Analytical approach to solve the problem of aircraft passenger boarding during the coronavirus pandemic
PPreprint - Article
Analytical approach to solve the problem of aircraftpassenger boarding during the coronavirus pandemic
Michael Schultz and Majid Soolaki Institute of Logistics and Aviation, Dresden University of Technology School of Mechanical and Materials Engineering, University College Dublin† These authors contributed equally to this work.Academic Editor: nameReceived: date; Accepted: date; Published: date
Abstract:
The handling processes at the airport will be significantly changed as a consequence of thecorona epidemic. We expect pandemic requirements will be establish as permanent as the inherentrequirements for safety and security in aviation. During the aircraft boarding, passengers are near eachother, which requires both an effective rule to guarantee physical distances and an efficient procedureto obtain appropriate boarding times. We design an optimal group boarding method using a stochasticcellular automata model for passenger movements, which is extended by a virus transmission approach.Furthermore, a new mathematical model is developed to determine an appropriate seat layout forgroups. The proposed seating layout is based on the idea that group members are allowed to have closecontact and that groups should have a distance among each other. The sum of individual transmissionrates is taken as the objective function to derive scenarios with a low level transmission risk. Afterthe determination of an appropriate seat layout, the cellular automata is used to derive and evaluate acorresponding boarding sequence aiming at both short boarding times and low risk of virus transmission.We find that the consideration of groups in a pandemic scenario will significantly contribute to a fasterboarding (reduction of time by about 60%) and less transmission risk (reduced by 85%), which reachesthe level of boarding times in pre-pandemic scenarios.
Keywords: aircraft boarding, virus transmission, COVID19, pandemic requirements, cabin operations,passenger groups, optimization model, genetic algorithm
1. Introduction
Air transportation offers international mobility in a globally well-connected network. In view of thecurrent pandemic situation caused by the new coronavirus SARS-CoV2, air transportation is part of thetransmission chains and is sustainably affected [1]. Along the passenger journey, the cabin operationsdemand to share a constricted environment with other passengers during boarding, flight, and deboarding.Thus, these operations hold the potential for virus transmissions between passengers and require anappropriate seat allocation and strategies to reduce the transmission risk significantly. The constraintsfrom pandemic situations lead to changes in passenger handling. Currently, airlines aim to protectpassengers and crews from Covid-19 and see face covering as mandatory for passengers onboard. Aphysical distance between passengers during aircraft boarding and deboarding is also part of airlines riskmitigation strategies. With our contribution we provide a customer-oriented solution for both airlinesand passengers, which enables a situative approach to establish appropriate seat layout and aircraft entrysequences considering minimum interactions between groups of passengers. a r X i v : . [ phy s i c s . s o c - ph ] J u l reprint - submitted to journal - Michael Schultz and Majid Soolaki The SARS outbreak in 2002 emphasizes the important role of air transportation in pandemic situations[2]. The climate control system of aircraft seems to reduce the spreading of airborne pathogens byfrequently recirculating the cabin air through high efficiency particulate air (HEPA) filters [3,4]. Thesefilters are designed to filter at least 99.95% of aerosols and are capable of removing viruses and bacteriaattached to droplets. But the transmission of infectious diseases is likely to be more frequent than reportedfor several reasons, such as a much shorter flight times than incubation periods. When considering thepassenger path to and in the aircraft cabin, upstream and downstream processes can also lead to infections(e.g. baggage handling, security checks). To minimize physical interactions, current handling approachesaim at a contactless passenger journey through the airport terminal. In a post-pandemic scenario thiscontactless journey could include biometric scans or the use of personal mobile devices for services orinflight entertainment. Studies on reported in-flight transmissions emphasize that proximity to the indexcase increases the risk of transmission [2,5]. The simulation of transmissions during flight, based on actualpassenger behaviors in single-aisle aircraft, indicate a low probability of direct transmission to passengersnot seated in close proximity to an infectious passenger [6]. An investigation of a long-haul flight indicatesa low risk of pandemic influenza transmission close to infected passengers with symptoms [7]. Thecalculation of the spatial and temporal distributions of droplets in an aircraft cabin showed a reducedinfluenza transmission risk, if respirator masks are used by the passengers [8]. The documentation of asymptomatic SARS-CoV2 index case flying a 15 hour trip in economy class shows that all 25 passengersbeing seated within a range of 2 m from the index case were tested negative for SARS-CoV2 [9]. Twoother case studies reports 11 transmissions [10] and one potential infection during a flight [11]. A briefintroduction about the understanding of SARS-CoV2 in the context of passenger boarding is given at [12].
