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Solve the linear programming problem. Maximize P = 40x + 50y Subject to 2x+ys 14 x+y ≤ 9 x + 2y ≤ 16 x, y ≥ 0

Solve the linear programming problem. Maximize P = 40x + 50y Subject to 2x+ys 14 x+y ≤ 9 x + 2y ≤ 16 x, y ≥ 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Linear Programming Problem --- **Objective:** Maximize \( P = 40x + 50y \) **Constraints:** 1. \( 2x + y \leq 14 \) 2. \( x + y \leq 9 \) 3. \( x + 2y \leq 16 \) 4. \( x, y \geq 0 \) --- ### Explanation of Constraints: 1. The first constraint \( 2x + y \leq 14 \) indicates that the sum of twice the value of \( x \) and the value of \( y \) must be at most 14. 2. The second constraint \( x + y \leq 9 \) indicates that the sum of \( x \) and \( y \) cannot exceed 9. 3. The third constraint \( x + 2y \leq 16 \) indicates that the sum of \( x \) and twice the value of \( y \) must be at most 16. 4. The fourth constraint \( x, y \geq 0 \) signifies that both \( x \) and \( y \) must be non-negative; they cannot be less than zero. ### Graphical Interpretation (for Explanation): To solve this problem graphically: 1. **Plot each constraint on a graph:** - \( 2x + y = 14 \) - \( x + y = 9 \) - \( x + 2y = 16 \) 2. **Identify the feasible region**: - This is the region where all the constraints overlap and \( x \) and \( y \) are non-negative. 3. **Find the corner points (vertices) of the feasible region**. 4. **Evaluate the objective function \( P = 40x + 50y \) at each corner point** to find which provides the maximum value. The graphical method provides a visual understanding, which can be supplemented with algebraic methods like the Simplex method for more complex or higher-dimensional problems.
Transcribed Image Text:### Linear Programming Problem --- **Objective:** Maximize \( P = 40x + 50y \) **Constraints:** 1. \( 2x + y \leq 14 \) 2. \( x + y \leq 9 \) 3. \( x + 2y \leq 16 \) 4. \( x, y \geq 0 \) --- ### Explanation of Constraints: 1. The first constraint \( 2x + y \leq 14 \) indicates that the sum of twice the value of \( x \) and the value of \( y \) must be at most 14. 2. The second constraint \( x + y \leq 9 \) indicates that the sum of \( x \) and \( y \) cannot exceed 9. 3. The third constraint \( x + 2y \leq 16 \) indicates that the sum of \( x \) and twice the value of \( y \) must be at most 16. 4. The fourth constraint \( x, y \geq 0 \) signifies that both \( x \) and \( y \) must be non-negative; they cannot be less than zero. ### Graphical Interpretation (for Explanation): To solve this problem graphically: 1. **Plot each constraint on a graph:** - \( 2x + y = 14 \) - \( x + y = 9 \) - \( x + 2y = 16 \) 2. **Identify the feasible region**: - This is the region where all the constraints overlap and \( x \) and \( y \) are non-negative. 3. **Find the corner points (vertices) of the feasible region**. 4. **Evaluate the objective function \( P = 40x + 50y \) at each corner point** to find which provides the maximum value. The graphical method provides a visual understanding, which can be supplemented with algebraic methods like the Simplex method for more complex or higher-dimensional problems.
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