Solve the following LP scenario. Maximize p a + 2y+ z subject to T+ y+z<3 2x + 3y + 9z < 14 r>0 y 20 z > 0
Solve the following LP scenario. Maximize p a + 2y+ z subject to T+ y+z<3 2x + 3y + 9z < 14 r>0 y 20 z > 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.4: Linear Programming
Problem 15E
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![**Linear Programming Problem**
This lesson focuses on solving a Linear Programming (LP) scenario using the simplex method. Here's the problem we will work with:
**Objective Function:**
Maximize \( p = x + 2y + z \)
**Subject to Constraints:**
\[
\begin{align*}
x + y + z & \leq 3 \\
2x + 3y + 9z & \leq 14 \\
x, y, z & \geq 0 \\
\end{align*}
\]
**Simplex Table Setup**
Below is a tableau representation used in the simplex method. This tableau will help identify the pivot elements that guide the iterations toward an optimal solution.
| | \( x \) | \( y \) | \( z \) | \( s \) | \( t \) | ? |
|----|---------|---------|---------|--------|--------|-----|
| \( s \) | | | | | | |
| \( t \) | | | | | | |
| \( u \) | | | | | | |
**Note:** The variables \( s \), \( t \), and \( u \) represent slack variables included in the inequalities to convert them into equalities.
**Pivot Element Identification**
The pivot is located in the ? column and the ? row. It has a value of _______.
This tableau format is essential for determining which variables to enter and leave the basis as we proceed toward the optimal solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb01ce655-4372-4af4-9606-6aca74ff4552%2Feb6fd03c-5577-435d-895c-c131d1f78e09%2Fa5zgmt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Programming Problem**
This lesson focuses on solving a Linear Programming (LP) scenario using the simplex method. Here's the problem we will work with:
**Objective Function:**
Maximize \( p = x + 2y + z \)
**Subject to Constraints:**
\[
\begin{align*}
x + y + z & \leq 3 \\
2x + 3y + 9z & \leq 14 \\
x, y, z & \geq 0 \\
\end{align*}
\]
**Simplex Table Setup**
Below is a tableau representation used in the simplex method. This tableau will help identify the pivot elements that guide the iterations toward an optimal solution.
| | \( x \) | \( y \) | \( z \) | \( s \) | \( t \) | ? |
|----|---------|---------|---------|--------|--------|-----|
| \( s \) | | | | | | |
| \( t \) | | | | | | |
| \( u \) | | | | | | |
**Note:** The variables \( s \), \( t \), and \( u \) represent slack variables included in the inequalities to convert them into equalities.
**Pivot Element Identification**
The pivot is located in the ? column and the ? row. It has a value of _______.
This tableau format is essential for determining which variables to enter and leave the basis as we proceed toward the optimal solution.
![The text appears to be about performing operations on matrices or tableaus, likely related to solving linear programming problems using the Simplex method.
---
**Text Transcription:**
The pivot is in the \( \ \ \) column and the \( \ \ \) row. It has value \( \ \ \).
To pivot, we need to multiply on the left by
\[
M =
\begin{bmatrix}
\ \ & \ \ & \ \ \\
\ \ & \ \ & \ \ \\
\ \ & \ \ & \ \
\end{bmatrix}
\]
After pivoting, we have our second tableau.
\[
\begin{bmatrix}
\ \ & \ \ & \ \ & \ \ & \ \ & \ \ \\
\ \ & \ \ & \ \ & \ \ & \ \ & \ \ \\
\ \ & \ \ & \ \ & \ \ & \ \ & \ \
\end{bmatrix}
\]
We need to pivot again if there are negative entries in the \( p \)-row. Analyzing the second tableau, we can see that we \( \ \) need to
---
**Explanation of Graphs/Diagrams:**
1. **Matrix/ Tableau for Pivoting:**
- The matrix labeled \( M \) seems to represent a transformation matrix used in the pivot operation of the Simplex method. It appears to be a 3x3 matrix.
- Another tableau is shown after pivoting. This tableau seems to be part of an iterative process in linear programming to reach the optimal solution. It looks like a 3x6 matrix with further instructions for determining if another pivot is necessary, especially to handle negative entries in what is labeled as the \( p \)-row.
This content is useful for students or individuals learning about matrix operations or optimization techniques in linear programming. The pivot process is a fundamental aspect of the Simplex method, which is widely used in operations research to find optimal solutions to linear inequality constraints.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb01ce655-4372-4af4-9606-6aca74ff4552%2Feb6fd03c-5577-435d-895c-c131d1f78e09%2Ftzn7t4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The text appears to be about performing operations on matrices or tableaus, likely related to solving linear programming problems using the Simplex method.
---
**Text Transcription:**
The pivot is in the \( \ \ \) column and the \( \ \ \) row. It has value \( \ \ \).
To pivot, we need to multiply on the left by
\[
M =
\begin{bmatrix}
\ \ & \ \ & \ \ \\
\ \ & \ \ & \ \ \\
\ \ & \ \ & \ \
\end{bmatrix}
\]
After pivoting, we have our second tableau.
\[
\begin{bmatrix}
\ \ & \ \ & \ \ & \ \ & \ \ & \ \ \\
\ \ & \ \ & \ \ & \ \ & \ \ & \ \ \\
\ \ & \ \ & \ \ & \ \ & \ \ & \ \
\end{bmatrix}
\]
We need to pivot again if there are negative entries in the \( p \)-row. Analyzing the second tableau, we can see that we \( \ \) need to
---
**Explanation of Graphs/Diagrams:**
1. **Matrix/ Tableau for Pivoting:**
- The matrix labeled \( M \) seems to represent a transformation matrix used in the pivot operation of the Simplex method. It appears to be a 3x3 matrix.
- Another tableau is shown after pivoting. This tableau seems to be part of an iterative process in linear programming to reach the optimal solution. It looks like a 3x6 matrix with further instructions for determining if another pivot is necessary, especially to handle negative entries in what is labeled as the \( p \)-row.
This content is useful for students or individuals learning about matrix operations or optimization techniques in linear programming. The pivot process is a fundamental aspect of the Simplex method, which is widely used in operations research to find optimal solutions to linear inequality constraints.
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