Chapter 3 solutions

# Chapter 3 solutions - Linear Programming Sensitivity...

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Linear Programming: Sensitivity Analysis and Interpretation of Solution Chapter 3 Linear Programming: Sensitivity Analysis and Interpretation of Solution Solutions: 1. a. b. The same extreme point, A = 7 and B = 3, remains optimal. The value of the objective function becomes 5(7) + 2(3) = 41 c. 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. B Optimal Solution 2 4 6 8 10 2 4 6 8 10 A A = 7, B = 3 A = 4, B = 6 3(7) + 2(3) = 27

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Chapter 3 2. a. 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 price 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 price 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. 4. a. b. The value of the optimal solution to the revised problem is 8(2.5) + 12(2.5) = 50. Compared to the original problem, the value of the optimal solution has increased by 50 - 48 = 2. However, this is a minimization problem and the increase of 2 is not an improvement in the value of the optimal B Optimal Solution 2 4 6 8 10 2 4 6 8 10 A A = 6.5, B = 4.5 3(6.5) + 2(4.5) = 28.5 Enlarged Feasible Region Y Optimal Solution 2 4 6 8 10 2 4 6 8 10 X X = 2.5, Y = 2.5 8(2.5) + 12(2.5) = 50
Linear Programming: Sensitivity Analysis and Interpretation of Solution solution. In fact, the value of the optimal solution is worse by 2. Thus, the dual price is -2. c. The right-hand side range for constraint 1 is 5 to 11. As long as the right-hand side stays within this range, the dual price of -2 is applicable. Since increasing the right-hand side does not improve the value of the optimal solution, decreasing the right-hand side of constraint 1 would b desirable. d. As long as the right-hand side of constraint 2 is between 9 and 18, a unit increase in the right-hand side will cause the value of the optimal solution to worsen by 3. 5. a. Regular Glove = 500 Catcher’s Mitt = 150 Value = 3700 b. The finishing and packaging and shipping constraints are binding. c.

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Chapter 3 solutions - Linear Programming Sensitivity...

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