lve the problem. 10) A stereo manufacturer makes three types of stereo systems,

Question

lve the problem. 10) A stereo manufacturer makes three types of stereo systems, L, II, and II, with profits of $20, $30, and $40, respectively. No more than 100 type-I systems can be made per day Type-I systems require 5 man-hours, and the corresponding numbers of man-hours for types II and III are 10 and 15, respectively. If the manufacturer has available 2000 man-hours per day, determine the number of units from each system that must be manufactured in order to maximize profit. Compute the corresponding profit.

Explanation / Answer

Let the number of stereo system I,II,and III, that are manufactured per day be x, y and z respectively. Then the daily profit is P(x) = 20x+30y+40z.

Also, since no more than 100 type-I units can be made per day, hence x ? 100…(1).

Since type I,II,and III systems require 5,10 ansd 15 man hours respectively, and since a maximum of 2000 man hours are available per day, hence 5x+10y+15z ? 2000 or, x+2y+3z ? 400…(2)

A scrutiny of the profit function P(x) = 20x+30y+40z reveals that P(x) will be maximum when z is maximum and x is minimum.Further, since x ? 100, we can assume x = 0 also. Then we have 2y+3z ? 400. Since both y and z have to be positive integers, the maximum possible value of z is 132. Then y = 2. Thus, the profit is maximum when the number of stereo system I,II,and III, that are manufactured per day are 0,2 and 132 respectively. Then the maximum daily profit is 20*0+2*30+132*40 = $ 5340.

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