A Small Scale Manufacturer produces two types of Products A and B which gives a
profit margin of AED 70 and AED 50 respectively. There are two plants I and II,
each having a capacity of 240 and 100 hours. Cycle time of A and B are 4 hours
and 3 hours in plant I and 2 hour and 1 hour in Plant II. How many products of
A and B to be manufactured? Formulate the problem

To formulate this problem as a **linear programming** (LP) problem, we aim to determine the number of products \( x_1 \) (Product A) and \( x_2 \) (Product B) that maximize profit, subject to the time constraints in two plants (I and II).

### Variables:
- Let \( x_1 \) be the number of Product A manufactured.
- Let \( x_2 \) be the number of Product B manufactured.

### Objective Function (Maximizing Profit):
The profit from Product A is AED 70, and the profit from Product B is AED 50. Therefore, the total profit \( Z \) can be written as:

\[
\text{Maximize} \quad Z = 70x_1 + 50x_2
\]

### Constraints:
The constraints are based on the time limitations in Plant I and Plant II.

#### Time Constraints for Plant I:
- Product A requires 4 hours per unit in Plant I.
- Product B requires 3 hours per unit in Plant I.
- The total available time in Plant I is 240 hours.

This gives the first constraint:
\[
4x_1 + 3x_2 \leq 240
\]

#### Time Constraints for Plant II:
- Product A requires 2 hours per unit in Plant II.
- Product B requires 1 hour per unit in Plant II.
- The total available time in Plant II is 100 hours.

This gives the second constraint:
\[
2x_1 + x_2 \leq 100
\]

#### Non-negativity Constraints:
The number of products \( x_1 \) and \( x_2 \) cannot be negative. Hence:
\[
x_1 \geq 

A Small Scale Manufacturer produces two types of Products A and B which gives a profit margin of AED 70 and AED 50 respectively. There are two plants I and II, each having a capacity of 240 and 100 hours. Cycle time of A and B are 4 hours and 3 hours in plant I and 2 hours and 1 hour in Plant II. How many products of A and B to be manufactured? Formulate the problem.

Learn how to formulate and solve a linear programming problem for a manufacturer producing two products with a profit margin of AED 70 and AED 50. The problem involves time constraints for two plants and seeks to maximize profit using linear equations and optimization techniques.

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