SIT718 Real world AnalyticsAssessment Task 3 2020 T3Total Marks = 100, Weighting – 25%11. A clothing garment factory produces T-shirts and shorts for the chain of Coles supermarkets. The contract is such that quality control is done before shipping and all productssupplied to Coles satisfy the quality requirements would be accepted by the chain. Thefactory employs 20 workers in the cutting department, 52 workers in the sewing department, and 14 workers in the packaging department. The garment factory works 8productive hours a day (no idle time during these 8 hours). There is a daily demand forat least 200 T-shirts. Each worker can participate only in one activity- the activity towhich they are assigned. The table below gives the time requirements (in minutes) andprofit per unit for the two garments. Amount (minutes) per operationCutting Sewing PackagingUnit Cost ($)T-shirtsShorts40 40 2020 100 201220 a) Explain why a Linear Programming (LP) model would be suitable for this case study.[5 marks]b) Formulate a LP model to help the factory management to determine the optimal dailyproduction schedule that maximises the profit while satisfying all constraints.[5 marks]c) Use the graphical method to find the optimal solution. Show the feasible region andthe optimal solution on the graph. Annotate all lines on your graph. What is the optimaldaily profit for the factory?[10 marks]Note: you can use graphical solvers available online but make sure that your graph isclear, all variables involved are clearly represented and annotated, and each line is clearlymarked and related to the corresponding equation.d) Find the range for the profit ($), of a T-shirt (if any), that can be changed withoutaffecting the optimal point of part (c)?[5 marks]2. A food factory makes three types of cereals, A, B and C, from a mix of several ingredients:Oates, Apricots, Coconuts and Hazelnuts. The cereals are packaged in 2kg boxes. Thefollowing table provides details of the sales price per box of cereals and the productioncost per ton (1000 kg) of cereals respectively. Sales price per box($)Production cost per tonCereal ACereal BCereal C2.502.003.504.002.803.00. The following table provides the purchase price per ton of ingredients and the maximumavailability of the ingredients in tons respectively. IngredientsPurchase price ($) per tonMaximum availability in tonsOatesApricotsCoconutsHazelnuts1001208020010522 The minimum daily demand (in boxes) for each cereal and the proportion of the Oates,Apricots, Coconut and Hazelnuts in each cereal is detailed in the following table, Proportion ofOates Apricots Coconuts AlmondsMinimum demand (boxes)Cereal ACereal BCereal C10007007500.80.650.50.10.20.10.050.050.10.050.10.3 a) Let xij ≤ 0 be a decision variable that denotes the number of kg of ingredient i, wherei could be Oates, Apricots, Coconuts, Hazelnuts, used to produce Cereal j, here j isone of A,B,C, (in boxes). Formulate an LP model to determine the optimal productionmix of cereals and the associated amounts of ingredients that maximises the profit, whilesatisfying the constraints.[15 Marks]b) Solve the model in R/R Studio. Find the optimal profit and optimal values of thedecision variables.[10 Marks]3. Two mining companies, Fox and Trot, bid for the right to drill a field. The possible bidsare $ 15 Million, $ 25 Million, $ 35 Million, $ 45 Million and $ 50 Million. The winner isthe company with the higher bid.The two companies decide that in the case of a tie (equal bids), Fox is the winner andwill get the field.Fox has ordered a geological survey and, based on the report from the survey, concludesthat getting the field for more than $ 45 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[5 Marks](b) Considering all possible combinations of bids, formulate the payoff matrix for thegame.[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 Company Trot in this game.[5 Marks](e) Produce an appropriate code to solve the linear programming model in part (d).[5 Marks](f) Solve the game for Trot using the linear programming model and the code you constructed in parts (d) and (e). Interpret your solution.[5 Marks]4. Consider two factories, Factory A and Factory B, producing the same model of iPads.The demand for the iPads produced by Company A is DA, and the demand for the iPadsproduced by Company B is DB. The demands are described by the following functions: DA = 200 – PA – (PA – P¯)DB = 200 – PB – (PB – P¯)(1)(2) where PA and PB are the prices of iPad for Factory A and Factory B respectively, and P¯is the average price over the prices PA and PB. For each company, the cost for producingone iPad is C = 20. Suppose that each company can only choose one of the three pricesf60; 70; 80g for a sale.(a) Compute the profits of each factory under all sale price combinations and producethe payoff matrix for each company.[Hint: the profit = the demand for the cellphones × the profit of one cellphone aftersale.][10 Marks](b) Find the Nash equilibrium of this game. What are the profits at this equilibrium?Explain your reason clearly.[5 Marks](C) If the cost C = 30, would the Nash equilibrium from part (b) change? Give clearreasons.[5 Marks]

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