The idea, step by step
Optimal within the model
An optimum is best under stated assumptions and an objective. A minimum tree may minimise cable cost but provide no backup connection. A shortest-time model may ignore queues. A stock-cost product may assume all items are available at the same price. Name the objective before recommending a result.
A solution violating a hard constraint is infeasible, regardless of its attractive cost. Examples include a closed road, a negative physical stock count or flow above capacity. Eliminate infeasible options before comparing totals.
Make the conclusion useful
State the selected route, edges or quantities and the calculated total with units. Explain why it solves the intended problem. Give a specific limitation rather than writing only that the model is unrealistic. Identify information to collect or a constraint to add.
Sensitivity checks whether a plausible change would alter the decision. Test alternatives when estimates are uncertain. Two totals only slightly apart may not establish a reliable winner. Retain precision in calculations and round the reported answer appropriately.
A common trap
The smallest computed number is always the best real choice.
Instead: Check objective, feasibility and omitted factors first.
See the thinking
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.
- The tree gives low installation cost but lacks alternative routes.
- The additional link raises the total to 54,000 dollars.
- Its value depends on the cost and likelihood of an outage.
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?
- The nominal difference is only 1 min.
- A could take up to 22 min.
Check your understanding
Should a stock count of −3 be rounded to zero and accepted?
Show answer
No.
Reconcile the model or records; rounding would hide infeasibility.
Your turn
6 questions · 12 marks · Work at your own pace.
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 ↗