Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Linear Programming Problem Using Simplex Method
**Problem Statement:**
Maximize \( P = 18x_1 + x_2 \)
**Subject to:**
\[
\begin{align*}
4x_1 + x_2 &\leq 12 \\
x_1 + 8x_2 &\leq 12 \\
x_1, x_2 &\geq 0
\end{align*}
\]
### Part (A) Using Slack Variables
To convert the inequalities into equalities by adding slack variables \( s_1 \) and \( s_2 \):
1. **First Constraint:**
\[
4x_1 + x_2 + s_1 = 12
\]
2. **Second Constraint:**
\[
x_1 + 8x_2 + s_2 = 12
\]
3. **Objective Function:**
\[
P = 18x_1 + x_2
\]
The variables \( x_1, x_2, s_1, s_2 \geq 0 \).
### Part (B) Simplex Tableau
Fill in the blanks in the simplex tableau as follows:
\[
\begin{array}{c|cccc|c}
& x_1 & x_2 & s_1 & s_2 & \text{P} \\ \hline
s_1 & 4 & 1 & 1 & 0 & 12 \\
s_2 & 1 & 8 & 0 & 1 & 12 \\
\text{P} & -18 & -1 & 0 & 0 & 0 \\
\end{array}
\]
Determine the pivot element, entering variable, and exiting variable:
- **Pivot Column:** The most negative value in the objective function row is in the \( x_1 \) column.
- **Pivot Row:** Determine using the smallest positive ratio of the right-hand side to the key column, 12/4 for \( s_1 \) and 12/1 for \( s_2 \). The smallest ratio is 12/1, so pivot row is row 2.
Therefore, the pivot element is in the \( x_1 \) column and the \( s_2 \)](https://content.bartleby.com/qna-images/question/d0cd4ae9-86e1-45b4-8824-a362b78dd084/533b6d41-e1ae-47c2-a14e-0d9fb5cdcaba/w0fgn3p_processed.png)
Transcribed Image Text:### Linear Programming Problem Using Simplex Method
**Problem Statement:**
Maximize \( P = 18x_1 + x_2 \)
**Subject to:**
\[
\begin{align*}
4x_1 + x_2 &\leq 12 \\
x_1 + 8x_2 &\leq 12 \\
x_1, x_2 &\geq 0
\end{align*}
\]
### Part (A) Using Slack Variables
To convert the inequalities into equalities by adding slack variables \( s_1 \) and \( s_2 \):
1. **First Constraint:**
\[
4x_1 + x_2 + s_1 = 12
\]
2. **Second Constraint:**
\[
x_1 + 8x_2 + s_2 = 12
\]
3. **Objective Function:**
\[
P = 18x_1 + x_2
\]
The variables \( x_1, x_2, s_1, s_2 \geq 0 \).
### Part (B) Simplex Tableau
Fill in the blanks in the simplex tableau as follows:
\[
\begin{array}{c|cccc|c}
& x_1 & x_2 & s_1 & s_2 & \text{P} \\ \hline
s_1 & 4 & 1 & 1 & 0 & 12 \\
s_2 & 1 & 8 & 0 & 1 & 12 \\
\text{P} & -18 & -1 & 0 & 0 & 0 \\
\end{array}
\]
Determine the pivot element, entering variable, and exiting variable:
- **Pivot Column:** The most negative value in the objective function row is in the \( x_1 \) column.
- **Pivot Row:** Determine using the smallest positive ratio of the right-hand side to the key column, 12/4 for \( s_1 \) and 12/1 for \( s_2 \). The smallest ratio is 12/1, so pivot row is row 2.
Therefore, the pivot element is in the \( x_1 \) column and the \( s_2 \)
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