A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day. the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?

Let the manufacturer produce x pedestal lamps and y wooden shades everyday.
Let P be the profits
Table

We are to maximise
P = 5x + 3y
subject to constraints
2x + y ≤ 12
3x + 2y ≤ 20
x ≥ 0, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
First, we draw the graph of 2x + y = 12
For x = 0, y = 12
For y = 0, 2 x = 12 or x = 6
∴  line meets OX in A(6, 0) and OY in L(0, 12).
              Again we draw the graph of 3x + 2y = 20
               For x = 0,  2y = 20   or  y = 10
               For y = 0,  3x = 20   or 
.
 Since feasible region satisfies all the constraints.

∴ OACM is the feasible region.
The corner points are O(0, 0), A(6, 0), C(4, 4), M(0, 10)
At O(0, 0), P = 5 × 0 + 3 × 0 = 0 + 0 = 0
At A(6, 0), P = 5 × 6 + 3 × 0 = 30 + 0 = 30
At C(4, 4), P = 5 × 4 + 3 × 4 = 20 + 12 = 32
At M(0, 10), P = 5 × 0 + 3 × 10 = 0 + 30 = 30
∴  maximum value = 32 at (4, 4)
∴  4 pedestal lamps and 4 wooden shades should be produced for maximum profit of Rs. 32.