Comprehensive overviews are provided for aircraft ground operations, passenger boarding, andcorresponding economic impact [13–20]. A common goal of simulation-based approaches for passengerboarding is to minimize boarding time. Thus, the efficiency of different boarding strategies was focus ofthe research activities [21–28]. These models are based on cellular automaton or analytical approaches, butalso other models were developed: mixed integer linear programs [29,30], statistical mechanics [31], powerlaw rule [32,33], cellular discrete-event system specification [34], stochastic approach covering individualpassenger behavior and aircraft/airline operational constraints [15,26].The quantity and quality of hand luggage determine the duration of boarding significantly. Thus,research was conducted with a particular focus on the physique of passengers (maximum speed), thequantity of hand luggage, and individually preferred distance [35], seats assigned to passengers withregards to hand luggage [28,36–38]. Furthermore, the fact that passengers travel in groups has an impacton the boarding efficiency [15,39]. Other research aims at the evaluation of pre-boarding areas [40,41],consideration of passenger expectations [42], use of apron busses [43], and real-time seat allocation [44,45].The impact of different aircraft cabin layouts on passenger boarding were focused on the following studies:aircraft interior design (seat pitch and passengers per row) [46], aircraft seating layouts and alternativedesigns single and twin-aisle configuration [47,48], impact of aircraft cabin modifications [49], novel aircraftconfigurations and seating concepts [50,51], and dynamic change of the cabin infrastructure [52]. Onlyfew experimental tests have been conducted to provide data for the calibration of input parameters andvalidation of simulation results: using a mock Boeing 757 fuselage [53], time to store hand luggage itemsin the overhead compartments [54], small-scale laboratory tests [55], evaluation of passenger perceptionsduring boarding/deboarding [56], operational data and passenger data from field trial measurements reprint - submitted to journal - Michael Schultz and Majid Soolaki [57,58], field trials for real-time seat allocation in connected aircraft cabin [44], and using a B737-800mock-up (1/3 size) to explore the factors effecting the time of luggage storage [59].There are two new research contributions available, which set a focus on behaviors during pandemicsituations and their impact on the aircraft boarding procedures. The first research addresses the quantityand quality of passenger interactions [60] and the second research additionally develops and implementsa transmission model to provide a more detailed evaluation [12]. With a focus on airport operations,the impact of physical distances on the performance of security control lanes was analyzed to provide areliable basis for appropriate layout adaptations [61].
We provide in this contribution an approach for aircraft boarding considering pandemic scenarios.These scenarios are mainly driven by the requirement of physical distance between passengers to ensurea minimal virus transmission risk during the boarding, flight, and deboarding. We consider passengergroups as an important factor to derive an appropriate seat layout and boarding sequence. The main ideabehind the group approach is that members of one group are allowed to be close to each other, as they arealready in close contact with each other before boarding, while different groups should be separated as farapart as necessary. Deboarding is not explicitly considered in our contribution. The paper is structuredas follows. After the introduction (Sec. 1), we present a stochastic cellular automata approach, whichis used for modeling the passenger movements in the aircraft cabin (Sec. 2). A transmission model isimplemented to evaluate the virus transmission risk during passenger movements and applied to evaluatestandard boarding procedures. In Sec. 3, we introduce a problem description and optimization strategiesconsidering passenger groups. The results of the optimization model are presented in Sec. 4, where we usea genetic algorithm for solving the complex problem. The achieved seat layouts are used as input for thepassenger movement model to derive an appropriate boarding sequence with a minimized transmissionrisk during boarding. Finally, our contribution ends with conclusion and outlook (Sec. 5).
2. Passenger boarding model using operational, individual, pandemic constraints
The initial model for movements of pedestrians was developed to provide a stochastic approachcovering short (e.g. avoid collisions, group behavior [62]) and long-range interactions (e.g. tacticalwayfinding) of human beings [63]. This cellular automata model is based on an individual transitionmatrix, which contains the transition probabilities to move to adjacent positions around the currentposition of the passenger [64].
