Evaluating Classical Airplane Boarding Methods for Passenger Health during Normal Times

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The work focuses on analyzing airplane boarding methods used in normal times and evaluating them by applying lessons learned during the COVID-19 pandemic. As a result, the airplane boarding methods used in practice by the airlines are analyzed through a series of health risks indicators and an operational indicator—the boarding time.

Abstract

The COVID-19 pandemic has produced changes in the entire aviation industry, including adjustments by airlines to keep the middle seats of airplanes empty to reduce the risk of disease spread. In this context, the scientific literature has introduced new metrics related to passengers’ health when comparing airplane boarding methods in addition to the previous objective of minimizing boarding time. As the pandemic concludes and the aviation industry returns to the pre-pandemic situation, we leverage what we learned during the pandemic to reduce the health risk to passengers when they are not social distancing. In this paper, we examine the performance of classical airplane boarding methods in normal times but while considering the health metrics established during the pandemic and new metrics related to passenger health in the absence of social distancing. In addition to being helpful in normal times, the analysis may be particularly helpful in situations when people think everything is normal but an epidemic has begun prior to being acknowledged by the medical scientific community. The reverse pyramid boarding method provides favorable values for most health metrics in this context while also minimizing the time to complete boarding of the airplane.

1. Introduction

The occurrence of the novel coronavirus COVID-19 has produced changes in the indicators used for evaluating the airplane boarding methods [1,2,3,4]. In addition to the classical metric represented by the boarding time—namely the amount of time since the first passenger enters the aircraft and the last passengers occupies his/her assigned seat—highly used in non-pandemic situations (to which we are going to refer in the following as “normal times”), a series of health metrics have been proposed in the research literature for the times characterized by COVID-19, for the purpose of better evaluating the appropriateness of each boarding method for the new situation the world-wide air transportation industry was facing [1,5,6]. These metrics refer to the calculus of a series of health risk indicators through which the risk of contracting the disease by the passengers involved in an airplane boarding process can be evaluated [1,7,8,9,10,11,12].

As mentioned above, the research literature associated with the airplane boarding process pre-COVID-19 has been mainly focusing on answering the question: What is the fastest airplane boarding method that will ensure the minimization of the airplane turnaround time? [9]. To answer this question, a series of assumptions have been made regarding passengers boarding, such as the passengers’ movement inside the aircraft [13,14], passengers’ personal characteristics [15,16,17], the presence or absence of passenger groups [18], the type and the quantity of the hand luggage [13,17,18,19,20], the occurrence of boarding interferences—namely the situations in which either the passengers are blocking the aisle while loading luggage in the overhead compartment (also called aisle interference) or the passengers are interacting with other passengers already seated on the same row as their allocated seat, when they need to make space to allow the other passengers to occupy their allocated seats (known as type-1, type-2, type-3, or type-4 seat interferences—as presented later on in this paper) [16,18,21,22,23]. In addition to these assumptions, some research papers have extracted data from field trials [24,25], which have been used in calibrating the models for comparing different boarding methods in terms of boarding time. The boarding time has been the performance metric employed in the pre-COVID-19 situations in which improvements of the classical boarding methods have been proposed in the scientific literature [13,18,19,20,22,23,26,27,28,29].

Throughout the COVID-19 pandemic, the research question accompanying the scientific literature dedicated to airplane boarding process has been: What is the safest airplane boarding method that ensures a short boarding time? [4,9,11]. For answering this question, the airplane boarding methods used by the airline companies have been tested while accounting for the imposed social distancing measures represented by leaving the middle seat empty [4,11] or by keeping only 50% of the seats occupied in the airplane [1], and imposing a minimum 1–2 m social distance among the passengers while walking down the aisle to their assigned seats [1,6,12]. The performance evaluation of the boarding methods used by airplanes or proposed in the research literature for the COVID-19 times has been made through the use of health risk indicators (e.g., aisle seats risk, window seat risk, number of seat interferences, transmission risk, etc.) [2,5,6,30].

As the signals regarding the recovery of the global aviation industry have been pointed out in the recent research literature [31,32], there is an expected return to business-as-usual after COVID-19 in the near future [33,34]. In this context, the lessons learned from the COVID-19 pandemic should not be left behind as the occurrence of another epidemic, pandemic, or endemic situations might arise unexpectedly any time. As in the case of COVID-19, in the future, it also might take some time between the moment an epidemic arises and the time it is well acknowledged by the medical scientific community. To be prepared for such situations, which might arise unexpectedly, the airlines could try to proactively address these cases by selecting and using in practice boarding methods that consider the passengers’ health risk along with a reduced boarding time. Furthermore, even without a pandemic, epidemic, or endemic situation, there are advantages in providing passengers with less exposure to other passengers who might have any number of contagious illnesses (e.g., influenza, common cold).

As a result, the present paper analyzes some of the most common airplane boarding methods employed in normal times (with all seats occupied and no imposed aisle social distance of 1–2 m) from the perspective of the passengers’ health risk. To provide a proper comparison between the selected boarding methods, the risk indicators used for the evaluation of the boarding methods in times of COVID-19 are adapted for the normal times. Additionally, some new health risk indicators are proposed, which have been especially created for the case of normal times in which the middle seats are not kept empty. In addition to the health risk indicators, the boarding time is provided for each of the considered airplane boarding methods for the purpose of offering the airlines the possibility to consider both the risk and the boarding time when making their decision related to the use of a particular boarding method in practice. Furthermore, the results of the present research can be used in epidemic, pandemic, or endemic situations by the airlines that decide not to consider the imposed social distancing measures—as it has happened even in the case of COVID-19, when some airlines have not respected the imposed social distancing measures inside the aircraft—e.g., America Airlines, which has announced that their seats will be filled as much as they can in these times, not keeping the middle seats empty [9,10].

For comparing the considered airplane boarding methods, we created an agent-based model in NetLogo and calibrated it using the measurements made in the literature from field trials [24,25]. A narrow airplane, Airbus A320, with 30 rows and 6 seats per row is used in the simulations as suggested by [21,23,29,35,36,37]. Even though the principles and conclusions in this paper apply to more airplanes than the A320 aircraft, we have chosen this airplane model for the same reasons mentioned by Bazargan [21]: namely, it is the most common type of airplane used by the airlines and discussed in the scientific literature, enabling a proper comparison among the results obtained in various scientific papers. We assume that the airplane is occupied at its maximum capacity, being able to accommodate 180 passengers, as presented in Figure 1.

The remainder of the paper is organized as follows: Section 2 provides the schemes of the considered airplane boarding methods to be simulated and compared through the use of the metrics and scenarios provided in Section 3 and through the simulations conducted with the agent-based model described in Section 4. The results of the simulations are discussed in Section 5, while the concluding remarks and further research directions are included in Section 6. The paper is accompanied by videos representing simulations of the airplane boarding methods for the selected scenarios.

