Dr. Eitan Bachmat uses abstract math to reduce airplane boarding time (Photo: Bigphoto.com).Anyone who travels frequently by air knows the routine.

At the gate inside the airport, you stand in the midst of an impatient crowd waiting to board the plane. When boarding time comes, the flight attendants methodically announce which groups of passengers are allowed to board, starting with the back rows and moving forward in five-row increments. Finally, you are allowed to get on – and are immediately stuck behind a slow-moving family as they make their way to their seats in the back.

While it’s all being done in the name of efficiency, you can’t help thinking as you finally sink into your seat that for a procedure that is supposed to make things move more smoothly, it seems to make boarding take an unnecessarily long time.

Israeli researcher Dr. Eitan Bachmat agrees with you.

Bachmat and his team of researchers from Ben-Gurion University of the Negev have proved mathematically what computer simulation models by airline companies – and the gut feelings of passengers – have already demonstrated. Back-to-front row boarding is not the best way to fill airplanes with passengers.

According to their work, the most efficient plan, which is commonly used on US budget airlines like Southwest and EasyJet, is letting the quickest customers get on first and choose their own seats – with unassigned seating.

But even when seats are assigned, “the best thing for the airlines to do in terms of telling passengers to board would be to do nothing,” says Bachmat of the faculty of computer science. Because boarders are frequently held up while those ahead of them block the aisles, the row-by-row system makes the process longer, not shorter.

“Back-to-front boarding is bad because it is designed for cardboard-thin passengers, or for the spacious surroundings of the first-class compartment,” he told ISRAEL21c.

It sounds like a common-sense conclusion, but it’s the result of complicated mathematical equations related to Einstein’s theory of relativity – something one wouldn’t normally associate with the problem of boarding a plane.

It all began with Bachmat’s unusual academic focus on systems research. Formerly a mathematician, he now applies math skills to practical problems. His work tends to be very interdisciplinary – “it’s not quite physics and not quite industrial engineering.”

He describes his academic detour with a generous supply of humor.

“I have come to be a systems researcher because I was a terrible mathematician,” he says. “To summarize my relations with mathematics, I love mathematics, but it does not love me back. Given this situation I had to leave this relationship at some point. As revenge, I am exploiting mathematics in my new role as a systems researcher.”

His academic focus has been on improving the way that disk drives in computers operate – the most efficient way of processing input and output requests. Bachmat works closely with the computer storage industry.

He was describing his computer work to a colleague, who mentioned that the math he was using was similar to the equations for airplane boarding. This inspired the research, which also included Bachmat’s colleagues Dr. Daniel Berend and Dr. Luba Sapir from BGU, and Dr. Steven Skiena from the New York State University at Stony Brook.

“It somehow dawned on us that we could use the same geometry that appears in relativity theory. There is a kind of geometry called Lorentzian or space-time geometry which models relativity theory. Nobody used it for anything else outside physics, It just kind of stood there for 100 years. Well, it turned out that this airplane boarding problem and my disk drive work are modeled by the same mathematics, the same type of geometry,” he said.

Bachmat and his team decided to use the model to calculate the most efficient way to board airplanes by determining how long boarding would take with different systems. They posed the problem in terms of permutations: the different orders in which passengers might board, which determines your chances of getting blocked by someone seated closer to the door.

The team examined how the physical relation between events in space-time corresponds to the relation of passengers blocking each other along the airplane aisle. Using the complex Lorentzian or space-time geometry, the researchers were able to analyze and compare various boarding policies. The results showed that the success of a given policy depended heavily on parameters such distance between rows, number of passengers per row, amount of personal luggage, and, not surprisingly, the average passenger waistline.

For years, the airline companies have examined this issue using simulations that imitate what goes on in the real boarding process. The existing simulation studies stem from the desire of the airlines to cut boarding times, because they often determine how quickly a plane can be turned around for another flight. Since 1970, boarding times have been steadily increasing, which not only costs the airlines money, but affects customer decisions regarding which airline to fly.

To try to improve the efficiency of the boarding process, the companies have used computer simulations in which each passenger takes their seat after performing activities such as helping family members and stowing carry-on luggage.

After comparing his results with the airline results, Bachmat admits that he was “utterly surprised” to find that his mathematical calculations accurately reflected the results of the airline’s simulations.

“We thought our study was a nice intellectual exercise, but we didn’t think it would model something realistically,” said Bachmat.

Their mathematical model predicted boarding times that were remarkably similar to the simulations – even though the latter include complicating factors such as slow-moving passengers, full overhead bins, and people sitting in the wrong place.

He says that according to his models, the fastest way of boarding a plane would be open seating.

When seats are assigned, “letting people join the line randomly would be the best idea. If there were to be any kind of controlled boarding to try to make it quicker, the only thing that might help would be getting people who have seats closest to the window to board first. But there is no need to play with the rows. It doesn’t matter which rows get on first.”

What Bachmat says is exciting about the exercise – with all due respect to travelers – is not the practical application, but the implications as a scientific exercise.

“Lorentzian geometry was invented for the sole purpose of describing relativity theory,” says Bachmat. “This seems to be the first application of this theory outside physics.”

He says it is a great demonstration of how looking at an everyday procedure can be explained using “profound mathematical and physical theories. This example has become the way that I explain the theory of relativity to many people.”

Bachmat knows a lot about plane travel – he has been shuttling between Israel and North America much of his life. Born in Israel, he spent several years in Canada as a child. After earning his bachelor’s and master’s degrees in mathematics at Hebrew University, he received his Ph.D. in math at MIT, and did post-doctoral work at Brown University. He made the switch from pure math to computer science while a visiting professor at BGU, where he later joined the faculty.

While he has derived great satisfaction from the plane boarding research, he knows it will be a long time before it makes any real impact on the airline industry – human behavior and psychology would have to be factored into his mathematical equations before making any substantive changes.

In the meantime, it has made his own travel more interesting.

“Every time I fly, I watch people to see if my theory works,” he says. “It’s fun – and waiting for the plane to board and take off is so boring, it’s good to have something to do.”

# Israeli researchers fly by the numbers

Dr. Eitan Bachmat uses abstract math to reduce airplane boarding time (Photo: Bigphoto.com).Anyone who travels frequently by air knows the routine. At the gate inside the airport, you stand in the midst of an impatient crowd waiting to board the …

By Allison Kaplan Sommer December 18, 2005, No Comments