the following question needs to be solved using linear programming method.

12. (a) An organisation makes three types of shed, denoted Standard, Extra and Ulti- mate. The following table provides their expected profit per shed, the number of units of raw material A and B required to make each type of shed and the number of hours needed to make each type of shed. Profit () Raw Material A Raw Material B Hours 30 Standard 2 4 2 Extra 40 1 2 Ultimate 60 3 6 4 The company wishes to maximise their profit given they have 75 units of raw material A and 120 units of raw material B. There are 50 hours available. In addition the total number of Ultimate sheds should not make up more than 50% of the total number of sheds made per month. Produce the linear programming formulation of this problem. Note that you are not required to solve this problem. [8] (b) Consider the following linear program: Maximise -212 + 4x3 s.t. 21-22 +213 8 212-13 28 I+I+I3 = 9 11, 12, 13 20 i. Explain why artificial variables are needed when the problem includes > constraints [2] ii. Solve the problem using the Simplex Tableau method. [10] iii. Either find a further optimal solution or explain why there are no other optimal solutions [5] 7