1. A cheese factory is making a new cheese from mixing two products A and B, each made of three different types of milk - sheep, cow and goat milk. The compositions of A and B and prices ($/kg) are given as follows, Amount (litres) per 1000 kg of A and B Sheep Cow A 30 60 B 80 40 Goat 40 70 Cost ($/kg) 5 8 The recipes for the production of the new cheese require that there must be at least 45 litres Cow milk and at least 50 litres of Goat milk per 1000 kg of the cheese respectively, but no more than 60 litres of Sheep milk per 1000 kg of cheese. The factory needs to produce at least 60 kg of cheese per week. a) Explain why a linear programming model would be suitable for this case study. [5 marks] b) Formulate a Linear Programming (LP) model for the factory that minimises the total cost of producing the cheese while satisfying all constraints. [10 marks] c) Use the graphical method to find the optimal solution. Show the feasible region and the optimal solution on the graph. Annotate all lines on your graph. What is the mini- mal cost for the product? [10 marks] Note: you can use graphical solvers available online but make sure that your graph is clear, all variables involved are clearly represented and annotated, and each line is clearly marked and related to the corresponding equation. d) Is there a range for the cost ($) of A that can be changed without affecting the opti- mum point obtained above? Hint: This question does not require conversion of litres into kilograms. [5 marks]