Linear programming: formulating problems

Decision variables

Start with a sentence: 'Let x be the number of chairs made each day and y the number of tables made each day.' Say what is being counted and over what period.

Constraints

Turn each limit into an inequality in x and y, for example 2x + 3y ≤ 60 (hours). Include the non-negativity constraints x ≥ 0, y ≥ 0. Write each condition as its own inequality and simplify (divide through by common factors) where you can.

Comparative and percentage constraints

• 'At least twice as many chairs as tables': x ≥ 2y.
• 'Tables are at least 25% of the total': y ≥ 0.25(x + y), which simplifies to 3y ≥ x.
• 'At most 40 items in total': x + y ≤ 40.

Objective function

Write what is maximised or minimised, for example 'maximise P = 30x + 40y'. Always include the word maximise or minimise. Convert units first (for example, pence to pounds, or minutes to hours). Simplex is not needed in D1: problems are solved graphically.

Worked example

A firm makes x chairs and y tables. A chair takes 2 hours of labour and a table 3 hours; 60 hours are available. A chair uses 4 m of wood and a table 2 m; 64 m are available. At least 4 tables are made. Profit is £30 per chair and £40 per table.

1. Labour: 2x + 3y ≤ 60
2. Wood: 4x + 2y ≤ 64, which simplifies to 2x + y ≤ 32
3. y ≥ 4, x ≥ 0
4. Maximise P = 30x + 40y

Answer: Maximise P = 30x + 40y subject to 2x + 3y ≤ 60, 2x + y ≤ 32, y ≥ 4, x ≥ 0

Check yourself

1. Let x be the number of red bulbs and y the number of white bulbs. 'There must be at least twice as many red bulbs as white bulbs.' Which inequality is correct?
   • 2x ≥ y
   • x ≥ 2y
   • y ≥ 2x
   • x + y ≥ 2
   Show answer

   x ≥ 2y. Red must be at least 2 × white: x ≥ 2y. Test: y = 5 needs x ≥ 10.

2. Let x and y be the numbers of two products. 'At least 30% of the total must be product y.' Which simplified inequality is correct?
   • 3y ≥ 7x
   • y ≥ 30
   • y ≥ 0.3x
   • 7y ≥ 3x
   Show answer

   7y ≥ 3x. y ≥ 0.3(x + y) gives 0.7y ≥ 0.3x, so 7y ≥ 3x.

3. A chair uses 4 m of wood and a table uses 2 m; at most 64 m are available. Write the wood constraint 4x + 2y ≤ 64 in its simplest form, giving the left-hand side only, with x and y as above.
   Show answer

   2x + y. Divide 4x + 2y ≤ 64 by 2: 2x + y ≤ 32.

4. Labour time: product X takes 20 minutes and product Y takes 45 minutes. 15 hours of labour are available. Writing the constraint as 20x + 45y ≤ k (in minutes), find k.
   Show answer

   900. 15 hours = 900 minutes.

5. Profit is 80p per x and £1.20 per y, to be given in pounds. What is the objective?
   • Maximise P = 0.8x + 1.2y
   • Minimise P = 0.8x + 1.2y
   • Maximise P = 80x + 1.2y
   • P = 0.8x + 1.2y
   Show answer

   Maximise P = 0.8x + 1.2y. Convert 80p to £0.80 and include the word maximise.

6. Which constraints are almost always needed but are often forgotten?
   • x ≤ 0 and y ≤ 0
   • x + y ≥ 1
   • x ≥ 0 and y ≥ 0
   • x = y
   Show answer

   x ≥ 0 and y ≥ 0. You cannot make a negative number of items.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Decision Mathematics 1 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.