Maximize f = 2x + 9y subject to 9х + бу S 42 -5x + y 2 4 x2 0, у20. (a) State the given problem in a form from which the simplex matrix can be formed (that is, as a maximization problem with < constraints). f = < 42 S-4 х, у 20 (b) Form the simplex matrix. S2 first constraint second constraint objective function Determine the first pivot entry. The first pivot is in row column
Maximize f = 2x + 9y subject to 9х + бу S 42 -5x + y 2 4 x2 0, у20. (a) State the given problem in a form from which the simplex matrix can be formed (that is, as a maximization problem with < constraints). f = < 42 S-4 х, у 20 (b) Form the simplex matrix. S2 first constraint second constraint objective function Determine the first pivot entry. The first pivot is in row column
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 and the Simplex Method**
**Problem Description:**
Maximize \( f = 2x + 9y \) subject to:
- \( 9x + 6y \leq 42 \)
- \(-5x + y \geq 4 \)
- \( x \geq 0, \, y \geq 0 \)
**Instructions:**
(a) **Convert the Constraints:**
Reformulate the second constraint in a form suitable for the Simplex method (as a maximization problem with \(\leq\) constraints).
- Given Objective Function: \( f = \)
- Constraint 1: \( \leq 42 \)
- Constraint 2: \( \leq -4 \)
Ensure \( x, y \geq 0 \).
(b) **Formulate the Simplex Matrix:**
Create the simplex matrix corresponding to the system of inequalities.
| | \( x \) | \( y \) | \( s_1 \) | \( s_2 \) | \( f \) |
|-------|---------|---------|-----------|-----------|---------|
| First Constraint | | | | | |
| Second Constraint | | | | | |
| Objective Function | | | | | |
Determine the first pivot entry.
- The first pivot is in row ___, column ___.
---
**Minimization Problem Solution:**
**Simplex Matrix** for the minimization problem:
\[
\begin{array}{cccccc|c}
x & y & z & s_1 & s_2 & s_3 & -f \\
\hline
1 & 0 & 0 & 2 & 2 & 0 & 70 \\
0 & 1 & 1 & 1 & 11 & 0 & 16 \\
0 & 0 & 1 & -9 & 11 & 0 & 30 \\
0 & 0 & 0 & 5 & 1 & 2 & -1300 \\
\end{array}
\]
**Solution:**
\( x = \)
\( y = \)
\( z = \)
\( f = \)
**Note:** Fill in the \( x \), \( y \), \( z \), and \( f \) values after](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7878796-4071-4397-9fae-1cbeb3b69837%2F3524b71e-b641-4e51-8d62-34fcb0e4be10%2Fos6eh1_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Programming and the Simplex Method**
**Problem Description:**
Maximize \( f = 2x + 9y \) subject to:
- \( 9x + 6y \leq 42 \)
- \(-5x + y \geq 4 \)
- \( x \geq 0, \, y \geq 0 \)
**Instructions:**
(a) **Convert the Constraints:**
Reformulate the second constraint in a form suitable for the Simplex method (as a maximization problem with \(\leq\) constraints).
- Given Objective Function: \( f = \)
- Constraint 1: \( \leq 42 \)
- Constraint 2: \( \leq -4 \)
Ensure \( x, y \geq 0 \).
(b) **Formulate the Simplex Matrix:**
Create the simplex matrix corresponding to the system of inequalities.
| | \( x \) | \( y \) | \( s_1 \) | \( s_2 \) | \( f \) |
|-------|---------|---------|-----------|-----------|---------|
| First Constraint | | | | | |
| Second Constraint | | | | | |
| Objective Function | | | | | |
Determine the first pivot entry.
- The first pivot is in row ___, column ___.
---
**Minimization Problem Solution:**
**Simplex Matrix** for the minimization problem:
\[
\begin{array}{cccccc|c}
x & y & z & s_1 & s_2 & s_3 & -f \\
\hline
1 & 0 & 0 & 2 & 2 & 0 & 70 \\
0 & 1 & 1 & 1 & 11 & 0 & 16 \\
0 & 0 & 1 & -9 & 11 & 0 & 30 \\
0 & 0 & 0 & 5 & 1 & 2 & -1300 \\
\end{array}
\]
**Solution:**
\( x = \)
\( y = \)
\( z = \)
\( f = \)
**Note:** Fill in the \( x \), \( y \), \( z \), and \( f \) values after
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