Assignment Task : 

1. 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 5 B 8 4 6 6 

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. 

b) Formulate a Linear Programming (LP) model for the factory that minimises the total cost of producing the beverage 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 mini- mal cost for the product? 

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

 

2. Supposing there are three players, each player is given a bag and asked to contribute in his own money with one of the three amount {$0,$3,$6}. A referee collects all the money from the three bags and then doubles the amount using additional money. Finally, each player share the whole money equally. For example, if both Players 1 and 2 put $0 and Player 3 puts $3, then the referee adds another $3 so that the total becomes $6. After that, each player will obtain $2 at the end. Every player want to maximise his profit, but he does not know the amount contributed from other players. [Hint: profit = money he obtained - money he contributed.] 

(a) Compute the profits of each player under all strategy combinations and make the payoff matrix for the three players. [Hint: you can create multiple payoff tables to demonstrate the strategy combinations. The referee is not a player and should not be in the payoff table.] 

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

 

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  • Posted on : June 06th, 2019
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