This problem involves setting up a linear programming model to maximize the company’s profit under certain constraints on labor hours and production requirements. Here’s how to approach it:

Step 1: Define Variables

Let:

• $x$ be the number of units of Product A.
• $y$ be the number of units of Product B.
• $z$ be the number of units of Product C.

Step 2: Set Up the Objective Function

The company wants to maximize profit, which can be defined as: $\text{Maximize } P = 18x + 12y + 15z$ where 18, 12, and 15 are the profits per unit of Products A, B, and C, respectively.

Step 3: Define Constraints Based on Labor Hours

The labor-hour constraints for Departments I, II, and III are given as follows:

1. Department I: $2x + y + 2z \leq 1020$
2. Department II: $3x + y + 2z \leq 1200$
3. Department III: $2x + 2y + z \leq 960$

Step 4: Additional Constraint on Production

According to the problem, the number of units of Product B (y) must be equal to or greater than the sum of the units of Products A and C: $y \geq x + z$

Step 5: Solve the Linear Program

With these equations, you can now solve this problem using linear programming methods (e.g., the Simplex method or by using a tool like Excel Solver).

Let me know if you’d like further details on how to proceed with calculations or if you have specific questions about setting up this problem.

Here are some questions to extend this problem:

1. What happens if the profit per unit changes for one of the products?
2. How would the solution change if there were a maximum production limit for each product?
3. How would you approach this problem if the production of Product B was limited to be no more than twice the production of Product C?
4. How would the labor-hour constraints change if Department II had additional overtime hours available?
5. What other constraints could affect production if raw materials were limited?

Tip: In linear programming, always start by clearly defining variables and constraints to simplify solving the problem.