For each problem, make an Excel model, simulate via YASAI, and hand in the standard printouts for a simulation problem.  On the simulation output reports, be sure to indicate the answer to each question posed in the corresponding problem.  Use a sample size of 1000 for each problem.
 

Q1.  (100 points)   You operate a maintenance shop that repairs aircraft engine compressors for a large airline.  The number of compressors you receive to be repaired each day is Poisson with a mean value of 8.1.  The crucial step in each repair operation is realigning the compressor's blades.  The machine used to align these blades has a part that wears out rapidly, as follows:

On days when the part does not fail, you can process 10 compressors.  On days when it does fail, you are equally likely to be able to process 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 compressors.  After a failure, you must replace the part and wait until the next day to resume work.

On any given day, if you have more compressors to repair than you are able to fix, you set the leftover compressors aside in a queue, and try to fix them the next day.  You estimate that each day each compressor waits in this queue costs you $80.

Your policy is to replace the alignment machine part after n days of use or when it fails, whichever comes first.  You perform the replacement just before the start of each day's work.  Replacement costs $500, whether done intentionally, or because of failure.  (Occasionally, you will have to replace a part twice in one day: intentionally at the start of the day, and then later in the day because of a part failure.)

What value of n gives you the lowest total cost?  Simulate a 100-day period with a sample size of 1000, and try n = 2, 3, 4, 5, and 6.  For the best choice of n, estimate the expected value of the following over the 100-day period: (a) the average number of compressors in the queue at the end of each day, (b) the number of scheduled part replacements (that is, the number of times you replace a part that has not failed) over the 100-day period, and (c) the number of times the part fails over the 100-day period. 

Finally, you are also interested in the probability of the event that, at any time during the 100 day period, the number of compressors in queue at the end of the day is 15 or larger.  (d) For the best choice of n, estimate the probability of this event.

Hints:

1. You can construct the spreadsheet model by combining features of the REPAIRSHOP and BELTS models covered in class.
    
2. A quick and easy way to generate an integer that is equally likely to take the values 0, 1, ..., 9 is via the formula INT(GENUNIFORM(0,10)).  Each value x created by  GENUNIFORM(0,10) will have 0 < x < 10, and the "INT" function rounds down to an integer.  This technique is similar to one used in the "3-door" example.