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 die 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?

Explanation

Let the number of pedestal lamps be x
and the number of wooden shades be y



Cutting Machine
Time on Pedestal lamps `to` 2 hrs
Time on Wooden shades `to` 1 hrs
Maximum Time `to` atmost 12 hrs
`therefore`
`2x+y<=12`
`x,y>=0`

Sprayer
Time on Pedestal lamps `to` 3 hrs
Time on Wooden shades `to` 2 hrs
Maximum Time `to` atmost 20 hrs
`therefore`
`3x+2y<=20`
`x,y>=0`

As, we want to maximize the profit, hence the function used here is Maximize `Z`

Profit on Pedestal lamps `to` Rs 5
Profit on wooden shades `to` Rs 3
Maximize `Z= 5x+3y`

Combining all Constraints:
Maximize `Z= 5x+3y`

Subject to Constraints
`2x+y<=12`
`3x+2y<=20`
`x>=0,y>=0`





`therefore` The profit will be maximum if,
` " " " ` Number of lamps `to` 4
` " " " ` Number of Wooden Shades `to` 4
` " " " `Maximum profit `to` Rs 32