What does 'linear' mean in linear programming?

Each term is a constant times one variable; no products or powers of variables.

How is a resource constraint built?

Sum of (amount per unit \times number of units) \le amount available.

How do you form an objective function for profit?

Sum of (profit per unit \times number made); then state maximise.

Which non-negativity conditions are normally stated?

x\ge0 and y\ge0 (replaced by a stronger lower limit if one is given).

When must variables be integers?

When the items are indivisible, such as tables, parcels or people.

What is a feasible point?

One that satisfies every constraint.

How do you test a plan?

Substitute it into every constraint, including non-negativity.

How do you write 'at least 25% of the total are y'?

y\ge0.25(x+y), which simplifies to 3y\ge x.

Formulating linear programsEdexcel International A Level Maths: Flashcards

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What are the three parts of a formulated linear program?

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What are the three parts of a formulated linear program?: Decision variables, an objective function, and constraints.
What does 'linear' mean in linear programming?: Each term is a constant times one variable; no products or powers of variables.
How do you write 'at most 80 hours' in symbols?: The hours used $\le80$ .
How do you write 'must provide at least 15 units' in symbols?: The units provided $\ge15$ .
How do you write 'at least twice as many $y$ as $x$ '?: $y\ge2x$
How do you write 'the number of $y$ must not exceed the number of $x$ '?: $y\le x$
How is a resource constraint built?: Sum of (amount per unit $\times$ number of units) $\le$ amount available.
How do you form an objective function for profit?: Sum of (profit per unit $\times$ number made); then state maximise.
Which non-negativity conditions are normally stated?: $x\ge0$ and $y\ge0$ (replaced by a stronger lower limit if one is given).
When must variables be integers?: When the items are indivisible, such as tables, parcels or people.
What is a feasible point?: One that satisfies every constraint.
How do you test a plan?: Substitute it into every constraint, including non-negativity.
How do you write 'at least 25% of the total are $y$ '?: $y\ge0.25(x+y)$ , which simplifies to $3y\ge x$ .

Exam questions on Formulating linear programs

A bakery makes two types of bread. Each batch of loaf A needs 3 kg of flour and 2 hours of oven time. Each batch of loaf B needs 2 kg of flour and 4 hours of oven time. Each day there are 60 kg of flour and 80 hours of oven time available. The profit is £20 per batch of A and £30 per batch of B. The bakery makes $x$ batches of A and $y$ batches of B each day, and wants to maximise its daily profit.
Write down the objective function, saying whether it is to be maximised or minimised, and write down the non-negativity conditions.2 marks
A farmer mixes two feeds, X and Y, for cattle. Feed X costs £0.40 per kg and contains 3 units of protein and 1 unit of vitamin per kg. Feed Y costs £0.60 per kg and contains 1 unit of protein and 2 units of vitamin per kg. Each day the mixture must provide at least 15 units of protein and at least 12 units of vitamin, and the farmer can store at most 20 kg of mixture. The farmer uses $x$ kg of X and $y$ kg of Y each day, and wants to minimise the daily cost.
Write down the inequality for the vitamin requirement and the inequality for the storage limit.2 marks
A workshop makes $x$ tables and $y$ chairs each week. A table needs 6 hours of carpentry and 1 hour of painting. A chair needs 2 hours of carpentry and 1 hour of painting. There are 120 hours of carpentry and 40 hours of painting available each week. The workshop must make at least 4 tables and at least three times as many chairs as tables. The profit is £50 per table and £15 per chair, and the workshop wants to maximise its weekly profit. Only whole tables and chairs can be made.
Write down the inequalities for the hours of carpentry and painting, and for the non-negativity of the variables.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).