Mathematics Standard • Year 12 • Module 6 • Lesson 9
Scheduling with Constraints — Problem Set
Apply lag, lead and resource-constraint reasoning to real construction, manufacturing and event-planning scenarios.
Problem 1 — Concrete slab with curing lag
A residential builder is laying a concrete slab. Activities (days): Site prep (2, −), Pour slab (1, Site prep), Cure slab (lag 3 days after pour finishes — no work happens, but framing cannot start), Frame walls (4, after cure), Roof (3, Frame walls).
Set up: What are we solving for?
(i) Calculate the ES and EF of every activity, treating the 3-day cure as a lag (not an activity with workers). 2 marks
(ii) State the total project duration in days. 1 mark
(iii) The builder asks whether adding extra concrete workers could shorten the 3-day cure. Explain in one sentence why crashing has no effect on a curing lag. 2 marks
Stuck? Revisit lesson § Lags and Leads — a chemical or physical lag is a hard constraint that cannot be crashed.Problem 2 — Apartment tower with floor-to-floor lead
A construction crew is building a 3-floor apartment block. Activities (days): Foundation (4, −), Floor 1 frame (5, Foundation), Floor 2 frame (5, Floor 1 frame with lead 2), Floor 3 frame (5, Floor 2 frame with lead 2), Roofing (3, Floor 3 frame).
Set up: What are we solving for?
(i) Calculate ES and EF for each activity using the lead rule (each upper floor starts 2 days before the lower floor finishes). 3 marks
(ii) What is the project duration with leads applied? 1 mark
(iii) What would the project duration be WITHOUT the leads (each floor waits for the one below to finish)? Compare the two figures and state the days saved by the lead strategy. 2 marks
Stuck? Revisit lesson § Lags and Leads — calculate both schedules and subtract.Problem 3 — Sydney café fit-out with limited tradies
A café fit-out has 5 activities for one week (5 working days). Each activity can run in parallel from a precedence point of view, but each needs 1 tradie and the contractor only has 3 tradies on site.
Day 1–3: Demolition (D) — critical, no float
Day 1–4: Plumbing rough (P) — float 1
Day 1–4: Electrical rough (E) — float 1
Day 1–5: Painting (Pa) — float 0
Day 1–5: Tiling (T) — float 2
Set up: What are we solving for?
(i) On day 1, all five activities would normally start. How many tradies are required if all five run? 1 mark
(ii) Identify which 3 activities should run on day 1 (use float to delay the right ones). Justify your choice. 3 marks
(iii) Does this resource-smoothing plan extend the project beyond day 5? Explain. 2 marks
Stuck? Revisit lesson § Resource Smoothing — start the float-zero activities first, delay the highest-float activity.Problem 4 — Outback farm fence with weather window
A farmer is fencing a paddock. The work crew is on contract and weather constrains the schedule. Activities (days): Mark out (1, −), Dig post-holes (2, Mark out), Set posts in concrete (1, Dig holes), Curing lag of 4 days, Stretch wire (2, after cure), Final inspection by farm manager available 7 days after Mark out begins (external dependency).
Set up: What are we solving for?
(i) Compute ES and EF for each work activity, treating the cure as a 4-day lag and ignoring the inspection for now. 2 marks
(ii) The inspection is an EXTERNAL dependency — it cannot happen before day 7. Does this extend the project, and if so, by how many days? 3 marks
(iii) State the total project duration including inspection. 1 mark
Stuck? Revisit lesson § External Dependency — the inspection cannot start before day 7 even if work is finished sooner.Problem 5 — Event setup with shared sound rig
An events company runs two stages at a community festival. Activities (hours): Truss assembly Stage A (3, −), Sound check Stage A (1, Truss A) — uses sound rig; Truss assembly Stage B (3, − ) — runs in parallel, no float; Sound check Stage B (1, Truss B) — uses sound rig.
The sound rig is a shared resource — only ONE sound check at a time.
