To solve this problem, let's define the variables and set up a linear programming model:

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.

Objective Function

The company wants to maximize profit. The profit per unit for each product is:

• Product A: $18
• Product B: $12
• Product C: $15

The objective function, therefore, is: $\text{Maximize } P = 18x + 12y + 15z$

Constraints

The time requirements in each department set constraints on the number of products that can be produced:

1. Department I (960 hours available): $2x + y + 2z \leq 960$

2. Department II (1020 hours available): $3x + y + 2z \leq 1020$

3. Department III (840 hours available): $2x + 2y + z \leq 840$

4. Production Constraint for Product B: Management requires that the number of units of Product B produced must be greater than or equal to the number of units of Products A and C combined: $y \geq x + z$

5. Non-negativity Constraints: $x, y, z \geq 0$

Solution Method

This problem can be solved using linear programming techniques, such as the Simplex method or a computational tool to find the optimal values of $x$, $y$, and $z$ that maximize $P$.

I'll compute the optimal solution using these constraints to determine the values of $x$, $y$, and $z$, along with the maximum profit.The optimal solution for maximizing profit, given the constraints, is as follows:

• Product A: Produce 60 units
• Product B: Produce 200 units
• Product C: Produce 320 units

The maximum profit achievable under these conditions is $8,280.

Would you like more details on the solution steps or have any questions?

Here are some related questions to deepen understanding:

1. What changes in constraints would affect the maximum profit?
2. How would the result differ if the profit per unit for Product B increased?
3. What if there was an additional constraint on the maximum units of Product C?
4. How would a decrease in available hours for Department II impact production?
5. Could this method work for cases with more than three products or departments?

Tip: In linear programming, always double-check constraint inequalities to ensure they accurately reflect real-world limitations.