2. Airplane Boarding Methods

The airplane boarding methods considered are some of the most common boarding methods used world-wide by the airlines before the pandemic, either on all the flights or only on selected flights, as listed in

Table 1. Boarding methods used by various airlines [6,9,23,27,38,39].

Boarding Method	Airlines
Open seating method	Southwest
Random with assigned seats	Air France; EasyJet; KLM; Hainan Airlines; Ryanair; Tarom; WizzAir
WilMA	Lufthansa; United Airlines; Swiss
Back-to-front-by-group	Air Canada; Air China; Alaska; American Airlines; British Airways; Cathay Pacific; Eva Air; Frontier; Japan Airlines; Korean Air; Spirit; Virgin Atlantic
Back-to-front-by-row	Delta
Reverse pyramid	JetBlue

.

Most of the boarding methods are based on seat assignment where prior to boarding, the passengers know their allocated seat, usually being stamped on the boarding ticket. An exception is the open seating method where the passengers choose their seats after entering the airplane based on their preference [38,40].

When seats are assigned, in the scientific literature, three types of airplane boarding methods emerge based on the passengers’ entrance into the aircraft, namely random, by group, and by seat. In the case of random boarding with assigned seats, the passengers know the position of their seats in the aircraft and are called for boarding in a random manner, without considering their seat [41,42]. The by group boarding methods divide the passengers prior to boarding into several groups depending on their seat positions in the aircraft and are called in each group for boarding according to some given rules. Some of the most-known by group methods are: back-to-front, WilMA, reverse pyramid, modified optimal method, rotating zone, and non-traditional method [29,41,43,44]. Last, the by seat boarding methods call the passengers one-by-one to board into the airplane based on their seat position, usually starting from the back of the airplane and by following some given rules. The methods included in this category are: Steffen, Variation in Steffen method, and By seat descending order [13,43].

As it can be observed from Table 1, in practice, only random (random with assigned seats) and by group (WilMA, back-to-front-by-group, back-to-front-by-row, reverse pyramid) methods are used by airlines. The by seat methods are hard to implement in practice due to the additional support staff needed for airlines to invite the passengers one-by-one to board.

The rules accompanying the methods in Table 1 are presented in the following for each airplane boarding method.

2.1. Open Seating Method

When the open seating method is used, the passengers do not have assigned seats. Once the passengers arrive inside the aircraft, they select from the unoccupied seats based upon their preference.

2.2. Random with Assigned Seats

In the random with assigned seats method, the passengers know their seats prior to boarding. The seats are either allocated by the system when the check-in is performed online or in the airport or they are selected by the passengers based on their preferences when completing the check-in. The passengers admission into the airplane is completed in a random manner. Once arrived inside the aircraft, each passenger proceeds to his/her assigned seat.

2.3. WilMA

The WilMA boarding method derives its name from the initials of the three groups in which the passengers are divided prior to boarding: window (W), middle (M), and aisle (A). The method is also known as Outside-in as the three groups are formed based on the seat positions (near the window, middle seats, or near the aisle seats) and are called for boarding starting from the seats located as close as possible to the margins of the airplane (window) to the seats located near the center of the airplane (aisle). Figure 2 depicts the scheme for this boarding method.

2.4. Back-to-Front-by-Group Boarding Method

The back-to-front-by-group boarding method divides the passengers into groups based on the rows of their seats. The first group of passengers to board is formed by the passengers having their seats located closest to the rear of the airplane. The next group is composed of the passengers with seats as close to the rear of the airplane that have not been allocated to the first group. The last group to board has the seats located nearest to the front of the airplane.

Depending on the airline boarding strategy and the length (number of rows) of the airplane, the back-to-front-by-group boarding method may use from three to five groups. Figure 3 presents the groups formed when the back-to-front-by-group method features the division of the passengers into three groups, Figure 4 shows a division into four groups, and Figure 5 shows a division into five groups. Once the groups are formed, the passengers with seat rows within the mentioned groups are called-in for boarding in a random manner within each group.

2.5. Back-to-Front-by-Row Boarding Method

In the back-to-front-by-row boarding method, the first group of passengers to board are located in the row closest to the rear of the airplane. The next group has the seats in the second-to-last row of the airplane. The final group to board has the seats in the first row of the airplane. Figure 6 presents the boarding scheme for this method, wherein 30 groups (one per row) are called to board in sequence. Each of these groups has six passengers.

2.6. Reverse Pyramid

The original reverse pyramid boarding method was proposed by Van den Briel et al. [29] and divides the passengers into five groups based on their seat locations. Over time, the method has proven its efficiency in diminishing the airplane boarding time as the passengers interfere with each other at a minimum level when taking their assigned seats.

The reverse pyramid method used in this paper is a variation of the original reverse pyramid boarding scheme; in our paper and other works, the method involves four boarding groups of passengers [45]. The first group to board has window seats in the rear half of the airplane, while the last group to board has aisle seats in the front half of the airplane. The remaining groups (2 and 3) board in a diagonal scheme: each group fills the remaining seats closest to the window and in the rear half of the airplane for one seat type (middle for group 2 and aisle for group 3). This boarding scheme is illustrated for our case in Figure 7.

3. Health Metrics for Normal Times and Scenarios

The performance evaluation of the considered boarding methods is made through the use of six health metrics as presented in the following. In addition to these methods, the average boarding time will be reported in the simulations and results section to ensure a proper comparison, being known that the airlines are interested, besides the safety of their passengers, in the operational indicator, trying to keep at a minimum the costs generated by the airplane turnaround time [27,28]. In addition to the health metrics, the considered scenarios for the simulations are presented in the second part of this section.

3.1. Health Metrics

The health metrics considered in this paper are inspired by the health metrics used in the recent research literature in times of COVID-19 [2,5,6,30] by either using them as they are in the literature, adapting them, or creating new metrics that follow the same line of thinking.

