SIT718 Real world Analytics Assessment Task 3 Total Marks = 100, Weighting - 30%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) Al 2 6 4 5 B| 7 4 8 15 The customer requires that there must be at least 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 150 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 total cost of producing the beverage while satisfying all constraints. [5 marks] 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. [5 marks] Note: you can use graphical solvers available online but make sure that vour graph is clear, all variables involved are clearly represented and annotated, and each line is clearly marked and related to the corresponding equation. d) What is the range for the cost ($) of A that can be changed without affecting the optimum solution obtained above? [5 marks] 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. Sales price Production cost Purchase price Spring $60 $5 Cotton 530 Autumn $55 $3 Wool $45 Winter $60 $5 Silk $50 The 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 proportion Spring 3600 55% 30% Autumn 3300 45% 40% Winter 4000 30% 50% a) Formulate an LP model for the factory that maximises the profit, while satisfying the demand and the cotton and wool proportion constraints. There is no penalty for the shortage. [20 Marks] b) Solve the model using R/R Studio. Find the optimal profit and optimal values of the decision variables. [20 Marks] Hints: You may refer to Week 8.7 Example - Blending Crude Qils into Gasolines. For ex- ample, let x;; > 0 be a decision variable that denotes the number of tons of products j for j {1 = Spring,2 = Autumn,3 = Winler} to be produced from Materials i {C=Cotton, W=Wool, S=Silk}. 3. Consider the following parlor game to be played between two players. Each player begins with three chips: one red, one white, and one blue. Each chip can be used only once. To begin, each player selects one of her chips and places it on the table, concealed. Both players then uncover the chips and determine the payoff to the winning player. In particular, if both players play the same kind of chip, it is a draw; otherwise, the following table indicates the winner and how much she receives from the other player. Next, each player selects one of her two remaining chips and repeats the procedure, resulting in another payoff according to the following table. Finally, each player plays her one remaining chip, resulting in the third and final payoft. Winning Chip | Payoff () Red beats white 250 White beats blue 100 Blue beats red 50 Matching colors (a) Formulate the payoff matrix for the game and identify possible saddle points. (10 Marks]| (b) Construct a linear programming model for each player in this game. (10 Marks]| (c) Produce an appropriate code to solve the linear programming model for this game. (10 Marks]| (d) Solve the game for both players using the linear programming model. (10 Marks]| [Hint: Each player has the same strategy set. A strategy must specify the first chip chosen, the second and third chips chosen. Denote the white, red and blue chips by W, R and B respectively. For example, a strategy \"WRB\" indicates first choosing the white and then choosing the red, before choosing blue at the end.] Sensitivity of Product A's Cost: 1. Currently, the cost of A is $5 per litre, and the cost of B is $15 per litre. 2. The range of the cost of A where the solution remains optimal is determined by comparing it to the cost of B. If the cost of A rises too much, the factory would shift toward using more of product B, changing the optimal solution. Range for the Cost of A: The cost of product A can increase up to the cost of product B ($15) before it affects the optimal solution. On the lower side, the cost of product A can decrease to any positive value, as a lower cost of A would still favor using more of A without violating the constraints. Thus, the range of the cost of A that can change without affecting the optimal solution is: 0