American Airlines (AA) runs daily flights from New York to LA. AA incurs a fixed cost of KRW 80 million per flight in terms of the crew salary, fuel, etc. Suppose each one way ticket generates a net revenue of KRW 500,000.

AA's jets can carry 300 passengers. From the past experience, AA knows that about 10% of ticket holders will not show up on average. Assume unused tickets are fully refundable.

To protect against such no-shows, AA decided to overbook its flight; i.e., it will sell more than 300 tickets although the jet capacity is just 300. In case a ticketed passenger is "bumped" due to overbooking, AA plans to give a compensation of KRW 250,000 to each bumped passenger and provide an alternative transportation (the cost of which just wipes out the KRW 500,000 revenues)

Solve the problem by using the normal approximation; i.e., use the normal distribution to simulate the number of no- shows (or number of shows) from 330 ticket holders, where no-show probability for each ticket holder is 0.1. Assume all other factors remain identical as the before.

1. Characterize the information of the normal distribution suitable for this problem.

2. Run the simulation based on the above normal distribution. To provide robustness in our simulation, let's set the number of simulation to 5,000! What is the optimal solution; i.e., optimal number of tickets to sell and the corresponding revenue?

3. Compare the normal approximation simulation result with the original simulation (using binomial distribution with 5,000 runs). Is this a reasonable approximation? What do you think?