3.1.1. Aisle Seat Risk

The aisle seat risk measures the overall risk experienced by the passengers already seated in the aisle seats due to possible exposure to an airborne disease carried by the passengers who are moving down the aisle to their assigned seats. As a prolonged exposure to any airborne contagious disease is connected to a higher health risk for the passengers seated in the aisle seats due to their direct face-to-face exposure, to consider a boarding method safer than another one, the value of this indicator should be as small as possible. The aisle seat risk is measured in seconds. The formula is provided in the following [4,5,30]:

$$AisleSeatRisk={\displaystyle \sum }_p{\displaystyle \sum }_{r\le RowSi{t}_p}\left(RowTim{e}_{pr}\ast {\displaystyle \sum }_{p^{\prime }<p}AisleSea{t}_{p^{\prime }r}\right)$$

where

• p = passenger advancing toward his/her seat
• r = row index
• RowSit_p = row in which passenger p has an assigned seat
• RowTime_pr = time that passenger p spends in row r (this duration begins when passenger p begins to enter row r and concludes when passenger p begins to leave row r; this convention is chosen because a passenger’s nose and mouth are at the front of the passenger)
• p’ = passenger boarding before passenger p
• AisleSeat_p’r = 1 if passenger p’ has an aisle seat in row
• r = 0 otherwise

3.1.2. Middle Seat Risk

Similar to the aisle seat risk, the middle seat risk measures the overall risk experienced by the passengers with middle seats, already seated, when the other passengers are moving down the aisle to their assigned seats. The measurement unit is seconds, and the formula for calculating the middle seats risk is:

$$MiddleSeatRisk={\displaystyle \sum }_p{\displaystyle \sum }_{r\le RowSi{t}_p}\left(RowTim{e}_{pr}\ast {\displaystyle \sum }_{p^{\prime }<p}MiddleSea{t}_{p^{\prime }r}\right)$$

where

• MiddleSeat_p’r = 1 if passenger p’ has a middle seat in row r = 0 otherwise.

3.1.3. Window Seat Risk

In the same manner as the other two seat risks presented above, the window seat risk measures the overall risk experienced by the window seat passengers when the other passengers are moving along the aisle to their assigned seats. The window seat risk formula follows [4,5,30]:

$$WindowSeatRisk={\displaystyle \sum }_p{\displaystyle \sum }_{r\le RowSi{t}_p}\left(RowTim{e}_{pr}\ast {\displaystyle \sum }_{p^{\prime }<p}WindowSea{t}_{p^{\prime }r}\right)$$

where

• WindowSeat_p’r = 1 if passenger p’ has a window seat in row r = 0 otherwise.

3.1.4. Total Number of Seat Interferences

The seat interferences are the situations in which a passenger with a seat near the window or a middle seat arrives at his/her allocated seat after one or two passengers with seats closer to the aisle in the same row and on the same side of the aisle have already taken their seats. In this situation, the passenger(s) with middle/aisle seats need to clear the path for the arriving passenger with the window/middle seat by exiting the row, waiting for the passenger with window/middle seat to take his/her seat, and returning to their seat.

Depending on the position of the allocated seat for the passenger who arrives after the other passenger(s) with seats in the same row are already seated, four types of seat interferences are acknowledged in the scientific literature, namely: type-1, type-2, type-3, and type-4 seat interferences [46].

Type-1 seat interference is encountered when a passenger with the window seat arrives near the allocated seat after the other two passengers with the middle and aisle seats in the same row and same side of the airplane are already seated. As a result, the seated passengers need to clear the way for the passenger with the window seat and then return to their seats. This type-1 seat interference is presented in Figure 8.

Type-2 seat interference is characterized by the presence of a passenger in the middle seat in the row in which a passenger with window seat arrives, while type-3 seat interference refers to a similar situation with the only exception that the passenger previously seated has a seat near the aisle (Figure 8).

Type-4 seat interference refers to the situation in which the passenger with a middle seat arrives after the passenger with the aisle seat has already taken his/her assigned seat (Figure 8).

The occurrence of any type of seat interferences increases the health risk for the involved passengers as during the interference the distance among the passengers is at its minimum and, if one of the passengers carries an airborne virus, the probability of spreading an airborne disease increases.

As in type-1 seat interference, the number of involved passengers is equal to 3, while in all the other types of seat interferences (type-2, type-3 and type-4), there are 2 involved passengers. Two metrics related to the total number of seat interferences are used to shape the health risk that might emerge from this situation: total number of type-1 seat interferences and total number of type-2-3-4 seat interferences. Both indicators measure the number of seat interferences of the given types that occur in a complete boarding process.

3.1.5. Aisle Standing Risk

The aisle standing risk metric [47] is new and inspired from the pandemic times in which the recommended social distance has been at least 1 m among the passengers while walking down the aisle [48,49]. Any violation of this rule is considered to increase the aisle standing risk. Depending on how much the 1 m aisle social distance—among any two passengers who walk down the aisle—is infringed, the value of the aisle standing risk changes, as we apply weights to heavily penalize the distances among the passengers that are further away the recommended 1 m aisle social distance. We consider a threshold of 0.4 m as the personal distance among the passengers needed to ensure sufficient personal space to allow for their comfort when traveling down the aisle. Consequently, the closer the aisle passengers are to each other than 1 m, the greater the weight applied in this metric. The formula for the aisle standing risk is:

$$AisleStandingRisk={\displaystyle \sum }_p{\displaystyle \sum }_{r\le RowSi{t}_p}{\displaystyle \sum }_{d{ }^{ }}Weigh{t}_d\ast RowTim{e}_{pr{p}^{\prime }d}$$

where

• p = passenger advancing toward his/her seat
• r = row index
• p’ = passenger boarding before passenger p
• RowSit_p = row in which passenger p has a seat
• d = amount that the distance between two adjacent aisle passengers is less than 1 m. For example, if the distance between two consecutive aisle passengers is 0.8 m, then d would be equal to 1 m − 0.8 m = 0.2 m.
• Weight_d = weight to apply when passenger p’ is d m closer to p than 1 m. We set to Weight_d to have a value of d.
• RowTime_prp’d = time that passenger p spends in row r while passenger p’ is standing in the aisle by d m closer to p than 1 m.

3.1.6. Individual Boarding Time

Considering the name of this health risk metric, one might be tempted to view it as being the same as the operational metric, “boarding time”. It is different. While boarding time refers to the time to complete the boarding of all passengers, the individual boarding time measures the time needed by a passenger to take his/her assigned seat after entering the airplane’s aisle. The higher the value of an individual passenger’s boarding time, the longer it waits for earlier boarding passengers to clear out of the way. Those waiting times involve the passenger being a mere 0.4 m away from the adjacent passenger in front, which is an unsafe distance if either passenger is contagious. Consequently, the risk to passengers walking or standing in the aisle stemming from higher individual boarding times (higher waiting times) is high due to the greater risk of contagion while standing in the aisle at the unsafe distance. If all other factors are equal (including the risks to seated passengers), the longer the average passenger spends standing in the aisle (whether walking or stationary), the higher the risk of contracting an airborne disease while standing in the aisle.

The average value of this metric is determined by summing the individual boarding times for all the passengers and dividing the resulting value by the number of passengers. The measurement unit is time expressed in seconds.

