Create schedules from dependencies
Use durations, prerequisites and shared resources to find a feasible earliest finishing time.
Section 26-2.1 · 2026–2027 · 2.1 · Choose operations and apply models
Original StudyForA scenarios and formative model answers; no Cambridge marks, predicted grades or certified mastery. Alternative reasoned answers may be valid. Full-length unseen papers, a wider mixed-problem bank and calibrated specialist teacher marking remain outstanding.
Before you begin
- Read tables and instructions in English
- Use arithmetic, ratios and whole-number constraints
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before calculating, what should you identify?
What you will learn
- Schedule dependent and parallel tasks
- Explain a lower bound on completion time
Separate duration from dependency
Write each task’s duration and what must finish before it starts. Parallel work is allowed only when the resources and stated rules permit it. Earliest start is the latest finish among the task’s prerequisites; earliest finish adds its duration.
Two workers: A takes 20 minutes; after A, B takes 30 and C takes 15 and can run in parallel. D takes 10 after both B and C. A runs 0–20, B 20–50, C 20–35 and D 50–60. Adding every duration gives 75, which ignores the permitted overlap.
Check shared resources and prove earliest
The A–B–D chain takes 60 minutes, so no compliant schedule can finish earlier. The schedule reaches that bound, proving earliest completion. If B and C need the same machine, their overlap is impossible even with two workers, and they must be scheduled serially.
For real clock times convert consistently, including journeys across midnight. State whether setup, rest and travel count in the duration; do not assume omitted time is zero when the question supplies it elsewhere.
Sources for this lesson
Pause and explain
When can D start if it needs B and C finished?
Worked example
Find earliest completion of A, B, C and D with the stated parallel resources.
Show the worked solution
- Finish A at minute 20.
- Run B and C in parallel; B finishes last at 50.
- D then finishes at 60, matching the A–B–D lower bound.
Answer Earliest completion is 60 minutes. The long dependency chain requires sixty, and the proposed schedule meets every prerequisite without a resource conflict.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Guided: If B and C share one machine, find completion time.
Give me a hint
They must run one after another.
Compare my reasoning
- A finishes at 20.
- B and C take another 45 serial minutes.
- D adds ten minutes.
Completion takes 75 minutes. B and C cannot overlap on one machine, so A, both middle tasks and D form a serial workload of 20+30+15+10.
Look for these in your work
- Recognises the conflict
- Serialises correctly
- Includes D
Independent: Tasks P (12 minutes) and Q (18 minutes) start together on separate machines. R (7 minutes) starts after both. Find the earliest finish.
Give me a hint
R waits for the slower prerequisite.
Compare my reasoning
- P ends at 12 and Q at 18.
- R starts at 18.
- Add seven minutes.
Earliest finish is minute 25. Q–R requires 18+7=25 minutes, and running P alongside Q attains that lower bound.
Look for these in your work
- Allows permitted overlap
- Uses the later finish
- Justifies earliest
Application: A shuttle departs 23:50, travels 35 minutes, then needs a 10-minute walk. What is the venue arrival time, and can it meet a 00:30 deadline?
Give me a hint
Carry the date across midnight.
Compare my reasoning
- The shuttle arrives at 00:25 the next day.
- The walk ends at 00:35.
- Compare with 00:30 on that next day.
Venue arrival is 00:35 the next day, five minutes late. Ignoring the day change or the final walk would produce an invalid deadline check.
Look for these in your work
- Crosses midnight
- Includes walking
- Checks the correct day
Common mistakes
- Adding durations that can overlap
- Overlapping tasks that share a resource
- Calling a feasible schedule optimal without a lower bound
What can you explain now?
When can D start if it needs B and C finished?
Compare with the explanation
After the later of their finish times
Every prerequisite must be completed.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
When can D start if it needs B and C finished?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.