IA idea · Optimisation & linear programming

What is the cheapest way to hire coaches for a school trip?

AA SLAI SLAA HLAI HL Accessible Also in: Finance

Research question

Given the real prices and seat counts of two or three vehicle sizes, a limit on the number of drivers and the size of the year group, which combination of vehicles carries everyone at the lowest cost, and how does the answer change as the group grows?

Adapt it: change the place, the data or the comparison until the question is yours.

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Why it makes a good exploration

It is a real decision your school makes every year, the numbers are easy to get, and the twist (you can't hire 2.6 coaches) forces you to think about the difference between the best continuous answer and the best whole-number one.

The mathematics you'll need

Linear inequalities and their graphs
Feasible regions and vertices (linear programming, new: explain it)
Objective functions and iso-cost lines
Integer solutions near the continuous optimum
Piecewise cost as the group size changes

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Ask your school office or two or three local coach firms for real quotes (seats, price, driver limits). No public dataset is needed; record the date of each quote.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Define the variables and write every constraint in words, then in symbols.
Draw the feasible region and find the vertices by solving pairs of equations.
Evaluate the cost at each vertex and slide an iso-cost line to confirm the optimum.
Find the best whole-number solution and compare it with the continuous one.
Repeat for different group sizes and reflect on the jumps in cost per student.

Pitfalls that cost marks

Treating the continuous optimum as the answer when vehicles come in whole numbers.
Forgetting a constraint such as staff seats or a maximum number of drivers.
Using made-up prices; real quotes make the decision yours.

Showing personal engagement

Use your own school's real trip and real quotes.
Predict the cheapest mix before calculating, then explain why you were wrong (or right).
Find the group size at which an extra minibus becomes worth it.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Linear inequalities and their graphs; Feasible regions and vertices (linear programming, new: explain it)
AA HL	Fits, but add an HL technique	Linear inequalities and their graphs; Feasible regions and vertices (linear programming, new: explain it)
AI SL	Good fit	Linear inequalities and their graphs; Feasible regions and vertices (linear programming, new: explain it)
AI HL	Fits, but add an HL technique	Linear inequalities and their graphs; Feasible regions and vertices (linear programming, new: explain it)

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Optimise a decision that is really yours or your school's (a timetable, a budget, a delivery), gather the real constraints yourself, and say which ones you chose to ignore and why.

Reflection (D)

Compare the mathematical optimum with what people actually do, and test how sensitive the optimum is: which constraint, if relaxed a little, would change the answer most? For this idea, start with: treating the continuous optimum as the answer when vehicles come in whole numbers — say how it affects your answer.

Use of mathematics (E)

SL: An objective function and constraints set up from the context, solved correctly (graphically for two variables, or with differentiation), the optimum checked and interpreted, and any new method such as linear programming explained in your own words.

HL: Optimisation with two or more variables, a justified numerical search, a proof that the optimum lies at a vertex, or a sensitivity analysis with calculus, used because the problem needs it.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Add a third vehicle type (three variables) and explain why the corner principle still holds, or compare the cost per student across group sizes as a piecewise function.

Extending it for HL

Add a second variable or a non-linear constraint, use a numerical search where calculus alone is not enough, and analyse how the optimum moves as a parameter changes.

See a complete IA, marked

Our annotated exemplar Should I lease or buy my first car? (AI SL) asks a different question, but shows how a complete optimisation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).