SIT718-Real World Analytics

Questions:

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, 

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. 

b) Formulate a Linear Programming (LP) model for the factory that minimises the total cost of producing the cheese while satisfying all constraints. 

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? [

d) Is there a range for the cost ($) of A that can be changed without affecting the optimum solution obtained above?

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.

The maximal demand (in tons) for each product, the minimum cotton and wool proportion in each product is as follows:

a) Formulate an LP model for the factory that maximises the profit, while satisfying the demand and the cotton and wool proportion constraints.

b) Solve the model using R/R Studio. Find the optimal profit and optimal values of the decision variables.

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).

(a) State reasons why/how this game can be described as a two-players-zero-sum game

(b) Considering all possible combinations of bids, formulate the payoff matrix for the game.

(c) Explain what is a saddle point. Verify: does the game have a saddle point?

(d) Construct a linear programming model for Company 1 in this game.

(e) Produce an appropriate code to solve the linear programming model in part (d).

(f) Solve the game for Company 1 using the linear programming model you constructed in part (e). Interpret your solution.

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-P)

Q2-180-P2-(P2-P)

where P1 and P2 are the prices of cellphone for Company 1 and Company 2 respectively, and 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.

(a) Compute the profits of each company under all sale price combinations and produce the payoff matrix for each company

(b) Find the Nash equilibrium of this game. What are the profits at this equilibrium? Explain your reason clearly.

(C) If the cost C = 30, would the Nash equilibrium from part (b) change? Give clear reasons