Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![# Understanding Dual Optimization in Linear Programming
In this educational resource, we will explore the concept of dual optimization in linear programming. Specifically, we will write the dual maximization problem to solve a given minimization problem using the Simplex Method. We'll also demonstrate the step-by-step solution process.
### Problem Statement
Given a minimization problem defined as follows:
**Objective Function:**
\[ \text{Minimize } g = 2y_1 + 10y_2 \]
**Subject to the constraints:**
\[ 2y_1 + y_2 \geq 11 \]
\[ y_1 + 3y_2 \geq 11 \]
\[ y_1 + 4y_2 \geq 16 \]
### Steps to Write the Dual Maximization Problem
To convert this minimization problem into its dual maximization problem, we follow these steps:
1. **Identify the primal problem's constraints and coefficients.**
2. **Convert the inequalities of each constraint.**
3. **Formulate the dual problem with new variables corresponding to the primal constraints.**
For the provided problem, let's formulate its dual:
**Dual Variables:**
Let \( x_1 \), \( x_2 \), and \( x_3 \) be the dual variables corresponding to the constraints \( 2y_1 + y_2 \geq 11 \), \( y_1 + 3y_2 \geq 11 \), and \( y_1 + 4y_2 \geq 16 \), respectively.
**Objective Function for the Dual Problem:**
\[ \text{Maximize } z = 11x_1 + 11x_2 + 16x_3 \]
**Subject to the constraints:**
1. \( 2x_1 + x_2 + x_3 \leq 2 \)
2. \( x_1 + 3x_2 + 4x_3 \leq 10 \)
3. \( x_1, x_2, x_3 \geq 0 \)
### Explanation
The dual maximization problem derived from the given minimization problem outlines the relationships between the dual variables and the primal constraints. The objective function and constraints should now be solved using the Simplex Method to find the optimal values of the dual variables \( x_1 \), \( x_2 \), and \(](https://content.bartleby.com/qna-images/question/adc2ecb0-6ee7-47e4-a42f-0ca26f0dbdf5/f5619900-8f8b-4186-89a3-293efce43c94/jo1ne7h.png)
Transcribed Image Text:# Understanding Dual Optimization in Linear Programming
In this educational resource, we will explore the concept of dual optimization in linear programming. Specifically, we will write the dual maximization problem to solve a given minimization problem using the Simplex Method. We'll also demonstrate the step-by-step solution process.
### Problem Statement
Given a minimization problem defined as follows:
**Objective Function:**
\[ \text{Minimize } g = 2y_1 + 10y_2 \]
**Subject to the constraints:**
\[ 2y_1 + y_2 \geq 11 \]
\[ y_1 + 3y_2 \geq 11 \]
\[ y_1 + 4y_2 \geq 16 \]
### Steps to Write the Dual Maximization Problem
To convert this minimization problem into its dual maximization problem, we follow these steps:
1. **Identify the primal problem's constraints and coefficients.**
2. **Convert the inequalities of each constraint.**
3. **Formulate the dual problem with new variables corresponding to the primal constraints.**
For the provided problem, let's formulate its dual:
**Dual Variables:**
Let \( x_1 \), \( x_2 \), and \( x_3 \) be the dual variables corresponding to the constraints \( 2y_1 + y_2 \geq 11 \), \( y_1 + 3y_2 \geq 11 \), and \( y_1 + 4y_2 \geq 16 \), respectively.
**Objective Function for the Dual Problem:**
\[ \text{Maximize } z = 11x_1 + 11x_2 + 16x_3 \]
**Subject to the constraints:**
1. \( 2x_1 + x_2 + x_3 \leq 2 \)
2. \( x_1 + 3x_2 + 4x_3 \leq 10 \)
3. \( x_1, x_2, x_3 \geq 0 \)
### Explanation
The dual maximization problem derived from the given minimization problem outlines the relationships between the dual variables and the primal constraints. The objective function and constraints should now be solved using the Simplex Method to find the optimal values of the dual variables \( x_1 \), \( x_2 \), and \(
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