Answer :
To solve the given linear programming problem, we need to maximize the objective function while satisfying all given constraints. Let's walk through the solution step by step:
### Step 1: Define the Objective Function
The objective is to maximize:
[tex]\[ z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5 \][/tex]
### Step 2: Define the Constraints
You have the following constraints that need to be satisfied:
1. [tex]\( 16x_1 + 19x_2 + 23x_3 + 15x_4 + 21x_5 \leq 36,000 \)[/tex]
2. [tex]\( 15x_1 + 10x_2 + 19x_3 + 23x_4 + 10x_5 \leq 26,000 \)[/tex]
3. [tex]\( 9x_1 + 16x_2 + 14x_3 + 12x_4 + 11x_5 \leq 25,000 \)[/tex]
4. [tex]\( 18x_1 + 20x_2 + 15x_3 + 17x_4 + 19x_5 \leq 31,000 \)[/tex]
5. [tex]\( x_1 \geq 0, x_2 \geq 0, x_3 \geq 0, x_4 \geq 0, x_5 \geq 0 \)[/tex] (non-negativity constraints)
### Step 3: Solve the Linear Programming Problem
Using appropriate linear programming methods, we determine the values of decision variables [tex]\(x_1, x_2, x_3, x_4, \)[/tex] and [tex]\(x_5\)[/tex] that maximize the objective function while satisfying all the constraints.
### Step 4: Find the Optimal Solution
The resulting optimal values from solving the problem are:
- [tex]\(x_1 = 544.79\)[/tex]
- [tex]\(x_2 = 0.00\)[/tex]
- [tex]\(x_3 = 600.94\)[/tex]
- [tex]\(x_4 = 0.00\)[/tex]
- [tex]\(x_5 = 641.04\)[/tex]
The maximum value of the objective function [tex]\( z \)[/tex], achieved with these variable values, is:
[tex]\[ z = 64,227.27 \][/tex]
### Step 5: Determine the Slack Variables
Slack variables show how much "slack" or "unused capacity" is left in each of the inequality constraints:
- Slack for Constraint 1: [tex]\(0.00\)[/tex]
- Slack for Constraint 2: [tex]\(0.00\)[/tex]
- Slack for Constraint 3: [tex]\(4632.35\)[/tex]
- Slack for Constraint 4: [tex]\(0.00\)[/tex]
This means Constraint 3 has some unused capacity, while the other constraints are fully utilized at the optimum point.
### Conclusion
The maximum value of the objective function is 64,227.27 when the values of [tex]\(x_1 = 544.79\)[/tex], [tex]\(x_2 = 0.00\)[/tex], [tex]\(x_3 = 600.94\)[/tex], [tex]\(x_4 = 0.00\)[/tex], and [tex]\(x_5 = 641.04\)[/tex]. The slack values indicate the unused potential within the constraints. Rounding has been done to the nearest hundredth as required.
### Step 1: Define the Objective Function
The objective is to maximize:
[tex]\[ z = 37x_1 + 34x_2 + 36x_3 + 30x_4 + 35x_5 \][/tex]
### Step 2: Define the Constraints
You have the following constraints that need to be satisfied:
1. [tex]\( 16x_1 + 19x_2 + 23x_3 + 15x_4 + 21x_5 \leq 36,000 \)[/tex]
2. [tex]\( 15x_1 + 10x_2 + 19x_3 + 23x_4 + 10x_5 \leq 26,000 \)[/tex]
3. [tex]\( 9x_1 + 16x_2 + 14x_3 + 12x_4 + 11x_5 \leq 25,000 \)[/tex]
4. [tex]\( 18x_1 + 20x_2 + 15x_3 + 17x_4 + 19x_5 \leq 31,000 \)[/tex]
5. [tex]\( x_1 \geq 0, x_2 \geq 0, x_3 \geq 0, x_4 \geq 0, x_5 \geq 0 \)[/tex] (non-negativity constraints)
### Step 3: Solve the Linear Programming Problem
Using appropriate linear programming methods, we determine the values of decision variables [tex]\(x_1, x_2, x_3, x_4, \)[/tex] and [tex]\(x_5\)[/tex] that maximize the objective function while satisfying all the constraints.
### Step 4: Find the Optimal Solution
The resulting optimal values from solving the problem are:
- [tex]\(x_1 = 544.79\)[/tex]
- [tex]\(x_2 = 0.00\)[/tex]
- [tex]\(x_3 = 600.94\)[/tex]
- [tex]\(x_4 = 0.00\)[/tex]
- [tex]\(x_5 = 641.04\)[/tex]
The maximum value of the objective function [tex]\( z \)[/tex], achieved with these variable values, is:
[tex]\[ z = 64,227.27 \][/tex]
### Step 5: Determine the Slack Variables
Slack variables show how much "slack" or "unused capacity" is left in each of the inequality constraints:
- Slack for Constraint 1: [tex]\(0.00\)[/tex]
- Slack for Constraint 2: [tex]\(0.00\)[/tex]
- Slack for Constraint 3: [tex]\(4632.35\)[/tex]
- Slack for Constraint 4: [tex]\(0.00\)[/tex]
This means Constraint 3 has some unused capacity, while the other constraints are fully utilized at the optimum point.
### Conclusion
The maximum value of the objective function is 64,227.27 when the values of [tex]\(x_1 = 544.79\)[/tex], [tex]\(x_2 = 0.00\)[/tex], [tex]\(x_3 = 600.94\)[/tex], [tex]\(x_4 = 0.00\)[/tex], and [tex]\(x_5 = 641.04\)[/tex]. The slack values indicate the unused potential within the constraints. Rounding has been done to the nearest hundredth as required.