To reflect operational conditions of aircraft and airlines (e.g. seat load factor, conformance to theboarding procedure) as well as the non-deterministic nature of the underlying passenger processes (e.g.hand luggage storage) the stochastic model was developed [26,47] and calibrated [57,58]. The model willbe used for the passenger movements during the aircraft boarding. The passenger boarding is modeledwith a cellular automata approach based on a regular grid (Fig. 1). This regular grid consists of equal cellswith a size of 0.4 x 0.4 m, whereas a cell can either be empty or contain exactly one passenger.The boarding progress consists of a simple set of rules for the passenger movement: a) enter theaircraft at the assigned door (based on the current boarding scenario), b) move forward from cell to cellalong the aisle until reaching the assigned seat row, and c) store the luggage (aisle is blocked for otherpassengers) and take the seat. The storage time for the hand luggage depends on the individual numberof hand luggage items. The seating process depends on the constellation of already used seats in thecorresponding row. A scenario is defined by the seat layout, the number of passengers to board, the arrival reprint - submitted to journal - Michael Schultz and Majid Soolaki front door rear door1 3 5 7 29272523... ...seat rowseat aisle
Figure 1.
Grid-based model - Airbus A320 with 29 seat rows and 6 seats per row (reference layout). frequency of the passengers at the aircraft, the number of available doors, the boarding strategy and theconformance of passengers in following the current strategy. Further details regarding the model and thesimulation environment are available at [15].In the simulation environment, the boarding process is implemented as follows. Depending on theseat load, a specific number of randomly chosen seats are used for boarding. For each seat, an agent(passenger) is created. The agent contains individual parameters, such as the number of hand luggageitems, maximum walking speed in the aisle (set for all agents to 0.8 m/s [44,58]), seat location, time tostore the hand luggage and arrival time at the aircraft door (both stochastically distributed). The agents aresequenced with regard to the active boarding strategy. From this sequence, a given percentage of agentsare taken out of the sequence (non-conformant behavior) and inserted into a position, which contradictsthe current strategy (e.g. inserted into a different boarding block).A waiting queue at the aircraft door is implemented and each agent enters this queue at the arrivaltime. In each simulation step, the first agent of the queue enters the aircraft by moving to the entry cell ofthe aisle grid (aircraft door), if this cell is free. Then, all agents located in the aisle move forward to the nextcell, if possible (free cell and not arrived at the seat row), using a shuffled sequential update procedure(emulate parallel update behavior [63,64]). If the agent arrives at the assigned seat row, the correspondingcell at the aisle is blocked until the hand luggage is stored. Depending on the seat row condition (e.g.blocked aisle or middle seat or both), additional time is added to perform the seating process (seat shuffle).When the seating process is finished the aisle cell is set free. Each boarding scenario is simulated 125,000times, to achieve statistically relevant results defined by the average boarding time (starts when the firstpassenger arrives the aircraft and finished when the last passenger is seated) and the standard deviation ofboarding times.Boarding strategies are derived from three major approaches: boarding per rows (aggregated toblocks), boarding per seat (window, middle, aisle), and sequences of specific seats. Fig. 2 (left) depicts howthe boarding strategies and operational constraints are implemented in the boarding model. The seatsare color-coded to emphasize the order of aircraft seats in the boarding sequence. Six different boardingstrategies are generally considered: random, back-to-front (based on 2 blocks), optimized block (based on6 blocks), outside-in (window seats first, aisle seats last), reverse pyramid (back-to-front plus outside-inwith 6 blocks), and individual seating.Thus, boarding strategies range from random boarding without a specific order to individual boarding,which is a specific solution of the optimized block (alternating seats) and the outside-in strategy (each blockcontains only one seat). Fig. 2 (right) illustrates how the operational constraints of 1 st class seats, passengerconformance, seat load factor, and the existence of passenger groups are covered by the boarding model. The fundamental cellular automata developed for the stochastic passenger movements is extendedby an approach to evaluate the risk of a virus transmission during the boarding process. We are notconsidering face masks in our approach. The transmission risk can be defined by two major input factors:distance to the index case and reduction of contact time. A straight forward approach is to count both the reprint - submitted to journal - Michael Schultz and Majid Soolaki frontrear random businessclass seats non-compliantbehavior seat loadfactor passengergroups major boarding strategies operational constraints back-to-front(2 blocks) optimizedblock (6 blocks) outside-in reversepyramid individual(staggered) frontrear
Figure 2.
