This problem can be solved using linear programming to maximize profit given the constraints on labor hours and the condition on Product B. Here’s the setup and solution outline.

1. Define Variables

Let:

• $x$ = units of Product A produced
• $y$ = units of Product B produced
• $z$ = units of Product C produced

2. Objective Function

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

3. Constraints

a) Labor Hour Constraints

Each department has limited labor hours:

• Department I: $2x + y + 2z \leq 840$
• Department II: $3x + y + 2z \leq 1020$
• Department III: $2x + 2y + z \leq 780$

b) Production Requirement for Product B

The number of units of Product B must be at least as many as the number of units of Products A and C combined: $y \geq x + z$

c) Non-negativity Constraints

$x \geq 0, \quad y \geq 0, \quad z \geq 0$

Solution Method

This problem can be solved using the Simplex Method or a computational tool, as it involves multiple constraints and variables. I'll outline the general steps:

1. Convert each inequality into standard form for a linear programming solver.
2. Apply the Simplex algorithm or use a solver to find the optimal values for $x$, $y$, and $z$ that maximize $P$ while satisfying all constraints.

Would you like a detailed step-by-step solution with calculations or verification using a specific linear programming solver? Alternatively, I can provide a computational answer directly.

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Follow-up Questions

1. How does changing the profit per unit for one of the products affect the maximum profit?
2. What would happen if the constraint on Product B’s quantity were removed?
3. How can we interpret the shadow prices of each constraint in this problem?
4. What if the labor hours in one of the departments were increased by 10%? How would that impact production and profit?
5. How could a graphical approach apply to solving this problem with two products instead of three?

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Tip

When dealing with linear programming problems, always start by clearly defining the decision variables and constraints to organize information effectively.