Do you get more seating room on “free choice seating” airlines (like Southwest) or “assigned seating” airlines (like United)?

[In this module, we will use an agent-based computational model to determine if you get more seating room flying with a free-choice-seating “passenger chooses their seat” airline like Southwest, or a traditional assigned-seating airline like United airlines.  Also, what is the optimal boarding position for a “passenger chooses” flight?]

Background

Recently I flew Southwest airlines (Editorial note: this post was written in 2014).  I’ve flown Southwest four times in the last 12 months, and each time, for some reason, I end up with a boarding position at the very end of the boarding for the plane.  I’ve never figured out… what do all the other passengers know that I don’t? Clearly airline loyalty or frequency of flying it are making no difference. I really don’t like flying Southwest airlines, just for this reason alone.

long_line

(shown: me, yet again, standing lonely, last in the long line behind all these people waiting to board)

Anyway, on this flight, I got boarding position C27.  There are 143 seats on the plane I was flying, and the A and B boarding positions total to 2*60=120.  You do the math…

Distressed that I was going to be the very last person to board a full flight (and what this would imply for the amount of seating and luggage  storage room I would have available), I decided to pay the extra $40 to board in the first A line.  Along with the other passengers at the front of the lineup, I boarded and happily took an aisle seat near the back of the plane; you are more likely to survive a plane crash at the back of the plane… this sometimes weighs upon my mind when travelling by air (Editorial note: now, in 2020, in the maturity of my years, I have come to realise that if the plane goes down we’re all going to die, so I no longer have a preference for sitting in the back of the plane, and just sit where ever, and try to enjoy the flight).

As the plane filled up, aisle seats went first, then the window seats, and finally only the dreaded “middle seats” were left:

 (shown: the dreaded middle seats)

I am not small, but I do fit into an airline seat with no trouble, but not much room to spare on the sides… to put things in perspective, I’m larger than the average supermodel, but smaller than the average Hell’s Angel (Editorial note: today, in 2020, I may be about the same size as the average Hell’s Angel… I haven’t done an exact quantitative comparison lately, but I do weigh more than I did in 2014).

hells_angel

(shown: not me)

Unfortunately for me, I ended up with a really-really-not-small person choosing the middle seat, who in turn was also sitting next to an also-not-small person in the window seat. This  led me to wonder; is your probability of sitting next to a not-small person less (or more) on a Southwest flight where you can chose your own seat, compared to an assigned seating flight? (I had time to wonder about this, because I had accidentally put the book I wanted to read in the overhead compartment, so I’d have to find something else to do until we reached cruising altitude).

This seemed worth studying as a matter of public health (more seating room means more room to move, and less chance of developing problems like deep vein thrombosis, and/or stress induced social behavioural disorders).

stress_and_crowding

Sources of Data

Sources of data used in this study included data collected while I was wandering around the cabin, on the pretense that I was on my way to the restroom, but really I was looking at the other passengers, their girth, and how much room they seemed to have.

Some people noticed my too-prolonged attention, and I got some dirty looks, upon which I had to hurry on a couple more rows before I could begin data collection again.

Based on my observations, about 70% of the passengers “fit nicely” into their seats in the sense that it looked like they had at least an inch or two of room on either side before the armrests.  Approximately 20% of passengers looked “bigger than me”, as evidenced by spillage over the armrests to invade the space of the seat next to them.  It turns out, my BMI (at the time) was in ~ the 75th percentile for women, so my observations on the plane about my relative rank in the population were approximately correct.

A small fraction of the passengers were kids, who appeared to take up approximately half the seat.  Lucky kids.

I returned to my seat, and measured the width of my area.  I had no tape measure available, but I was able to determine that my available seating width+ 50% of the armrests on either side, was approximately equal to the width of my 15″x13″ aluminum case 2013 Macbook pro, plus the width of the softcover Jared Diamond book “Collapse”

 

    macbook
+

collapse

=


(by this time we had reached cruising altitude, and I was able to retrieve my book, but I was so interested in this analysis by that point that I never did end up reading it until several days later)

It later turned out that Southwest says its seats are 17″ across for the seat itself, plus another couple of inches for armrest room.  This pretty much matches my observation. Data collection complete, I was ready to do my agent-based computer simulations!

Simulations: seated girth and calculation of seat roominess

The agent-based simulations began by taking a cadre of 144 passengers (ie; a full plane), and randomly choosing “seat-width-girth” for them from the Normal distribution.  I truncated the Normal distribution at 10″, and choose the parameters such that roughly 20% of the distribution had girth greater than 19″.  Here is the distribution of girths that I used (mean=17.5 and sigma=2… this plot was produced by the R script girth_distribution.R):

girth_distribution

Now I needed a metric to measure “how much room” people have once they are in their seats. I make some assumptions here:

  • People are centered in their seats
  • They are infinitely compressible to the side (thus two not-small people sitting side by side are touching, and I assume that the force they exert on each other is a linear function of the amount of “spillover” they would have had if they were sitting in a seat alone without someone beside them).
  • I also assume that not-small people in the aisle and window seats can have spillover to the side of their seat not adjacent to the middle seat, and this spillover room counts as extra room for them.

So, based on the above assumptions, here are the schematic for the amount of room that each person gets (swidth is the width of the seat)

girth_seating_plan
The total net room for the person in each set is the sum of the net room they have on each half of their seat.  Zero net room means that they are just touching the people beside them, and negative net room means they are being squished by the person beside them.