3.2. Scenarios

The scenarios considered for the airplane boarding methods’ simulations are related to the quantity and type of hand luggage carried by the passengers inside the aircraft. Seven different scenarios are used, featuring various combinations of small and large bags, that cover various situations from a heavy hand luggage load (scenario S₁) to a no-hand luggage situation (scenario S₇). The percentages accompanying each type of hand luggage in the envisioned scenarios are presented in

Table 2. The scenarios based on the quantity and type of hand luggage.

Scenario	Percentages of Bags Carried by the Passengers
	0 Bag	1 Small Bag	2 Small Bags	1 Large Bag	1 Large and 1 Small Bag
S1	10%	10%	0%	10%	70%
S2	15%	20%	5%	10%	50%
S3	25%	20%	10%	15%	30%
S4	35%	25%	10%	15%	15%
S5	60%	10%	10%	10%	10%
S6	80%	5%	5%	5%	5%
S7	100%	0%	0%	0%	0%

[6].

4. Description of the Agent-Based Model

For modeling the passengers’ behavior while boarding an airplane, an agent-based modeling approach has been used [3,5,45,50,51,52]. This type of modeling has been considered appropriate as it offers the needed tools for representing the passengers involved in the airplane boarding process and their interactions, while providing at the same time a graphical user interface which makes it easy to understand and observe the boarding process as a whole.

Compared to other approaches, such as cellular automata, the agent-based modeling approach offers several types of agents with specific functions that can fulfill a series of given tasks. Considering the scientific literature, it has been observed that the agent-based modeling in general, and NetLogo software in particular, has been a common choice for the researchers who have proposed models that involve the embedding of the human behavior [53,54,55,56,57].

We implemented the agent-based model using NetLogo 6.2.1 [58]. Figure 9 presents the graphical user interface of the model (GUI).

4.1. The Agent-Based Environment

The agent-based environment that represents the interior of the airplane has been built up through the use of patch agents. These agents have a square shape and cover the entire area in which the turtle agents—used for depicting the passengers involved in the boarding process—are moving.

Essentially, the inside area of the airplane is represented as a grid of squares composed of patch agents, each of them being uniquely described by a coordinate on Ox axes, called pxcor, and a coordinate on Oy axes, called pycor. Figure 10 presents the patch with the coordinates 42 1, marked in red, both in the context of the airplane (on the left) and in a specific monitor (on the right) in which all the properties of the patch can be observed. In addition to the pxcor and pycor variables, each patch has three other characteristics: pcolor—the color of the patch (in our case is either light gray for window and aisle seats, gray for middle seats, and dark blue for the aisle and the space in front of the seats), isseat?—a Boolean variable that takes either a true or a false value depending on whether the patch represents a seat in the model or not, and seat-row—a variable that retains the corresponding row of seats the patch belongs to; this variable takes values between 1 and 30 in our case, as the airplane model considered in this paper has 30 rows of seats. Based on the field trials made in the scientific literature, it has been determined that the corresponding size of a patch in real life is a 0.4 m × 0.4 m area [1,25,59].

4.2. Passengers’ Movement

The turtle agents model the passengers’ behavior, have a human shape, and move according to the given rules on the grid of patches. When moving, they can take different positions in the model, being either seated or in the aisle, which are positions that are not necessarily in the middle of a patch. Figure 10 presents some examples of turtles (in yellow).

4.2.1. Agents’ Characteristics for Seat Assignment Boarding Methods

For shaping the behavior of the turtles when boarding according to the rules of one of the airplane boarding methods—random with assigned seats, WilMA, back-to-front-by-group, back-to-front-by-row, and reverse pyramid—a series of variables have been associated with these types of agents: speed—which ranges between 0 and 1 patch/tick (tick is the time unit in NetLogo, which corresponds to 1.2 s in real life as determined in the scientific literature [25,26,60]), the speed is influenced by the amount of hand luggage and by the speed of the passenger-ahead, who may have stopped to store luggage; luggage?—a Boolean variable taking a true value when the agent has hand luggage and false when it is traveling without luggage; large-luggage-no—an integer variable taking as a value the number of large hand luggage; small-luggage-no—an integer variable storing the number of small hand luggage; seated?—a Boolean variable taking a true value when the agent has reached its assigned seat and has taken it; agent-seat-row—taking values between 1 and 30, representing the row in which the agent has a seat; agent-seat-column—taking either A, B, C, D, E, or F as values, corresponding to the letter used by the airlines for uniquely identify the seats on a particular row; comfort-distance—the personal space needed by an agent while moving down the aisle, which has a value of 1 patch; and time-to-sit—equal to 1 tick, representing the time needed for a passenger not involved in any type of seat interferences to take their seat.

Additionally, all the turtle agents possess a luggage-store-time variable that accounts for the time needed to store the luggage (if any) in the overhead bin compartment. The value of this variable is dynamically determined in the moment in which the agent arrives near its allocated seat, and it takes into account the type and quantity of luggage the agent needs to store, and the type and the quantity of luggage already stored in the overhead compartment by the agents who have arrived in the same row prior to the current agent. The formula for the Luggage-store-time variable is inspired from [61] and has been previously used by [3,4,26,45,47]:

$$\begin{array}{cc}\hfill Luggage\_store\_time& =((NbinLarge+0.5 NbinSmall+NpassengerLarge\hfill \\ \hfill & +0.5 NpassengerSmall)\ast (NpassengerLarge\hfill \\ \hfill & +0.5 NPassengerSmall)/2)\ast Trow\hfill \end{array}$$

where

• NbinLarge is the number of large bags in the overhead bin prior to the agent’s arrival;
• NbinSmall is the number of small bags in the overhead bin prior to the agent’s arrival;
• NpassengerLarge is the number of large bags carried by the agent;
• NpassengerSmall is the number of small bags carried by the agent;
• Trow is the time for an agent to walk from one row to the next (when not delayed by another agent in front).

4.2.2. Agents’ Characteristics for Open Seating Method

In the case of the open seating boarding method, modeling the manner in which the passengers choose their seats might be a difficult task as mentioned by Qiang et al. [40], since human selection behavior needs to be included in the decision-making process. Considering a study conducted by Skyscanner.com (https://www.skyscanner.net/press-releases/skyscanner-reveals-perfect-seat-6a, accessed on 9 August 2021) on 1000 regular air travel passengers, it has been observed that approximatively 60% of the passengers prefer the window seat, while 40% opt for an aisle seat and less than 1% opt for a middle seat. Using a questionnaire with a 10-point Likert scale on 387 passengers, Delcea et al. [38] extracted the passengers’ preference to window, middle, and aisle seats, their preference to front, middle, and rear of the airplane, and their preference for the crowded areas in the airplane. Based on the importance given to each of the three categories, they observed that most of the passengers prefer seats placed in non-crowded areas in the middle–front part of the airplane.

