Problem 3 (Team): Maintenance Shop Operation

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 has a part that wears out rapidly, as follows:

Day of Use     Chance of Part Failure

1                             2%

2                             4%

3                             10%

4                             12%

5                             15%

6                             20%

7 or higher            25%

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 500, 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), and (c) the number of times the part fails.