IA idea · Optimisation & linear programming

Where should a shared mast go? Minimising total distance to villages

AA HLAI HLAA SL Ambitious Also in: Geometry & Voronoi, Numerical methods

Research question

For three or more real villages, where should a single shared mast or depot be placed to minimise the total cable or travel distance, how does the answer compare with the centroid, and how quickly does an iterative method find it?

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

For three points there is a geometric answer (the 120° Fermat point); for more points there is no formula and you need an iteration. Moving from one to the other is a genuine piece of exploration.

The mathematics you'll need

Distance function of two variables
The Fermat point and its 120° property (proved with geometry or vectors)
Weiszfeld's iteration for the geometric median (new: explain it)
Comparison with the centroid (which minimises the sum of squared distances)
Convergence of the iteration

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

Read coordinates of real villages, schools or farms from OpenStreetMap and state your scale and origin.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
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

Choose real sites and set up coordinates.
Solve the three-site case and verify the 120° property.
Introduce the iteration for more sites and show one step by hand.
Compare the result with the centroid and with a grid search.
Reflect on roads, terrain and weighting by population.

Pitfalls that cost marks

Confusing the centroid with the point of minimum total distance.
Running an iteration without checking it converges or explaining why.
Ignoring that real cables follow roads.

Showing personal engagement

Use places you know.
Weight each village by its population and see the point move.
Check your answer with a quick experiment (string and weights on a map, if you can).

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Fits — ambitious at SL	Distance function of two variables; The Fermat point and its 120° property (proved with geometry or vectors)
AA HL	Good fit	Distance function of two variables; The Fermat point and its 120° property (proved with geometry or vectors)
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Distance function of two variables; The Fermat point and its 120° property (proved with geometry or vectors)

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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: confusing the centroid with the point of minimum total distance — 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

Prove that the centroid minimises the sum of squared distances, or explore weighted versions and when the optimum sits exactly on one village.

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 What is the quickest way to deliver a leaflet to every house on my estate? (AI HL) 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

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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).