Including the human behavior in the models that consider the open seating assignment has been made in the scientific literature through different approaches. One approach features the use of a seat attraction map built upon some fixed rules (e.g., Qiang et al. [40] consider all the passengers homogenous and having the same preference for the seats), while another approach gives a certain energy to each seat and updating the energy of the remained unselected seats upon each seat selection (e.g., the Steffen approach [14] based on the principles of statistical mechanics). Recently, an approach based on creating a fitness function for each passenger based on the preference to different parts of the airplane—front/middle/rear, window/middle/aisle, crowded/non-crowded area—and on the importance given to each type of category has been used in the research literature (e.g., Delcea et al. [38] determine a fitness function for each passenger through which the seats in the airplane form a preference map, each passenger having his/her own preference map based on the values extracted from the questionnaire).

In this paper, we have used the approach featuring the calculation of a fitness function as it creates a personalized preference map for each agent based on the data extracted from questionnaires, benefiting from the heterogenous characteristic of the agents in agent-based modeling, rather than using a set of rules on homogenous agents. The fitness function is the following [38]:

$$Fitness=iRow\ast pRow+iColumn\ast pColumn+iAgglomeration\ast AgglomerationState$$

where

• $pRow$ and $pColumn$ are passenger preference indicators, having values between 1 and 10;
• $iRow, iColumn$, and $iAgglomeration$ are passenger importance indicators, having values between 1 and 10;
• $AgglomerationState$ is a variable taking 0 or 10 as possible values.

$$pRow=\{\begin{array}{c}pRowFront, \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{o}\mathrm{w} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{t} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{a}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t}\\ pRowMiddle, \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{o}\mathrm{w} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{m}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{l}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{a}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t} \\ pRowBack, \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{r}\mathrm{o}\mathrm{w} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{a}\mathrm{i}\mathrm{r}\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{f}\mathrm{t}\end{array}$$

$$pColumn=\{\begin{array}{c}pColumnWindow, \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{w}\\ pColumnMiddle, \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{m}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{l}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{n} \\ pColumnAisle, \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{e}\mathrm{a}\mathrm{t} \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{a}\mathrm{i}\mathrm{s}\mathrm{l}\mathrm{e}\end{array}$$

$$AgglomerationState=\{\begin{array}{c}0, if one or both seats located in the half-row are occupied \hfill \\ 10, if the seats in the same half-row are both empty \hfill \end{array}$$

The values for the fitness function have been determined considering the responses to the questionnaire, as provided in [38], where a 10-point Likert scale has been used for respondents’ answers. Based on them, the agent-seat-row and the agent-seat-column are determined for each agent. The rest of the variables used for the turtle agents are the same as described in Section 4.2.1. For more information on how the preference for the row and column and the importance given by the respondents relate to the agglomeration, row, and column have been extracted, we invite the reader to see [38].

4.3. Steps Involved in Simulations

At the beginning of each simulation run, at time zero, the first turtle agent enters the airplane and heads toward its allocated seat, moving down the aisle at a speed determined based on the type and quantity of hand luggage carried by the agent. Upon arriving near its allocated seat, the agent places the hand luggage (if any) in the overhead compartment, blocking the aisle for any other passengers for a time equal to the luggage-store-time plus the time-to-sit. The last action of the agent before changing its state to seated is to occupy its assigned seat. While moving down the aisle, the agent right behind the previous agent adapts its speed to the speed of the agent in front, not being allowed to overpass the agent ahead, always keeping a comfort-distance to the agent ahead. When confronted with a seat interference, the agent causing this interference blocks the aisle by an additional time given by the type of interference: approximatively 18 ticks for type-1 (ranging between 16 and 21 ticks uniformly distributed), 10 ticks for type-2 (ranging between 8 and 11 ticks uniformly distributed), 8 ticks for type-3 and type-4 (ranging between 7 and 10 ticks uniformly distributed) [25,52]. The boarding process continues until the last agent occupies its assigned seat.

5. Simulation Results

Each airplane boarding method has been simulated 10,000 times for each luggage scenario, resulting a number of 560,000 simulations for which the results have been reported in the following by using the average value, rounded to the nearest integer. For performing the simulations, the BehaviourSpace tool provided by NetLogo [51,62] has been used both for setting up the experiments and for running them. The results are reported for each health metric. A discussion is made based on the health metrics results and the boarding time reported for each airplane boarding method in a given luggage scenario.

5.1. Numerical Results for Aisle Seat Risk

The overall risk experienced by the passengers with aisle seats who have already taken their seats while other passengers are still moving down the aisle to their assigned seats is measured through the aisle seat risk indicator.

Table 3. Aisle seat risk.

Boarding Method	Aisle Seat Risk
	S1	S2	S3	S4	S5	S6	S7
Open seating method	15,636	14,544	13,552	12,565	11,692	10,715	9318
Random with assigned seats	18,323	17,413	16,357	15,358	14,518	13,473	12,252
WilMA	4312	3991	3681	3385	3114	2769	2126
Back-to-front-by-3-groups	9068	8380	7772	7208	6750	6319	6019
Back-to-front-by-4-groups	7675	7034	6467	6031	5599	5276	5046
Back-to-front-by-5-groups	6748	6207	5734	5289	4953	4639	4446
Back-to-front-by-row	2786	2538	2316	2135	2016	1937	1885
Reverse pyramid	2244	2021	1825	1652	1481	1304	1045

presents the average aisle seat risk for each of the boarding methods as a function of the luggage scenario. These results are based on agent-based simulations (rounded to the nearest integer and expressed in seconds). For all the boarding methods, this risk decreases as the quantity of luggage brought in the airplane by the passengers decreases from the high-luggage scenario S₁ to the no-luggage scenario S₇.

For each of the seven luggage scenarios, the best and third best-performing methods according to this criterion are the reverse pyramid and WilMA boarding methods, respectively. (We highlight the best value of each luggage scenario in bold font within the table.) These two methods share a common attribute of boarding all aisle seat passengers after all window and middle seat passengers have boarded. This means that their aisle seat passengers have aisle seat risk only from other aisle seat passengers. Observe that with WilMA, all 60 aisle seat passengers board in one group whereas with reverse pyramid, there are 30 aisle seat passengers in group 3 and the other 30 aisle seat passengers in group 4. With reverse pyramid, the passengers in group 3 have aisle seat risk only from other group 3 passengers, and similarly, its group 4 passengers have aisle seat risk only from other group 4 passengers. The division of aisle seat passengers into two groups essentially halves the number of passengers potentially causing aisle seat risk. For each luggage scenario, the aisle seat risk of reverse pyramid is approximately half that of WilMA. The relationship is not exact because aisle seat risk is a time-based metric and thus influenced by the passengers’ boarding dynamics over time and not solely by the number of potential interactions. The implication of boarding dynamics is illustrated by the fact that for each of these two boarding methods, the high-luggage scenario S1 results in about twice as much aisle seat risk as the no-luggage scenario S7.

