The idea, step by step
A correct calculation is not the whole recommendation
Scheduling and assignment models simplify a real situation. Before using a result, check that durations or costs are comparable, dependencies are complete, resources permit the proposed overlaps, and each pairing is available. State whether the objective is elapsed time, total labour time, cost or profit.
A clear recommendation gives the actual activity sequence or pairings and the objective value with units. It also identifies an important limitation. 'Minimum total $9 if the listed technicians are available and each performs one independent job' is more useful than 'the answer is 9'.
Test the decisions that matter
Use a sensible sensitivity check: increase a likely-to-vary duration, remove an unavailable worker or alter a disputed cost, then solve again. Do not assume the old optimum survives. A small change can switch the critical path or assignment even when the overall objective changes only slightly.
A forbidden pairing can be excluded directly. If software needs a large substitute cost, explain why it is safely more expensive than a known feasible complete assignment; an arbitrary number can give a false solution. When predictions have uncertainty, distinguish the calculated optimum from a guaranteed real-world outcome.
A common trap
Treating an optimum computed from estimated inputs as a guaranteed delivery date or cost.
Instead: State the assumptions, units and uncertainty, and test changes that are plausible in the context.
See the thinking
An unavailable technician
In the technician table, Ali cannot do job 2. Find the minimum feasible assignment.
| Technician | Job 1 | Job 2 | Job 3 |
|---|---|---|---|
| Ali | 9 | 2 | 7 |
| Bea | 6 | 4 | 3 |
| Chen | 5 | 8 | 1 |
- Exclude all allocations using Ali→2.
- The remaining original totals are 14,20,21 and 16.
- The minimum is 14 for Ali→1, Bea→2, Chen→3.
State the resource limitation
A network minimum is 13 days. Its schedule requires two machines on day 4, but only one is available. How should the result be reported?
- The forward scan solved the dependency-only model.
- The real resource restriction prevents at least one modelled simultaneous activity.
- The 13-day minimum is a lower bound until a resource-feasible schedule is constructed.
Check your understanding
An assignment minimises total driving hours. Does it necessarily minimise the latest arrival time when all drivers leave together?
Show answer
No.
Total hours is a sum objective. Latest arrival depends on the maximum individual time.
Your turn
6 questions · 16 marks · Work at your own pace.
Read the questions online
- The technician minimum is $9. Write a recommendation that includes the pairings and one assumption.3 marks
Cost ($) of assigning one technician to each job. Technician Job 1 Job 2 Job 3 Ali 9 2 7 Bea 6 4 3 Chen 5 8 1 - Ali cannot do job 2. Find and interpret the new technician optimum.3 marks
Cost ($) of assigning one technician to each job. Technician Job 1 Job 2 Job 3 Ali 9 2 7 Bea 6 4 3 Chen 5 8 1 - Why is $1,000 a safe substitute for the forbidden entry in the preceding problem when all other costs are nonnegative and a feasible $14 allocation is known?3 marks
- A manager asks for the 'best assignment' but supplies both times and profits. What must be clarified before optimisation?2 marks
- A project finish moves from 20 to 21 days when an uncertain activity is extended by 3 days. What does this tell you?3 marks
- An optimal assignment leaves one job with a dummy worker. Write the limitation that must accompany the recommendation.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- The technician minimum is $9. Write a recommendation that includes the pairings and one assumption.3 marks
Cost ($) of assigning one technician to each job. Technician Job 1 Job 2 Job 3 Ali 9 2 7 Bea 6 4 3 Chen 5 8 1 Identify the chosen entries, not only the total.
State a relevant feasibility/additivity assumption.
Assign Ali to 2, Bea to 1 and Chen to 3 for $9, assuming all are available and each does exactly one independent job. - Ali cannot do job 2. Find and interpret the new technician optimum.3 marks
Cost ($) of assigning one technician to each job. Technician Job 1 Job 2 Job 3 Ali 9 2 7 Bea 6 4 3 Chen 5 8 1 Exclude the two assignments using Ali→2.
Among the remaining totals 14,20,21,16, choose 14.
$14: Ali→1, Bea→2, Chen→3. - Why is $1,000 a safe substitute for the forbidden entry in the preceding problem when all other costs are nonnegative and a feasible $14 allocation is known?3 marks
The bound depends on nonnegative remaining costs.
The large value is justified by an actual feasible alternative.
Any assignment using the forbidden entry would cost at least $1,000, so it cannot beat the feasible $14 assignment. - A manager asks for the 'best assignment' but supplies both times and profits. What must be clarified before optimisation?2 marks
Different objectives can select different pairings.
A single optimisation requires an explicit criterion.
The objective and any constraints: for example minimise total time, maximise profit, or meet a time limit then maximise profit. - A project finish moves from 20 to 21 days when an uncertain activity is extended by 3 days. What does this tell you?3 marks
The delay in the overall finish is only 1 day.
Not every task-duration change transfers fully to project duration.
The network absorbed two days of that change before it affected the finish, but the revised critical structure must still be checked. - An optimal assignment leaves one job with a dummy worker. Write the limitation that must accompany the recommendation.2 marks
Identify the uncompleted job explicitly.
If every job is mandatory, the original capacity model must be revised.
That real job remains unfilled, with the stated penalty or allowed omission; the model has not created an extra worker.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: discrete models, project scheduling and assignment. Activity-on-arrow conventions are used for project networks; technology may verify calculations but does not replace interpreting the model.
Official curriculum source ↗