SIT718 Real world AnalyticsAssessment Task 3Total Marks = 100, Weighting - 30%Your nal submission should consist of:1. \name-report.pdf": A pdf le (created in any word processor) with up to 8 pages, con-taining the solutions of the questions, labelled with your name;2. \name-code.R": Two codes combined in one with your R le, labelled with yourname.R,with lp models for Questions 2 and Questions 3.Your assignment will not be assessed if we cannot reproduce your results with your Rcode.Reference style: Harvard.11. A food factory is making a beverage for a customer from mixing two di erent existingproducts A and B. The compositions of A and B and prices ($/L) are given as follows,Amount (L) in /100 L of A and BLime Orange Mango Cost ($/L)A 3 6 4 5B 8 4 6 6The customer requires that there must be at least 4.5 Litres (L) Orange and at least5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no morethan 6 Litres of Lime concentrate per 100 Litres of beverage. The customer needs atleast 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 totalcost of producing the beverage while satisfying all constraints.[10 marks]c) Use the graphical method to nd the optimal solution. Show the feasible region andthe 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 isclear, all variables involved are clearly represented and annotated, and each line is clearlymarked and related to the corresponding equation.d) Is there a range for the cost ($) of A that can be changed without a ecting the opti-mum solution obtained above?[5 marks]2. A factory makes three products called Spring, Autumn, and Winter, from three materialscontaining 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 priceSpring $60 $5 Cotton $30Autumn $55 $4 Wool $45Winter $60 $5 Silk $50The 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 proportionSpring 4800 55% 30%Autumn 3000 45% 40%Winter 3500 30% 50%a) Formulate an LP model for the factory that maximises the pro t, while satisfying thedemand and the cotton and wool proportion constraints.[10 Marks]b) Solve the model using R/R Studio. Find the optimal pro t and optimal values of thedecision variables.[10 Marks]Hints:1. Let xij 0 be a decision variable that denotes the number of tons of productsj for j 2 f1 = Spring; 2 = Autumn; 3 = Winterg to be produced from Materialsi 2 fC=Cotton, W=Wool, S=Silkg.2. The proportion of a particular type of Material in a particular type of Product can becalculated as:e.g., the proportion of Cotton in product Spring is given by:xC1xC1 + xW1 + xS1.3. Helen and David are playing a game by putting chips in two piles (each player has twopiles P1 and P2), respectively. Helen has 6 chips and David has 4 chips. Each playerplaces his/her chips in his/her two piles, then compare the number of chips in his/hertwo piles with that of the other player's two piles. Note that once a chip is placed in onepile it cannot be moved to another pile. There are four comparisons including Helen'sP1 vs David's P1, Helen's P1 vs David's P2, Helen's P2 vs David's P1, and Helen's P2vs David's P2. For each comparison, the player with more chips in the pile will score 1point (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 nal score of the game isthe sum score over the four comparisons. For example, if Helen puts 5 and 1 chips in herP1 and P2, David puts 3 and 1 chips in his P1 and P2, respectively. Then Helen will get1 (5 vs 3) + 1 (5 vs 1) - 1 (1 vs 3) + 0 (1 vs 1) = 1 as her nal score, and David will gethis nal score of -1.(a) Give reasons why/how this game can be described as a two-players-zero-sum game.[5 Marks](b) Formulate the payo matrix for the game.[5 Marks](c) Explain what is a saddle point. Verify: does the game have a saddle point?[5 Marks](d) Construct a linear programming model for each player in this game;[5 Marks](e) Produce an appropriate code to solve the linear programming model in part (c).[5 Marks](f) Solve the game for David using the linear programming model you constructed inpart (d). Interpret your solution.[5 Marks][Hint: To record the number of chips in each pile for each player you may use the notation(i; j), where i is the number of chips in P1 and j is the number of chips in P2, for example(2,4) means two chips in P1 and four chips in P2.]4. Supposing there are three players, each player is given a bag and asked to contribute inhis own money with one of the three amount f$0; $3; $6g. A referee collects all the moneyfrom the three bags and then doubles the amount using additional money. Finally, eachplayer share the whole money equally. For example, if both Players 1 and 2 put $0 andPlayer 3 puts $3, then the referee adds another $3 so that the total becomes $6. Afterthat, each player will obtain $2 at the end. Every player want to maximise his pro t,but he does not know the amount contributed from other players. [Hint: pro t = moneyhe obtained - money he contributed.](a) Compute the pro ts of each player under all strategy combinations and make thepayo matrix for the three players. [Hint: you can create multiple payo tables todemonstrate the strategy combinations. The referee is not a player and should not be inthe payo table.][10 Marks](b) Find the Nash equilibrium of this game. What are the pro ts at this equilibrium?Explain your reason clearly.[10 Marks]