*This week’s theme on the Hospitality Blog is overbooking. Here’s an exercise taken directly from my eCornell course Overbooking Practices in Hotel Revenue Management. The overbooking-ratio method is a very, very popular and successful method that many hotels use for their overbooking policy. When you’ve completed this exercise, be sure to download this free step-by-step guide to overbooking. *

Now, sometimes people talk about overbooking as being evil. Why would anyone ever overbook, because that’s not fair to the customer? Let’s think about this. If you never overbook—you say that I’m a good and pure person, and I would never, ever do anything like that—what happens if you have a 5% no-show rate?

You have those rooms sitting there empty—it actually might end up costing you more than if you overbook just a little bit. And so when we try to look at overbooking, we try to come up with the least expensive overbooking policy, realizing that whatever we do with this, we’re still going to be making a guess. It’s kind of like gambling—we’re going to come up with a forecast of how many no-shows, and come up with our best bet on what the overbooking policy should be. So, when we look at coming up with an overbooking policy, there are more than several factors to consider.

#### The Cost of Walking

First of all, we need to know how much it’s going to cost to walk somebody, and there are all sorts of different costs associated with this, such as you might have to give the person a free meal, you might have to pay for transportation for them, you might want them to come back to your hotel—so you give them a voucher.

And most importantly is an intangible cost: the cost of ill will. There’s been research showing that 70% of customers who have been walked don’t want to go back to the hotel again. So, this is something you should definitely consider. And then the cost of an empty room is the opportunity cost associated with the room that went empty.

Let’s go through a simple example here. We’re going to talk about the cost of walking. Let’s say the cost of walking is 300 dollars. Let’s say the cost of an empty room is 100 dollars. We’re going to use these two pieces of information—the cost of walking and the cost of an empty room—in conjunction with the no-show distribution, to come up with an overbooking policy. I’m going to call this an overbooking-ratio method. It’s basically a little equation with a few tricks thrown into it. It’s going to help you come up with the cheapest and best approach for overbooking. It’s going to take into consideration the distribution of no-shows, also the cost of an empty room, and also the cost of walking.

The way we’re going to calculate this overbooking ratio is you’re going to take the cost of walking—so, in this case, it would be 300 dollars—and you’re going to divide that by the sum of the cost of walking plus the cost of an empty room, so 300 plus 100 is 400. So, our overbooking ratio here would be 300 divided by 400 is equal to .75. Now, you might be wondering, “What in the world am I going to be doing with that .75?” Well, let’s go through and figure this out. I’m going to give you an example, and then we’ll go back through and talk about exactly what we did with this. Let’s go to a hotel called the Duck Hotel. The Duck Hotel can sell each of their rooms for an average of 120 dollars, with a variable cost of 20 dollars. So, it sounds like their cost of an empty room is 100 dollars. If they have to walk a guest, they estimate that it costs them about 300 dollars. So, our cost of walking is 300 dollars.

They’ve kept track of the Thursday no-shows for the past six months, and found the following distribution. So, let’s look at this and see if we can come up with the best overbooking policy for them. If you look at this, 10% of the time they don’t have any no-shows, 15% of the time they have one no-show, down to 5% of the time they have five no-shows.

What we’re going to do with this now is, first of all, what we have to do is look at these probabilities. Probabilities are like a chance. And what we’re going to do is come up with something called the cumulative probability. That might sound a little bit complicated, but it really isn’t.

What we’re going to be looking at is, What’s the chance that you have at least a certain number of no-shows? So, for example, if I look at what’s the chance that I have at least zero no-shows? Well, that would be 100%, because you can’t exactly have negative no-shows. So, my chance of having at least zero no-shows is 100%. What’s my chance of having one or more no-shows? Well, there are two ways you can calculate this: one is, you can say, 100% minus 10%, the chance of zero no-shows, is 90%. Or, you can always go through and look at, well, what’s the chance of one no-show—15%—plus the chance of two no-shows, and all the way up through five no-show—that adds up to 90%. So, 90% chance of having one or more no-show.

