To determine how many of each type of chair the Donaldson Furniture Company should make to be operating at full capacity, let's go through a detailed step-by-step solution.
1. Identify the Variables:
- Let \( x_1 \) be the number of children's chairs produced.
- Let \( x_2 \) be the number of standard chairs produced.
- Let \( x_3 \) be the number of executive chairs produced.
2. Formulate the Objective Function:
- The objective is to maximize the total number of chairs produced. Thus, the objective function is:
[tex]\[
\text{Maximize } Z = x_1 + x_2 + x_3
\][/tex]
3. Formulate the Constraints:
- The constraints are based on the available hours for cutting, construction, and finishing. These constraints limit the number of chairs that can be produced.
- For cutting hours:
[tex]\[
5x_1 + 4x_2 + 7x_3 \leq 174
\][/tex]
- For construction hours:
[tex]\[
3x_1 + 2x_2 + 5x_3 \leq 110
\][/tex]
- For finishing hours:
[tex]\[
2x_1 + 2x_2 + 4x_3 \leq 86
\][/tex]
- Additionally, the number of chairs produced cannot be negative:
[tex]\[
x_1 \geq 0, \quad x_2 \geq 0, \quad x_3 \geq 0
\][/tex]
4. Solve the Linear Programming Problem:
Based on calculations using the constraints given above, the optimal solution has been determined to be:
- The number of children's chairs, \( x_1 \), is 2.0.
- The number of standard chairs, \( x_2 \), is 41.0.
- The number of executive chairs, \( x_3 \), is 0.0.
Thus, the company should produce:
- 2 children's chairs,
- 41 standard chairs, and
- 0 executive chairs.
Therefore, the final answers would be:
- The number of executive chairs the company should make is \(0\).
- The number of standard chairs the company should make is \(41\).
- The number of children's chairs the company should make is [tex]\(2\)[/tex].
