Reference no: EM132533136
SIT718 - Real World Analytics Assignment - Deakin University, Australia
Q1. 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 ($/L) are given as follows,
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Amount (L) in /100 L of A and B
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Cost ($/L)
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Lime
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Orange
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Mango
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A
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3
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6
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4
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5
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B
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8
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4
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6
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6
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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 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?
Q2. 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.
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Sales price
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Production cost
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Purchase price
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Spring
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$60
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$5
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Cotton
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$30
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Autumn
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$55
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$4
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Wool
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$45
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Winter
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$60
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$5
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Silk
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$50
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The maximal demand (in tons) for each product, the minimum cotton and wool proportion in each product is as follows:
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Demand
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min Cotton proportion
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min Wool proportion
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Spring
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4800
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55%
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30%
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Autumn
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3000
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45%
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40%
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Winter
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3500
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30%
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50%
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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.
Q3. Helen and David are playing a game by putting chips in two piles (each player has two piles P1 and P2), respectively. Helen has 6 chips and David has 4 chips. Each player places his/her chips in his/her two piles, then compare the number of chips in his/her two piles with that of the other player's two piles. Note that once a chip is placed in one pile it cannot be moved to another pile. There are four comparisons including Helen's P1 vs David's P1, Helen's P1 vs David's P2, Helen's P2 vs David's P1, and Helen's P2 vs David's P2. For each comparison, the player with more chips in the pile will score 1 point (the opponent will lose 1 point). If the number of chips is the same in the two piles, then nobody will score any points from this comparison. The final score of the game is the sum score over the four comparisons. For example, if Helen puts 5 and 1 chips in her P1 and P2, David puts 3 and 1 chips in his P1 and P2, respectively. Then Helen will get 1 (5 vs 3) + 1 (5 vs 1) - 1 (1 vs 3) + 0 (1 vs 1) = 1 as her final score, and David will get his final score of -1.
(a) Give reasons why/how this game can be described as a two-players-zero-sum game.
(b) 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 each player in this game;
(e) Produce an appropriate code to solve the linear programming model in part (c).
(f) Solve the game for David using the linear programming model you constructed in part (d). Interpret your solution.
Q4. Supposing there are three players, each player is given a bag and asked to contribute in his own money with one of the three amountf$0;$3;$6g. 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.
(a) Compute the profits of each player under all strategy combinations and make the payoff matrix for the three players.
(b) Find the Nash equilibrium of this game. What are the profits at this equilibrium? Explain your reason clearly.
Attachment:- Real World Analytics Assignment File.rar