A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to product A and product B. Each unit of product A requires 2, 1 and 1 running feet and each unit of product B requires 1, 2 and 1 running feet of teak wood, plywood and rosewood respectively. If product A is sold for Rs. 48 per unit and product B is sold for Rs. 40 per unit, how many units of product A and product B should be produced and sold by the carpenter, in order to obtain the maximum gross income? Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph.
Solution
| Product | A | B | |
| Teak wood | 2 | 1 | ≤90 |
| plywood | 1 | 2 | ≤80 |
| Rosewood | 1 | 1 | ≤50 |
Selling price per unit Rs. 48/- for product A and for product B Rs. 40/Mathematical formation :
`Z_max = 48x_1 + 40x_2` (objective function)
s.t. 2x1 + x2 ≤ 90
x1 + 2x2 ≤ 80:
x1 + x2 ≤ 50;
x1,x2 ≥ 0
2x1 + x2 = 90 ...........(1)
x1 + 2x2 = 80 ..........(2)
X1 + x2 =50 ........(3)
from eq (3) and (2)
x2 = 30
x1 = 20
from eq. (1) & (3)
`x1 = 40`
`x2 = 10`
0(0,0)
| A(0,40) | Zmax = 48 × x1 + 40 × x2 | ZA = 1600 |
| B(20,30) | = 48 × 20 + 40 × 30 | ZB = 2160 |
| C(40,10) | = 48 × 40 + 40 +10 | Zc = 2320 |
| D(45,0) | = 48 × 45 + 40(0) | ZD = 2160 |
40, 10), ZC = 2320.
Hence the optimal solution is x1 = 40, x2 = 10 (Product A = 40 units, Product B = 10 units)