Overview of different boarding strategies (darker seats are boarded first; black - blue - green) andimplementation of operational constraints in the cellular automata model. individual interactions (passengers located in adjacent cells) and the duration of these contacts in aisle andduring the seating process. However, counting the individual contacts will only provide a first indicationabout potential ways of infections.We derived a more comprehensive approach, which is based on the transmission model [65] definingthe spread of SARS-CoV2 as a function of different public distancing measures [66]. The probability of aperson n to become infected in a time step t is described in Equation (1). P n , t = − exp (cid:0) − θ ∑ SR m , t i nm , t t nm , t (cid:1) (1)defined by: P n , t the probability of the person n to receive an infectious dose. This shall not be understoodas “infection probability”, because this strongly depends on the immune response by theaffected person. θ the calibration factor for the specific disease.SR m , t the shedding rate, the amount of virus the person m spreads during the time step t . i nm , t the intensity of the contact between n and m , which corresponds to their distance. t nm , t the time the person n interacts with person m during the time step t .In our approach, we define the shedding rate SR as a normalized bell-shaped function (Eq. 2) with z ∈ ( x , y ) for both longitudinal and lateral dimensions, respectively (see [12]).SR xy = ∏ z ∈ ( x , y ) (cid:32) + | z − c z | a z b z (cid:33) − (2) reprint - submitted to journal - Michael Schultz and Majid Soolaki The parameters are a (scaling factor), b (slope of leading and falling edge), and c (offset) to determinethe shape of the curve. The parameters have been initially set to a x = b x = c x = a y = b y = c y =
0. This generates a slightly smaller footprint in y-direction (lateral to moving direction)than in x-direction (longitudinal to moving direction). Additionally, the spread in x-direction is higher infront of the index case than behind it (see Fig. 3). Consequently, the moving direction is changed by 90degrees with a heading to the aircraft window, when the passenger arrives his seat row. [ P \ P S U R E D E L O L W \ [ P S U R E D E L O L W \ f ( y ;0.6,2.5,0.25) \ P S U R E D E L O L W \ f ( y ;0.65,2.7,0.0) Figure 3.
Transmission probability for longitudinal (x) and lateral (y) components and as two-dimensionalprobability field (right).
Finally, the individual probability for virus transmission P n is corresponds to Θ , the specific intensity(dose) per time step (Eq. 3). We set Θ to , which means a passenger reaches a probability of P n = xy = α ∈ {
1, 2 } is 1and changed to 2 when the passenger stores the luggage or enters the seat row. This doubled sheddingrate reflects the higher physical activities within a short distance to surrounding passengers. P n = Θ SR xy α (3) We introduce a baseline setup to depict the results for the evaluation of transmission risks, consideringa seat load factor of 85%, a conformance rate of 85%, and an inter-arrival time of 3.7 s (exponentialdistributed) [15]. Tab. 1 shows the comprehensive evaluation of transmissions around one infectedpassenger, which is randomly seated in the aircraft cabin. Two different scenarios are evaluated againstthe reference implementation (R) of the boarding strategies: (A) applying a minimum physical distancebetween two passengers of 1.6 m, and (B) additionally to the physical distance, the amount of handluggage items is reduced by 50%. Furthermore, the use of two aircraft doors in the front and at the rearis evaluated (A2 and B2) using the transmission risk and boarding time as indicators. In particular, theback-to-front strategy (2 blocks: front block with rows 1-15 , rear block with rows 16-29) exhibits lowervalues for the transmission probability than the optimized block strategy (using 6 blocks of aggregatedseat rows) (see [15]). When passengers board (block-wise) from the back to the front, the chance to pass aninfected person is reduced to a minimum, which is confirmed by the reduced transmission probabilityexhibited in Tab. 1. This effect is also a root cause of the low transmission risks of the outside-in, reversepyramid, and individual boarding strategy. reprint - submitted to journal - Michael Schultz and Majid Soolaki
Table 1.
