To solve this optimization problem, we can set it up as a linear programming (LP) problem. We need to maximize the profit while satisfying constraints based on labor-hour availability in each department and additional constraints on production quantities.

Step 1: Define the Decision Variables

Let:

• $x$ = number of units of Product A to produce
• $y$ = number of units of Product B to produce
• $z$ = number of units of Product C to produce

Step 2: Objective Function

The goal is to maximize the total profit: $\text{Profit} = 18x + 12y + 15z$

Step 3: Constraints

1. Labor-hour Constraints

Based on the labor-hours available in each department:

• Department I: $2x + y + 2z \leq 1020$
• Department II: $3x + y + 2z \leq 1140$
• Department III: $2x + 2y + z \leq 960$

2. Production Requirement Constraint

The management requires that the number of units of Product B manufactured must equal or exceed the number of units of products A and C manufactured: $y \geq x + z$

3. Non-negativity Constraints

All variables must be non-negative: $x \geq 0, \quad y \geq 0, \quad z \geq 0$

Step 4: Solve the Linear Programming Problem

Using these constraints and objective function, we can apply a method such as the Simplex algorithm or use software like Python's SciPy library or Excel's Solver to find the optimal solution.

Would you like me to solve this problem using a computational approach to provide the exact values for $x$, $y$, and $z$?

---

Questions for Further Exploration

1. How would the solution change if the labor-hours for any department increased?
2. What if the profit per unit for Product B increased—how would it affect production?
3. How can we determine if there are multiple optimal solutions to this problem?
4. How does the restriction $y \geq x + z$ influence the feasible region in this LP problem?
5. Could this problem be solved graphically if only two products were considered?

Tip:

In linear programming, always check the feasibility of constraints, as an infeasible problem has no solution.