A company produces regular and deluxe ice cream at three plants. Per hour of ope

Question

A company produces regular and deluxe ice cream at three plants. Per hour of operation, Plant A produces 40 gallons of regular ice cream and 20 gallons of deluxe ice cream. Plant H produces 20 gallons of regular ice cream and 40 gallons of deluxe ice cream, and Plant M produces 40 gallons of regular and 40 gallons of deluxe. It costs $60 per hour to operate Plant A, $99 per hour to operate Plant H, and $130 per hour to operate Plant M. The company must produce at least 310 gallons of regular ice cream and at least 290 gallons of deluxe ice cream each day How many hours per day should each plant operate in order to produce the required amounts of ice cream and minimize the cost of production? Select the correct choice below and fil in any answer boxes within your choice. A To minimize the cost of production, Plant A should operate for hours per day, Plant H should operate for hours per day, and Plant M should operate for hours per day (Round to the nearest tenth as needed.) O B. There is no optimal solution

Explanation / Answer

Let the company operate the plants A,H and M for x, y, and z hours respectively. The quantity of regular ice cream produced is 40x+20y+40z so that 40x+20y+40z=310 or,4x+2y+4z=31 …(1) . Also, the quantity of deluxe ice cream produced is 20x+40y+40z so that 20x+40y+40z= 290 or, 2x+4y+4z = 29…(2). The augmented matrix for the 1st and the 2nd equations is P =

4

2

4

31

2

4

4

29

The RREF of P is

1

0

2/3

11/2

0

1

2/3

9/2

Hence x+2z/3 = 11/2 or, x = 11/2-2z/3 and y+2z/3 = 9/2 or, y = 9/2+2z/3.

Also, the operating cost is C(x) = 60x+99y+130z….(1)

It is apparent that the operating cost of the plant M is the highest. Thus, to minimize C(x), z should be minimized. Since the minimum value of z is 0, we have x = 11/2 and y = 9/2.

Thus, to minimize the operating cost, plant A should be operated for 11/2 = 5.5 hours per day and the plant H should be operated for 9/2 = 4.5 hours per day . The plant M may be kept shut.

4

2

4

31

2

4

4

29

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