# Evaluating Classical Airplane Boarding Methods for Passenger Health during Normal Times
## 5 Simulation Results
### Numerical Results for Middle Seat Risk  
The middle seat risk indicator measures the overall risk experienced by the passengers with middle seats as a result of potentially contagious passengers proceeding down the aisle to their assigned sets. The values obtained through simulations in all the luggage scenarios are presented in Table 4.  
While boarding aisle seat passengers last helped the reverse pyramid and WilMA with aisle seat risk, it worsened their performance with middle seat risk. As indicated in Table 4, the back-to-front-row method results in the smallest values of middle seat risk followed by back-to-front-by-5-groups, back-to-front-by-4-groups, and back-to-front-by-3-groups respectively. The variations in the value of this risk for the back-to-front-by-row method resulted in 60.38% more middle seat risk with the high-luggage scenario S1 than with the no-luggage scenario S7.  
The reverse pyramid method performed better than WilMA again, this time due to its middle seat passengers being divided into two boarding groups as opposed to the one group of WilMA. For this metric, WilMA's performance was the second worst for the no-luggage scenario S7 and the third worst for the other luggage scenarios, swapping places in the open seating method, which was the third and second worst method for those scenarios, respectively.  
The highest values for the middle seat risk results from the random with assigned seats method. For this method, we observed that its middle seat risk is lower for each luggage scenario than its corresponding aisle seat risks. We explain why as follows. Within a particular row and side of the airplane, there is an equal chance of any particular passenger (aisle, middle, or window) boarding first, second, or third. However, the different types of seat interferences influence the risk to aisle seat passengers differently than middle  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Boarding Method**} & \multicolumn{6}{c}{**Middle Seat Risk**} \\ \cline{2-7}  & \(\mathbf{S_{1}}\) & \(\mathbf{S_{2}}\) & \(\mathbf{S_{3}}\) & \(\mathbf{S_{4}}\) & \(\mathbf{S_{5}}\) & \(\mathbf{S_{6}}\) & \(\mathbf{S_{7}}\) \\ \hline Open seating method & 12,090 & 11,259 & 10,417 & 9668 & 8978 & 8165 & 7129 \\ \hline Random with assigned seats & 18,086 & 17,077 & 16,097 & 15,084 & 14,234 & 13,269 & 12,032 \\ \hline WilMA & 13,073 & 12,062 & 11,102 & 10,243 & 9418 & 8440 & 6452 \\ \hline Back-to-front-by-3-groups & 8762 & 8108 & 7466 & 6936 & 6464 & 6034 & 5738 \\ \hline Back-to-front-by-4-groups & 7352 & 6744 & 6179 & 5727 & 5319 & 4991 & 4747 \\ \hline Back-to-front-by-5-groups & 6439 & 5923 & 5417 & 4981 & 4627 & 4333 & 4136 \\ \hline Back-to-front-by-row & **2502** & **2240** & **2022** & **1827** & **1705** & **1612** & **1560** \\ \hline Reverse pyramid & 8900 & 8126 & 7426 & 6784 & 6173 & 5489 & 4283 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Middle seat risk.  
seat passengers. Aisle seat passengers need to leave their seats for type-1, type-3, and type-4 seat interferences whereas middle seat passengers need to leave their seats only for type-1 and type-2 seat inferences. The time spent clearing the path for a later boarding passenger is included within the aisle and middle seat risks. Consequently, the aisle seat passengers incur a greater duration risk for a subsequent passenger in the aisle of that row than the middle seat passengers. In the same manner, the window seat risk (which will be discussed in the following subsection) is lower than the middle seat risk. That is because a passenger seated next to the window never needs to leave the seat to clear a path for any later boarding passengers. Consequently, the window seat passengers incur increased risk duration only for the type-4 seat interference.