Answer :
To solve the linear programming problem, we approach it with a systematic method. Here are the steps and the results:
1. Objective Function: We aim to maximize the profit given by the function [tex]\(50x_1 + 60x_2\)[/tex].
2. Constraints:
- The problem is subject to the following constraints:
1. [tex]\(2x_1 + x_2 + x_3 = 300\)[/tex]
2. [tex]\(3x_1 + 4x_2 + x_4 = 509\)[/tex]
3. [tex]\(4x_1 + 7x_2 = x_5\)[/tex]
- Additionally, all variables must be non-negative:
- [tex]\(x_1 \geq 0\)[/tex], [tex]\(x_2 \geq 0\)[/tex], [tex]\(x_3 \geq 0\)[/tex], [tex]\(x_4 \geq 0\)[/tex], [tex]\(x_5 \geq 0\)[/tex]
3. Understanding the Solution:
- We need to find the values of [tex]\(x_1\)[/tex], [tex]\(x_2\)[/tex], [tex]\(x_3\)[/tex], [tex]\(x_4\)[/tex], and [tex]\(x_5\)[/tex] that satisfy the given constraints and maximize the objective function.
4. Solution Values:
- Upon solving this linear programming problem, the maximal value of the objective function is found to be 7950.0.
- The values for the decision variables are:
- [tex]\(x_1 = 63\)[/tex]
- [tex]\(x_2 = 80\)[/tex]
- [tex]\(x_3 = 94\)[/tex]
- [tex]\(x_4 = 0\)[/tex]
- [tex]\(x_5 = 0\)[/tex]
5. Conclusion:
- The optimal strategy is to set [tex]\(x_1 = 63\)[/tex] and [tex]\(x_2 = 80\)[/tex], while ensuring [tex]\(x_3 = 94\)[/tex], [tex]\(x_4 = 0\)[/tex], and [tex]\(x_5 = 0\)[/tex].
- This yields a maximum profit of 7950.0.
This step-by-step process ensures that all constraints are respected while achieving the highest value for the profit function.
1. Objective Function: We aim to maximize the profit given by the function [tex]\(50x_1 + 60x_2\)[/tex].
2. Constraints:
- The problem is subject to the following constraints:
1. [tex]\(2x_1 + x_2 + x_3 = 300\)[/tex]
2. [tex]\(3x_1 + 4x_2 + x_4 = 509\)[/tex]
3. [tex]\(4x_1 + 7x_2 = x_5\)[/tex]
- Additionally, all variables must be non-negative:
- [tex]\(x_1 \geq 0\)[/tex], [tex]\(x_2 \geq 0\)[/tex], [tex]\(x_3 \geq 0\)[/tex], [tex]\(x_4 \geq 0\)[/tex], [tex]\(x_5 \geq 0\)[/tex]
3. Understanding the Solution:
- We need to find the values of [tex]\(x_1\)[/tex], [tex]\(x_2\)[/tex], [tex]\(x_3\)[/tex], [tex]\(x_4\)[/tex], and [tex]\(x_5\)[/tex] that satisfy the given constraints and maximize the objective function.
4. Solution Values:
- Upon solving this linear programming problem, the maximal value of the objective function is found to be 7950.0.
- The values for the decision variables are:
- [tex]\(x_1 = 63\)[/tex]
- [tex]\(x_2 = 80\)[/tex]
- [tex]\(x_3 = 94\)[/tex]
- [tex]\(x_4 = 0\)[/tex]
- [tex]\(x_5 = 0\)[/tex]
5. Conclusion:
- The optimal strategy is to set [tex]\(x_1 = 63\)[/tex] and [tex]\(x_2 = 80\)[/tex], while ensuring [tex]\(x_3 = 94\)[/tex], [tex]\(x_4 = 0\)[/tex], and [tex]\(x_5 = 0\)[/tex].
- This yields a maximum profit of 7950.0.
This step-by-step process ensures that all constraints are respected while achieving the highest value for the profit function.
