Answer :
To solve this linear programming problem, we want to maximize the objective function while satisfying a set of constraints. Let's walk through the process step-by-step:
### Objective Function:
The objective function we want to maximize is:
[tex]\[ z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5 \][/tex]
### Constraints:
We have four inequality constraints and non-negativity constraints for the variables:
1. [tex]\( 16x_1 + 19x_2 + 23x_3 + 15x_4 + 21x_5 \leq 35,000 \)[/tex]
2. [tex]\( 15x_1 + 10x_2 + 19x_3 + 23x_4 + 10x_5 \leq 29,000 \)[/tex]
3. [tex]\( 9x_1 + 16x_2 + 14x_3 + 12x_4 + 11x_5 \leq 23,000 \)[/tex]
4. [tex]\( 18x_1 + 20x_2 + 15x_3 + 17x_4 + 19x_5 \leq 31,000 \)[/tex]
All variables must be non-negative:
[tex]\[ x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0, \, x_4 \geq 0, \, x_5 \geq 0 \][/tex]
### Solution:
After analyzing the problem, the values for the variables that maximize the objective function given the constraints are:
- [tex]\( x_1 = 930.48 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( x_3 = 679.14 \)[/tex]
- [tex]\( x_4 = 0 \)[/tex]
- [tex]\( x_5 = 213.90 \)[/tex]
The maximum value of the objective function is:
[tex]\[ z = 66,363.64 \][/tex]
### Summary:
- The maximum value of [tex]\( z \)[/tex] occurs when [tex]\( x_1 = 930.48 \)[/tex], [tex]\( x_2 = 0 \)[/tex], [tex]\( x_3 = 679.14 \)[/tex], [tex]\( x_4 = 0 \)[/tex], and [tex]\( x_5 = 213.90 \)[/tex].
- The value of the maximum objective function is [tex]\( 66,363.64 \)[/tex] (rounded to two decimal places).
By keeping the constraints in mind and optimizing the allocation of resources (the variables [tex]\( x_1 \)[/tex] to [tex]\( x_5 \)[/tex]), this solution achieves the highest possible outcome for the given problem.
### Objective Function:
The objective function we want to maximize is:
[tex]\[ z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5 \][/tex]
### Constraints:
We have four inequality constraints and non-negativity constraints for the variables:
1. [tex]\( 16x_1 + 19x_2 + 23x_3 + 15x_4 + 21x_5 \leq 35,000 \)[/tex]
2. [tex]\( 15x_1 + 10x_2 + 19x_3 + 23x_4 + 10x_5 \leq 29,000 \)[/tex]
3. [tex]\( 9x_1 + 16x_2 + 14x_3 + 12x_4 + 11x_5 \leq 23,000 \)[/tex]
4. [tex]\( 18x_1 + 20x_2 + 15x_3 + 17x_4 + 19x_5 \leq 31,000 \)[/tex]
All variables must be non-negative:
[tex]\[ x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0, \, x_4 \geq 0, \, x_5 \geq 0 \][/tex]
### Solution:
After analyzing the problem, the values for the variables that maximize the objective function given the constraints are:
- [tex]\( x_1 = 930.48 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( x_3 = 679.14 \)[/tex]
- [tex]\( x_4 = 0 \)[/tex]
- [tex]\( x_5 = 213.90 \)[/tex]
The maximum value of the objective function is:
[tex]\[ z = 66,363.64 \][/tex]
### Summary:
- The maximum value of [tex]\( z \)[/tex] occurs when [tex]\( x_1 = 930.48 \)[/tex], [tex]\( x_2 = 0 \)[/tex], [tex]\( x_3 = 679.14 \)[/tex], [tex]\( x_4 = 0 \)[/tex], and [tex]\( x_5 = 213.90 \)[/tex].
- The value of the maximum objective function is [tex]\( 66,363.64 \)[/tex] (rounded to two decimal places).
By keeping the constraints in mind and optimizing the allocation of resources (the variables [tex]\( x_1 \)[/tex] to [tex]\( x_5 \)[/tex]), this solution achieves the highest possible outcome for the given problem.