While those two methods have an advantage with aisle seat risk from their aisle seat passengers boarding last, the second-best performing method, back-to-front-by-row, has an advantage in that its aisle seat risks result only from passengers assigned seats in the same row. Due to the boarding dynamics, those passengers in the same row may be standing in that row’s aisle a long time due to seat interferences. Overall, the back-to-front-by row method provides an increased aisle seat risk compared to reverse pyramid by up to 80.38%.

The back-to-front-by-5-groups, back-to-front-by-4-groups, and back-to-front-by-3-groups are the 4th, 5th, and 6th best-performing methods for aisle seat risk, respectively. Their relative performance stems from the same advantage that reverse pyramid has over WilMA: namely, the fewer number of passengers who can breathe near each other, the lower that risk.

Even higher values for this risk are reported for the open seating and random with assigned seats methods. The latter method has the worst aisle seat risks. The increased risks vary from 716.53% (with S1) to 1072.44% (with S7) compared to the same luggage scenarios when the reverse pyramid method is used.

5.2. 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. Middle seat risk.

Boarding Method	Middle Seat Risk
	S1	S2	S3	S4	S5	S6	S7
Open seating method	12,090	11,259	10,417	9668	8978	8165	7129
Random with assigned seats	18,086	17,077	16,097	15,084	14,234	13,269	12,032
WilMA	13,073	12,062	11,102	10,243	9418	8440	6452
Back-to-front-by-3-groups	8762	8108	7466	6936	6464	6034	5738
Back-to-front-by-4-groups	7352	6744	6179	5727	5319	4991	4747
Back-to-front-by-5-groups	6439	5923	5417	4981	4627	4333	4136
Back-to-front-by-row	2502	2240	2022	1827	1705	1612	1560
Reverse pyramid	8900	8126	7426	6784	6173	5489	4283

.

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 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.

5.3. Numerical Results for Window Seat Risk

Back-to-front-by-row provides the smallest value for the window seat risk in all the luggage scenarios, as illustrated in

Table 5. Window seat risk.

Boarding Method	Window Seat Risk
	S1	S2	S3	S4	S5	S6	S7
Open seating method	25,070	23,499	21,947	20,531	19,021	17,436	15,014
Random with assigned seats	17,744	16,825	15,746	14,756	13,896	12,849	11,663
WilMA	20,784	19,418	18,046	16,859	15,592	14,076	10,770
Back-to-front-by-3-groups	8401	7693	7132	6559	6102	5664	5340
Back-to-front-by-4-groups	6974	6388	5828	5364	4967	4613	4370
Back-to-front-by-5-groups	6066	5526	5052	4582	4273	3968	3761
Back-to-front-by-row	2113	1860	1631	1448	1316	1219	1156
Reverse pyramid	16,840	15,621	14,490	13,449	12,402	11,134	8610

. For this method, the window seat risk for the high-luggage scenario S1 is 82.79% higher than the no-luggage scenario S7. The middle seat risk with this method varies between 18.41% and 34.95% higher than the corresponding window seat risk depending on the luggage scenario. As implied by the discussion of the previous subsection, seat interferences contribute to the differences.

The airplane boarding methods featuring a back-to-front-by-group approach score the second, third, and fourth best values for the window seat risk, in the same sequence as with the other two seat risks. The reverse pyramid method has the fourth worst results for window seat risk. WilMA provides the third worst results for luggage scenario S7 and the second worst results for the other six scenarios, the reverse being true for the random with assigned seats method.

The open seating method provides the highest values of the window seat risk for all luggage scenarios. This is due to the passengers’ preferences for window seats located near the front door of the airplane. After sitting down in those window seats, the passengers will be passed by many later boarding passengers.

5.4. Numerical Results for Seat Interferences

We first discuss the results for the total number of seat interferences that involve two passengers (namely the aisle and middle seat passengers) needing to leave their seat to clear space for a later boarding passenger (namely the window seat passenger) to sit. This is the type-1 seat interference. Secondly, we will discuss the total number of seat inferences that involve exactly one passenger (either the aisle or the middle seat passenger) needing to leave their seat to clear space for a later boarding passenger. These are the type-2, type-3, and type-4 seat interferences that we will total because they all involve one passenger leaving a seat to clear space for a later boarding passenger.

5.4.1. Numerical Results for Total Number of Type-1 Seat Interferences

The total number of type-1 seat interferences is zero for reverse pyramid and WilMA—

Table 6. Total number of type-1 seat interferences.

Boarding Method	Total Number of Type-1 Seat Interferences
	S1	S2	S3	S4	S5	S6	S7
Open seating method	0	0	0	0	0	0	0
Random with assigned seats	20	20	20	20	20	20	20
WilMA	0	0	0	0	0	0	0
Back-to-front-by-3-groups	20	20	20	20	20	20	20
Back-to-front-by-4-groups	20	20	20	20	20	20	20
Back-to-front-by-5-groups	20	20	20	20	20	20	20
Back-to-front-by-row	20	20	20	20	20	20	20
Reverse pyramid	0	0	0	0	0	0	0

. This situation occurs because the window seat passengers of a given row always board before any other passengers in that row. For the open seating method, the values appear to be zero in Table 6, because the results have been rounded to the nearest integer; however, if decimal places had been included in the table, then the reported values would have been near zero. This is due to it being rare that a passenger will prefer a middle seat on the airplane and furthermore due to the predilection for passengers to prefer window seats most often when the aisle and middle seats are empty. The rare condition would occur when a passenger strongly prefers a window seat in a row and side of the airplane where the aisle and middle seat passengers are already seated.

For the other boarding methods, in each row and side of the airplane, it is equally likely that the aisle, middle, or window seat passenger is the last to board the airplane. The means that $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ of the time, that final passenger of the three to board will have the window seat. With 30 rows and two sides of the airplane, there would be an average of 30 × 2 × $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ = 20 type-1 seat interferences for those other boarding methods, which matches the results in Table 6 for each luggage scenario. Another way to view this is illustrated in Figure 11, which shows the six possible sequences (cases) in which the three passengers for each side of an airplane’s row would board the airplane.

Each of the six sequences (cases) has a probability of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$ of occurring when the sequence is randomly generated such as during a computer simulation run. Among these cases, only C₁ and C₃ involve type-1 seat interferences. As a result, $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ of the six cases on each side of a row results in a type-1 seat interference. So, again, we have 30 row × 2 sides × $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ = 20 type-1 seat interferences for those other boarding methods.

