Edexcel A Level Further Maths · Decision 1
Linear Programming (LP) Problems
Big idea: Linear programming is just a smart way of turning a real-world "how do I get the best result within my limits?" story
into maths — you write down what you're allowed to do (the constraints), what you're trying to achieve (the objective),
and then optimise it.
Quick Summary
- Linear programming (LP) solves problems where you need to maximise or minimise something, while staying inside a set of restrictions (constraints).
- Decision variables (usually $x$, $y$, $z$) are the "things" you're allowed to vary — like the number of chairs or tables made.
- The objective function is the thing you're actually trying to optimise — profit ($P$) to maximise, or cost ($C$) to minimise — written in terms of the decision variables.
- Constraints are the real-world limits (time, materials, space) turned into inequalities involving the decision variables.
- The non-negativity constraint ($x, y \geq 0$) should almost always be included, since decision variables represent quantities of real things — you can't make −3 chairs!
- Formulating an LP problem always follows the same 4 steps: define variables → write constraints → write objective function → present formally.
1. What Is Linear Programming?
Think about a furniture manufacturer who makes chairs and tables. Every day they have to decide:
They can't just make infinite chairs and tables — they're limited by
how much timber they have, how many hours their workshop runs, and how long the varnish takes to dry.
At the same time, they want to make as much profit as possible.
That's exactly what linear programming is for. It's a mathematical method for problems that:
- involve working within a set of constraints (restrictions/limits), and
- require some quantity to be maximised or minimised.
LP shows up constantly in finance and manufacturing — anywhere a business has limited resources
(time, money, raw materials, staff) and wants the best possible outcome (most profit, least cost).
Analogy
Imagine you have £20 to spend at a sweet shop and you want to buy the combination of sweets that
makes you happiest. Your "constraint" is the £20 budget. Your "objective" is maximising happiness.
LP is the maths that finds your best combination — that's genuinely all it is, just formalised.
2. Decision Variables
What are they?
Decision variables are the quantities in the problem that can be varied — the things you actually get to choose.
They're usually labelled $x$, $y$, $z$ (in that order, for as many variables as you need).
For our furniture manufacturer, they could choose to make $x$ chairs and $y$ tables per day. Changing $x$ and $y$
changes everything else in the problem — most importantly, it changes the profit.
For example: making 12 chairs and 3 tables might give £200 profit, while making 3 chairs and 12 tables
might only give £160 profit. Same "total effort," very different outcome — that's why choosing the right
values of $x$ and $y$ actually matters, and why we need LP to find the combination rather than
just guessing.
3. The Objective Function
objective function
4. Constraints & Inequalities
restrictions
inequality