The Optimal Way to Board Plane Passengers

"Zone 4 is now boarding."

With that announcement, one lines up, walks past the ticket-taker, down the ramp, and eventually enters the plane to witness a few people valiantly crunching their ungainly carry-on bags into the overhead compartments and most others impatiently waiting in the aisles to do the same. Those already in the aisle seats are casting a wary eye on the next backpack threatening to attack them when its owner unthinkingly pivots in the aisle.

Is the standard boarding method really the best way to load an airplane? There are variations, but almost all methods currently in use board by seat or zone number so that those whose seats are near the back of the plane board first and hence don't block passengers boarding after them. That's the theory, at least.

Alas, since the people in a single row can't all stow their bags simultaneously, there are always people blocking the aisle as they try to stow their bags. Moreover, those sitting in the aisle seats are not only dodging passing luggage, but must rise and let later-arriving passengers sitting in the window and middle seats enter their row. It's a time-consuming, spirit-sapping mess.

Better Boarding Procedures: Separating Passengers

Enter Jason Steffen, an astrophysicist at Fermi Lab near Chicago. Using computer simulations, he has examined different plane boarding strategies with an eye toward their speed and efficiency.

For simplicity, his model assumes a plane with 120 passengers seated in 40 rows, each with a central aisle having three seats to the left and three seats to the right of it. It also assumes that each of the 120 passengers has an assigned seat number and carry-on baggage and that they move forward if and only if no one is directly in front of them.

So, given these reasonable assumptions, what's the best way to board? It seems intuitive that the worst way is to load those passengers seated in the front of the plane first and then those a bit further back and so on. And this is, in fact, the worst way to board passengers.

So, it might seem almost as intuitive that the standard way -- loading the back rows first and then gradually rows nearer the front -- should be among the best ways to board, but Steffen's simulations indicate that this is the second-slowest way. Even random boarding is faster.

The Algorithm

After many simulations allowing for different sets of passenger quirks and luggage-stowing times, it turns out that the best method (one of several more or less equivalent methods) calls for passengers in even-numbered window seats near the back of the plane to board first.

Passengers hefting their carry-ons into the overhead compartments are less likely to get in each other's way if there's an empty row between them. Moreover, they can step into the empty row if someone seated further back needs to pass.

After these passengers have boarded, passengers in even-numbered window seats in the middle of the plane board, and they are followed by those in even-numbered window seats near the front of the plane. Next, the same procedure is followed for those in the even-numbered middle seats and then for those in the even-numbered aisle seats.

Finally, after the even-numbered passengers have boarded, the same procedure (window, middle, aisle from back to front) is followed for passengers in the odd-numbered seats. These passengers may not always have an empty row to step into, but they will still be separated from entering passengers by a row of already seated even-numbered passengers.

It appears that the reason the protocol is faster is that it allows multiple passengers to simultaneously stow their baggage, the most time-consuming component of the boarding process.

This and other similar schemes Steffen discusses may seem too complicated for passengers to master, but passengers needn't remember the seating order algorithm. They can each be assigned a zone consistent with it and enter by zones, as they presently do.

The outcome is fairly robust in the sense that it's relatively insensitive to deviations from it, say, because of couples or families being seated together.

Airlines should, of course, supplement these theoretical conclusions with empirical investigations.

The simulations suggest that using something like the above protocol would reduce boarding time to 1/6th of that required by the standard procedure! Multiply the average number of passengers per plane trip by the approximate number of trips by the number of minutes saved per boarding, and the number of man hours saved would be more than considerable.

There must be faster, more rational ways to go through security, as well, but, alas, "Zone 4 is now boarding."

John Allen Paulos, a professor of mathematics at Temple University, is the author of the best-sellers "Innumeracy" and "A Mathematician Reads the Newspaper," as well as of the just-released "Irreligion: A Mathematician Explains Why The Arguments for God Just Don't Add Up." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.

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