A firm makes two types of furniture: chairs and tables. The contribution for each product as calculated by the accounting department is 20 birr per chair and 30 birr per table. Both products are processed on three machines M1, M2 and M3. the time required by each product and total time available per week on each machine are as follows;
Machines
Chair
Table
Available hrs
M1
3
3
36
M2
5
2
50
M3
2
6
60
How should the manufacturer schedule his production in order to maximize contribution? Formulate the above problem as a linear programming model.
Let's denote the number of chairs produced as x and the number of tables produced as y.
The contribution for each chair is 20 birr per chair, and the contribution for each table is 30 birr per table.
To maximize the contribution, we need to maximize the total contribution, which can be calculated as follows:
Total Contribution = 20x + 30y
Now, let's consider the constraints:
1. Machine M1:
The time required by each chair on machine M1 is 3 hours, and the time required by each table on machine M1 is also 3 hours. The total available hours on machine M1 per week is 36 hours. Therefore, the constraint for machine M1 is:
3x + 3y ≤ 36
2. Machine M2:
The time required by each chair on machine M2 is 5 hours, and the time required by each table on machine M2 is 2 hours. The total available hours on machine M2 per week is 50 hours. Therefore, the constraint for machine M2 is:
5x + 2y ≤ 50
3. Machine M3:
The time required by each chair on machine M3 is 2 hours, and the time required by each table on machine M3 is 6 hours. The total available hours on machine M3 per week is 60 hours. Therefore, the constraint for machine M3 is:
2x + 6y ≤ 60
4. Non-negativity constraint:
Since we cannot produce negative numbers of chairs or tables, the non-negativity constraint is:
x ≥ 0, y ≥ 0
Now, we can formulate the linear programming model:
Maximize: 20x + 30y
Subject to:
3x + 3y ≤ 36
5x + 2y ≤ 50
2x + 6y ≤ 60
x ≥ 0, y ≥ 0