This is one of those problems, that while it sounds deceptively simple to solve, and is, given a perfect world, becomes horribly unmanageable to solve once you factor in real life. Having done a significant amount of research on this problem for the Mathematical Contest in Modeling last year, there are many more things that need to be considered then would first appear.
Some of the things that need to be considered with this problem:
To board in any sort of weird order, you have to spend time beforehand ordering people into the groups that you want them to be in. I'll let you think about this for a second. You're trying to get about 100-300 people, who are tired, irritated, with scawling babies, and rather disinterested to organize into some sort of coherent and well planned out structure... WITHOUT using a loudspeaker. Those of you that have been in marching band can contest to how difficult this is.
Most airplanes only have a single aisle, and generally, it is narrow enough that only one person can stand in it at a time. If you try to board one row at a time, maybe, if you're lucky, two people will be taking their seat at a time, while everyone else in the plane queues up behind them. Add in the fact that if the window seat arrives after the aisle and middle seats, they both generally have to get out of their seats to let the window sit down, creating even more delays.
Luggage. If you've ever flown, you've probably seen people trying to cram all of their bags into the overhead bins. If your bin is full, the most natural thing to do is to go and look for one that isn't. Once again though, due to the narrowness of the aisles, this means that you will be holding up many more people while you stuff your things above everyone's heads.
Families, especially those with multiple small children, really don't want to be broken into individual members just to be seated. They all have seats together, generally, so they want to get to those seats together, the boarding order of the airline be damned. And really, can you blame them? What 7 year old wants to have to wait in line, alone, with a bunch of really tall adults, and then have to find a chair in an unfamiliar environment?
First class can generally be ignored in the problem. Because the number of first class fliers compared to the number of coach fliers is very small, they have minimal issues boarding and the time it takes to board them is likewise minimal.
The end results that we came up with is that on average, letting people board randomly, in whatever order they please, beat out every other model that we simulated. On average. Although we didn't have the time and/or resources to come up and run a model, it would be very interesting to model the way Southwest boards their flights, where they have no preassigned seating. But because that factors in the decision making capabilities of people, it would be a far more difficult task to model then simply what order they line up in. That being said though, Southwest does generally have boarding times that are better then other major airlines.
Anyways, stuff to think about.
