Question

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5,760 to invest and has space for at the most 20 items. A fan costs him Rs 360 and a sewing machine Rs 240. He expects to sell a fan at a profit of Rs 22 and a sewing machine for a profit of Rs 18. Assuming that he can sell all the items that he buys, how should he invest his money to maximise his profit? Solve it graphically.

OR

A dealer in a rural area wishes to purchase some sewing machines. He has only Rs 57,600 to invest and has space for at most 20 items. An electronic machine costs him Rs 3,600 and a manually operated machine costs Rs 2,400. He can sell an electronic machine at a profit of Rs 220 and a manually operated machine at a profit of Rs 180. Assuming that he can sell all the machines that he buys, how should he invest his money in order to maximise his profit? Make it as an LPP and solve it graphically.

Answer 1

Let the dealer purchases x fans and y sewing machines, then cost of x fans and y sewing machines is given by

360x + 240y

∴ 360x + 240y ≤ 5, 760

⇒ 3x + 2y ≤ 48

As, he has space for at most 20 items,

∴ x + y ≤ 20

Now, profit earned by the dealer on selling x fans and y sewing machines is = 22x + 18y

Hence, our LPP is

Maximise, Z = 22x + 18y …(i)

Subject to the constraints:

3x + 2y ≤ 48 …(ii)

x + y ≤ 20 …(iii)

x, y ≥ 0 …(iv)

Let us evaluate, Z = 22x + 18y at each corner point.

The region satisfying inequalities (ii) to (iv) is shown (shaded) in the figure.

Thus, maximum value of Z is 392 at B (8, 12).

Hence the profit is maximum i.e., Rs 392 when he buys 8 fans and 12 sewing machines.

OR

Solve yourself as above solution. Here Z = 220x + 180y is objective function.