1 hour, BD by 2 hours).
(iii) Project duration = 14 hours = doors-open target. Yes — but only just: any slippage on SP, SB, SS, CC or DO pushes the doors-open time past 14 hours.
Problem 4 — Hospital ward fit-out
Set up. Both scans, then decide whether MG's 1-day delay affects sign-off.
(i) Forward: AP(0,2), DM(2,5), RW(5,9), RP(5,8), MG(9, 11), DW(max(9,8)=9, 14), FP(max(14,11)=14, 16), SG(16, 17). Project duration = 17 days.
(ii) Backward (SG LF=17): SG LS=16. FP LF=16, LS=14. DW LF=14, LS=9. MG LF=14, LS=12. RW LF=min(9,12)=9, LS=5. RP LF=9, LS=6. DM LF=min(5,6)=5, LS=2. AP LF=2, LS=0.
(iii) MG slack = LS − ES = 12 − 9 = 3 days. A 1-day delay (1 ≤ 3) is absorbed by MG's slack. No — sign-off is not delayed.
Problem 5 — Highway resurfacing
Set up. Single-chain network, fast scans, compare to 14-hour window.
(i) Forward: CL(0,1), MO(1,4), SW(4,5), TC(5,6), LA(6,10), RL(10,12), LM(12,14), RP(14, 15). Project duration = 15 hours.
(ii) Backward (RP LF=15): RP LS=14. LM LF=14, LS=12. RL LF=12, LS=10. LA LF=10, LS=6. TC LF=6, LS=5. SW LF=5, LS=4. MO LF=4, LS=1. CL LF=1, LS=0.
For every activity, LS = ES. Reason: the network is a single chain with no parallel paths, so every activity lies on the only path from start to finish and has zero slack.
(iii) Project duration = 15 hours > 14-hour window. No — the crew cannot finish in the window with the activities as written; they would need to crash a critical activity (or change the schedule) to gain at least 1 hour.