Quail Resorts (QR) is a premier resort company, operating several ski resorts throughout California. The company was hit during the past few seasons by lack of snow, and thus the executives decided to take a closer look at their operations, aiming to increase profitability for the coming season. As expected, the main parameter driving QR’s profits is snowfall. The company’s team of meteorologists used historical data and an extensive analysis of recent climate patterns to estimate that snowfall this year will follow a triangular distribution, with a minimum of 140 inches, a maximum of 200 inches, and a likeliest value of 175 inches. Furthermore, the marketing team data suggests that the company’s resorts face an annual demand that, as a function of the total snowfall and the price of a lift ticket, is uniformly distributed (assume fractional demand is okay),

with Minimum = 5500 (customers/inch)*snowfall (inches) – 6000 (customers/$)*price ($)

Maximum = 6200 (customers/inch)*snowfall (inches) – 6000 (customers/$)*price ($)

For instance, if the lift ticket price is 80$, and the total snowfall for the year is 150 inches, then the demand would be uniformly distributed between 345,000 and 450,000 customers. During one season, QR can accommodate at most 600,000 customers cumulatively, across all the resorts that it manages.

For the following questions, if you feel that some modeling element is ambiguous, and you need to make additional assumptions, please state them explicitly in your spreadsheet and proceed.

1. Build a simulation model to predict the mean revenue when the lift ticket price is $80. In addition to giving the mean revenue, turn in a histogram of the revenue.

2. Using the model in (1), what is the probability of making at least $35 million in revenue?

3. The first two questions involved only the company’s revenues. In QR’s experience, the operating costs are also somewhat uncertain, and depend on the realization of the annual demand. The marketing department came up with the information in Table 4 below to describe the change in operating costs as a function of the total annual demand. For instance, if the annual demand is below 350,000 customers, the operating costs are uniformly distributed between $2.5M and $3.5M. Embed the costs in the model you constructed for Question 1, and answer the following questions:

a) What is the mean profit (i.e., revenues minus costs) when the lift ticket price is $80?

b) What is the lift ticket price that maximizes Quail Resort’s mean profit? [Please give an answer within $1 of the best possible price.]

For the following question, use the model you constructed in Question 3 as a starting point (we are interested in maximizing total profit).

4. Assume the QR team has decided to fix the lift ticket price at $80. Instead of changing the price, they came up with a very interesting proposition. If the snowfall during the season proves to be low, a special snow making system could be purchased. The system is quite expensive, at roughly $7,500,000, but it can increase the total demand to the resorts by roughly 100,000 customers. It can be placed on special order, and received and installed within a matter of days. The management would like to follow a very simple “rule of thumb” to decide when to order the system: if snowfall falls below a certain threshold value, they would order the system. If snowfall is above the threshold, however, they proceed without ordering. They need your help in determining the optimal threshold to use. Update the model in Question 3 to address this issue.

For the purpose of this question, you may assume that QR completely observes the amount of snowfall, before having to decide whether to order the system. 

       

Overall Realized Demand	Operating Costs
Below 350,000	Uniformly distributed between $2,500,000 and $3,500,000
Between 350,000 and 450,000	Uniformly distributed between $4,000,000 and $5,000,000
Above 450,000	Uniformly distributed between $4,500,000 and $5,200,000

Table 4: Operating costs as a function of the total annual demand