A merchant plants to sell two types of personal computers - a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 unite. Determine the number of units of each type of computers which the merchant should stock to get maximum profit, if he does not want to invest more than Rs. 70 lakhs and if his profit on the desktop model is Rs. 4500 and on portable model is Rs. 5000.

.: Let the merchant stocks x desktop computers and y portable computers.

Clearly $x \ge 0,y \ge 0.$

Since the total monthly demand of computers will not exceed 250 units,

so, $x + y \le 250.$

Also, cost of desktop model will cost Rs.

25,000 and portable model Rs. 40,000

and the merchant does not want to invest more than Rs. 70 lakhs.

$\therefore$ $25000x + 40000y \le 7000000$

Total profit Z on desktop model and portable model is $Z = 4500x + 5000y$

Hence, the mathematical formulation of the problem is

Maximize $Z = 4500x + 5000y$ ...(1)

subject to the constraints
$x + y \le 250$ ...(2)

$5x + 8y \le 1400$ ...(3)

$x,y \ge 0$ ...(4)

let ${l_1}:x + y = 250$

${l_2}:5x + 8y = 1400$

Let us graph the inequalities (2) to (4).

The feasible region determined by the system is shown in the graph.

Here again, observe that the feasible region is bounded.

Let us evaluate Z at the corner points A(250, 0), E(200, 50) and D(0, 175).

We find that maximum value of Z is 1150000 at E(200, 50).

Hence the merchant should stock 200 units of desktop model and 50 units

of portable model to realise maximum profit and maximum

profit is Rs. 1150000.