Linear programming graphical method

2 Consider the following linear programming problem.
Maximise $$\mathrm { P } = 6 x + 7 y$$ subject to $$\begin{aligned} & 2 x + 3 y \leqslant 9
& 3 x + 2 y \leqslant 12
& x \geqslant 0
& y \geqslant 0 \end{aligned}$$

1. Use a graphical approach to solve the problem.
2. Give the optimal values of \(x , y\) and P when \(x\) and \(y\) are integers.

3 Solve the following LP problem graphically.
Maximise \(2 x + 3 y\)
subject to \(\quad x + y \leqslant 11\) $$\begin{aligned} 3 x + 5 y & \leqslant 39
x + 6 y & \leqslant 39 . \end{aligned}$$

6 A company manufactures two types of potting compost, Flowerbase and Growmuch. The weekly amounts produced of each are constrained by the supplies of fibre and of nutrient mix. Each litre of Flowerbase requires 0.75 litres of fibre and 1 kg of nutrient mix. Each litre of Growmuch requires 0.5 litres of fibre and 2 kg of nutrient mix. There are 12000 litres of fibre supplied each week, and 25000 kg of nutrient mix. The profit on Flowerbase is 9 p per litre. The profit on Growmuch is 20 p per litre.

1. Formulate an LP to maximise the weekly profit subject to the constraints on fibre and nutrient mix.
2. Solve your LP using a graphical approach.
3. Consider each of the following separate circumstances.
   (A) There is a reduction in the weekly supply of fibre from 12000 litres to 10000 litres. What effect does this have on profit?
   (B) The price of fibre is increased. Will this affect the optimal production plan? Justify your answer.
   [0pt] (C) The supply of nutrient mix is increased to 30000 kg per week. What is the new profit? [1]

3 Use a graphical approach to solve the following LP. $$\begin{aligned} & \text { Maximise } \quad 2 x + 3 y
& \text { subject to } \quad x + 5 y \leqslant 14
& \quad x + 2 y \leqslant 8
& \quad 3 x + y \leqslant 21
& \quad x \geqslant 0
& y \geqslant 0 \end{aligned}$$ Section B (48 marks)