CV. KayuSejati is a family-owned business that makes handcrafted dining room tables and chairs.
They obtain the oak from a local tree farm, which ships them 2,500 pounds of oak each month. Each table uses 50 pounds of oak while each chair uses 25 pounds of oak.
The family builds all the furniture itself and has 480 hours of labor available each month. Each table or chair requires 6 hours of labor. Each table nets Oak Works $400 in profit, while each chair nets $100 in profit. Since chairs are often sold with the tables, they want to produce at least twice as many chairs as tables.
CV. KayuSejati would like to decide how many tables and chairs to produce so as to maximize profit.
Formulate and solve a linear programming model for this problem, then answer questions below:
a. How many tables should be produced to maximize total profit?

This is a classic linear programming problem that can be solved by defining a set of variables, constraints, and an objective function.

### Step 1: Define the Decision Variables
Let:
- $ x_1 $ = the number of tables produced.
- $ x_2 $ = the number of chairs produced.

### Step 2: Write the Objective Function
The objective is to **maximize profit**. Each table nets $400, and each chair nets $100. So the total profit function is:

\[
\text{Maximize } Z = 400x_1 + 100x_2
\]

### Step 3: Define the Constraints
1. **Oak constraint**: The total amount of oak used cannot exceed 2,500 pounds. Each table uses 50 pounds of oak, and each chair uses 25 pounds. Therefore:

\[
50x_1 + 25x_2 \leq 2500
\]

2. **Labor constraint**: The total labor available is 480 hours. Each table or chair requires 6 hours of labor. Therefore:

\[
6x_1 + 6x_2 \leq 480
\]

3. **Chair to table ratio constraint**: They want to produce at least twice as many chairs as tables:

\[
x_2 \geq 2x_1
\]

4. **Non-negativity constraint**: The number of tables and chairs produced cannot be negative:

\[
x_1 \geq 0, \quad x_2 \geq 0
\]

### Step 4: Solve the Linear Programming Problem

We now have the following system to solve:

\[
\text{Maximize } Z = 400x_1 + 100x_2
\]
subject to:
\[
50x_1 + 25x_2 \leq 2500
\]
\[
6x_1 + 6x_2 \leq 480
\]
\[
x_2 \geq 2x_1
\]
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

#### Let's solve this system to find the values of $ x_1 $ and $ x_2 $. I will now calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you through the steps to solve it manually or provide additional help. Let me know how you'd like to proceed!

Here are 5 questions to expand on the information:
1. How can linear programming be applied to other resource allocation problems in businesses?
2. What happens if the oak supply increases to 3,000 pounds? How will the solution change?
3. What is the impact of adjusting the labor availability on the optimal solution?
4. How can adding more products (e.g., benches) affect this linear programming model?
5. How would a change in profit per chair or table alter the results?

Tip: Always double-check constraints when solving linear programming problems manually to ensure you're accurately modeling the situation.

CV. KayuSejati is a family-owned business that makes handcrafted dining room tables and chairs. They obtain the oak from a local tree farm, which ships them 2,500 pounds of oak each month. Each table uses 50 pounds of oak while each chair uses 25 pounds of oak. The family builds all the furniture itself and has 480 hours of labor available each month. Each table or chair requires 6 hours of labor. Each table nets Oak Works $400 in profit, while each chair nets $100 in profit. Since chairs are often sold with the tables, they want to produce at least twice as many chairs as tables. CV. KayuSejati would like to decide how many tables and chairs to produce so as to maximize profit.

Learn how CV. KayuSejati, a family-owned business, can maximize profit by optimizing their production of handcrafted tables and chairs using linear programming. This detailed solution covers constraints on labor and materials, including profit calculations for each item. Suitable for advanced high school or college-level students.

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