IA idea · Optimisation & linear programming

How many of each? Planning a charity bake sale with linear programming

AA SLAI SL Accessible Also in: Finance

Research question

With limited oven time, ingredient budget and preparation time, how many batches of each of two or three bakes should a charity stall make to maximise profit, and which constraint is worth relaxing first?

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

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A small, real business decision with every number under your control. The method (draw the constraints, find the corners) is new but accessible, and the question of which constraint to relax gives a natural second layer.

The mathematics you'll need

Linear inequalities and feasible regions
Vertices found by simultaneous equations
Objective function and iso-profit lines
Sensitivity: how the optimum moves when a constraint 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

Time your own baking, price the ingredients and use the real oven capacity; run the stall if you can and record sales.

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

Measure times and costs for each bake.
Write and graph the constraints; find the vertices.
Find the maximum profit and check it with an iso-profit line.
Relax each constraint in turn and measure the gain.
Compare the plan with what actually sold.

Pitfalls that cost marks

Assuming you can sell everything you make; consider a demand constraint.
Rounding the optimum without checking feasibility.
Too few constraints, so the answer is obvious without the method.

Showing personal engagement

Run the stall and compare plan with reality.
Ask friends how many they'd buy at a given price to set a demand limit.
Decide which constraint you would pay to relax (an extra oven hour?).

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 feasible regions; Vertices found by simultaneous equations
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Linear inequalities and feasible regions; Vertices found by simultaneous equations
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

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: assuming you can sell everything you make; consider a demand constraint — 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

Let the selling price vary and model demand with a linear function, so the profit becomes quadratic; compare the two approaches.

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.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

Get the free checklist

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).