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July 24, 2014

Posted by **Willoby** on Saturday, September 3, 2011 at 7:30pm.

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?

- Math OPTIMIZATION -
**MathMate**, Sunday, September 4, 2011 at 6:01amThe 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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