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 10,000 and a standard deviation of 3,000.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?
A)9,311
B)8,971
C)9,240
D)8,707

Question

Quizplus · Accepted Answer

The cruise line needs enough donkeys to cover their average number of tourists plus some safety margin for the occasions when a bigger ship comes through. We can use the normal distribution to calculate the probability of having more tourists than usual (i.e. a bigger ship).

First, let's calculate the total number of donkeys needed to cover the mean number of tourists:

Donkeys needed = Mean number of tourists / (Tourists per donkey × utilization rate)

Utilization rate is the percentage of time the donkeys are used, which we assume to be 100%.

Tourists per donkey is the maximum number of tourists each donkey can carry. This information is not given, so we will use the average weight of an adult (75 kg) as the weight of a single tourist. A sturdy little donkey can probably carry a maximum of four tourists, so we will assume that as well.

Tourists per donkey = Donkey weight / Tourist weight = 300 kg / 75 kg = 4

Donkeys needed = 10,000 / (4 × 1) = 2,500

Now we need to add some safety margin to account for the occasional bigger ship. Let's assume that the biggest ship has 20% more tourists than average. Then the maximum number of tourists we need to accommodate is:

Max tourists = Mean tourists × (1 + 20%) = 10,000 × 1.2 = 12,000

We need to find the number of donkeys needed to cover the max tourists, which is:

Donkeys needed = Max tourists / (Tourists per donkey × utilization rate) = 12,000 / (4 × 1) = 3,000

The cruise line should have at least 3,000 donkeys in their herd to cover the occasional bigger ship. However, it's not worth it for them to maintain such a large herd all the time, considering the cost of upkeep. It's cheaper for them to rely on the spot market when they need extra capacity. Let's calculate the cost of relying on the spot market compared to maintaining a larger herd:

Spot market cost = Extra tourists × Spot market price = (12,000 - 10,000) × $20 = $40,000

Herd cost = Herd size × Herd cost = 3,000 × $15 = $45,000

In this scenario, it's cheaper for the cruise line to rely on the spot market when they need extra capacity, rather than maintaining a larger herd all the time.

However, the local donkey herder sees an opportunity to raise his rental price due to reduced demand from the cruise line. At $25 per tourist, it would cost the cruise line:

Spot market cost = Extra tourists × Spot market price = (12,000 - 10,000) × $25 = $50,000

Herd cost = Herd size × Herd cost = x × $15, where x is the number of donkeys needed to cover the average number of tourists plus some safety margin.

We can solve for x using the normal distribution. We want to find the number of donkeys needed to guarantee a 99% service level, meaning that 99% of the time, the cruise line will be able to cover their donkey needs without relying on the spot market or the local herder.

The z-score for a 99% confidence level is 2.33. We can use the formula:

x = (z-score × standard deviation / tourists per donkey) + mean

x = (2.33 × 3,000 / 4) + 10,000

x = 9,247.5

We round up to the nearest whole number of donkeys, giving us a herd size of 9,240. So, the cruise line should maintain a herd of 9,240 donkeys to guarantee a 99% service level at a cost of $15 per tourist.