Welcome to our last lecture of this module on Discrete Random Variables and Probability Distribution. In this last lecture I'm going to talk about a very important application of the binomial distribution and probabilistic thinking in general. Airlines, airlines overbook their flights. Which means, let's say, there is a flight from Zurich to London that has 200 seats, airlines may actually sell 210 tickets or 220 tickets. Why are they doing it? What are the benefits and the costs of doing this? Airlines face a big problem, last minute cancellations or total no-shows. People may cancel a flight, just a couple hours before it's supposed to take off, and they still may get a substantial refund. At this point, the aircraft may leave with empty seats. That's bad for the airlines, because there's lost revenue. They could have sold a seat, they could have made more money.

And as we know, airlines are in a tough competition, tough business, they really need that money.

It is also said that airline seats are the most perishable consumption good that exist. If you think about tomatoes in your fridge, they decay overtime. They first get wrinkles and eventually they get moldy and you have to toss them away, it's a process, not so with airlines. An airline seat is fresh, meaning you can buy it, you can buy it until five minutes before the departure, the door closes and it's dead.

The case from one second to a next, it was available for sale and it's gone, and the airline cannot sell it anymore. It went bad, it went moldy, sort of in an instance and that's very bad. So, the airlines wants to get extra revenues and one way is to do airline over booking, that's the benefit. The cost on the other hand is now when too many people show up. You have only 200 seats, you sold 210 tickets, suddenly 205 people show up. Now you have to buy some people out of traveling, either by giving them vouchers. Or in the worst case, if you don't have enough volunteers who take vouchers, you have to kick off people. That cost you money directly, you may really annoy your customers, and they say, I never fly with that airline again. So you may have serious loss in customer goodwill, and that's difficult to put Swiss Franc or Dollar figure on that. And in general, you have customer dissatisfaction.

So the question now is, what's the best behavior of airlines and how can we can understand their decisions? So, we want to look at how do they trade off the benefits on potentially more revenue, against the cost of disgruntled customers and giving out vouchers or perhaps have to pay for hotel nights for customers. This is actually an example of a very large and an important area, in what business schools call Operations Management, and this is the area of Revenue Management, how to increase revenues.

Now, for the sake of this discussion, we make a simplifying assumption. All people make the decision of whether to show up or not at the airplane independently, versus same probability p.

This independence of course is not quite correct. Whenever you have groups travelling, the family of four, parents with their kids maybe all go to a vacation trip, visiting relatives or nobody goes. So clearly there is some dependents or some business people either all go to visit the customer, or nobody goes. Here, we simplify from this and say all the people make the decision, independently with a probability p. And voila, we have a binomial setup, and we can use the binomial distribution. I put together a large spreadsheet for you on airline overbookings. So let's have a look at this spreadsheet. Here but for yellow background, I provided some parameters, that we just take as given. We have an airliner with 200 seats. The price of a ticket is $300. The bump cost is $500, which means in addition to reimbursing an overbooked passenger who cannot fly with us, the $300 for the price, there is also a $500 voucher or hotel cost, or you name it. And we assume a probability of no shows of 0.07. I've found this type of number in some reports on airline overbooking. Here now is the number of tickets sold. Let's say we sell 200 tickets, now look at this. Under the assumption that 7% don't show up, we assume 14 people won't show up. So on average only 186 people show up. So on average, our airliner there is not full. We don't like this, and we see here that revenues would be in this case $55,800. Now let's say we sell 205 tickets.

Now the expected value of people showing up is still nowhere close to 200, tt's now 190.65. Notice the expected revenue went up. How did I calculate those? Now let's go deep into the spreadsheet here. If we sell 205 tickets, then in the worst case 205 people show up. My revenue, however, is only 60,000. Why, because only 200 people will actually leave with us, 200 times 300. Then I have the over cost of 2,500. That means five people get over cost of 500, five people get 500 each, 2500, in addition to the 300 that I don't get, makes revenue minus cost 57,500. Now how likely is it that that happens? Here, look at this probability, it's a binormal distribution of having exactly 200 people show up. If I sell 205 people show up, if I sell 205 tickets the probability of 93% showing up, that's essentially 0. This worst case scenario just doesn't happen. Let's look at now this case, 202 people show up. Again the revenue's the same, 60,000, I can only take 60,000, make only that because I can take only 200 with me, 2 people I have to bump, that's 500 in overage cost. So I make net only 59,000, not 60,000, only 59,000. What's the probability of that happening? Still very tiny, and again I use a binomial distribution. 202 people show up, if I sell 205 tickets, but for probability of showing up of 93%, that's one minus the 7%. And so, now I have all the possible revenue minus cost. Here I have all the possible probabilities, and then I can calculate the expected value, which I do here in the column J. I add up all these numbers. Now, I can tell you the truth, I don't actually do this for all the possible numbers. Why, because at some point they're all zero, the probabilities are so tiny. And here, I get the round number to two digits, 57,193. Now, let's play with this, let's say we sell five more tickets, 210. What happens to revenue minus cost? It's still goes up. Let's try 214, it still goes up! Now let's go crazy, 230, not a good idea, it goes way down. Why, because, suddenly it's very likely that we have to bump many people. For example, if we sell now 230 tickets, the probability of say 214 showing up is 10%. That's rather high, so clearly it's not a good idea to sell 230 tickets. Now how can you find the optimal number? Here with some trial and error, you can play around with this number, and it's around 214, 215 is the optimal number. Now for those of you who know a little bit more Excel, if you have access to the Excel server, I have hooked up an Excel server to this Excel sheet, and with this server we can actually find the optimal solution here. However that requires some optimization, which is not part of this class. But for any people who have played around with the server before, you may find that interesting. Now let me summarize what we have seen here in the slides. So, and move back to the slides.

So let me summarize what we just learned. The objective of the airline overbooking problem is to find the optimal trade-off between the benefit of selling more tickets than you have seats on an airplane. And the cost of perhaps having to push off

customers from the airplane who had booked a regular ticket.

And in the spreadsheet we saw how we can use probabilities to sort of find the optimal, the maximized expected payoff to the airline. Now clearly some assumptions went into our calculation, but I think there's one very robust takeaway. It is not optimal for airlines to only sell as many tickets as they have seats. It really is necessary for them to overbook their airlines, their flights. To summarize, we have now seen in our last lecture in this module, an application of the binomial distribution, namely airline overbooking. This is just one example of this large area called Revenue Management. And this concludes the module on Discrete Random Variables and Discrete Probability Distributions. In our next module now, we will take a look at continuous distributions. We will see the famous bell curve and the normal distribution with some cool applications. So please stay with us and come back for the next module of An Intuitive Introduction to Probability, Decision Making in an Uncertain World. Thank you.

Application: Airline Overbooking

An Intuitive Introduction to Probability

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