2. Given the general linear programming problem, write the equations (with slack that would be used for the simplex method. Write the initial tableau and give the Minimize: c = 3x + 2y + 4z Subject to: 3x + 2y +4z > 20 2x - y + 3z < 18
2. Given the general linear programming problem, write the equations (with slack that would be used for the simplex method. Write the initial tableau and give the Minimize: c = 3x + 2y + 4z Subject to: 3x + 2y +4z > 20 2x - y + 3z < 18
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 and Simplex Method
**Problem Statement:**
2. Given the general linear programming problem, write the equations (with slack/surplus variables) that would be used for the simplex method. Write the initial tableau and give the current solution.
**Objective Function:**
Minimize: \( c = 3x + 2y + 4z \)
**Constraints:**
1. \( 3x + 2y + 4z \geq 20 \)
2. \( 2x - y + 3z \leq 18 \)
3. \( x, y, z \geq 0 \)
**Equations with Slack/Surplus Variables:**
To convert the inequalities into equations usable in the simplex method, we introduce slack and surplus variables.
1. For \(3x + 2y + 4z \geq 20\), add a surplus variable \(s_1\):
\[
3x + 2y + 4z - s_1 = 20
\]
2. For \(2x - y + 3z \leq 18\), add a slack variable \(s_2\):
\[
2x - y + 3z + s_2 = 18
\]
3. \(x, y, z, s_1, s_2 \geq 0\)
**Initial Tableau:**
The tableau for the Simplex method is set up using the equations above, typically showing coefficients for each variable including the objective function row and solution column.
Since the tableau is not part of the image, create a typical form for the problem:
\[
\begin{array}{c|cccc|c}
\text{Basic Variables} & x & y & z & s_1 & \text{Solution} \\
\hline
s_1 & 3 & 2 & 4 & -1 & 20 \\
s_2 & 2 & -1 & 3 & 0 & 18 \\
\hline
z & -3 & -2 & -4 & 0 & 0 \\
\end{array}
\]
**Current Solution:**
In the initial tableau, the basic feasible solution can be derived by setting non-basic variables (here, \(x, y, z\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb83496cd-9baf-4218-91af-6a9eb71001cb%2Febb91ff8-b8ef-464b-901a-003fd281bfb1%2F0jnu2q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Programming Problem and Simplex Method
**Problem Statement:**
2. Given the general linear programming problem, write the equations (with slack/surplus variables) that would be used for the simplex method. Write the initial tableau and give the current solution.
**Objective Function:**
Minimize: \( c = 3x + 2y + 4z \)
**Constraints:**
1. \( 3x + 2y + 4z \geq 20 \)
2. \( 2x - y + 3z \leq 18 \)
3. \( x, y, z \geq 0 \)
**Equations with Slack/Surplus Variables:**
To convert the inequalities into equations usable in the simplex method, we introduce slack and surplus variables.
1. For \(3x + 2y + 4z \geq 20\), add a surplus variable \(s_1\):
\[
3x + 2y + 4z - s_1 = 20
\]
2. For \(2x - y + 3z \leq 18\), add a slack variable \(s_2\):
\[
2x - y + 3z + s_2 = 18
\]
3. \(x, y, z, s_1, s_2 \geq 0\)
**Initial Tableau:**
The tableau for the Simplex method is set up using the equations above, typically showing coefficients for each variable including the objective function row and solution column.
Since the tableau is not part of the image, create a typical form for the problem:
\[
\begin{array}{c|cccc|c}
\text{Basic Variables} & x & y & z & s_1 & \text{Solution} \\
\hline
s_1 & 3 & 2 & 4 & -1 & 20 \\
s_2 & 2 & -1 & 3 & 0 & 18 \\
\hline
z & -3 & -2 & -4 & 0 & 0 \\
\end{array}
\]
**Current Solution:**
In the initial tableau, the basic feasible solution can be derived by setting non-basic variables (here, \(x, y, z\
Expert Solution
Step 1
SIMPLEX METHOD: The Simplex method is a method for manually solving linear programming models with slack variables, tableaus, and pivot variables to identify the best solution to an optimization problem. A linear program is a way to get the optimal result from a maximum or minimum equation with linear constraints.
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