Maths IA · Modelling · topic 38 of 40

Budget allocation: how does the best plan change? · IB Maths IA idea: guide, sample graph and step-by-step plan

good choice. the sensitivity is the maths!

Same budget, several ways to spend it. The best plan sits on a corner, and a small price rise can send it to another corner. Your job is to see it coming.

Difficulty●●○ Medium
KitReal prices and limits, a spreadsheet or GDC
TimeAbout 12 to 15 hours
FitsAI, SL and HL
  • ✓ you like decisions with clear numbers
  • ✓ you can find real prices and limits
  • ✓ you enjoy asking 'what if the price changes?'

The big idea

A fixed budget, several options and limits on how much of each: that is linear programming. The feasible region is a polygon and the best plan sits at a corner. The real maths is the question that follows: what happens when costs or benefits change? Sensitivity analysis shows which inputs the answer depends on and where the best plan jumps to another corner. Pick a real decision, set it up carefully, and do more than draw a graph. Linear programming is not a named topic in the course, so show the course maths inside it: systems of equations, inequalities, functions and, at HL, matrices.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Use a real decision

A club budget, a party, a school trip, a small shop's stock order.

→ Real prices and limits, and a reason to care about the answer.
Insist on whole numbers

You cannot buy 3.4 tables: restrict to whole numbers and compare.

→ The corner may not be allowed; you search whole-number plans and discuss the difference.
Test the sensitivity

Change one cost or benefit at a time and track the best value and the best plan.

→ A graph of best value against the changed input, with kinks where the best plan changes.
Make the budget the variable

Find the best value for every budget from zero up to some maximum.

→ A piecewise function whose gradients are the value of one more dollar.
Solve by algebra

Find the corners by solving pairs of constraint equations, or with a matrix inverse.

→ The course maths becomes visible. At HL, matrices are a natural link.
Add a third option

Work in three dimensions, or reduce it back with a clear argument.

→ The picture changes; justify how you cope without a flat graph.

Your Maths IA research question: run the check

write it FIRST, then check it

Write your own question first. Then make sure it ticks every box:

  • Names the real decision, the options and where the prices and limits come from.
  • States the objective you maximise (benefit, enjoyment, items) and how you measure it in numbers.
  • Names the input you change in the sensitivity analysis, and over what range.
  • Says whether the plans must be whole numbers.
  • Shows maths beyond a graph: equations solved, corners checked, a justification.

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The picture to draw

Each constraint cuts the plane. The shaded region holds every affordable plan. Slide the objective line outwards until it just touches the last corner. Draw yours with your own constraints.

yxfeasible region:every plan you can affordbudget line: slides as B changesobjective line:slide to the last corneroptimum is a corner: check its neighbours

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; the decision and why it matters to you; what you expect before solving.
The decision and the data~2 pages
Options, prices, limits and their sources; any simplifications and why.
Setting up the model~2.5 pages
Variables defined; the objective and each constraint as an inequality, with units.
Solving~3 pages
The feasible region; the corners found by solving equations; the best corner checked against its neighbours. HL: matrices.
Sensitivity~3 pages
Change one input at a time; where the best plan changes; the value of an extra dollar.
Whole numbers or another twist~1.5 pages
The same problem with a restriction or an extra option, and what changes.
Conclusion~1 page
The recommended plan, how sure you are, and which input to check first.
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

mark the kinks ✓

Made-up numbers to show the form of a strong graph, not the answer: the best total benefit for each budget, with two options. Yours will look different. example only

0102030405060700102030405060708090100110Budget B / $Best total benefit V (points)original costscost of option y up 25%optima you solved forthe kink: the best planswitches option hereILLUSTRATIVE · MADE-UP DATA
  • Budget on the x-axis, best benefit on the y-axis; say what one 'point' of benefit is.
  • Each marked point is an optimum you solved for, not a measurement: say how you solved it.
  • The slope of each straight piece is the value of one more dollar. Explain why it changes.
  • A kink is where the best plan changes corner: find it by algebra, not by eye.
  • Another graph to consider: best value against the cost of one option, with the changes of plan marked.

Which change moves the best value most?

rank them: don't just list

Change one input at a time, re-solve, and rank by the effect on the best value. These numbers are made up; with your own, the ranking tells you which price or limit to check most carefully.

Benefit of option y down 25%14.5%Cost of option y up 25%13.2%Budget down 10%8.8%Benefit of option x down 20%5.3%check this input first% change in the best total benefit at B = 40ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

solve a small version first

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Choose the decisionPick a real decision with real prices. List the options, the limits and the budget.
  2. PredictBefore solving, say which option you think wins and which input the answer is most sensitive to.
  3. Trial run (1 lesson)Solve a two-option version by drawing. Do the constraints behave? Is the objective measurable in numbers?
  4. Set up the modelDefine variables, write each constraint as an inequality, and check the units agree.
  5. Solve with algebraFind the corners by solving pairs of equations. Test each in the objective, including the neighbours of your best one.
  6. Check for senseDoes the answer fit the real decision? Which constraints are binding, and why?
  7. Test the sensitivityChange one input at a time over a range. Find where the best plan changes, and the value of an extra dollar.
  8. Go furtherSL: whole numbers or a second change. HL: matrices, a proof, or a third option handled with care.
  9. Write, then cutWrite in layout order. One worked solve, then a table of results.
  10. Check before your teacher's draftCheck notation, units, which constraints are binding, and reflection all the way through.

Maths you'll need

Traps to avoid in this Maths IA

  • Drawing the graph and reading off the answer: that is mostly prior learning, so show the algebra.
  • Objective and constraints in different units (dollars in one, hours in another) with no conversion.
  • A sensitivity analysis that changes three things at once, so you cannot say which mattered.
  • A non-integer answer for things you cannot split.
  • Constraints chosen to make the answer neat, not because the real decision has them.
  • Not saying which inequalities are binding at the optimum.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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