1. a. 10 8 6. Optimal Solution A=7, B= 3 4 A = 4, B = 6 2 4 6. 8 10 b. The same extreme point, A = 7 and B = 3, remains optimal. The value of the objective function becomes 5(7) + 2(3) = 41 A new extreme point, A = 4 and B = 6, becomes optimal. The value of the objective function becomes 3(4) + 4(6) = 36. . d. The objective coefficient range for variable A is 2 to 6. Since the change in part (b) is within this range, we know the optimal solution, A = 7 and B = 3, will not change. The objective coefficient range for variable B is 1 to 3. Since the change in part (c) is outside this range, we have to re-solve the problem to find the new optimal solution. 3(7) + 2(3)= 27 2.

2. a. B 10 8 3(6.5) + 2(4.5) = 28.5 6 Optimal Solution A = 6.5, B =4.5 4 Enlarged Feasible Region 2 2 4 6 8 10 b. The value of the optimal solution to the revised problem is 3(6.5) + 2(4.5) = 28.5. The one-unit increase in the right-hand side of constraint 1 has improved the value of the optimal solution by 28.5 - 27 = 1.5. Thus, the dual value for constraint 1 is 1.5. c. The right-hand-side range for constraint 1 is 8 to 11.2. As long as the right-hand side stays within this range, the dual value of 1.5 is applicable. d. The improvement in the value of the optimal solution will be 0.5 for every unit increase in the right- hand side of constraint 2 as long as the right-hand side is between 18 and 30. 2. a. B 10 8 3(6.5) + 2(4.5) = 28.5 6 Optimal Solution A = 6.5, B =4.5 4 Enlarged Feasible Region 2 2 4 6 8 10 b. The value of the optimal solution to the revised problem is 3(6.5) + 2(4.5) = 28.5. The one-unit increase in the right-hand side of constraint 1 has improved the value of the optimal solution by 28.5 - 27 = 1.5. Thus, the dual value for constraint 1 is 1.5. c. The right-hand-side range for constraint 1 is 8 to 11.2. As long as the right-hand side stays within this range, the dual value of 1.5 is applicable. d. The improvement in the value of the optimal solution will be 0.5 for every unit increase in the right- hand side of constraint 2 as long as the right-hand side is between 18 and 30