Simulations: average seat roominess with assigned seating

Simulating the passengers in a plane with assigned seating is easy.  You just randomly assign the passengers a girth from the girth distribution above, then randomly assign them a seat.  Then calculate the amount of room each passenger has. I do this in the R script airline_assigned_seating.R. To run the script in R, download it to your working directory, in the R interactive window type

setwd("")
source("airline_assigned_seating.R")

The script produces the following plot for the distribution of room that window/aisle and middle seats have:

net_room_assigned
No wonder everyone hates the middle seats!  The aisle and window seats are much more likely to be very roomy, and less likely to be very squished.

The script tells me that on average the aisle and window seats have 1.6 inches of spare room, with standard deviation 1.4 inches.  The middle seats, on the other hand, have on average 1.5 inches of spare room, with standard deviation 1.2 inches.

In both aisle/window and middle seats there is an approximate 11% probability you will be squished, but in a middle seat you are more likely to be excessively squished. The results of the simulation of course depend on the assumptions we made about the distribution of the passenger girths.  If you are seated in a plane with a high proportion of not-small people, the probability of having negative room goes up.

Simulations: average seat roominess with passengers allowed to choose their seating

This simulation is a bit more complicated.  We will use an agent-based model, with each passenger being an agent that boards the plane one after the other. Essentially we will let the people board who board first take the aisle and window seats, and assume that they just pick their seats randomly.  After all, at that point they don’t have to worry about how much room they have!

I assume that the airplane has 48 rows of three seats each, which describes the majority of the planes in the Southwest fleet:

southwest_fleet
After the aisle and window passengers board, we will let the middle seat passengers board one at a time.  Each of them gets to pick the seat that optimizes the net amount of room that they have.

I do this simulation in the R script airline_free_choice_seating.R Here are the results, showing the distribution of the room per passenger for people in the aisle and window seats, and people in the middle seats:

free_seating_results

Look at this plot:  the distribution of room for aisle/window and middle seats exactly matches that of the assigned seating plan!
 Why do you think that is?

So, is there any advantage, when it comes to seating room, of flying a free-choice seating airline? Here is the amount of room per person vs their boarding position:

room_vs_boarding_number_all
Position B27 is the boarding position at which there are no more aisle and window seats left, and passenger B27 has to take a middle seat.  Notice that as the middle seats fill up, there is less and less room available per passenger. In fact, the people in the B27 position have, on average, 3 inches of spare room;  compare this to the people in the aisle and window seats, who have on average only 1.6 inches of spare room!  Unfortunately for the people in the last boarding position, they have, on average, zero spare room. It turns out the average amount of spare space you get in the middle seats is at least that of the aisle and window seats as long as you are in boarding positions B27 to B57.

Everyone after that, B58, B59 B60 and all of the C boarding positions, has on average less spare space than people in the aisle and window seats.

Here is the amount of spare room per person vs boarding number from the simulations, for the people in the upper and lower 10th percentile in girth:

room_vs_boarding_number

An interesting thing about this plot is that even if you are in the upper 10th percentile in girth, the B27 position assures you are likely to have at least some room.

Conclusions of the study

To summarise, people in the aisle and window seats on free-choice seating airlines on average will get just as much space as they would on an assigned seating airline.  The best boarding position to have on free-choice seating airlines is 2/3 of the way down the lineup, when now only middle seats are left, and you can choose the seat between the two little 80 lb grannies.  But things rapidly degrade from there, and for people with boarding positions in the C lineup, they are much more likely to have a miserable flight.

Designing an optimal seating plan

We can use our agent based model to examine other seating plans.  For instance, what it we sort passengers by size, and let the smallest passengers board first, and not-smallest people board last.  Once on the plane, I assume that the aisle seats get filled first, then the windows, then middle seats, and that all the people in the agent based model choose the seats that optimize the space available to them.

The R script airline_optimal_packing.R simulates this, and produces the following histogram showing the room available to the passengers:

optimal_packing_small_first
Notice that the distribution of the amount of room that all people get is a bit worse that what they would have gotten if they had flown an assigned seating airline. Here is the plot of room vs girth (boarding order) of the passengers (remember, the smallest passengers board first, and the not-smallest passengers board last).

small_people_board_first
How about if we board the plane not-smallest first, and smallest last? Just change the line in the airline_optimal_packing.R script that says

vgirth = sort(vgirth,decreasing=F)

to

vgirth = sort(vgirth,decreasing=T)

and rerun the script in R. This produces the following distribution of amount of room for the passengers:

optimal_packing_not_small_first
The distribution of the amount of room vs boarding position is:

not_small_people_board_first
Here is a summary of all the results for the average room per passenger (the number in brackets is the probability of having negative room) Assigned seating (or free-choice seating):

  • Aisle/window seats: 1.63 inches (11%)
  • Middle seats: 1.50 inches (11%)
  • All seats: 1.59 inches (11%)

Small people board first:

  • Aisle/window seats: 1.77 inches (4%)
  • Middle seats: 0.96 inches (15%)
  • All seats: 1.50 inches (8%)

Not-small people board first:

  • Aisle/window seats: 1.42 inches (0.007%)
  • Middle seats: 2.04 inches (0%)
  • All seats: 1.63 inches (0.005%)

When not-small people board first, the average amount of room that all passengers get is the same as the average amount of room that aisle/window people get in assigned seating flights!  Plus there is an extremely low probability of anyone getting negative room!

Here is a paper I have written summarizing these studies.  I worked on the paper over the course of a year or so, while waiting in airport lounges.  Sadly, despite several attempts, it has proven difficult to get this paper published.  Editors of transportation journals appear to think it is a joke.  Besides which, the field of transportation studies is entirely centered on optimisation of transport systems for revenue… the field does not care about passenger comfort.

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