We will formulate the mathematical model for this optimization problem to minimize the total cost of production while meeting the demand for 2000 units of product $A$. Here's the model:

Variables:

Let:

• $x_1$: Hours Machine 1 operates
• $x_2$: Hours Machine 2 operates
• $x_3$: Hours Machine 3 operates
• $R_1$: Pounds of raw material purchased at $4/lb
• $R_2$: Pounds of raw material purchased at $5.50/lb
• $L_o$: Overtime labor hours

Objective Function:

Minimize the total cost: $\text{Total Cost} = 4R_1 + 5.5R_2 + 8L_r + 12L_o$ Where:

• $L_r = 900$: Regular labor hours available
• $L_o$: Overtime labor hours (at $12/hour)

Constraints:

1. Production Requirement: The total output must meet or exceed the demand for 2000 units of $A$: $37x_1 + 43x_2 + 52x_3 \geq 2000$

2. Labor Hour Constraints:

   • Total labor hours used by machines: $16x_1 + 35x_2 + 33x_3 \leq 900 + L_o$

3. Raw Material Constraints:

   • The raw material used must match machine usage: $80x_1 + 50x_2 + 76x_3 = R_1 + R_2$
   • First raw material limit (up to 1 ton or 2000 lbs at $4/lb): $R_1 \leq 2000$

4. Non-Negativity: $x_1, x_2, x_3, R_1, R_2, L_o \geq 0$

This linear programming model can now be solved using techniques like the simplex method or software like Excel Solver, Python (with libraries such as PuLP or scipy.optimize), or any optimization tool.

Would you like a step-by-step explanation on solving this model or help setting it up in software? Here are some additional questions you might find useful:

1. What is the role of each machine in determining cost-effectiveness?
2. How do changes in labor hours affect the production output?
3. What happens if the cost of raw materials changes?
4. How can this model accommodate a higher demand, say 3000 units of $A$?
5. What constraints are most restrictive for minimizing costs?

Tip: Ensure to validate that all input values (e.g., labor hour limits and raw material costs) are realistic before solving the model.