No flights when you want them? Throw some math at the problem!

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    a portion of the equations from "Airline Timetable Development and Fleet Assignment Incorporating Passenger Choice" by Keji Wei, Vikrant Vaze, Alexandre Jacquillat Courtesy—

  • FILE- In this Nov. 27, 2019, file photo an American Airlines flight arrives at O'Hare airport in Chicago. Summer will be here soon, so it's a good time to start planning that vacation trip. Experts have plenty of tips for saving money on flights, lodging and other expenses. (AP Photo/Nam Y. Huh, File) Nam Y. Huh

Monitor staff
Published: 2/3/2020 6:36:47 PM

Recently I flew to D.C. to visit my family and had to get up at 4:30 a.m. to make my flight in Manchester because the next flight was way too late. Between yawns, I muttered deprecations about American Airlines all day.

Vikrant Vaze, an assistant professor of engineering at Dartmouth College, thinks I should be more sympathetic. Scheduling airline flights, he said, is actually quite hard.

“It’s a very challenging problem. It’s the most difficult of the strategic problems that the airlines face,” he said.

Vaze should know. Along with operations research analyst Keji Wei at Dartmouth’s engineering school and MIT professor Alexandre Jacquillat, he has been applying mathematics and computer algorithms to find a better way for airlines to do this.

Spoiler alert: They haven’t found it yet, but they think they’re getting there. Their study, using 2016 data from Alaska Airlines, was published in Transportation Science, a leading journal in the field of transportation analysis, suggests that their approach could help. “Some of the most critical decision-making steps taken by airlines across the world rely on tools that do not fully incorporate passengers’ preferences and the dynamics of flight scheduling, resulting in missed profits and unsatisfied passengers,” they say.

Vaze and colleagues didn’t take this issue on because they were irritated by the difficulty of getting to academic conferences. They tackled it because it’s an interesting example of an optimization problem – trying to get the best result when dealing with lots of often-conflicting variables.

I talked to Vaze because it struck me as an interesting and unexpected example of using abstract mathematics to create models that deal with not-at-all-abstract problems.

This process takes two steps, he said – one with math and one with computers.

“We want to take this real-world problem and convert it into a bunch of equations somehow that will capture all of these things. That is the mathematical formulation,” Vaze said.

What sort of math did they use? Integer linear optimization, of course! (A reminder: “integer” is math-speak for “whole number.”)

“They are integer in the sense that some of the values that we want to get out of this need to be non-fractions. If you have to decide between scheduling a Boeing 737 or Airbus 320, the answer cannot be that we will have half of this and half of that,” Vaze said.

This part boggles most folks’ minds because the resulting equations look like nothing we’ve ever seen, full of Greek letters and weird symbols. (I have an entire book about the history of mathematical symbols – it’s more than 500 pages long.) Unless you’ve spent the time to learn what everything means, and most people haven’t, it’s intimidating gobbledygook.

Yet, said Vaze, “the writing of the mathematical formulation is, I would argue, not terribly complicated. People are pretty good at it these days. What was really problematic was: What do we do with the formulation now?”

What they did was use computer software to do calculations with the formulas and then translate the result back into reality.

This part, Vaze said, involves “some really cool software solvers – optimization solvers. There are powerful tools available both commercially and open-source.”

The art and science of the process is deciding which formulations to pursue, which real-world issues to incorporate, which solvers to employ and how often to reiterate. To most of us this can seem like mumbo-jumbo, which is why you’ll often hear people hiding their unsupported beliefs behind the excuse “you can prove anything with a model.”

Yet business owners, the most practical of people, use mathematical models all the time to take zillion-dollar decisions. We ignore them at our peril.

“This has been around, you can argue, for the past 50 years or so, but every passing day there are new approaches. Beyond airlines there are a ton of applications that are used,” said Vaze.

If I find that there’s a nice convenient flight next time I go home, I know who to thank. Mathematics, that’s who.

Need more geek in your week?

If you want more geek in your week, subscribe to David Brooks’s free weekly newsletter at You can also listen to him talk about his stories on the GraniteGeek podcast,, or talk with Chris Ryan on WKXL radio at

David Brooks bio photo

David Brooks is a reporter and the writer of the sci/tech column Granite Geek and blog, as well as moderator of Science Cafe Concord events. After obtaining a bachelor’s degree in mathematics he became a newspaperman, working in Virginia and Tennessee before spending 28 years at the Nashua Telegraph . He joined the Monitor in 2015.

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