IA idea · Optimisation & linear programming

How few staff can a café roster? An integer programming rota

AI SLAI HLAA SL Solid Also in: Finance, Statistics

Research question

Given the number of staff a café needs in each hour and shifts that must be a fixed length, what rota uses the fewest staff-hours, and how much would allowing shorter shifts save?

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

Scheduling is one of the biggest real uses of optimisation. Hourly demand you count yourself turns the constraints into your own data, and comparing shift rules gives a clear decision.

The mathematics you'll need

Variables for the number of staff starting each hour
Covering constraints (one per hour) and an objective
Integer solutions; solving with a spreadsheet solver, explained
Lower bounds and checking optimality
Comparing scenarios

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

Count customers each hour at a café, canteen or shop over several days (or ask a manager for typical staffing needs); convert counts to staff needed with a stated rule.

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

Collect hourly demand and set a staffing rule.
Write the covering constraints for a fixed shift length.
Solve by hand for a short day, then with a solver.
Compare shift-length rules and cost them.
Reflect on breaks, absence and the rule that turned customers into staff.

Pitfalls that cost marks

Inventing demand instead of counting it.
Not explaining how the solver finds the answer.
Ignoring legal breaks or minimum shift lengths.

Showing personal engagement

Count the customers yourself on different days.
Talk to a manager about the rules they actually use.
Find the cheapest rule that still covers the busiest day.

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	Variables for the number of staff starting each hour; Covering constraints (one per hour) and an objective
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	Variables for the number of staff starting each hour; Covering constraints (one per hour) and an objective
AI HL	Good fit	Variables for the number of staff starting each hour; Covering constraints (one per hour) and an objective

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: inventing demand instead of counting it — 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

Make demand random (fit a Poisson model to your counts) and choose the rota that covers demand on, say, 90% of days.

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.

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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).