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 20 gallons of regular ice cream and 10 gallons of deluxe ice cream. Plant H produces 10 gallons of regular ice cream and 20 gallons of deluxe ice cream and Plant M produces 20 gallons of regular and 20 gallons of deluxe. It costs $70 per hour to operate Plant A, $84 per hour to operate Plant H, and $130 per hour to operate Plant M. The company must produce at least 440 gallons of regular ice cream and at least 420 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?

To minimize production, plant A should operate for ? hours per day. Plant H should operate for ? hours per day. Plant M should operate for ? hours per day.

Explanation / Answer

x = number of hours the A plant is in production

y = hours the H plant is in production

z = hours the M plant is in production

Regular Ice Cream production in gallons =

20 x gal + 10 y gal + 40 z gal ? 440 gal

Deluxe Ice cream production in gallons =

10 x gal + 20 y gal + 20 z gal ? 420 gal

Cost = $70 x + $84 y + $130 z

You have three variables and only two equations, one for regular and the other for premium, the cost function is NOT a equation you can use to solve the system because you have no data for it. You must first solve the system in order to have the data to evaluate, and minimize, the cost function. The set of all answers to the system is the domain of the cost function.

Fortunately, we can eliminate the complicating feature of this problem.

Notice that if plants A and H both work for an hour, they will produce 30 gal of Regular and 30 gal of Premium Ice Cream for a cost of $154. For plant M to meet that order it would have to operate for 1.5 hours for a cost of $195.

Thus, the optimal solution will not use plant M at all because it cannot complete with A and H working together.

So now our two equations, in two variables, are:

20 x + 10 y ? 440

10 x + 20 y ? 420

this two equation can be written as

2 x + 1 y ? 44

1 x + 2 y ? 42

these lines will intersect at

x=15.33 and y=13.335

Our cost equation is now

C= $70x + $84y

Our boundary values are (44, 0) (0, 42) and (15.33, 13.33)

C(44, 0) = $70 (44) + $84 (0) = $3080

C(0,42) = $70 (0) + $84 (42) = $3528

C(15.33, 13.335) = $70 (15.33) + $84 (13.335) = $2193.24

So 15.33 hours at plant A and 13.335 at plant H is the lowest-cost production schedule.

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