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.