OK, math whizzes. This is a test. Only one question on it. And while that question doesn’t require calculations about trains speeding toward each other in the night, it does involve another popular conveyance and a situation with which you probably have experience:
Airlines are free to seat passengers waiting to board an aircraft in any order. But from the airline’s point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips a plane can make in a day. Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85–210), midsize (210–330), and large (450–800).
Pencils ready? You may begin.
That’s what three UPS students did for five days in mid-February, when they participated in the 2007 Mathematical Contest in Modeling; 949 teams representing institutions from 12 countries participated. When it was over, judges chose 14 Outstanding Winners, and the UPS crew of Sara Beck ’08, Spencer K’Burg ’07, and Alex Twist ’07 was one of them. Among other winning schools: Duke, Harvard, MIT, the University of Washington, and two universities from China. The 14 “outstanding” solution papers will be published in The UMAP Journal.
But back to the problem of the best way to get people on and off an airplane. Got you stumped? Here’s the solution our young mathematicians came up with:
We performed a careful analysis of boarding techniques currently practiced in the airline industry, as well as a novel technique not currently in use. The boarding techniques we examined included several variations of a back-to-front, outside-in (passengers board window seats first, aisle seats last), random assignment, and a new process we called “roller coaster” for its resemblance to the process of boarding similarly named theme-park rides.
We designed a simulation that replicates the behavior of passengers boarding airplanes of different sizes. To accurately model the different sizes and shapes of the interior of the planes, we used an Airbus 320 to represent small aircraft, a Boeing 747 to represent midsize aircraft, and an alternate configuration of the Boeing 747 to represent a large aircraft.
We physically modeled and observed common interactions to accurately reflect boarding time. Variables in our model included walking time, stowage time, and seating time. Boarding delays were measured as the sum of these variables. We ran 500 simulations for various combinations of airplane sizes and boarding plans.
In our simulations, the roller coaster boarding method, in which passengers line up in groups before they board the plane, performed the best. We estimated that with this method, airlines can board smaller planes (162 passengers) in approximately ----3 1/2 minutes, 67 percent faster than the next-best option; midsize (288 passengers) planes in approximately 5 3/4 minutes, 37 percent faster than the next-best option; and large planes (550 passengers) in approximately 10 1/2 minutes, 35 percent faster than the next-best option.
These time estimates are only the amount of time it takes to move passengers onto the plane; they do not reflect the time it takes to order passengers into roller coaster-type lines before boarding. Because putting passengers into an order takes time, the next step would be to investigate what resources would be required to implement this plan (e.g., metal bars or chains similar to those used at amusement parks to guide lines of people). If the ordering process takes a significant amount of time, it may be undesirable to ask passengers to arrive at the gate earlier, or to wait until everyone is in the roller coaster line before boarding.
As to the deboarding process, we considered the common practice on today’s airplanes, which is to allow passengers to deboard at their convenience. Typically passengers are very anxious to get off the plane. The very moment (or sometimes even slightly before) passengers are given the OK from the pilot and flight crew, they are already unbuckled and ready to start deboarding. It would be a poor customer relations move to try to tell passengers to sit still and wait for other passengers to get off of the plane first. Further, previous studies have shown that it is generally the boarding process that acts as the primary constraint in decreasing plane turnaround time. Airlines must be seen as supportive of passenger comfort. From both a customer service perspective and a practicality perspective, our best option regarding deboarding methods is to allow the passengers to exit the airplane as they please.