# Unit Learning Outcome (ULO) Graduate Learning Outcome (GLO)

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Your final submission must include the following two files:

• 1. "name-report.pdf"A report, in pdf format (created in any word processor), covering all of the items in above (where “name” is replaced with your name -you can use your surname or first name). With plots and tables, it should be up to 8 pages.
• 2. "name-code.R"The R code file (that you have written to produce your results) named(where “name” is replaced with your name - you can use your surname or first name).

• 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 Goat Cost (\$/kg) A 30 60 40 5 B 80 40 70 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.
1. a) Explain why a linear programming model would be suitable for this case study. [5 marks]
2. 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]
3. 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.

4. d) Is there a range for the cost (\$) of A that can be changed without affecting the optimum solution obtained above? [5 marks]

Hint: This question does not require unit conversion.

• 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 propor- tion in each product is as follows:

Demand min Cotton proportion min Wool proportion Spring 4200 50% 40% Autumn 3200 60% 40% Winter 3500 40% 50%

1. a) Formulate an LP model for the factory that maximises the profit, while satisfying the demand and the cotton and wool proportion constraints. [10 Marks]
2. b) Solve the model using R/R Studio. Find the optimal profit and optimal values of the decision variables. [10 Marks]

Hints: 1 Let xij ≥ 0 be a decision variable that denotes the number of tons of products j for j ∈ {1 = Spring,2 = Autumn,3 = W inter} to be produced from Materials i ∈ {C=Cotton, W=Wool, S=Silk}.

• 2. The proportion of a particular type of Material in a particular type of Product can be calculated as: e.g., the proportion of Cotton in product Spring is given by: xC1 xC1 + xW 1 + xS1.
•
• 3. Two mining companies, Company 1 and Company 2, bid for the right to drill a field. The possible bids are \$ 10 Million, \$ 20 Million, \$ 30 Million, \$ 40 Million and \$ 50 Million. The winner is the company with the higher bid.

In case of a tie (equal bids) Company 1 is the winner and will get the field.

For Company 1 getting the field for more than \$ 40 Million is as bad as not getting it (assume loss), except in case of a tie (assume win).

1. (a) State reasons why/how this game can be described as a two-players-zero-sum game [5 Marks]
2. (b) Considering all possible combinations of bids, formulate the payoff matrix for the game. [5 Marks]
3. (c) Explain what is a saddle point. Verify: does the game have a saddle point?[5 Marks] (d) Construct a linear programming model for Company 1 in this game. [5 Marks]
4. (e) Produce an appropriate code to solve the linear programming model in part (d). [5 Marks]
5. (f) Solve the game for Company 1 using the linear programming model you constructed in part (e). Interpret your solution. [5 Marks]
• 4. Consider two companies, Company 1 and Company 2, producing the same model of cellphones. The demand for the cellphones produced by Company 1 is Q1, and the demand for the cellphones produced by Company 2 is Q2. The demands are described by the following functions:

Q1 = 180 − P1 − (P1 − Ì„P) (1)

Q2 = 180 − P2 − (P2 − Ì„P) (2)

where P1 and P2 are the prices of cellphone for Company 1 and Company 2 respectively, and Ì„P is the average price over the prices P1 and P2. For each company, the cost for producing one cellphone is C = 20. Suppose that each company can only choose one of the three prices {60,70,80} for a sale.

1. (a) Compute the profits of each company under all sale price combinations and produce the payoff matrix for each company. [Hint: the profit = the demand for the cellphones × the profit of one cellphone after sale.] [10 Marks]
2. (b) Find the Nash equilibrium of this game. What are the profits at this equilibrium? Explain your reason clearly. [5 Marks]
3. (C) If the cost C = 30, would the Nash equilibrium from part (b) change? Give clear reasons. [5 Marks]

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