Let the number of padestal lamps and wooden shades manufactured by cottage industry be x and y respectively.
Here profit is the objective function Z.
∴ Z =5x + 3y … (i)
We have to maximise Z subject to the constrains
2x + y ≤ 12 … (ii)
3x + 2y ≤ 20 … (iii)
x ≥ 0 and y ≥ 0 … (iv)
On plotting graph of above constraints or inequalities (ii), (iii) and (iv) we get shaded region having corner point A, B, C as feasible region.
Since (0, 0) Satisfy 3x + 2y ≤ 20
⇒ Graph of 3x + 2y ≤ 20 is that half plane in which origin lies.
The shaded area OABC is the feasible region whose corner points are O, A, B and C.
For coordinate B.
Equation 2x + y =12 and 3x+ 2y = 20 are solved as
3x + 2 (12 - 2x) = 20
⇒ 3x + 24 - 4x = 20 ⇒ x = 4
⇒ y =12 - 8 = 4
Coordinate of B = (4, 4)
Now we evaluate objective function Z at each corner
Hence maximum profit is ` 32 when manufacturer produces 4 lamps and 4 shades.