The Donaldson Furniture Company produces three types of rocking chairs: the children's model, the standard model, and the executive model. Each chair is made in three stages: cutting, construction, and finishing.

\begin{tabular}{l|c|c|c}
\hline
Stage & Children's & Standard & Executive \\
\hline
Cutting & [tex]$5 \, hr$[/tex] & [tex]$4 \, hr$[/tex] & [tex]$7 \, hr$[/tex] \\
\hline
Construction & [tex]$3 \, hr$[/tex] & [tex]$2 \, hr$[/tex] & [tex]$5 \, hr$[/tex] \\
\hline
Finishing & [tex]$2 \, hr$[/tex] & [tex]$2 \, hr$[/tex] & [tex]$4 \, hr$[/tex] \\
\hline
\end{tabular}

The time needed for each stage of each chair is given in the chart. During a specific week, the company has available a maximum of 174 hours for cutting, 110 hours for construction, and 86 hours for finishing. Determine how many of each type of chair the company should make to be operating at full capacity.

The number of executive chairs the company should make is [tex]$\square$[/tex].

The number of standard chairs the company should make is [tex]$\square$[/tex].

The number of children's chairs the company should make is [tex]$\square$[/tex].



Answer :

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].

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