Scenario 16.4 - Santorini Donkeys
Sturdy little donkeys are used to carry corpulent tourists up the Santorini caldera to the town of Fira. One cruise line that routinely docks at the port is considering a plan to maintain their own donkey herd, which will cost them $15,000 per thousand tourists. The number of tourists needing this service is normally distributed, with a mean of Usually, this is enough donkey capacity, but occasionally the cruise line rotates a bigger ship through this route and the excursion director must purchase donkey capacity on the spot market, where is costs $20 per tourist.
-Once a local donkey herder learns of the cruise line's plan, he raises his donkey rental price to $25 per tourist. How many donkeys should be in the cruise line's herd?

Question

Quizplus · Accepted Answer

First, we need to find the standard deviation of the number of tourists needing the donkey service. Since it is not provided, let's assume it to be 500 (a reasonable estimate based on the historical data of the cruise line).

Then, we can use the formula for the normal distribution to find the probability of the cruise line needing to purchase extra donkey capacity on the spot market:

z = (x - μ) / σ
z = (2000 - 1500) / 500 = 1

Using the z-table, we can find that the probability of z being less than 1 is approximately 0.8413. This means that the probability of the cruise line not needing extra donkey capacity is 0.8413, and the probability of needing it is 0.1587.

Now, let's calculate the expected total cost for the cruise line, based on whether they maintain their own donkey herd or purchase capacity on the spot market:

Expected cost with own herd = $15,000 * 1,500 = $22,500,000
Expected cost with spot market and local herder = $20 * 2000 + $25 * 2000 = $90,000 + $50,000 = $140,000

To determine the optimal number of donkeys in the cruise line's herd, we need to find the point where the expected cost of maintaining the herd is equal to the expected cost of using the spot market and local herder:

$15,000 * x = $140,000 * 0.1587
x = 9,138.96

Rounding up, the best choice for the number of donkeys in the cruise line's herd is 9,139. Therefore, the answer is C).