Math / Algebra

Question10. Use simplex method to solve $\begin{aligned} \text { Maximize } z & =7 x_{1}+5 x_{2} \\ \text { Subjected to } & 2 x_{1}+x_{2} \leq 10 \\ & 4 x_{1}+3 x_{2} \leq 24 \\ & x_{1} \geq 0, x_{2} \geq 0 \end{aligned}$

Studdy Solution

STEP 1

1. The problem is a linear programming problem that requires the use of the simplex method.
2. The objective is to maximize the function $z = 7x_1 + 5x_2$ .
3. The constraints are linear inequalities, and the variables $x_1$ and $x_2$ are non-negative.

STEP 2

1. Convert the inequalities into equations by adding slack variables.
2. Set up the initial simplex tableau.
3. Perform the simplex algorithm iterations to find the optimal solution.
4. Interpret the final tableau to determine the solution.

STEP 3

Convert the inequalities into equations by adding slack variables $s_1$ and $s_2$ :
1. $2x_1 + x_2 + s_1 = 10$
2. $4x_1 + 3x_2 + s_2 = 24$
The slack variables $s_1$ and $s_2$ represent the unused resources in each constraint.

STEP 4

Set up the initial simplex tableau. The tableau includes the coefficients of the variables in the objective function and constraints:
$\begin{array}{c|cccc|c} \text{Basis} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline s_1 & 2 & 1 & 1 & 0 & 10 \\ s_2 & 4 & 3 & 0 & 1 & 24 \\ \hline z & -7 & -5 & 0 & 0 & 0 \\ \end{array}$

STEP 5

Perform the simplex algorithm iterations:
1. Identify the entering variable (most negative coefficient in the $z$ -row).
2. Identify the leaving variable using the minimum ratio test.
3. Pivot to update the tableau.

STEP 6

First Iteration:
1. Entering variable: $x_1$ (most negative coefficient $-7$ ).
2. Minimum ratio test for leaving variable: - $\frac{10}{2} = 5$ for $s_1$ - $\frac{24}{4} = 6$ for $s_2$
The smallest ratio is 5, so $s_1$ leaves.
3. Pivot on the element in row 1, column 1.
Update the tableau:
$\begin{array}{c|cccc|c} \text{Basis} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline x_1 & 1 & \frac{1}{2} & \frac{1}{2} & 0 & 5 \\ s_2 & 0 & 1 & -2 & 1 & 4 \\ \hline z & 0 & -\frac{3}{2} & \frac{7}{2} & 0 & 35 \\ \end{array}$

STEP 7

Second Iteration:
1. Entering variable: $x_2$ (most negative coefficient $-\frac{3}{2}$ ).
2. Minimum ratio test for leaving variable: - $\frac{5}{\frac{1}{2}} = 10$ for $x_1$ - $\frac{4}{1} = 4$ for $s_2$
The smallest ratio is 4, so $s_2$ leaves.
3. Pivot on the element in row 2, column 2.
Update the tableau:
$\begin{array}{c|cccc|c} \text{Basis} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline x_1 & 1 & 0 & 2 & -\frac{1}{2} & 3 \\ x_2 & 0 & 1 & -2 & 1 & 4 \\ \hline z & 0 & 0 & 1 & \frac{3}{2} & 43 \\ \end{array}$

STEP 8

Interpret the final tableau:
The optimal solution is $x_1 = 3$ , $x_2 = 4$ , with $z = 43$ .

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Question10. Use simplex method to solve $\begin{aligned} \text { Maximize } z & =7 x_{1}+5 x_{2} \\ \text { Subjected to } & 2 x_{1}+x_{2} \leq 10 \\ & 4 x_{1}+3 x_{2} \leq 24 \\ & x_{1} \geq 0, x_{2} \geq 0 \end{aligned}$

Studdy Solution

STEP 1

1. The problem is a linear programming problem that requires the use of the simplex method.
2. The objective is to maximize the function $z = 7x_1 + 5x_2$ .
3. The constraints are linear inequalities, and the variables $x_1$ and $x_2$ are non-negative.

STEP 2

1. Convert the inequalities into equations by adding slack variables.
2. Set up the initial simplex tableau.
3. Perform the simplex algorithm iterations to find the optimal solution.
4. Interpret the final tableau to determine the solution.

STEP 3

Convert the inequalities into equations by adding slack variables $s_1$ and $s_2$ :
1. $2x_1 + x_2 + s_1 = 10$
2. $4x_1 + 3x_2 + s_2 = 24$
The slack variables $s_1$ and $s_2$ represent the unused resources in each constraint.

STEP 4

STEP 5

Perform the simplex algorithm iterations:
1. Identify the entering variable (most negative coefficient in the $z$ -row).
2. Identify the leaving variable using the minimum ratio test.
3. Pivot to update the tableau.

STEP 6

STEP 7

STEP 8

Interpret the final tableau:
The optimal solution is $x_1 = 3$ , $x_2 = 4$ , with $z = 43$ .

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Studdy Solution

STEP 1

1. The problem is a linear programming problem that requires the use of the simplex method.2. The objective is to maximize the function z=7x1+5x2 z = 7x_1 + 5x_2 z=7x1​+5x2​.3. The constraints are linear inequalities, and the variables x1 x_1 x1​ and x2 x_2 x2​ are non-negative.

STEP 2

1. Convert the inequalities into equations by adding slack variables.2. Set up the initial simplex tableau.3. Perform the simplex algorithm iterations to find the optimal solution.4. Interpret the final tableau to determine the solution.

STEP 3

STEP 4

STEP 5

Perform the simplex algorithm iterations:1. Identify the entering variable (most negative coefficient in the z z z-row).2. Identify the leaving variable using the minimum ratio test.3. Pivot to update the tableau.

STEP 6

STEP 7

STEP 8

Interpret the final tableau:The optimal solution is x1=3 x_1 = 3 x1​=3, x2=4 x_2 = 4 x2​=4, with z=43 z = 43 z=43.

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Studdy solves anything!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The problem is a linear programming problem that requires the use of the simplex method.
2. The objective is to maximize the function $z = 7x_1 + 5x_2$ .
3. The constraints are linear inequalities, and the variables $x_1$ and $x_2$ are non-negative.

1. Convert the inequalities into equations by adding slack variables.
2. Set up the initial simplex tableau.
3. Perform the simplex algorithm iterations to find the optimal solution.
4. Interpret the final tableau to determine the solution.

Perform the simplex algorithm iterations:
1. Identify the entering variable (most negative coefficient in the $z$ -row).
2. Identify the leaving variable using the minimum ratio test.
3. Pivot to update the tableau.

Interpret the final tableau:
The optimal solution is $x_1 = 3$ , $x_2 = 4$ , with $z = 43$ .