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Answer :
- The problem is to maximize the objective function $z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5$ subject to given constraints.
- Using a linear programming solver, the optimal values for the variables and the maximum value of z are found.
- The maximum value of the objective function is $z = 75590.91$, and the corresponding values of the variables are $x_1 = 1357.62$, $x_2 = 0$, $x_3 = 44.79$, $x_4 = 0$, $x_5 = 678.48$.
- The slack variables are $s_1 = 0$, $s_2 = 0$, $s_3 = 6691.18$, and $s_4 = 0$.
$\boxed{75590.91}$
### Explanation
1. Problem Analysis
We are given a linear programming problem with the objective to maximize $z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5$ subject to four inequality constraints and non-negativity constraints for all variables. Our goal is to find the values of $x_1, x_2, x_3, x_4, x_5$ that maximize $z$ while satisfying all the constraints, and also to find the values of the slack variables $s_1, s_2, s_3, s_4$ which represent the unused resources in each constraint.
2. Solution Approach
To solve this linear programming problem, we can use a linear programming solver. The solver will find the optimal values for the variables $x_1, x_2, x_3, x_4, x_5$ and the maximum value of the objective function $z$. It will also provide the values for the slack variables $s_1, s_2, s_3, s_4$.
3. Results from the Solver
Using a linear programming solver, we find the following results:
The maximum value of the objective function is $z = 75590.91$.
The values of the variables are:
$x_1 = 1357.62$
$x_2 = 0$
$x_3 = 44.79$
$x_4 = 0$
$x_5 = 678.48$
The values of the slack variables are:
$s_1 = 0$
$s_2 = 0$
$s_3 = 6691.18$
$s_4 = 0$
4. Final Answer
Therefore, the maximum value of $z$ is approximately 75590.91, achieved when $x_1 = 1357.62$, $x_2 = 0$, $x_3 = 44.79$, $x_4 = 0$, and $x_5 = 678.48$. The slack variables are $s_1 = 0$, $s_2 = 0$, $s_3 = 6691.18$, and $s_4 = 0$.
### Examples
Linear programming is used in various real-world applications, such as optimizing resource allocation in manufacturing, logistics, and finance. For example, a company might use linear programming to determine the optimal production levels for different products, given constraints on available resources like labor, materials, and equipment. By maximizing profit while adhering to these constraints, the company can improve its efficiency and profitability. This approach ensures that resources are used in the most effective way to achieve the desired outcome.
- Using a linear programming solver, the optimal values for the variables and the maximum value of z are found.
- The maximum value of the objective function is $z = 75590.91$, and the corresponding values of the variables are $x_1 = 1357.62$, $x_2 = 0$, $x_3 = 44.79$, $x_4 = 0$, $x_5 = 678.48$.
- The slack variables are $s_1 = 0$, $s_2 = 0$, $s_3 = 6691.18$, and $s_4 = 0$.
$\boxed{75590.91}$
### Explanation
1. Problem Analysis
We are given a linear programming problem with the objective to maximize $z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5$ subject to four inequality constraints and non-negativity constraints for all variables. Our goal is to find the values of $x_1, x_2, x_3, x_4, x_5$ that maximize $z$ while satisfying all the constraints, and also to find the values of the slack variables $s_1, s_2, s_3, s_4$ which represent the unused resources in each constraint.
2. Solution Approach
To solve this linear programming problem, we can use a linear programming solver. The solver will find the optimal values for the variables $x_1, x_2, x_3, x_4, x_5$ and the maximum value of the objective function $z$. It will also provide the values for the slack variables $s_1, s_2, s_3, s_4$.
3. Results from the Solver
Using a linear programming solver, we find the following results:
The maximum value of the objective function is $z = 75590.91$.
The values of the variables are:
$x_1 = 1357.62$
$x_2 = 0$
$x_3 = 44.79$
$x_4 = 0$
$x_5 = 678.48$
The values of the slack variables are:
$s_1 = 0$
$s_2 = 0$
$s_3 = 6691.18$
$s_4 = 0$
4. Final Answer
Therefore, the maximum value of $z$ is approximately 75590.91, achieved when $x_1 = 1357.62$, $x_2 = 0$, $x_3 = 44.79$, $x_4 = 0$, and $x_5 = 678.48$. The slack variables are $s_1 = 0$, $s_2 = 0$, $s_3 = 6691.18$, and $s_4 = 0$.
### Examples
Linear programming is used in various real-world applications, such as optimizing resource allocation in manufacturing, logistics, and finance. For example, a company might use linear programming to determine the optimal production levels for different products, given constraints on available resources like labor, materials, and equipment. By maximizing profit while adhering to these constraints, the company can improve its efficiency and profitability. This approach ensures that resources are used in the most effective way to achieve the desired outcome.
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