Let the cottage industry manufacture x pedestal lamps and y wooden shades. Therefore, x≥0 and y≥0The given information can be compiled in a table as follows.The profit on a lamp is Rs 5 and on the shades is Rs 3 . Therefore, the constraints are,2x+y≤12 3x+2y≤20 Total profit, z = 5x + 3yThe mathematical formulation of the given problem isMaximize z = 5x + 3y .......(1)subject to the constraints,2x+y≤12 ......(2)3x+2y≤20 ......(3)x,y≥0 ......(4)The feasible region determined by the system of constraints is as follows.The corner points are A(6,0), B(4,4), and C(0,10). The values of Z at these corner points are as followsThe maximum value of Z is 32 at (4,4).Thus, the manufacturer should produce 4 pedestal lamps and 4 wooden shades to maximize his profits.