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Answer :
- The problem is to maximize $z = x_1 + 4x_2 + x_3 + 5x_4$ subject to $x_1 + 5x_2 + x_3 + x_4 \leq 70$ and $5x_1 + x_2 + 9x_3 + x_4 \leq 180$.
- Using the simplex method, the maximum value is found to be 350 when $x_1 = 0$, $x_2 = 0$, $x_3 = 0$, and $x_4 = 70$.
- Substituting these values into the constraint equations, we find the slack variables $s_1 = 0$ and $s_2 = 110$.
- The maximum is 350 when $x_1=0$, $x_2=0$, $x_3=0$, $x_4=70$, $s_1=0$, and $s_2=110$. $\boxed{350}$
### Explanation
1. Problem Analysis
We are given a linear programming problem and asked to solve it using the simplex method. The problem is to maximize $z = x_1 + 4x_2 + x_3 + 5x_4$ subject to the constraints $x_1 + 5x_2 + x_3 + x_4 \leq 70$, $5x_1 + x_2 + 9x_3 + x_4 \leq 180$, and $x_1, x_2, x_3, x_4 \geq 0$.
2. Applying the Simplex Method
To solve this linear programming problem, we can use the simplex method. After using a tool to perform the simplex method, we find that the maximum value of the objective function is 350, which occurs when $x_1 = 0$, $x_2 = 0$, $x_3 = 0$, and $x_4 = 70$.
3. Finding Slack Variables
To find the values of the slack variables $s_1$ and $s_2$, we substitute the values of $x_1, x_2, x_3,$ and $x_4$ into the constraint equations. The first constraint is $x_1 + 5x_2 + x_3 + x_4 + s_1 = 70$. Substituting the values, we get $0 + 5(0) + 0 + 70 + s_1 = 70$, which simplifies to $70 + s_1 = 70$. Therefore, $s_1 = 0$. The second constraint is $5x_1 + x_2 + 9x_3 + x_4 + s_2 = 180$. Substituting the values, we get $5(0) + 0 + 9(0) + 70 + s_2 = 180$, which simplifies to $70 + s_2 = 180$. Therefore, $s_2 = 110$.
4. Final Answer
Thus, the maximum value of the objective function is 350, which occurs when $x_1 = 0$, $x_2 = 0$, $x_3 = 0$, $x_4 = 70$, $s_1 = 0$, and $s_2 = 110$.
### 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 of different products to maximize profit while considering constraints such as available resources, production capacity, and demand. Similarly, logistics companies use linear programming to optimize delivery routes and minimize transportation costs. Financial institutions use it to optimize investment portfolios and manage risk.
- Using the simplex method, the maximum value is found to be 350 when $x_1 = 0$, $x_2 = 0$, $x_3 = 0$, and $x_4 = 70$.
- Substituting these values into the constraint equations, we find the slack variables $s_1 = 0$ and $s_2 = 110$.
- The maximum is 350 when $x_1=0$, $x_2=0$, $x_3=0$, $x_4=70$, $s_1=0$, and $s_2=110$. $\boxed{350}$
### Explanation
1. Problem Analysis
We are given a linear programming problem and asked to solve it using the simplex method. The problem is to maximize $z = x_1 + 4x_2 + x_3 + 5x_4$ subject to the constraints $x_1 + 5x_2 + x_3 + x_4 \leq 70$, $5x_1 + x_2 + 9x_3 + x_4 \leq 180$, and $x_1, x_2, x_3, x_4 \geq 0$.
2. Applying the Simplex Method
To solve this linear programming problem, we can use the simplex method. After using a tool to perform the simplex method, we find that the maximum value of the objective function is 350, which occurs when $x_1 = 0$, $x_2 = 0$, $x_3 = 0$, and $x_4 = 70$.
3. Finding Slack Variables
To find the values of the slack variables $s_1$ and $s_2$, we substitute the values of $x_1, x_2, x_3,$ and $x_4$ into the constraint equations. The first constraint is $x_1 + 5x_2 + x_3 + x_4 + s_1 = 70$. Substituting the values, we get $0 + 5(0) + 0 + 70 + s_1 = 70$, which simplifies to $70 + s_1 = 70$. Therefore, $s_1 = 0$. The second constraint is $5x_1 + x_2 + 9x_3 + x_4 + s_2 = 180$. Substituting the values, we get $5(0) + 0 + 9(0) + 70 + s_2 = 180$, which simplifies to $70 + s_2 = 180$. Therefore, $s_2 = 110$.
4. Final Answer
Thus, the maximum value of the objective function is 350, which occurs when $x_1 = 0$, $x_2 = 0$, $x_3 = 0$, $x_4 = 70$, $s_1 = 0$, and $s_2 = 110$.
### 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 of different products to maximize profit while considering constraints such as available resources, production capacity, and demand. Similarly, logistics companies use linear programming to optimize delivery routes and minimize transportation costs. Financial institutions use it to optimize investment portfolios and manage risk.
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