May 2006

# The Mathematics of Aircraft Boarding

What is the most efficient way to board a passenger aircraft? This question occurred to me as I shuffled down the aisle of a Boeing 777 at San Francisco Airport on my way to give a lecture in Italy at the start of this month. I felt sure that some mathematicians must have studied this problem, and sure enough, a quick Web search when I got to London revealed the recent paper Analysis of Airplane Boarding Times, by Eiten Bachmat, Daniel Berend, Luba Sapir, Steven Skiena, and Natan Stolyarov. Besides presenting the results of their own research, these co-authors describe the conclusions of several earlier studies.

The standard procedure adopted by all the airlines I regularly fly with is to group passengers and board the groups one at a time, starting with First Class, then Business, then finally the Coach Class in groups starting from the back of the plane. The inefficiencies in this procedure are obvious to anyone who flies with any regularity. For airplanes where the boarding door is at the front, the bulk of the passengers have to wait until the (often less agile) first and business class passengers have stashed away their bags and taken their seats, while for airplanes that board further down the plane, with first class to the left (front) of the boarding door and economy to the right (rear), the opportunity is lost of having the front and rear portions of the plane fill up at the same time.

It seems pretty clear that, from a purely mathematical point of view, the most efficient boarding process for an airplane that boards from the front would be to linearly order all passengers, starting with those at the back of the plane, ordered from outside (window seat) inwards (aisle seat last), and then working forward. But managing such a process would be a nightmare, and the confusions inevitably caused by trying to line the passengers up in this fashion would surely make matters even worse than they already are, to say nothing of making passengers feel even more like cattle being transported to market than is already the case. (British Airways staff used to make a fetish of treating passengers badly in that regard; things may have improved, but after a series of bad experiences with bossy BA staff back in the 80s, for the past twenty-five years I have avoided flying BA as much as possible, and taken most of my flying business to United. Others may have different experiences.)

Moreover, with First and Business Class passengers expecting (with considerable justification) preferential treatment in return for paying several times more for their ticket, and with high-mileage gold card flyers like myself also expecting a "special deal" in exchange for our many hours spent in the air, it would be business suicide for any airline to ignore the many human aspects of flying.

My suspicion as I stood patiently in line on United 930 at the gate in San Francisco, watching as those ahead of me sorted themselves (and their baggage) out, was that a purely random boarding process was probably more efficient than the current systems, since boarding by class or by airplane section leads to localized congestion. And indeed, that is one of the conclusions reached by Bachmat at al in their recent paper. It also seems believable, at least to my mathematical mind, that allowing as much randomness as possible into a system designed to give some preference to First Class, gold card flyers, families-flying-with-small-children, etc. would also speed up the boarding process. But the mathematics does become pretty complicated, once you realize the different variables that are involved. Besides the position of the door (or doors), other factors that affect the boarding process are the number and width of the aisles, the number of seats per row, and the separation between the rows. Each of these parameters is built in to the mathematical model that Bachmat and his colleagues developed.

In order to capture the various complexities of the problem, the researchers use some highly sophisticated mathematics to construct a computer model that can simulate different boarding strategies for different aircraft layouts. How sophisticated? How about Lorenzian geometry, developed to handle relativity theory?

Why that geometry? you may ask. Well, the boarding process is a space-time universe, and the total time it takes to board the aircraft is the length of the longest path in that universe. Obvious once it is pointed out, isn't it? A great example of how mathematics developed for one purpose can find a good use in an entirely different domain.

The conclusion? Summarizing runs of their computer model for various aircraft configuration, Bachmat et al observe that if the airlines were able to adopt a fairly rigid system, then outside-in boarding is much better than back-to-front, but that, overall, given passenger resistance to being strongly regimented, and recognizing the airlines' desire to continue their different passenger-pleasing preference policies, the most efficient method seems to be - are you listening United? - a policy that combines the various customer-satisfaction policies with a hefty dose of outside-in boarding and just a touch of back-to-front; otherwise as much randomness as possible.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs) was published recently by Thunder's Mouth Press.