Evaluation of transmissions risk assuming one SARS-CoV2 passenger in the cabin. The simulatedscenarios are: (R) reference implementation [15], (A) 1.6 m minimum physical distance between twopassengers, (B) additional reduction of hand luggage by 50%, (A2) and (B2) use of two door configuration[12]. transmission risk boarding time (%)boarding strategy R A B A2 B2 R A B A2 B2random 5.9 1.6 1.1 1.4 1.0 100 198 154 133 103back-to-front (2 blocks) 5.6 1.4 1.0 1.2 0.8 96 220 169 153 116optimized block (6 blocks) 6.5 2.3 1.5 1.5 1.0 95 279 210 166 125outside-in 3.5 0.4 0.2 0.3 0.1 80 161 116 107 77reverse pyramid 3.0 0.2 0.1 0.2 0.1 75 185 128 119 82individual 2.0 0.2 0.1 0.2 0.1 66 114 104 103 74deboarding 10.0 9.7 7.8 7.6 6.0 55 97 68 52 36
The use of two aircraft doors for boarding will provide an appropriate solution for a reducedtransmission risk inside and outside the cabin, if near apron stands could be used and passengers couldwalk from the terminal to the aircraft. This kind of walk boarding also prevents passengers from standing inthe badly ventilated jetway during the boarding. Deboarding is difficult to control by specific proceduresgiven that passengers demonstrated little discipline and high eagerness to leave the aircraft. More attentionshould be paid to this process and consideration should also be given to procedural or technical solutionsto provide passengers better guidance and control.
3. Optimized Boarding of Passenger Groups
In contrast to the prior analysed standard boarding procedures, we will provide a new model anda new optimization strategy which incorporates passenger groups and considers the requirements ofphysical distances in the aircraft cabin. In the following, we generally describe the mathematical problemand formulate the optimization model.
We develop a new mathematical model to determine an optimal strategy for assigning seats incabin under the objective to minimize the virus transmission risk. The idea to create an appropriate seatallocation for a pandemic situation includes three assumptions. The first one is that an airline couldassign just a percentage of the available seats (e.g. 50%) to reduce the virus transmissions in cabin andthis strategy will be the primary solution to face with the pandemic situation. The next assumption isabout minimizing passenger contacts or maximizing the distances between passengers in the cabin andguaranteeing at the same time that the confined space inside the aircraft is used efficiently.Looking at Fig. 4, passengers have maximized distances from each other respecting the limitationthat only 50% of the seats can be occupied. Each airline company could determine the seat load factor foreach flight individually also considering risk assessments or economic reasons. Indeed, many airlines arecurrently operating the generally accepted strategy of the empty middle seat.Although complex boarding strategies, such as outside-in, reverse pyramid and individual leadto better boarding times, there will be an issue. The boarding process is driven by the willingness ofpassengers to follow the proposed strategy. We will assume a group of four members (e.g. a family) to beseated. If one of these boarding strategies are applied, they will have just two options. The first one isseating near each other, therefore they have to split during the boarding (see Fig. 2). The next option isremaining as one group in the boarding sequence and as a result they have to seat in different rows. Both reprint - submitted to journal - Michael Schultz and Majid Soolaki front door1 3 5 7 29272523... ...seat rowoccupied seats aisle
Figure 4.
Fifty percent of the seats will be allocated to passengers during the pandemic situation accordingto a pattern with maximum physical separation. options are inconvenient for group members (families). Here we propose to look at the group members asa community, since they were already in close contact before boarding. The strategy that is used in Fig. 4depicts a general solution, but it could be improved considering groups. Without loss of generality, wecould suppose that the transmission rate for the members of each group is zero, which will result in abetter use of space and create a new pattern.The introduced concept of a shedding rate of infected passengers will be used here as well. If aninfected passenger was assigned to different columns, the several shedding rates must be counted basedon the location of the adjacent locations. Taking Fig. 5 as an example, when a passenger seated in row i =
21 and column C (aisle), we compute the shedding rate for the passenger from other groups that seatin the same row ( i =
21 at column A (window), B (middle), and D (aisle)) and previous row i − = F B E C D1 11 11 133333333 A1 2 2 22 2 215 54 4 4664
Figure 5.
Different types of interactions generated around the infected passengers (coded red).
