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OC. Min C at x₁

OC. Min C at x₁

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.8: Linear Programming
Problem 33E
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### Linear Programming - Dual Problem Method #### Problem Statement Solve the following linear programming problem by applying the simplex method to the dual problem. **Objective:** Minimize \( C = 3y_1 + 2y_2 \) **Subject to Constraints:** 1. \( y_1 + 2y_2 \geq 7 \) 2. \( 2y_1 + 7y_2 \geq 17 \) 3. \( 5y_1 + 9y_2 \geq 21 \) **Non-negativity Constraints:** \( y_1, y_2 \geq 0 \) #### Multiple Choice Answers A. \(\min C = 21 \text{ at } x_1 = \dfrac{21}{5}, x_2 = 0 \) B. \(\min C = 17 \text{ at } x_1 = \dfrac{17}{2}, x_2 = 0 \) C. \(\min C = 7 \text{ at } x_1 = 7, x_2 = 0 \) D. \(\min C = 3y_1 + 2y_2 \geq 21 \text{ with } y_2 = 0 \) #### Explanation Here, we have a typical linear programming problem that requires minimizing a cost function subject to a set of linear inequalities and non-negativity constraints. Each answer option provides a possible minimized value of \( C \) and the corresponding values of variables \( x_1 \) and \( x_2 \) for each constraint. In these types of problems, the simplex method can be used to determine the optimal solution by converting the problem into its dual form and then solving it using the simplex algorithm. The correct option will be the one that satisfies the given constraints and achieves the minimum value of \( C \). ### Notes: - **Simplex Method:** A powerful technique used in linear programming for finding the optimal solution to an optimization problem. - **Dual Problem:** A linear programming problem derived from another linear programming problem (the primal problem), which helps in finding the solution more efficiently. - **Constraints:** These are the conditions that the solution must satisfy. The transcription provided incorporates the problem statement, constraints, multiple-choice answers, and an explanation, suitable for an educational website focusing on understanding and solving linear programming problems using the
Transcribed Image Text:### Linear Programming - Dual Problem Method #### Problem Statement Solve the following linear programming problem by applying the simplex method to the dual problem. **Objective:** Minimize \( C = 3y_1 + 2y_2 \) **Subject to Constraints:** 1. \( y_1 + 2y_2 \geq 7 \) 2. \( 2y_1 + 7y_2 \geq 17 \) 3. \( 5y_1 + 9y_2 \geq 21 \) **Non-negativity Constraints:** \( y_1, y_2 \geq 0 \) #### Multiple Choice Answers A. \(\min C = 21 \text{ at } x_1 = \dfrac{21}{5}, x_2 = 0 \) B. \(\min C = 17 \text{ at } x_1 = \dfrac{17}{2}, x_2 = 0 \) C. \(\min C = 7 \text{ at } x_1 = 7, x_2 = 0 \) D. \(\min C = 3y_1 + 2y_2 \geq 21 \text{ with } y_2 = 0 \) #### Explanation Here, we have a typical linear programming problem that requires minimizing a cost function subject to a set of linear inequalities and non-negativity constraints. Each answer option provides a possible minimized value of \( C \) and the corresponding values of variables \( x_1 \) and \( x_2 \) for each constraint. In these types of problems, the simplex method can be used to determine the optimal solution by converting the problem into its dual form and then solving it using the simplex algorithm. The correct option will be the one that satisfies the given constraints and achieves the minimum value of \( C \). ### Notes: - **Simplex Method:** A powerful technique used in linear programming for finding the optimal solution to an optimization problem. - **Dual Problem:** A linear programming problem derived from another linear programming problem (the primal problem), which helps in finding the solution more efficiently. - **Constraints:** These are the conditions that the solution must satisfy. The transcription provided incorporates the problem statement, constraints, multiple-choice answers, and an explanation, suitable for an educational website focusing on understanding and solving linear programming problems using the
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