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A factory manufactures two products, A and B. Each product requires the use of three machines, Machine I, Machine II, and Machine III. The time requirements and total hours available on each machine are listed below. If product A generates a profit of $45 per unit and product B a profit of $50 per unit, how many units of each product should be manufactured to maximize profit, and what is the maximum profit? Fill in the blanks : To maximize profit, the factory should produce ________ units of product A and ________ units of product B. The maximum profit would be _________.

To solve this problem, we need to formulate a system of linear equations based on the given information and then solve for the number of units of product A and product B that should be manufactured to maximize profit.

### Let:
- $ x_1 $ = number of units of Product A
- $ x_2 $ = number of units of Product B

The factory constraints based on the table are as follows:

- Machine I: $ 1x_1 + 2x_2 \leq 72 $
- Machine II: $ 2x_1 + 2x_2 \leq 98 $
- Machine III: $ 4x_1 + 2x_2 \leq 180 $

### Objective:
Maximize profit $ P $, where:
\[
P = 45x_1 + 50x_2
\]

### Step 1: Set up the constraints
\[
1x_1 + 2x_2 \leq 72
\]
\[
2x_1 + 2x_2 \leq 98
\]
\[
4x_1 + 2x_2 \leq 180
\]

### Step 2: Solve the system
We can solve this linear programming problem using the graphical method, or by applying a method like the simplex algorithm. Let me compute the optimal solution for you using a simplex method approach.

I'll proceed with the necessary calculations to determine the values of $ x_1 $ and $ x_2 $.To maximize profit, the factory should produce approximately **26 units of Product A** and **23 units of Product B**. 

The maximum profit would be **\$2320**.

Would you like more details on any specific step or have further questions?

Here are 5 related questions to deepen your understanding:
1. How does the simplex method work in solving linear programming problems?
2. What would happen if the profit per unit for Product B increased?
3. How can sensitivity analysis be applied to this problem?
4. Could we use another method, like graphical analysis, to solve this?
5. How does changing machine availability impact the production quantities?

**Tip:** When using linear programming for optimization, always double-check that the constraints reflect real-world limitations, like availability of resources.

A factory manufactures two products, A and B. Each product requires the use of three machines, Machine I, Machine II, and Machine III. The time requirements and total hours available on each machine are listed below. If product A generates a profit of $45 per unit and product B a profit of $50 per unit, how many units of each product should be manufactured to maximize profit, and what is the maximum profit?

Solve a factory production problem using linear programming. This example shows how to maximize profit by producing the optimal number of two products (A and B) based on machine constraints. Includes detailed steps and application of the simplex method. Suitable for university-level operations research students.

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