Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![To solve the given linear programming problem, let's first introduce slack variables to convert the inequalities into equalities.
The original problem is:
Maximize:
\[ P = 3x_1 + 5x_2 + 2x_3, \]
Subject to:
\[
\begin{array}{rcl}
3x_1 + 4x_2 + 5x_3 & \leq & 10, \\
x_1 + 3x_2 + 10x_3 & \leq & 5, \\
x_1 - 2x_2 & \leq & 1, \\
x_1, x_2, x_3 & \geq & 0.
\end{array}
\]
### Step 1: Introduce Slack Variables
To convert each inequality constraint to an equality, we introduce slack variables \( s_1, s_2, s_3 \geq 0 \):
\[
\begin{array}{rcl}
3x_1 + 4x_2 + 5x_3 + s_1 & = & 10, \\
x_1 + 3x_2 + 10x_3 + s_2 & = & 5, \\
x_1 - 2x_2 + s_3 & = & 1, \\
x_1, x_2, x_3, s_1, s_2, s_3 & \geq & 0.
\end{array}
\]
### Step 2: Construct the Initial Simplex Tableau
Next, we construct the initial simplex tableau.
The objective function \( P \) is expressed in terms of all variables with slack variables included. We aim to maximize \( P \) by converting it into a minimization problem of the negative of \( P \):
\[
P = 3x_1 + 5x_2 + 2x_3 + 0s_1 + 0s_2 + 0s_3
\]
Converting to standard form for minimizing:
\[
-P + 3x_1 + 5x_2 + 2x_3 = 0
\]
The initial simplex tableau will look like this:
\[
\begin{array}{c|cccccc|c}
& x_1 & x_2](https://content.bartleby.com/qna-images/question/c20b48a5-43ea-4f47-9a73-2be9a8dcfa66/5d3389b9-f3b5-4449-b11b-e914d7e2d22f/xrveuzp.jpeg)
Transcribed Image Text:To solve the given linear programming problem, let's first introduce slack variables to convert the inequalities into equalities.
The original problem is:
Maximize:
\[ P = 3x_1 + 5x_2 + 2x_3, \]
Subject to:
\[
\begin{array}{rcl}
3x_1 + 4x_2 + 5x_3 & \leq & 10, \\
x_1 + 3x_2 + 10x_3 & \leq & 5, \\
x_1 - 2x_2 & \leq & 1, \\
x_1, x_2, x_3 & \geq & 0.
\end{array}
\]
### Step 1: Introduce Slack Variables
To convert each inequality constraint to an equality, we introduce slack variables \( s_1, s_2, s_3 \geq 0 \):
\[
\begin{array}{rcl}
3x_1 + 4x_2 + 5x_3 + s_1 & = & 10, \\
x_1 + 3x_2 + 10x_3 + s_2 & = & 5, \\
x_1 - 2x_2 + s_3 & = & 1, \\
x_1, x_2, x_3, s_1, s_2, s_3 & \geq & 0.
\end{array}
\]
### Step 2: Construct the Initial Simplex Tableau
Next, we construct the initial simplex tableau.
The objective function \( P \) is expressed in terms of all variables with slack variables included. We aim to maximize \( P \) by converting it into a minimization problem of the negative of \( P \):
\[
P = 3x_1 + 5x_2 + 2x_3 + 0s_1 + 0s_2 + 0s_3
\]
Converting to standard form for minimizing:
\[
-P + 3x_1 + 5x_2 + 2x_3 = 0
\]
The initial simplex tableau will look like this:
\[
\begin{array}{c|cccccc|c}
& x_1 & x_2
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