Understand the idea.
Optimal within the model
A mathematical optimum is best for a specified objective under the model’s constraints. A minimum tree minimises installation weight; a shortest route minimises its stated travel weight. Neither automatically optimises reliability, queues or other omitted concerns.
Check feasibility before ranking alternatives. A closed road, negative physical stock count or capacity violation disqualifies a result even if its computed cost is attractive. Hard constraints cannot be traded away without changing the problem.
Make the conclusion useful
A useful recommendation states the actual route, edges or quantities, its total and units, and the conditions behind it. Name a specific limitation and the information needed to assess it, rather than simply saying a model is unrealistic.
A sensitivity check varies an uncertain input and asks whether the decision changes. If a one-minute advantage can disappear under a plausible wait change, report that uncertainty. An extra reliability cost needs comparison with the consequences of the particular failure it prevents.
See the method in action.
Cost versus resilience
A four-link tree connects five clinics for 48,000 dollars. An additional link costs 6,000 dollars and backs up one bridge. Explain the choice.
Worked solution
- 1
Identify what the tree provides.
The tree connects all five clinics with no cycle, so each selected link is a bridge.
- 2
Calculate the cost with the backup link.
The proposed extra connection adds six thousand dollars to the installation cost.
- 3
Compare the objectives.
Keeping the tree costs less; adding the link provides an alternative route for the specified bridge but does not establish backup for every failure.
- 4
State what decides the recommendation.
Compare the extra cost with the likelihood and consequences of losing that connection; installation cost alone cannot determine the reliability choice.
Minimum installation cost and stronger reliability are different objectives.
Uncertain travel times
Route A is modelled at 17 min, including a wait that may increase by up to 5 min. Route B takes 18 min. Is A definitely faster?
Worked solution
- 1
Calculate the nominal advantage.
At the modelled times, route A is only one minute faster.
- 2
Apply the waiting uncertainty.
The stated possible wait increase extends A’s time from seventeen up to twenty-two minutes.
- 3
Compare with route B.
A wait increase above one minute makes A slower than B; an increase of one minute gives a tie.
- 4
Conclude about certainty.
A is not definitely faster. More information about the wait or a decision criterion for uncertainty is needed to recommend a route.
No; compare waiting uncertainty before recommending a route.
Try it yourself.
Check your understanding
Should a stock count of −3 be rounded to zero and accepted?
Show answer & reasoning
No.
No. A stock count of negative three violates a physical count constraint. Reconcile the records or model; replacing it by zero would conceal the discrepancy.
Keep practising
6 worksheet questions · 12 marks
Read the questions online
- A 12 dollars route uses a closed bridge. The cheapest permitted route costs 15 dollars. Which is recommended?2 marks
- A flow network uses L/s but demand is 720 L/min. Convert demand.2 marks
- A cable model weights edges by distance and ignores rocky terrain. Suggest a specific improvement.2 marks
- Trees cost 9,200 dollars and 9,250 dollars based on edge estimates rounded to 100 dollars. Explain the uncertain ranking.2 marks
- Maximum flow is 40 units/hour and demand is 45. Is the system adequate?2 marks
- A subtotal of 180 dollars excludes one fixed 25 dollars delivery fee. State the corrected total and assumption.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A 12 dollars route uses a closed bridge. The cheapest permitted route costs 15 dollars. Which is recommended?2 marks
The cheaper route is infeasible.
Optimise only over permitted routes.
The 15 dollars route. - A flow network uses L/s but demand is 720 L/min. Convert demand.2 marks
There are 60 seconds per minute.
.
12 L/s. - A cable model weights edges by distance and ignores rocky terrain. Suggest a specific improvement.2 marks
Equal distances can cost different amounts to excavate.
Costs better match a cost-minimisation objective.
Use terrain-dependent installation costs as edge weights. - Trees cost 9,200 dollars and 9,250 dollars based on edge estimates rounded to 100 dollars. Explain the uncertain ranking.2 marks
Several edge values may vary inside their rounding intervals.
Obtain better estimates or test sensitivity.
The 50 dollars difference may be smaller than estimation error. - Maximum flow is 40 units/hour and demand is 45. Is the system adequate?2 marks
.
Demand exceeds capacity.
No; it is short by 5 units/hour. - A subtotal of 180 dollars excludes one fixed 25 dollars delivery fee. State the corrected total and assumption.2 marks
.
Add the fixed fee once, not for every product.
205 dollars assuming one delivery and no other charges.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: matrices and networks. Original learning resources follow the approved Ascent Library chapter map.
Official curriculum source ↗