Revise Linear Programming (LP) problems for Further Decision Mathematics 1 WDM11 (AS Level) — revision notes and instant AI marking. Free to start.
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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.
Key Point
Decision variables = the "number of things" you can choose to produce/buy/use
The values they're to take are controlled entirely by the constraints — a manufacturer can't just pick $x = 1{,}000{,}000$ chairs and expect that to be valid!
3. The Objective Function
The objective function is the whole point of the problem — it's the quantity you're actually trying to optimise.
It's written as a function of the decision variables (so if your variables are $x$ and $y$, your objective function
will be some expression in $x$ and $y$).
If you're trying to maximise something (like profit), the objective function is usually called $P$.
If you're trying to minimise something (like cost), the objective function is usually called $C$.
Going back to our furniture example: if a chair makes £30 profit and a table makes £40 profit, and the manufacturer
makes $x$ chairs and $y$ tables, then their total daily profit is:
Objective Function Example
$P = 30x + 40y$
In plain English: "total profit equals £30 for every chair made, plus £40 for every table made." This is the expression you want to make as large as possible.
Why is it called "linear"?
Because the objective function and every constraint are straight-line (degree 1) expressions in the decision
variables — no $x^2$, no $xy$, no square roots. That's what makes the whole problem solvable with straight
lines and regions on a graph.
4. Constraints & Inequalities
Constraints are the restrictions the problem is bound by — the reasons you can't just pick huge values
for your decision variables. Mathematically, each constraint is written as an inequality involving the
decision variables.
For our furniture manufacturer, constraints might include: the amount of timber available, how many hours
the factory can run, and how much drying space there is for varnish.
The non-negativity constraint
Because decision variables almost always represent a "number of things" (chairs, tables, litres, staff hours),
they can't be negative — you can't make −2 chairs! So we almost always add:
Always Include
$x, y \geq 0$
This is called the non-negativity constraint. Include it even if the question doesn't explicitly mention it — examiners expect it.
Simplify where possible!
If a constraint comes out as $2x + 4y \leq 8$, always simplify it down to $x + 2y \leq 4$ (divide through by the
common factor). Examiners want the simplest form, and it makes graphing the constraint much easier later on.
5. How to Formulate an LP Problem (The 4 Steps)
Formulating an LP problem just means translating a wordy real-world scenario into clean, formal maths.
Every single question follows the same four steps, in the same order:
Step 1 — Define the decision variables. Read the question carefully and identify what "number of things" can be varied. State clearly what $x$ and $y$ (and $z$, if needed) represent, including the units/timeframe (e.g. "per day").
Step 2 — Write each constraint as an inequality. Go through the question's restrictions one at a time, translate each into an inequality in terms of the decision variables, and simplify where possible.
Step 3 — Determine the objective function. Identify what's being maximised (usually $P$, profit) or minimised (usually $C$, cost), and write it as an expression in the decision variables.
Step 4 — Write the problem out formally. Present everything in the standard layout, including the non-negativity constraint.
Standard Formal Layout
Maximise (or Minimise)
<objective function>
subject to
<constraint 1>
<constraint 2>
... (etc)
$x, y \geq 0$
Worked Example — Full Walkthrough
Furniture Manufacturer Problem
A furniture manufacturer makes chairs and tables. Due to timber availability, they cannot make more than
a total of 10 chairs and tables per day. A chair takes 1 hour to produce, a table takes 2 hours, and the
factory can run for a maximum of 18 hours per day. The varnish on a chair takes 3 hours to dry, a table's
varnish takes 2 hours, and there are four drying zones each providing 6 hours of drying time per day.
A chair makes £30 profit, a table makes £40 profit. The manufacturer wants to maximise daily profit.
Formulate this as an LP problem.
Step 1 — Decision variables: Let $x$ = number of chairs made per day. Let $y$ = number of tables made per day.
Step 2 — Constraints:
Total items limited by timber: $x + y \leq 10$
Production time (1 hr/chair, 2 hr/table, max 18 hrs): $x + 2y \leq 18$
Drying time (3 hr/chair, 2 hr/table, max $4 \times 6 = 24$ hrs): $3x + 2y \leq 24$
Maximise
$P = 30x + 40y$
subject to
$x + y \leq 10$
$x + 2y \leq 18$
$3x + 2y \leq 24$
$x, y \geq 0$
Practice Question 1
A bakery makes cakes ($x$ per day) and pies ($y$ per day). Each cake uses 2 eggs, each pie uses 3 eggs,
and the bakery has 60 eggs available per day. Each cake takes 30 minutes of oven time, each pie takes
45 minutes, and the oven is available for 12 hours (720 minutes) per day. The bakery cannot make more
than 20 items in total per day. A cake makes £4 profit and a pie makes £6 profit. Formulate this as an
LP problem to maximise profit.
Practice Question 2
A farmer wants to decide how many sacks of wheat feed ($x$) and corn feed ($y$) to buy weekly to minimise
cost, while meeting minimum nutritional needs for the animals. Each sack of wheat feed provides 4 units of
protein and costs £10. Each sack of corn feed provides 2 units of protein and costs £6. The animals need
at least 40 units of protein per week. The farmer also has storage space for at most 15 sacks in total.
Formulate this as an LP problem.
What to Memorise
Linear Programming (LP)
A method for maximising or minimising a quantity, subject to a set of constraints, where every relationship involved is linear.
Decision Variables
The quantities that can be varied (usually $x$, $y$, $z$) — typically a "number of things" produced, bought, or used.
Objective Function
The expression (in terms of the decision variables) that needs optimising — $P$ for maximising (e.g. profit), $C$ for minimising (e.g. cost).
Constraints
The restrictions of the problem, written as inequalities in terms of the decision variables (e.g. time, materials, budget limits).
Non-Negativity Constraint
$x, y \geq 0$ — included because decision variables represent real, non-negative quantities of things.
The 4-Step Formulation Process
1) Define variables → 2) Write constraints as inequalities → 3) Determine objective function → 4) Write the formal LP statement.
Concepts Checklist
Exam Tips & Common Mistakes
Forgetting the non-negativity constraint
Even if the question doesn't mention it, always add $x, y \geq 0$ at the end of your formal write-up. Examiners
expect it every time — it's an easy mark to lose.
Not simplifying constraints
If your inequality comes out as $2x + 4y \leq 8$, simplify it to $x + 2y \leq 4$. Leaving it unsimplified can
cost marks and makes later graphing steps harder.
Mixing up ≤ and ≥
"At most" / "no more than" / "cannot exceed" → use $\leq$. "At least" / "no less than" / "a minimum of" → use
$\geq$. Read the wording of each constraint carefully — this is one of the most common places marks are lost.
Not defining variables with units/timeframe
"Let $x$ = number of chairs" is incomplete. Always specify the timeframe too: "Let $x$ = number of chairs made
by the manufacturer per day." Examiners want this precision.
Always follow the same structure
Maximise/Minimise → objective function → "subject to" → list of constraints → non-negativity constraint.
Sticking to this exact layout every time makes your answers instantly clear to an examiner (and to future you,
revising!).
Double-check by "plugging in words"
Once you've written a constraint, try reading it back in plain English and check it matches the original
sentence in the question. This quick sanity check catches most translation errors.
Also in the full note
4. Constraints & Inequalities
Exam Tips & Common Mistakes
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