Understand the idea.
A correct calculation is not the whole recommendation
Before recommending a computed optimum, check that the model matches the decision. Confirm common units, complete dependencies, available pairings and enough resources for any overlap. Distinguish elapsed completion time, total labour time, cost and profit: optimising one does not automatically optimise the others.
Report the actual sequence or pairings, the objective value with units, and the assumptions that matter. A minimum calculated from listed costs is conditional on those costs and pairings; a network date assumes the specified durations and available parallel resources.
Test the decisions that matter
Test a plausible change that could affect the recommendation: increase an uncertain duration, exclude an unavailable pairing or correct a disputed cost, then solve again. The old critical path or allocation may cease to be optimal even after a small change.
Exclude forbidden pairings directly when possible. If software requires a substitute cost, justify it against a known feasible total so it cannot be chosen accidentally. A deterministic optimum is a result under assumptions, rather than a guarantee about uncertain real-world conditions.
See the method in action.
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 |
Worked solution
- 1
Apply the availability restriction first.
Every assignment with Ali doing job 2 is infeasible, even if its total would be lower.
- 2
Enumerate feasible completions with Ali on job 1.
With Ali fixed on job 1, Bea and Chen can fill the remaining two jobs in two orders; A, B and C abbreviate Ali, Bea and Chen.
- 3
Enumerate feasible completions with Ali on job 3.
Ali's only other permitted job is 3, and these are the remaining two one-to-one assignments.
- 4
Choose the least feasible total.
Assign Ali→1, Bea→2 and Chen→3, subject to those pairings being available; all permitted complete assignments have been compared.
Minimum $14, subject to those pairings being available.
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?
Worked solution
- 1
Identify what the scan established.
The network calculation permits simultaneous work whenever the precedence rules allow it.
- 2
Check the proposed schedule against machine capacity.
Two machines are needed at the same time on day 4, but only one exists, so that schedule is not resource-feasible.
- 3
State the defensible conclusion.
Thirteen days remains a lower bound. Include the one-machine restriction and construct a feasible schedule before claiming an achievable finish; the information given does not determine the revised duration.
Do not promise completion in 13 days. Revise the schedule to include the one-machine restriction.
Try it yourself.
Check your understanding
An assignment minimises total driving hours. Does it necessarily minimise the latest arrival time when all drivers leave together?
Show answer & reasoning
No.
No. Total driving hours minimises , whereas the latest arrival is when everyone starts together. A lower sum can still contain a longer individual journey, so the objectives can choose different assignments.
Keep practising
6 worksheet questions · 16 marks
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 ↗