Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Simplex Method for Linear Programming
For this educational example, we will demonstrate how to perform a pivot operation in the simplex method. Below is a simplex tableau that requires pivoting to proceed towards finding the optimal solution.
---
#### Initial Simplex Tableau:
\[
\begin{array}{c|cccccc|c}
& x_1 & x_2 & x_3 & s_1 & s_2 & z & \\
\hline
& 1 & 2 & 4 & 1 & 0 & 0 & 66 \\
& \textcolor{cyan}{2} & \textcolor{cyan}{2} & 1 & 0 & 1 & 0 & 54 \\
& -6 & -7 & -5 & 0 & 0 & 1 & 0 \\
\end{array}
\]
The entry in cyan (2) is chosen as the pivot element.
---
#### Completing the Simplex Tableau:
Following the standard simplex algorithm steps, we pivot around the selected element to form the new simplex tableau.
**Step-by-Step Explanation:**
1. **Pivot on the chosen element (2) at row 2, column 1 (x_1):**
- Divide the entire pivot row by the pivot element to make the pivot element equivalent to 1.
2. **Perform row operations to make all other entries in the pivot column (x_1) equal to 0:**
- Update the remaining rows by subtracting appropriate multiples of the new pivot row.
---
#### Resultant Simplex Tableau:
\[
\begin{array}{c|cccccc|c}
& x_1 & x_2 & x_3 & s_1 & s_2 & z & \\
\hline
& 1 & 2 & 4 & 1 & 0 & 0 & 66 \\
& 1 & 1 & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 27 \\
& -6 & -7 & -5 & 0 & 0 & 1 & 0 \\
\end{array}
\]
**Note:**
- The diagram above illustrates the initial simplex tableau and requires you to complete the tableau after performing the pivot operation.
- The tableau displays variables \( x_1](https://content.bartleby.com/qna-images/question/132027c3-2ed1-42c8-912e-c4831680baff/c3d8c476-1701-416f-8378-4e53b3890ead/b6ja15o_processed.png)
Transcribed Image Text:### Simplex Method for Linear Programming
For this educational example, we will demonstrate how to perform a pivot operation in the simplex method. Below is a simplex tableau that requires pivoting to proceed towards finding the optimal solution.
---
#### Initial Simplex Tableau:
\[
\begin{array}{c|cccccc|c}
& x_1 & x_2 & x_3 & s_1 & s_2 & z & \\
\hline
& 1 & 2 & 4 & 1 & 0 & 0 & 66 \\
& \textcolor{cyan}{2} & \textcolor{cyan}{2} & 1 & 0 & 1 & 0 & 54 \\
& -6 & -7 & -5 & 0 & 0 & 1 & 0 \\
\end{array}
\]
The entry in cyan (2) is chosen as the pivot element.
---
#### Completing the Simplex Tableau:
Following the standard simplex algorithm steps, we pivot around the selected element to form the new simplex tableau.
**Step-by-Step Explanation:**
1. **Pivot on the chosen element (2) at row 2, column 1 (x_1):**
- Divide the entire pivot row by the pivot element to make the pivot element equivalent to 1.
2. **Perform row operations to make all other entries in the pivot column (x_1) equal to 0:**
- Update the remaining rows by subtracting appropriate multiples of the new pivot row.
---
#### Resultant Simplex Tableau:
\[
\begin{array}{c|cccccc|c}
& x_1 & x_2 & x_3 & s_1 & s_2 & z & \\
\hline
& 1 & 2 & 4 & 1 & 0 & 0 & 66 \\
& 1 & 1 & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 27 \\
& -6 & -7 & -5 & 0 & 0 & 1 & 0 \\
\end{array}
\]
**Note:**
- The diagram above illustrates the initial simplex tableau and requires you to complete the tableau after performing the pivot operation.
- The tableau displays variables \( x_1
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