2.	A company produces two products (A and B) that are processed on two assembly lines. Each product requires 10 hours of processing time on line 1, while on line 2 product A requires 7 hours and product B requires 3 hours. Assembly line 1 has 100 available hours, and assembly line 2 has 42 available hours. The profit for product A is Birr 6 per unit, and the profit for product B is Birr 4 per unit. 
A.	Formulate a linear programming model for this problem.
B.	Solve this model by using graphical method


### A. Formulation of the Linear Programming Model

Let:
- \( x_1 \) be the number of units of product A produced.
- \( x_2 \) be the number of units of product B produced.

**Objective Function:**  
Maximize the profit, \( Z \):
\[
Z = 6x_1 + 4x_2
\]

**Subject to the following constraints:**

1. **Assembly Line 1 constraint (time available is 100 hours):**
   \[
   10x_1 + 10x_2 \leq 100
   \]

2. **Assembly Line 2 constraint (time available is 42 hours):**
   \[
   7x_1 + 3x_2 \leq 42
   \]

3. **Non-negativity constraints:**
   \[
   x_1 \geq 0, \quad x_2 \geq 0
   \]

### B. Solving the Model Using the Graphical Method

1. **Convert the inequalities into equations:**
   \[
   10x_1 + 10x_2 = 100
   \]
   \[
   7x_1 + 3x_2 = 42
   \]

2. **Plot the constraints:**

   - **Constraint 1:** \( 10x_1 + 10x_2 \leq 100 \)
     - Simplify to \( x_1 + x_2 \leq 10 \).
     - For \( x_1 = 0 \), \( x_2 = 10 \).
     - For \( x_2 = 0 \), \( x_1 = 10 \).
   
   - **Constraint 2:** \( 7x_1 + 3x_2 \leq 42 \)
     - For \( x_1 = 0 \), \( x_2 = 14 \).
     - For \( x_2 = 0 \), \( x_1 = 6 \).

3. **Identify the feasible region:**
   - The feasible region is where all the constraints overlap in the first quadrant (since \( x_1 \geq 0 \) and \( x_2 \geq 0 \)).

4. **Find the corner points of the feasible region:**
   - Intersection of \( x_1 + x_2 = 10 \) and \( 7x_1 + 3x_2 = 42 \).
   - Substitute \( x_2 = 10 - x_1 \) into the second equation:
     \[
     7x_1 + 3(10 - x_1) = 42
     \]
     Simplifying this gives:
     \[
     4x_1 = 12 \implies x_1 = 3
     \]
     Then \( x_2 = 10 - 3 = 7 \).

   - Thus, the corner points are \( (0, 10) \), \( (6, 0) \), \( (3, 7) \).

5. **Evaluate the objective function at each corner point:**

   - \( Z(0, 10) = 6(0) + 4(10) = 40 \)
   - \( Z(6, 0) = 6(6) + 4(0) = 36 \)
   - \( Z(3, 7) = 6(3) + 4(7) = 18 + 28 = 46 \)

6. **Determine the optimal solution:**

   - The maximum profit is \( Z = 46 \) at \( (x_1, x_2) = (3, 7) \).

### Conclusion:
The company should produce 3 units of product A and 7 units of product B to maximize profit, yielding a maximum profit of Birr 46.

Do you want more details or have any questions?

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Here are 5 related questions:

1. How would the solution change if the profit per unit of product B increased?
2. What would happen if the available hours on Assembly Line 1 were reduced?
3. How can the graphical method be extended to handle more than two variables?
4. What is the significance of the feasible region in linear programming?
5. How would the optimal solution change if the constraint for Assembly Line 2 was made stricter?

**Tip:** In linear programming, always ensure the constraints are properly formulated before attempting to solve the problem graphically or by other methods.

Explore how to maximize profit using linear programming with a detailed example of product production optimization. Learn about formulating objective functions, setting constraints, and solving graphically. Case study includes profit maximization for two products with specific assembly line constraints.

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