This problem can be formulated as a linear programming problem, where we maximize profit under the given constraints of cutting and assembly hours.

Step 1: Define Variables

Let:

• $x$ be the number of chairs produced per week.
• $y$ be the number of tables produced per week.

Step 2: Objective Function

The objective is to maximize profit, where:

• Profit per chair = $20
• Profit per table = $30

So, the objective function is: $\text{Maximize} \: Z = 20x + 30y$ Where $Z$ represents total profit.

Step 3: Constraints

The constraints are based on the availability of cutting and assembly time.

Cutting time:

• Chairs require 3 hours of cutting.
• Tables require 4 hours of cutting.
• The total cutting time is limited to 16 hours per week.

Thus, the constraint is: $3x + 4y \leq 16$

Assembly time:

• Chairs require 2 hours of assembly.
• Tables require 3 hours of assembly.
• The total assembly time is limited to 18 hours per week.

Thus, the constraint is: $2x + 3y \leq 18$

Step 4: Non-negativity Constraint

Since the number of chairs and tables cannot be negative: $x \geq 0, \: y \geq 0$

Step 5: Solve the Linear Programming Problem

We now have the following system to solve:

Maximize: $Z = 20x + 30y$ Subject to: $3x + 4y \leq 16$ $2x + 3y \leq 18$ $x \geq 0, \: y \geq 0$

We will solve this by identifying the corner points of the feasible region and evaluating the profit at each point.

Step 6: Find the Corner Points

The corner points are where the constraints intersect or where they meet the axes. We will solve this step-by-step using substitution or elimination to find the points where the constraints intersect. I'll calculate them.The solution to the linear programming problem is:

• Produce 0 chairs.
• Produce 4 tables.

This will result in a maximum profit of $120 per week.

Would you like more details on how this was calculated, or do you have any further questions?

Here are 5 related questions to consider:

1. What if the profit per chair was increased to $25? How would the solution change?
2. How would the result be affected if the cutting machine time was increased to 20 hours per week?
3. What is the maximum number of tables that can be produced if no chairs are made?
4. What is the shadow price for the cutting and assembly time constraints?
5. Can you graphically represent the feasible region of this problem?

Tip: When dealing with linear programming problems, understanding the corner points of the feasible region is crucial, as the optimal solution always occurs at one of these points.