A food factory is making a beverage for a customer from mixing two different existing products A and B. The compositions of A and B and prices ($/L) are given as follows, Amount (L) in /100 L of A and B Lime Orange Mango Cost ($/L) A 3 6 4 3 B 8 4 6 10 The customer requires that there must be at least 4.5 Litres (L) Orange and at least 5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more than 6 Litres of Lime concentrate per 100 Litres of beverage. The customer needs at least 100 Litres of the beverage 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 beverage 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 minimal 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 optimum solution obtained above? [5 marks] 2. A factory makes three products called Spring, Autumn, and Winter, from three materials containing Cotton, Wool and Silk. The following table provides details on the sales price, production cost and purchase cost per ton of products and materials respectively. Sales price Production cost Purchase price Spring $60 $5 Cotton $30 Autumn $55 $4 Wool $45 Winter $60 $5 Silk $50 The maximal demand (in tons) for each product, the minimum cotton and wool proportion in each product is as follows:
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