Overbooking
In many sectors of the leisure or transport industry, booking in advance for a precise date is a common practice when there is a constraint on the capacity.
This is true for seats, beds, rooms, cabins, car parking spaces and much else. The list is so long that we’ll keep things simple by focusing on booking a ‘seat’. On the due date, some of the seats booked are sometimes not occupied, because some people do not show up. This happens frequently in a world where uncertainty of schedule, traffic or work is everywhere. Most of the time, those ‘noshows‘ or cancellations are not declared in advance and the company has no other option than observing that some seats are empty.
It is too late to resell these seats
For the company, this is a waste of capacity and of revenue, since some other people may have been willing to pay for those seats if they were available. If the demand is high for that period, those seats may even have a higher value. Of course, the company has already made some money with those seats but at the end of the day, these are valuable assets that will not be utilised and money is left on the table as a result.
To avoid this situation, some companies practice overbooking and sell more than the actual capacity.
How to choose the overbooking rate – the percentage of additional seats available for sale relative to the total capacity – is then crucial. It depends not just on the noshow rate – the percentage of seats not occupied relative to the total capacity – but also on some other factors.
Empty seats are spoilage
Over the years, many companies have followed airlines practices and try to forecast their noshow rates. It is quite a difficult task: these rates vary greatly and are specific to each industry, each good, each date, each population of clients and reservation constraints.
Noshow rate can be important
For airlines, the noshow rate decreases with the length of the flight and varies with the type of demand: people travelling for business reasons are more likely to change flights at the last minute. The noshow rate can be as large as 15 to 20% of the capacity for the most indemand flights or to some destinations. This is all the more reason to try and sell seats that would otherwise remain empty.
A simulation exercise
Let’s work out the expected revenue of overbooking
Consider a 400 seats airliner departing from Paris to NewYork next Monday morning. On this flight, noshows are very likely but not certain. Should the company practice overbooking? To what extent?
All you need is demand and a good knowledge of the noshow probabilities.
Imagine that the airline sells one additional seat, expecting additional revenues for that seat. The expected revenue depends both on the ticket price and on the probability of selling an additional seat. The way this expected revenue is calculated is quite a complex process, which we won’t detail here.
Anyway, we imagine that the flight is quite demanded, so the probability to sell this seat is high, and the price few days before departure is quite high too. Let’s assume that the computed expected revenue here is €400 for this very seat, and that the probability of noshow for this seat is 0.9, which is quite high as we are talking of the first additional seat sold on a 400 seats plane.
Of course, there is still some chance that all passengers will finally present themselves for boarding. In that case the airline will have to find a passenger who will accept to be transferred to another flight upon financial compensation. Let’s assume that the compensation for a denied boarding is set at €300. We can also easily work out the probability of an occurrence of denied boarding for that seat. Indeed, considering one particular seat that has been sold twice, either we have a noshow (and the airline cashes the additional revenue of €400), or a denied boarding (and the airline has to pay the €300 compensation). Hence the probability of denied boarding is the complementary of the probability of noshow, that is 1  0.9 = 0.1 for that seat.
The overall additional revenue of overbooking for the additional seat is then calculated as the sum of the revenues in the case of a noshow multiplied by the probability that there is effectively a noshow, plus the losses (or here revenues) when there is a denied boarding multiplied by the probability of facing a denied boarding.
In our simulation, the calculus for the first additional seat clearly shows that the airline would improve its revenue by €370 if overbooking one seat.
Probability of a ‘noshow’  0.9 
If ‘Noshow’, Expected revenue of that additional seat  €400 
Probability of denied boarding  1  0.9 = 0.1 
If Denied boarding, compensation coast  €300 
Airline revenue/loss of that seat in case of denied boarding  400  300 = €100 
Overall additional revenue from overbooking one seat  0.9x400 + 0.1x100 = €370 
But how many seats should the airline overbook?
Let us repeat that computation seat by seat and do the calculation for the 20th seat we may overbook on the same plane, on the same flight, on the same day.
Things look a little different:

The probability of having a 20th noshow for this flight is of course much smaller than the probability of having one noshow: it is now 0.5.

At the same time, the probability of selling an additional seat, the 20th additional seat for that flight, is also lower. So, even if the price of that ticket few days before is the same as for the first seat to overbook, the additional revenue expected from the sale of this additional seat (the 420th seat for that flight) is much lower: it is €300 for that seat.

Of course, the financial compensation for a denied boarding must be the same for all the additional seats on which the airline practiced overbooking: it is still set at €300.
Using the same calculation table as above, could you work out the airline additional revenue from overbooking this 20th additional seat? (you will have the opportunity to check your result below)
A seat by seat computation
As with many calculationof this type, searching for the equilibrium between revenues and losses may be performed in an iterative way. Let us skip ahead and see what happens if the airline considers now proposing 30 additional seats for that flight.
The data for the 30th additional seat are as follows:

Probability of having a 30th noshow for this flight: 0.25

Expected revenue for that seat in case of noshow: €200

Financial compensation in case of denied boarding for that seat: €300
What is the overall additional revenue from overbooking a 30th seat?
Got it? Normally you should have the following results:
Additional seat  Overall additional revenue 

401st seat (1st additional)  €370 
420th seat (20th additional)  €150 
430th seat (30th additional)  €25 
If you did not get it right, check the solution in the DOWNLOADS section at the bottom of this page.
What do these results show?
First, we can see that even if noshows are unsure and the compensation cost is equal to the expected revenue of a 20th additional seat, it is still worthwhile for the company to overbook at least 20 seats. On the contrary, selling 30 seats more than the total capacity of the plane is detrimental to the airline, as it leads to a €25 loss.
So the airline should try to sell more than 20 additional seats but fewer than 30 seats on his flight. In this simulated case, the overbooking rate should be somewhere between 20/400 and 30/400, that is between 5% and 7.5%.
Note also that the compensation cost is set to €300, but in reality this depends on the length of flight, the delay incurred for the denied boarding passenger and other factors. Here, we do not take into account the fuzzy cost to the airline reputation that can be very important for some airlines.
To overbook or not to overbook?
These numbers are to be compared with the choice of not overbooking, leading, for sure, to no additional revenue and probably to some empty seats.
In many situations, not overbooking would not only be leaving money on the table, but also would leave people willing to travel (even at a high price) on the ground, while some seats remain empty!
© By ENAC  Christophe Bontemps CC BYNCSA 3.0