Based on the assumptions of the problem description, we list the sets, parameters, and decisionvariables for achievement of the purposes of the research. reprint - submitted to journal - Michael Schultz and Majid Soolaki
Notation Definition
Sets and Indexesi
Index set of row i ∈ {
1, 2, . . . , I} j Index set of column j ∈ {
1, 2, . . . ,
J } k Index set of group k ∈ {
1, 2, . . . , K} r Index set of interaction type r ∈ {
1, 2, . . . , R} ParametersT k Number of members in the group kSR r The related shedding rate for interaction rDecision Variablesx ijk
Binary variable, equals one if a passenger from group k is seated in a seatin row i and column j ; equals zero otherwise d ijk The summation of shedding rates that the passengers of other groups can causefor a passenger from group k who is seated in a seat in row i and column j The proposed a mixed-integer linear programming model for the problem is introduced as follows. min I ∑ i = J ∑ j = K ∑ k = d ijk (4) K ∑ k = x ijk ≤ ∀ i , j (5) I ∑ i = J ∑ j = x ijk = T k ∀ k (6)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x i ( j + ) k (cid:48) + SR x i ( j + ) k (cid:48) } ≤ d ijk ∀ i = j = k (7)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) SR { x i ( j − ) k (cid:48) + x i ( j + ) k (cid:48) } ≤ d ijk ∀ i = j =
2, 5, k (8)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x i ( j − ) k (cid:48) + SR x i ( j − ) k (cid:48) + SR x i ( j + ) k (cid:48) } ≤ d ijk ∀ i = j = k (9)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x i ( j − ) k (cid:48) + SR x i ( j + ) k (cid:48) + SR x i ( j + ) k (cid:48) } ≤ d ijk ∀ i = j = k (10)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x i ( j − ) k (cid:48) + SR x i ( j − ) k (cid:48) } ≤ d ijk ∀ i = j = k (11)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x ( i − ) jk (cid:48) + SR x ( i − )( j + ) k (cid:48) + SR x i ( j + ) k (cid:48) + SR x i ( j + ) k (cid:48) } ≤ d ijk ∀ i ≥ j = k (12)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x ( i − )( j − ) k (cid:48) + SR x ( i − ) jk (cid:48) + SR x ( i − )( j + ) k (cid:48) + SR x i ( j − ) k (cid:48) + SR x i ( j + ) k (cid:48) } ≤ d ijk ∀ i ≥ j =
2, 5, k (13) reprint - submitted to journal - Michael Schultz and Majid Soolaki
10 of 19 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x ( i − )( j − ) k (cid:48) + SR x ( i − ) jk (cid:48) + SR x ( i − )( j + ) k (cid:48) + SR x i ( j − ) k (cid:48) + SR x i ( j − ) k (cid:48) + SR x i ( j + ) k (cid:48) } ≤ d ijk ∀ i ≥ j = k (14)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x ( i − )( j − ) k (cid:48) + SR x ( i − ) jk (cid:48) + SR x ( i − )( j + ) k (cid:48) + SR x i ( j − ) k (cid:48) + SR x i ( j + ) k (cid:48) + SR x i ( j + ) k (cid:48) } ≤ d ijk ∀ i ≥ j = k (15)6 ( x ijk − ) + K ∑ k (cid:48) = k (cid:54) = k (cid:48) { SR x ( i − )( j − ) k (cid:48) + SR x ( i − ) jk (cid:48) + SR x i ( j − ) k (cid:48) + SR x i ( j − ) k (cid:48) } ≤ d ijk ∀ i ≥ j = k (16) x ijk ∈ {
0, 1 } , d ijk ≥ ∀ i , j , k (17)The summation of shedding rates of all passengers, as objective function, is minimized in equation (4).Constraints (5) guarantee that each seat would be assigned to not more than one passenger. The numberof group members are indicated by constraints (6). Constraints (7)-(11) correspond to the shedding ratesof passengers that are seated in the first row (i=1) in cabin. For instance, if a passenger was seated in aseat in column C (j=3), then the shedding rate for that passenger can be calculated based on the otherpassengers of different groups that were seated in columns A (j=1), B (j=2), and D (j=4) on constraints (9).Also, the shedding rates of passengers that are seated in other rows in cabin are computed by constraints(12)-(16). Here, we must consider the shedding rates not only for passenger in the same row, but also forthe previous row. For instance, if a seat in the second row which located in column C (j=3), was assignedto a passenger, then the shedding rates of the other passengers from different groups that were seated incolumn A (j=1), B (j=2), and D (j=4) of that row and columns B (j=2), C (j=3), and D (j=4) of the first row arecalculated based on constraints (14) as well. On the left hand side of the constraints (7)-(16), the first termtakes a value of zero if the seat (i, j) is assigned to group k and −
4. Application of the Model and Evaluation of the Results
We solved the mathematical model for a small size problem ( i , j , k = i , j , k = The GA has many applications in the optimization problems. In light of the NP-hard class of theseat layout and boarding problem, several methods were conducted to present the optimal/near optimalsolutions [24,30]. Therefore, we designed an GA to solve the problem. The proposed chromosome structureis represented as follows: reprint - submitted to journal - Michael Schultz and Majid Soolaki
