This is a Linear Programming Problem that can be solved to maximize the profit of the factory while adhering to the constraints of machine hours for moulding, grinding, and polishing. Here's how to structure the problem:

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Decision Variables:

Let:

• $x$: Number of units of product $A$ produced.
• $y$: Number of units of product $B$ produced.

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Objective Function:

Maximize profit: $Z = 30x + 50y$

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Constraints:

1. Moulding time: Each unit of $A$ requires 2 hours, and each unit of $B$ requires 3 hours. The available moulding time is 110 hours. $2x + 3y \leq 110$

2. Grinding time: Each unit of $A$ requires 2 hours, and each unit of $B$ requires 2 hours. The available grinding time is 150 hours. $2x + 2y \leq 150$

3. Polishing time: Each unit of $A$ requires 3 hours, and each unit of $B$ requires 4 hours. The available polishing time is 120 hours. $3x + 4y \leq 120$

4. Non-negativity constraints: Since production cannot be negative: $x \geq 0, \quad y \geq 0$

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Problem Summary:

Maximize: $Z = 30x + 50y$

Subject to: $2x + 3y \leq 110$ $2x + 2y \leq 150$ $3x + 4y \leq 120$ $x \geq 0, \quad y \geq 0$

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Would you like me to solve this problem graphically, by using the simplex method, or via any computational method? Let me know your preference!

Alternatively, if you'd like, I can explain the constraints in more detail or visualize them.

Further Questions:

1. Would you like to see a graphical representation of the feasible region?
2. Should we use computational tools (Python) for exact maximization?
3. Do you need a step-by-step explanation of the simplex method for this problem?
4. Are there any specific restrictions on producing certain quantities of $A$ or $B$?
5. Would you like insights into sensitivity analysis for the constraints?

Tip:

Always write down the constraints and objective function clearly when tackling optimization problems. This helps to structure the problem effectively!