SOLUTION: A printing press has 8 machines, each of which has a speed of 3 600 prints per hour. It costs P50 to set up each machine for a run and (100+60 n ) pesos to run n machines for 1 hou

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Question 542467: A printing press has 8 machines, each of which has a speed of 3 600 prints per hour. It costs P50 to set up each machine for a run and (100+60 n ) pesos to run n machines for 1 hour. How many printing press machines should be used to produce 50 000 copies of calendars most profitably?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
most profitably sounds like least cost, since we don't know the revenue.
the equation will be:
cost = 50*n + (100+60n)*h
n is the number of machines.
h is the number of hours.
this assumes that 50 dollars is setup costs for each machine regardless how long it runs.
this also assumes that 100 is fixed for any number of machines to be run for 1 hour and 60 is variable cost per machine to be run for 1 hour.
an example would be:
cost for 2 machines run for 2 hours would be:

50 pesos * 2 machines = 100 pesos setup cost regardless of the number of hours run by each machine.
this is the cost per machine.

2 hours * 100 pesos = 200 pesos run cost for 2 hours regardless of the number of machines.
this is the cost per run per hour.

2 hours * 60 pesos * 2 machines = 240 pesos run cost for 2 hours of 2 machines.
this is the cost per machine per run per hour.

total cost for the 2 machines to be run for 2 hours would then be 100 + 200 + 240 = 450 pesos.

i used excel to analyze each scenario and came up with a minimum cost solution at 5 machines.

5 machines would take 50,000 / (3600*5) = 2.78 hours to run.

the cost for 5 machines would be:

5*50 = 250 total set up cost.
2.78 * 100 = 278 total run cost per hour
2.78 * 5 * 60 = 834 total run cost per machine per hour
grand total = 250 + 278 + 834 = 1362

i was able to generate an eqaution for graphing purposes.
that equation is:

y = (1800x^2 + 30000x + 50000)/(36x)

the graph of that equation looks like this:



just draw a vertical line at x = 5 and you will see that it intersects the graph of the equation when y = 1362.

the results from excel indicated the following: 

number of machines
      number of hours		
                 total cost per machine
                          total cost per hour
                                        total cost per machine per hour
                                                     grand total

1     13.89      50       1388.89       833.33       2272.22	
2     6.94       100      694.44        833.33       1627.78	
3     4.63       150      462.96        833.33       1446.30	
4     3.47       200      347.22        833.33       1380.56	
5     2.78       250      277.78        833.33       1361.11	*****
6     2.31       300      231.48        833.33       1364.81	
7     1.98       350      198.41        833.33       1381.75	
8     1.74       400      173.61        833.33       1406.94


note that total cost per machine per hour is the same.
this effectively cancels it out as a criteria for least cost.
the reason for this is because the number of hours times the number of machines works out to be the same.

1 machine takes 50,000 / 3600 = 13.89 hours
8 machines take 50,000 / (3600*8) = 1.74 hours each for a total of 13.89 hours.
the total number of machine hours is the same, but the total number of hours spent is different because they are working in parallel to each other.