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To solve this problem, we need to approach it as a linear programming (LP) optimization problem, aiming to maximize profit under certain constraints.

### Variables
Define:
- \( x_A \): the number of units of Product \( A \).
- \( x_B \): the number of units of Product \( B \).
- \( x_C \): the number of units of Product \( C \).

### Objective Function
The profit function (to maximize) is:
\[
\text{Profit} = 18x_A + 12x_B + 15x_C
\]

### Constraints

#### Department Hours Constraints
From the given table, each product requires a certain number of hours from each department. The constraints based on available hours for each department are:

1. **Dept. I constraint**:
   \[
   2x_A + x_B + 2x_C \leq 1020
   \]

2. **Dept. II constraint**:
   \[
   3x_A + x_B + 2x_C \leq 1200
   \]

3. **Dept. III constraint**:
   \[
   2x_A + 2x_B + x_C \leq 960
   \]

#### Additional Constraint for Product Quantities
The problem specifies that the number of units of Product \( B \) must equal or exceed the combined total of units of Products \( A \) and \( C \). This constraint is expressed as:
\[
x_B \geq x_A + x_C
\]

#### Non-negativity Constraints
All production variables must be non-negative:
\[
x_A \geq 0, \quad x_B \geq 0, \quad x_C \geq 0
\]

### Solution Approach
We can use a linear programming solver or perform a systematic analysis to determine values of \( x_A \), \( x_B \), and \( x_C \) that maximize the profit function subject to the constraints. 

Would you like me to go through each step of solving this optimization problem or provide a solution with suggested values for \( x_A \), \( x_B \), and \( x_C \) that maximize profit? 

---

Here are some questions you may have:
1. How can we convert the constraints to make this easier to solve manually?
2. Could a graphical solution be applied for this problem?
3. What are the implications if the constraint \( x_B \geq x_A + x_C \) changes?
4. Is there an alternative approach to maximize profit if some department resources change?
5. How would introducing additional products affect the problem?

**Tip:** Linear programming problems like this one are often solved using software such as Excel Solver or Python's SciPy library to find an optimal solution efficiently.

A company manufactures products A, B, and C. Each product requires different amounts of time from three departments (I, II, III), and the company aims to maximize profit. Given the labor-hour constraints for each department and the condition that product B units produced must equal or exceed the total units of products A and C, how many units of each product should the company produce to maximize profit?

Learn to maximize profit in a production scenario involving products A, B, and C with department labor constraints and product quantity conditions. Explore linear programming for optimization, including constraints and profit calculation. Suitable for undergraduate-level business or operations research students.

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