Math OPTIMIZATION

A plastics firm has received an order from the city recreation department to manufacture 9,000 special Styrofoam kickboards for its summer swimming program. The firm owns 10 machines, each of which can produce 30 kickboards an hour. The cost of setting up the
machines to produce the kickboards is $20 per machine. Once the machines have been set up, the operation is fully automated and can
be overseen by a single production supervision earning $15 per hour.
a. How many of the machines should be used to minimize the cost of production?
b. How much will the supervisor earn during the production run if the optimal number of machines is used?
c. How much will it cost to set up the optimal number of machines?

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  1. The idea is to setup a cost function in terms of n, the number of machines used, and N, the number of units to be produced.

    Number of hours to run n machines
    H = N/(30n)
    Cost for setup = 20n
    Supervisor cost = 15H

    Total cost
    C(N,n)=20n + 15H
    =20n + 15N/(30n)

    Take N to be a constant, and differentiate with respect to n:
    ∂C(N,n)/∂n
    =20-N/(n²)
    Equate to zero and solve for n, reject negative roots:
    20-N/(2*n^2)=0
    n=(1/2)sqrt(N/10)
    =(1/2)sqrt(9000/10)
    =15
    Hence there are not enough machines for the optimal cost, use 10 = maximum the firm has.
    Check:
    C(9000,15)=600
    C(9000,10)=650
    (not optimal, but best effort)

    Substitute N=9000, n=10 to calculate the individual costs.

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