A manufacturer has three machines I, II and III installed in his factory. Machine I and II are capable of being operated for atmost 12 hours whereas machine III must be operated for atleast 5 hours a day. He produces only two items M and N each requiring the use of all the three machines.The number of hours required for producing 1 unit of M and N on three machines are given in the following table : Items Number of hours required       I II III M 1 2 1 N 2 1 1.5 He makes a profit of rupees 600 and rupees 400 on one unit of items M and N respectively. How many units of each item should he produce so as to maximize his profit assuming that he can sell all the items that he produced. What will be the maximum profit ?

Let x units of item M and y units of item Nare produced

According to the question:

$x+2 y \leq 12 \\$

$2 x+y \leq 12 \\$

$x+1.25 y \geq 5 \\$

$x \geq 0, y \geq 0$

By graph we will get Corner point A(0,4), B(5,0), C(6,0), D(4,4), E(0,6).

$\\ \text {Maximize Profit } \\ P=600 x+400 y$

At A(0,4) $P=600 \times 0 +400 \times 4 = 1600$

At B(5,0) $P=600 \times5 +400 \times 0 = 3000$

At C(6,0) $P=600 \times 6 +400 \times 0 = 3600$

At D(4,4) $P=600 \times 4 +400 \times 4 = 4000 \ \(Maximum)$

At E(0,6) $P=600 \times 0 +400 \times 6 = 2400$

Maximum profit is Rs 4000 at 4 units of M and 4 units of N.

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