Translate 'at most' and 'at least' into inequalities.

'At most' is \leq; 'at least' is \geq.

What is the feasible region?

The set of points satisfying every constraint.

Which side of a boundary line do you shade?

The unwanted side, so that the feasible region is left unshaded.

How do you find a vertex of the feasible region?

From an axis intercept, or by solving two boundary equations simultaneously.

How does the objective line method work for maximising?

Slide the line parallel to itself away from the origin to the last vertex it touches.

How does the objective line method work for minimising?

Slide the line parallel to itself towards the origin to the first vertex it touches.

Where does the optimal value of the objective always occur?

At a vertex of the feasible region.

Describe the vertex testing method.

Find every vertex, evaluate the objective at each, and choose the greatest or the least.

How do you check that a point is feasible?

Substitute it into every constraint, including x\geq0 and y\geq0.

What should a final answer include?

The values of the variables and the value of the objective, in context with units.

Formulating and graphical solution of LPsAQA A-Level Further Maths: Flashcards

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What are the four steps in formulating an LP?

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What are the four steps in formulating an LP?: Define the decision variables, write the objective function, write the constraints as inequalities, and add the non-negativity constraints.
Translate 'at most' and 'at least' into inequalities.: 'At most' is $\leq$ ; 'at least' is $\geq$ .
Write 'at least twice as many chairs ( $y$ ) as tables ( $x$ )'.: $y\geq2x$ .
What is the feasible region?: The set of points satisfying every constraint.
Which side of a boundary line do you shade?: The unwanted side, so that the feasible region is left unshaded.
How do you find a vertex of the feasible region?: From an axis intercept, or by solving two boundary equations simultaneously.
What is the gradient of the objective line $ax+by=k$ ?: $-\frac ab$ .
How does the objective line method work for maximising?: Slide the line parallel to itself away from the origin to the last vertex it touches.
How does the objective line method work for minimising?: Slide the line parallel to itself towards the origin to the first vertex it touches.
Where does the optimal value of the objective always occur?: At a vertex of the feasible region.
Describe the vertex testing method.: Find every vertex, evaluate the objective at each, and choose the greatest or the least.
How do you check that a point is feasible?: Substitute it into every constraint, including $x\geq0$ and $y\geq0$ .
What should a final answer include?: The values of the variables and the value of the objective, in context with units.

Exam questions on Formulating and graphical solution of LPs

A workshop makes $x$ tables and $y$ chairs each week. Each table needs 4 hours of carpentry and 2 hours of finishing. Each chair needs 2 hours of carpentry and 3 hours of finishing. There are 40 hours of carpentry and 30 hours of finishing available each week. The workshop must make at least twice as many chairs as tables. The profit is £50 per table and £30 per chair, and the workshop wants to maximise its weekly profit.
Write down the objective function and explain why $x\geq0$ and $y\geq0$ must be included as constraints.2 marks
A farmer mixes $x$ kg of feed $X$ and $y$ kg of feed $Y$ for each day. Each kilogram of $X$ has 3 units of protein and 1 unit of fibre and costs £3. Each kilogram of $Y$ has 2 units of protein and 2 units of fibre and costs £4. Each day the mixture must contain at least 24 units of protein and at least 12 units of fibre, and it must weigh at most 10 kg. The farmer wants to minimise the daily cost.
Explain why a mixture of 5 kg of $X$ and 4 kg of $Y$ is not allowed.2 marks
Maximise $P=3x+2y$ subject to $2x+y\leq14$ , $x+2y\leq10$ , $x\geq0$ and $y\geq0$ .
Find the coordinates of the vertices of the feasible region.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).