5.4.2. Numerical Results for Total Number of Type-2-3-4 Seat Interferences

The total number of type-2-3-4 seat interferences is zero for reverse pyramid and WilMA—

Table 7. Total number of type-2-3-4 seat interferences.

Boarding Method	Total Number of Type-2-3-4 Seat Interferences
	S1	S2	S3	S4	S5	S6	S7
Open seating method	53	53	53	53	53	53	53
Random with assigned seats	50	50	50	50	50	50	50
WilMA	0	0	0	0	0	0	0
Back-to-front-by-3-groups	50	50	50	50	50	50	50
Back-to-front-by-4-groups	50	50	50	50	50	50	50
Back-to-front-by-5-groups	50	50	50	50	50	50	50
Back-to-front-by-row	50	50	50	50	50	50	50
Reverse pyramid	0	0	0	0	0	0	0

. These methods board in the C₆ sequence, which involves no seat interferences.

All but one of the remaining boarding methods averages 50 seat interferences of type-2, type-3, and type-4. For these methods, type-2 seat interferences result from case C₅, and type-3 seat interferences result from case C₂, while type-4 seat interferences result from cases C₁, C₂, and C₄. This is a total of five case occurrences for each of the 30 rows and two sides of the airplane, resulting in 30 × 2 × $\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$ = 50 occurrences combined of type-2, type-3, and type-4 seat interferences, as shown for the average values in Table 7.

A higher value for window seat risk is reported for the open seating method, where 53 cases combined of type-2, type-3, and type-4 seat interferences are reported. The increased number of these seat interferences compared to random methods is a consequence of the fact that the passengers boarding in an open seating approach tend to choose more of the window seats, which decreases the number of type-1 seat interferences (C₁ and C₃), increases the number of C₂ and C₅ cases, and highly increases the number of C₄ and C₆ cases. The total number of type-2, type-3, and type-4 seat interferences is highly dependent on the passengers’ preference to the position of certain seats in the airplane and the agglomeration degree, and its value might change were the passengers to have different preferences.

5.5. Numerical Results for Aisle Standing Risk

The numerical results for the aisle standing risk are reported in

Table 8. Aisle standing risk.

Boarding Method	Aisle Standing Risk
	S1	S2	S3	S4	S5	S6	S7
Open seating method	6913	6523	6131	5768	5443	5063	4258
Random with assigned seats	7764	7368	6970	6585	6258	5879	5266
WilMA	5501	5202	4908	4670	4488	4207	2859
Back-to-front-by-3-groups	12,493	11,604	10,839	10,039	9388	8668	7756
Back-to-front-by-4-groups	13,496	12,534	11,539	10,717	9966	9222	8228
Back-to-front-by-5-groups	14,283	13,195	12,168	11,269	10,452	9632	8629
Back-to-front-by-row	21,834	19,426	17,312	15,370	13,940	12,452	11,007
Reverse pyramid	5798	5485	5189	4916	4720	4390	3034

. For this metric, the best-performing method is WilMA, which is followed closely by reverse pyramid. Both of these methods work well as they avoid seat interferences and spread the passengers of each group over a large number of rows. Those attributes lead to less congestion in the aisle, leading to fewer and briefer occurrences of a passenger proceeding down the aisle being impeded by the (slower) progress of the previous passenger to board. While the passengers of each group are assigned to a single seat type (window, middle, or aisle) with both methods, WilMA performs better than reverse pyramid. This situation occurs due to the fact that the passengers of each of the former method’s groups is spread across all 30 rows of the airplane, whereas that is true for two of the latter method’s four boarding groups with its first and last groups spread across half the rows of the airplane. The presence of hand luggage influences the value of this risk for both methods. For WilMA, the change in the quantity of luggage, from S₇ to S₁, increases the aisle standing risk by 92.41%. A similar increasement, of 91.10%, is reported for the reverse pyramid boarding method when comparing S₇ to S_1.

The third and fourth best methods for this risk are open seating and random with assigned seats, which have an increasement ranging between 20.35 and 48.93%, respectively, and between 39.44 and 84.19%, when compared to WilMA.

The highest values for this risk result from the back-to-front-by-row, back-to-front-by-5-groups, back-to-front-by-4-groups, and back-to-front-by-3-groups, and back-to-front-by-row, in that sequence. These methods have the least spread across rows per boarding group and the most congestion in the aisle. The worst performing back-to-front-by-row method has a value for this risk that varies between 195.98% to 296.91% higher than the best-performing method, WilMA.

5.6. Numerical Results for Individual Boarding Time

The best (and equally good) individual boarding times result from WilMA and reverse pyramid in each of the considered luggage scenarios—

Table 9. Individual boarding time.

Boarding Method	Individual Boarding Time
	S1	S2	S3	S4	S5	S6	S7
Open seating method	87	82	77	72	67	61	51
Random with assigned seats	97	92	86	81	77	71	62
WilMA	72	67	63	59	55	50	37
Back-to-front-by-3-groups	131	121	113	105	98	91	80
Back-to-front-by-4-groups	139	129	119	110	103	95	83
Back-to-front-by-5-groups	145	134	124	115	106	98	87
Back-to-front-by-row	212	189	168	150	136	122	106
Reverse pyramid	72	67	63	59	55	50	37

. The increasement in the quantity and type of hand luggage brings an increasement of this risk of at most 94.59% (S₇ vs. S₁). The low values of the individual boarding times for these methods is related to the absence of seat interferences.

The remaining boarding methods, from best to worst, are the following in sequence: open seating, random with assigned seats, back-to-front-by-row, back-to-front-by-5-groups, back-to-front-by-4-groups, and back-to-front-by-3-groups, and back-to-front-by-row. This sequence of relative performance is identical to the sequence for aisle standing risk. Consequently, individual boarding time may be used as a proxy for aisle standing risk when comparing boarding methods. We mention this because individual boarding time would be easier to calculate than aisle standing risk and has an added advantage in that it bears a relationship with customer satisfaction due to passengers preferring shorter individual boarding times.

5.7. Discussion

The relative performance of the boarding methods obtained for the considered health risk metrics differs depending on the metric. None of the methods is best for all of the health indicators. Before discussing the health performance, we show the total boarding times (average time to complete boarding of the airplane) for the methods and luggage scenarios in

Table 10. Boarding time.

Boarding Method	Boarding Time
	S1	S2	S3	S4	S5	S6	S7
Open seating method	802	747	694	643	598	548	454
Random with assigned seats	864	813	764	716	675	629	547
WilMA	605	559	514	473	437	391	279
Back-to-front-by-3-groups	842	782	729	679	638	594	516
Back-to-front-by-4-groups	862	801	740	688	645	599	524
Back-to-front-by-5-groups	885	820	757	704	657	608	532
Back-to-front-by-row	1227	1095	976	869	794	714	614
Reverse pyramid	530	491	454	420	390	353	247

.

