Final answer:
To calculate the under-stocking cost Cu, use the Z-score formula with X = 0, then multiply the probability by the high fare. To calculate the over-stocking cost Co, use the Z-score formula with X = capacity - 1, then multiply the probability by the low fare. The service level is obtained by subtracting the probability of under-stocking from 1.
Step-by-step explanation:
To calculate the under-stocking cost Cu, we need to find the probability of demand exceeding the airline's capacity. Given that the demand follows a normal distribution with mean (mu) = 52 and standard deviation (sigma) = 5, we can use the Z-score formula to find the probability. The Z-score is calculated as (X - mu) / sigma, where X is the capacity. Here, X = 0, since any demand above the capacity would result in understocking. We then use the Z-score to find the probability using the standard normal distribution table or a calculator. The under-stocking cost Cu is calculated as the probability multiplied by the high fare of $756.
To calculate the over-stocking cost Co, we need to find the probability of demand falling below the airline's capacity. Here, X = capacity - 1, as demand up to the capacity is met without any overstocking. We calculate the Z-score using this X value and find the probability using the standard normal distribution table or a calculator. The over-stocking cost Co is calculated as the probability multiplied by the low fare of $171.
The service level or critical fractile is the probability of meeting the demand without any stockouts. It is calculated as 1 minus the under-stocking cost Cu. We can use the Z-score formula with X = capacity to find the corresponding Z-score. The service level is then calculated as 1 minus the probability obtained from the standard normal distribution table or a calculator.