A company produces three products, X, Y, and Z. Each unit of Product X requires 2 hours of labor, 1 hour of machine time, and 3 hours of assembly time. Each unit of Product Y requires 1 hour of labor, 2 hours of machine time, and 2 hours of assembly time. Each unit of Product Z requires 3 hours of labor, 2 hours of machine time, and 1 hour of assembly time. The company has the following resources available: 200 hours of labor, 150 hours of machine time, 120 hours of assembly time The profit per unit for each product is as follows: Product X: Rs. 30/-, Product Y: Rs. 25/-, Product Z: Rs. 20/-. The company wants to maximize its total profit. d) The final simplex table for the above problem is given below. Note: s1, s2and s3 relates to products X, Y, and Z respectively. Basic Variables Quantity 30 25 20 0 0 0 X y Z $1 $2 S3 X 40 1 2/3 1/3 0 0 1/3 S1 40 1 -1 1 0 0 -1/3 S2 30 0 5 -4 0 1 -2/3 Zj 400 -15 5 0 0 10 Cj -Zj 45 20 20 0 0 -10 As per the above table; (i) What is the optimal solution? (2 marks) Interpret the shadow prices of labor hours, machine hours, and assembly hours. (3 marks) (111) Construct the dual for the problem. (5 marks) (iv) Find the optimal values for the dual variables. (2 marks) (v) Find the range within which the production of X can be changed without affecting the optimal solution. (vi) (3 marks) Find the range within which the production of Y can be changed without affecting the optimal solution. (3 marks) (vii) Find the range within which the production of Z can be changed without affecting the optimal solution. (3 marks) (viii) Identify the range within which labor hours can be varied without altering the optimal solution. (ix) (3 marks) Identify the range within which machine hours can be varied without altering the optimal solution. (x) (3 marks) Identify the range within which assembly hours can be varied without altering the optimal solution. (3 marks)