Question

6. Assume Adey Company produces two products; both of which are made on the same two machines. Since there are two different products, there are two independent activities producing product A and product B. producing a unit of A requires 1 hour on the first machine and 4 hours on second machine. Producing a unit of B requires 4 hours on the first machine and 1 hour on the second machine. Each week, there are 20 hours available on the first machine and 35 hours available on the second machine. The firm makes profit of 60 birr on a unit of product A and 90 birr on each unit of product B. it is assumed that all units produced will be sold. Required: Formulate this problem as a linear programming problem to determine as to how many units of each of the products should be produced per week so that the firm can earn the maximum profit.

Answer 1

To formulate this problem as a linear programming problem, we need to define the decision variables, the objective function, and the constraints based on the information provided.

### Step 1: Define the Decision Variables
Let:
- \( x \): the number of units of product A produced per week
- \( y \): the number of units of product B produced per week

### Step 2: Define the Objective Function
We want to maximize the total profit, which can be expressed as:
\[
\text{Maximize } Z = 60x + 90y
\]
where \( Z \) is the total profit.

### Step 3: Define the Constraints
Based on the production times and machine availability:

1. **Constraint for Machine 1 (First Machine)**:
Producing product A requires 1 hour per unit, and product B requires 4 hours per unit. The total time available on Machine 1 is 20 hours:
\[
1x + 4y \leq 20
\]

2. **Constraint for Machine 2 (Second Machine)**:
Producing product A requires 4 hours per unit, and product B requires 1 hour per unit. The total time available on Machine 2 is 35 hours:
\[
4x + 1y \leq 35
\]

3. **Non-negativity Restrictions**:
We must also ensure that the production quantities are non-negative:
\[
x \geq 0
\]
\[
y \geq 0
\]

### Step 4: Summary of the Linear Programming Model
Putting it all together, the linear programming model can be summarized as follows:

**Objective Function**:
\[
\text{Maximize } Z = 60x + 90y
\]

**Subject to**:
1. \( 1x + 4y \leq 20 \) (Machine 1 constraint)
2. \( 4x + 1y \leq 35 \) (Machine 2 constraint)
3. \( x \geq 0 \) (Non-negativity)
4. \( y \geq 0 \) (Non-negativity)

### Step 5: Solve the Model
You can solve this linear programming problem using graphical methods, the simplex method, or optimization software to find the optimal values of \( x \) and \( y \) that maximize the profit \( Z \) while satisfying the constraints.