The Brewery Problem We're making beer! Beer needs malt, hops and yeast. We hope to make 4 different kinds of beer- Light, Dark, Ale and Premium. The number of gallons of each type is defined to be X1, X2, X3 and x4 respectively. The amount of each ingredient, together with revenues and restrictions are given below, which result in the LP below. X1 X2 X3 X4 lbs avail Malt 1 1 0 3 | 2 1 2 1 Yeast | 1 1 1 4 Revenue | 6 5 3 7 Hops 2 150 max z = 6X1 + 5x2 + 3x3 + 7x4 st X1 + x2 + 324 0 The optimal tableau is given as: X1 X2 X3 0 0 0 0 1 0 1 0 0 0 0 1 X4 7 7 -4 1 S1 3 0 1 -1 S2 1 -1 1 0 S3 1 2 -2 1 rhs 380 10 40 30 (a) Use strong duality to show that the solution in the table is the correct one. (b) We had a shortage of hops- We could only get 100 lbs. Is the current basis still optimal? If not, set up the tableau we would need to work on to find the new solution, and describe how we would solve it (don't actually solve it). (c) Find the allowable range for changing each variable below, while still keeping the current solution optimal. i. For b2 (middle RHS coefficient): ii. For c (the coefficient of X): iii. For c4 (the coefficient of x4): The Brewery Problem We're making beer! Beer needs malt, hops and yeast. We hope to make 4 different kinds of beer- Light, Dark, Ale and Premium. The number of gallons of each type is defined to be X1, X2, X3 and x4 respectively. The amount of each ingredient, together with revenues and restrictions are given below, which result in the LP below. X1 X2 X3 X4 lbs avail Malt 1 1 0 3 | 2 1 2 1 Yeast | 1 1 1 4 Revenue | 6 5 3 7 Hops 2 150 max z = 6X1 + 5x2 + 3x3 + 7x4 st X1 + x2 + 324 0 The optimal tableau is given as: X1 X2 X3 0 0 0 0 1 0 1 0 0 0 0 1 X4 7 7 -4 1 S1 3 0 1 -1 S2 1 -1 1 0 S3 1 2 -2 1 rhs 380 10 40 30 (a) Use strong duality to show that the solution in the table is the correct one. (b) We had a shortage of hops- We could only get 100 lbs. Is the current basis still optimal? If not, set up the tableau we would need to work on to find the new solution, and describe how we would solve it (don't actually solve it). (c) Find the allowable range for changing each variable below, while still keeping the current solution optimal. i. For b2 (middle RHS coefficient): ii. For c (the coefficient of X): iii. For c4 (the coefficient of x4)