Reverse pyramid provides the fastest boarding time, followed by WilMA. The open seating method, random with assigned seats, and all three of the back-to-front-by-group methods provide similar boarding times with their results ranging between 454 and 885 s, depending on the quantity of hand luggage. The longest boarding process results from back-to-front-by-row, which provides a boarding time up to 148.58% higher than reverse pyramid.

Table 11. Methods ranking based on the considered indicators in the S1 luggage scenario.

Rank	Airplane Boarding Method
	Aisle Seat Risk	Middle Seat Risk	Window Seat Risk	Type-1 Seat Interferences
1.	Reverse pyramid	Back-to-front-by-row	Back-to-front-by-row	Reverse pyramid
2.	Back-to-front-by-row	Back-to-front-by-5-groups	Back-to-front-by-5-groups	WilMA
3.	WilMA	Back-to-front-by-4-groups	Back-to-front-by-4-groups	Open seating method
4.	Back-to-front-by-5-groups	Back-to-front-by-3-groups	Back-to-front-by-3-groups	Back-to-front-by-row
5.	Back-to-front-by-4-groups	Reverse pyramid	Reverse pyramid	Back-to-front-by-5-groups
6.	Back-to-front-by-3-groups	WilMA	Random with assigned seats	Back-to-front-by-4-groups
7.	Open seating method	Open seating method	WilMA	Back-to-front-by-3-groups
8.	Random with assigned seats	Random with assigned seats	Open seating method	Random with assigned seats
Rank	Airplane Boarding Method
	Type-2-3-4 Seat Interferences	Aisle Standing Risk	Individual Boarding Time	Boarding Time
1.	Reverse pyramid	WilMA	Reverse pyramid	Reverse pyramid
2.	WilMA	Reverse pyramid	WilMA	WilMA
3.	Back-to-front-by-row	Open seating method	Open seating method	Open seating method
4.	Back-to-front-by-5-groups	Random with assigned seats	Random with assigned seats	Back-to-front-by-3-groups
5.	Back-to-front-by-4-groups	Back-to-front-by-3-groups	Back-to-front-by-3-groups	Back-to-front-by-4-groups
6.	Back-to-front-by-3-groups	Back-to-front-by-4-groups	Back-to-front-by-4-groups	Random with assigned seats
7.	Random with assigned seats	Back-to-front-by-5-groups	Back-to-front-by-5-groups	Back-to-front-by-5-groups
8.	Open seating method	Back-to-front-by-row	Back-to-front-by-row	Back-to-front-by-row

provides an ordinal ranking of the considered boarding methods for all the metrics based on the results obtained in the S₁ luggage scenario.

Based on the ranking in Table 11, the reverse pyramid method performs best in five of the eight metrics, namely: aisle seat risk, type-1 seat interferences, type-2-3-4 seat interferences, individual boarding time, and time to complete the boarding process for all passengers. Furthermore, reverse pyramid performs nearly as well for aisle standing risk as that metric’s best method, WilMA. Moreover, we note that the two metrics for which reverse pyramid performs poorly, namely middle seat risk and window seat risk, may not be as important for health as aisle seat risk and the seat interferences given that the middle seat risk and window seat risk refer to risks of the passengers being further from other passengers than with the other health risks. As such, the reverse pyramid method should be seriously considered by airlines who prefer short times to complete the boarding of their airplanes while reducing health risks to their passengers.

Meanwhile, WilMA also performs well. It performs best for four of the metrics, namely, aisle standing risk, type-1 seat interferences, type-2-3-4 seat interferences, and individual boarding time. It performs second best for boarding time and third best for aisle seat risk. Its performance for middle seat risk and window seat risk is worse than reverse pyramid. Overall, WilMA performs well but not as well as reverse pyramid. WilMA may be the second-best method for most airlines.

Moderate results stem from the back-to-front-by-group methods. The back-to-front-by-5-groups airplane boarding method provides slightly better results than back-to-front-by-4-groups and back-to-front-by-3-groups for three risk indicators: aisle seat risk, middle seat risk, and window seat risk; meanwhile, they achieve the same results for type-1 seat interferences and type-2-3-4 seat interferences and worse results for the aisle standing risk, individual boarding time, and time to complete boarding of the airplane.

The back-to-front-by-row boarding method performs better than the back-to-front-by-group methods for aisle seat risk, middle seat risk, and window seat risk, the same for type-1 seat interferences and type-2-3-4 seat interferences, and worse for aisle standing risk, individual boarding time, and time to complete boarding of the airplane.

The open seating method performs better than the back-to-front methods for type-1 seat interferences, aisle standing risk, individual boarding time, and the time to complete boarding of the aircraft. The random with assigned seats method does not perform well enough on any metric to make it a good choice for airlines other than its ease of implementation.

6. Conclusions

As the restrictions for the airplane boarding process imposed by the occurrence of the novel coronavirus begin to wane, the challenge becomes how to reduce the health risks to passengers during normal (non-pandemic) times. In this paper, we analyzed through an agent-based approach the airplane boarding methods used in practice by the airlines in terms of the health metrics used during the pandemic situation and new health metrics that apply when there is no social distancing. In short, we provide an overview on the performance of the airplane boarding methods used in practice from a health perspective. Especially because epidemic situations might occur in the future before they are acknowledged by the medical scientific community, airlines may want to assess the health risks to their passengers when deciding upon an airplane boarding method.

Based on the numerical results of agent-based simulations, the reverse pyramid method provides favorable health outcomes for important metrics, while at the same time providing the shortest time to complete boarding of the airplane. Airlines interested in passenger health and short boarding times should seriously consider using the reverse pyramid method. The WilMA method may be the second-best boarding method, and random boarding with assigned seats may be the worst.

The best-performing airplane boarding method in Table 11, reverse pyramid, has not been commonly used in practice by the airlines. To successfully use some of these best-performing boarding methods in practice, the airports need to offer additional infrastructure and to better monitor the entire boarding process. We conjecture that with the advancement of technology, in the near future, these impediments can be overcome, and the airlines can opt for the best-performing airplane boarding methods determined based on the considered health indicators.

Future research may consider alternative airplane configurations, groups of passengers that board together (e.g., families), and the use of apron buses rather than jet bridges for passengers to reach the airplane. Epidemiological research may examine how different types of diseases spread within an airplane. Additional boarding methods and health metrics may emerge.

The paper is accompanied by videos made for the S₁ luggage scenario for all eight boarding methods. The videos can be accessed at the following link: https://github.com/liviucotfas/applied-sciences-normal-times (accessed on 31 May 2020).