IA idea · Optimisation & linear programming

Cutting shelves from planks with the least waste

AA SLAA HLAI SLAI HL Solid Also in: Pure maths

Research question

When shelves of several lengths must be cut from standard-length planks, which cutting patterns minimise the number of planks (and the waste), and how close does a simple 'longest first' rule get to the best possible answer?

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

The cutting-stock problem is used in real factories. A small version can be solved completely by listing patterns, which shows exactly why integer problems are harder than continuous ones, and lets you test a simple rule of thumb.

The mathematics you'll need

Listing cutting patterns systematically (counting)
Integer linear programming, explained on a small case
Lower bounds: total length ÷ plank length
Greedy algorithms and their worst cases

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

Use a real job: shelves for a room, a DIY project or the school's workshop, with real plank lengths from a hardware shop.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

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

State the order and plank length; find a lower bound.
List all efficient cutting patterns.
Find the optimum (by hand for a small order, then with a solver).
Test the 'longest first' rule against it on several orders.
Reflect on saw width, offcuts that can be reused and real cost.

Pitfalls that cost marks

Missing cutting patterns because the listing is not systematic.
Ignoring the width of the saw cut.
Testing the rule on only one order.

Showing personal engagement

Use a project you or your family are actually building.
Invent an order that makes the greedy rule look as bad as possible.
Check the plan by cutting card strips.

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	Listing cutting patterns systematically (counting); Integer linear programming, explained on a small case
AA HL	Good fit	Listing cutting patterns systematically (counting); Integer linear programming, explained on a small case
AI SL	Good fit	Listing cutting patterns systematically (counting); Integer linear programming, explained on a small case
AI HL	Good fit	Listing cutting patterns systematically (counting); Integer linear programming, explained on a small case

Level: Solid. Needs some independent work beyond class examples. 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: missing cutting patterns because the listing is not systematic — 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 a bound on how badly the greedy rule can do, or add a second plank length with a different price.

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 How far can a stack of books lean over the edge of a table? (AA 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

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.

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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).