What’s the chance of having two or more no-shows? The easiest way to calculate this, again, is to take the 90% minus the 15%, and that leaves you with a 75% chance of two or more no-shows. If you want to check yourself, look at the chance of two no-shows, three, four, and five and, yep, that adds up to 75%. What’s the chance of three or more no-shows? Well, that’s 75 minus 25—50%. And what’s the chance of 4 or more no-shows?…would be 50 minus 30 is 20%. And the chance, finally, of five or more no-shows is 20 minus 15%, or five.

And a quick tip on this, the cumulative probability of that last one that we came up with, the 5% chance of five or more no-shows—notice that that’s the same as the chance of five no-shows?

Those numbers should always be the same; if they’re not, you’ve made a math mistake. So, let’s go back and remember what that overbooking ratio was. In this case, it was the cost of walking—300 dollars—divided by the cost of walking plus the cost of an empty room. So, 300 plus 100 is 400. So, our overbooking ratio is 300 divided by 400, or .75.

What we want to do now—and this is where the trick comes in—is we go ahead and we look at this cumulative probability, and we look for the first number there that is either the same or larger than our overbooking ratio. And when I look at this, it comes out to be…well…I keep going along and there’s a 75%—that’s the first number that’s the same or larger than my overbooking ratio. And if I look back across the columns here, that shows me that I should overbook by two.

Now, let’s say that you’re getting into an argument with someone, and someone’s saying to you, “Well, wait a minute, I don’t think the cost of walking somebody is 300 dollars, I actually think it’s more like 500 dollars.” Well, what should you do? Well, it’s pretty simple. All you’re going to do now is go back and recalculate your overbooking ratio.

So, in this case it’s going to be 500 for walking, and then I’m going to be dividing that by 500 plus 100—so, 600. So, 500 divided by 600, that comes out to be .875. And then I’m going to compare that against my cumulative probabilities there, and again, it looks like the first number that’s bigger than that is 90%. And so I go across the table, and I should overbook by one room.

If you use this overbooking ratio, it will lead you to the most profitable overbooking policy. Notice that you’re not always going to be perfect. Sometimes you’re going to have more no-shows than you were expecting; sometimes you’re going to have fewer no-shows than you were expecting.

But still, the overbooking ratio gives you the best possible result in the long term. And some of the other things that you need to think about with this: this cost of walking is somewhat subjective. I mean, I might say that it costs 300 dollars to walk someone; someone else might argue that it’s 500 dollars; someone else might say it’s millions of dollars because the person was so upset—it was a regular customer. So this is something you need to bear in mind.

A good rule of thumb that some hotels use is they say the cost of walking is two times larger than the cost of an empty room. That might be something you might be able to use at your hotel. This also doesn’t consider multiple-night stays, so, let’s say somebody was planning on staying two nights, and they didn’t show up. But typically this isn’t that huge of an issue.

And the other thing, too, is it doesn’t consider the number of arrivals. And this definitely has an impact on the number of no-shows you’re going to have, because, like we’ve talked before earlier, if you don’t have any arrivals, you’re not going to have any no-shows. Still, the overbooking-ratio method is a very, very popular and successful method that many hotels use for their overbooking policy.

*Download your free step-by-step guide to overbooking here.*

#### Sheryl Kimes

#### Latest posts by Sheryl Kimes (see all)

- The Cheapest and Best Approach to Overbooking - April 8, 2013

The paragraph (What’s the chance that I have at least zero no-shows? Well, that would be 100%, because you can’t exactly have negative no-shows. So, my chance of having at least zero no-shows is 100%.) is somehow vague, especially (you can’t exactly have negative no-shows).

Anyway, it was fully clarified after reading the step-by-step guide.

Thank you.

Fascinating article, thank you very much Sheryl! How to approach the “art” of overbooking with a mathematical twist.

One addition I would do to your brilliant approach is the room type and upgrade rate. As much as you can use your No-Show rate to “predict” how likely your rooms will stay empty, you can also use your upgrade rate to determine how often you are able to overbook your standard rooms, using the power of upgrades (preferably paid, obviously) to yield your guests into the superior room types, usually not as booked as the standard ones.

Using this strategy, you can make sure that all room types are occupied (and therefore increase your occupancy %) without having to give away upgrades for free. An upgrade / upsell technology tool can help you determine and maximise your upgrades.