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Sec 4.7 1. Consider the following problem. Maximize Z = x₁ + 2x₂, subject to x₁ + 3x₂ ≤ 8 (resource 1) x₁ + x₂ ≤4 (resource 2) x₁ ≥ 0, x₂ ≥ 0. and The optimal solution is (x₁, x₂) = (2,2) with Z* = 6. (a) Is any of these two constraints binding constraint? (b) Use graphical analysis to determine the shadow prices for the respective resources. (c) Determine how many additional units of resource 1 would be needed to increase the optimal value by 1.5.

Sec 4.7 1. Consider the following problem. Maximize Z = x₁ + 2x₂, subject to x₁ + 3x₂ ≤ 8 (resource 1) x₁ + x₂ ≤4 (resource 2) x₁ ≥ 0, x₂ ≥ 0. and The optimal solution is (x₁, x₂) = (2,2) with Z* = 6. (a) Is any of these two constraints binding constraint? (b) Use graphical analysis to determine the shadow prices for the respective resources. (c) Determine how many additional units of resource 1 would be needed to increase the optimal value by 1.5.

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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**Sec 4.7** 1. **Consider the following problem.** Maximize \( Z = x_1 + 2x_2 \), subject to \( x_1 + 3x_2 \leq 8 \) (resource 1), \( x_1 + x_2 \leq 4 \) (resource 2) and \( x_1 \geq 0 \), \( x_2 \geq 0 \). The optimal solution is \( (x_1^*, x_2^*) = (2, 2) \) with \( Z^* = 6 \). (a) Is any of these two constraints a binding constraint? (b) Use graphical analysis to determine the shadow prices for the respective resources. (c) Determine how many additional units of resource 1 would be needed to increase the optimal value by 1.5. **Graph Explanation:** The graph on the right side shows two lines representing the constraints. The line with a negative slope (blue) represents the constraint \( x_1 + 3x_2 \leq 8 \), and the line with a less steep negative slope (green) represents \( x_1 + x_2 \leq 4 \). The feasible region, defined by the constraints and axes, is a quadrilateral area. The optimal solution, indicated by a large dot at the point (2, 2), lies on the intersection of the two lines, showing that both constraints are binding at this optimal point.
Transcribed Image Text:**Sec 4.7** 1. **Consider the following problem.** Maximize \( Z = x_1 + 2x_2 \), subject to \( x_1 + 3x_2 \leq 8 \) (resource 1), \( x_1 + x_2 \leq 4 \) (resource 2) and \( x_1 \geq 0 \), \( x_2 \geq 0 \). The optimal solution is \( (x_1^*, x_2^*) = (2, 2) \) with \( Z^* = 6 \). (a) Is any of these two constraints a binding constraint? (b) Use graphical analysis to determine the shadow prices for the respective resources. (c) Determine how many additional units of resource 1 would be needed to increase the optimal value by 1.5. **Graph Explanation:** The graph on the right side shows two lines representing the constraints. The line with a negative slope (blue) represents the constraint \( x_1 + 3x_2 \leq 8 \), and the line with a less steep negative slope (green) represents \( x_1 + x_2 \leq 4 \). The feasible region, defined by the constraints and axes, is a quadrilateral area. The optimal solution, indicated by a large dot at the point (2, 2), lies on the intersection of the two lines, showing that both constraints are binding at this optimal point.
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