Set up: What are we solving for?
(i) Without the sound rig constraint, both sound checks could run at hour 3 to 4. What is the project duration in that case? 1 mark
(ii) With the rig constraint, the two sound checks must run sequentially. What is the new project duration? 2 marks
(iii) If the company hires a second sound rig at $200, how much time is saved and is it worth it if the company values its time at $150 per hour? Show the cost-benefit calculation. 3 marks
Stuck? Revisit lesson § Resource Constraints — a shared resource forces sequential running.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Concrete slab with curing lag
Set up. We are computing ES/EF where a 3-day curing lag delays Frame walls from starting.
(i) Site prep: 0→2. Pour: 2→3. Cure lag = 3 days (Frame cannot start until day 3 + 3 = 6). Frame: 6→10. Roof: 10→13.
(ii) Duration = 13 days.
(iii) Curing is a chemical process — adding workers cannot speed it up. The lag is a hard physical constraint.
Problem 2 — Apartment tower lead
Set up. We are computing ES/EF using a 2-day lead between floors, then comparing total duration with and without the leads.
(i) Foundation: 0→4. F1 frame: 4→9. F2 frame: ES = 9 − 2 = 7, EF = 12. F3 frame: ES = 12 − 2 = 10, EF = 15. Roofing: 15→18.
(ii) With leads: 18 days.
(iii) Without leads: Foundation 4 + F1 5 + F2 5 + F3 5 + Roof 3 = 22 days. Days saved by leads = 22 − 18 = 4 days.
Problem 3 — Café fit-out, 3 tradies
Set up. We are smoothing day-1 demand from 5 tradies to 3 by using float on non-critical activities.
(i) Five activities × 1 tradie each = 5 tradies needed if all run on day 1.
(ii) Critical (float 0): Demolition (D), Painting (Pa). Both MUST run on day 1. That's 2 tradies. Need 1 more for the 3rd slot.
Options for the 3rd: P (float 1), E (float 1), or T (float 2). Pick the lowest-float-but-not-zero — P or E. Choose P (1 day float). Delay E by 1 day and T by up to 2 days.
Day 1 = D + Pa + P (3 tradies).
(iii) No — Pa (5 days, no float) still finishes at day 5. P, E, T all finish within their floats and the project completes on day 5 as planned. Project not extended.
Problem 4 — Farm fence with external dependency
Set up. We are computing ES/EF with a curing lag, then checking whether an external 7-day-after-start inspection constraint extends the project.
(i) Mark out: 0→1. Dig holes: 1→3. Set posts: 3→4. Cure lag = 4 days (Stretch wire cannot start until day 4 + 4 = 8). Stretch wire: 8→10.
(ii) External inspection earliest day = 7 (counted from Mark out start). Stretch wire finishes at day 10, so the work itself completes after day 7. Inspection can start at day 10 (the earlier of "after work" and "after day 7"). Since day 10 > day 7, the external dependency is NOT the binding constraint — the work schedule is. No extra delay from the external rule.
(iii) Total project duration = 10 days (work finishes day 10, inspection runs day 10 onward — typically 1 day, so finish = day 11 if inspection takes 1 day; the question asks for duration including inspection so state day 11 if a single-day inspection is assumed, or day 10 if instantaneous).
Problem 5 — Stage setup with shared sound rig
Set up. We are comparing project duration with and without the rig constraint, then doing a cost-benefit on hiring a second rig.
(i) Without constraint: Trusses run 0–3 in parallel. Sound checks run 3–4 in parallel. Duration = 4 hours.
(ii) With constraint: Trusses run 0–3 in parallel. Sound checks must run sequentially. Sound check A: 3–4. Sound check B: 4–5. Duration = 5 hours.
(iii) Time saved by second rig = 5 − 4 = 1 hour. Value of time saved = 1 × $150 = $150. Cost of second rig = $200. Cost ($200) > Benefit ($150). NOT worth it (loses $50 net).