statistics probbality data analysis

Question

An airline is deciding how much to overbook a flight. The plane has 175 seats. Historically, 95 percent of those with a reservation will actually show up. The airline charges $500 for each seat on the flight, but incurs $25 in fuel and other costs (e.g. pretzels) for each customer. However, if the airline overbooks and more than 175 customers show up, they will need to make other arrangements for these guests. The cost of placing these guests on other flights is $950 per person. Build a model to show how many tickets the airline should sell for this particular flight to maximize their profit

Answer 1

For number of tickets sold of x >= 175,The revenue can be shown to equal

475 * .95 * x - (950+475) * the sum i > 175 (i - 175)Px(i)

The advantage of this formulation is that we can see that there will be a maximum, as the marginal revenue is decreasing and will become negative.

If we are only looking for the optimal number of tickets, we can divide through by 475 and have

.95x - 3 * the sum i > 175 (i - 175)Px(i)

Px(i) = C(x,i).95^i*.05^(x-i)

We expect optimal x to be between175 and 175/.95

It turns out that the optimal x is 183, and the net revenue is

475 * .95 * 183 - (950+475) * the sum i=176 to 183 (i - 175)C(183,i).95^i*.05^(183-i) = 81642.97

(Note that 175/.95 = 184.21)