11 of 19 C = y y y y y y y y y y y y ... ... ... ... ... ... y y y y y y , y i , j = k , if x i , j , k =
1, otherwise zero.The value of each array of the matrix such as y i , j is the group’s number of related decision variable x i , j , k , which is k if the seat (row i , column j ) is assigned to a passenger group k , otherwise it takes zero. Thepopulation of chromosomes for the first generation are created based on the structure above. Also, weevaluate each chromosome with it’s fitness function value which is determined by the value of the originalobjective function. After that, we implement several operators as follows to create the next generationsfrom the current generation: selection, crossover, mutation, migration, and elitism operator.The selection operator (roulette wheel) ensures that each chromosome with a lower fitness functionvalue is more likely to be selected. New offsprings are created by a recombination of parental genes.Therefore, the group’s numbers are divided into two separate sets. The first offspring receives their genes(the value of arrays in Matrix C ) of the first set from the first parent and the second part from the secondparent and vice versa for the second offspring (see Fig. 6 (left)). Here, we explain with colors to clarify theimplication of operation. For example, the first offspring receives their genes which are colored with lightgreen, light blue, purple, and red from the first parent and receives the genes which are colored with darkgreen, navy, pink, orange, and yellow from the second parent. If there is overlap between a gene’s locationof the first and second parent, then we use a random strategy to select another array in matrix and valueit (e.g. for the first offspring, instead of y =
31, we randomly set y =
31 because the array (5,5) wascolored with light green before or y = A B C D E F 1234 A B C D E F
First parent Second parent Second offspring First offspring
First Mutation Operator Second Mutation Operator Third Mutation Operator
Figure 6.
Crossover operator (left) and mutation operators (right).
Therefore, we always create feasible solutions. The mutation operator is used to maintain the diversityof solutions. Therefore, we designed several operators to change a number of genes in a chromosome tocreate a new chromosome for the next generation (see Fig. 6 (right)). In the first mutation operator, wechange the seat location of a passenger ( y = y = y =
31 and y =
24 ). Therefore,after implementation of the operator, we have: y =
24 and y =
31. Finally, the arrays of two randomrows (the third and fourth rows) are changed in the last mutation operator. A low percentage of each reprint - submitted to journal - Michael Schultz and Majid Soolaki
12 of 19 generation is randomly transferred to the next generation (migration operator). The elitism operator selectsthe best chromosomes in terms of fitness function value, and transfer them form the current generation tothe next generation. The following parameters were used for executing the code: initial population = 1000,number of generations = 1000, crossover rate = 0.55, mutation rate = 0.35, elitism = 0.075, and migrationrate = 0.025.We consider 8 groups with one member (i.e. G1 to G8 which coded with green color), 9 groups oftwo members (i.e. G9 to G17 which coded with blue color), 5 groups of three members (i.e. G18 to G22which coded with purple color), 3 groups of four members (i.e. G23 to G25 which coded with ping color),3 groups of five members (i.e. G26 to G28 which coded with red color), 2 groups of six members (i.e. G29to G30 which coded with orange color), and finally a group of seven members (i.e. G31 which codedwith yellow color). Fig. 7 depicts an optimized solution for the seating layout based on the designed GAgenerated for the 31 groups (87 passengers).
Figure 7.
Best layout to seat 31 groups (87 passengers, 50% seat load) solved by GA approach.
The run time for GA is 1805 s, the value of the objective function for the best solution is 9.1916. Theevolutionary diagram concerning the GA is shown in Fig. 8. The fitness function of the elite and the meanof each generation demonstrate the increasing quality of generated solutions (decreasing fitness function).
Number of generation F i t n e ss F un c t i o n V a l u e Best Fitness Function ValueMean of Fitness Function Values
Figure 8.
Progress of GA fitness function.
To understand the impact of our group approach, we introduce five scenarios and their relatedsolutions based on the assumptions below and compare them by the values of the objective function andthe number of passengers considered. • Scenario 1: Aircraft seats are assigned randomly to passengers with a maximum distance and a seatload of 50% (87 passengers). • Scenario 2: Similar with scenario 1, while the group members are seated close to each other in thesame area. reprint - submitted to journal - Michael Schultz and Majid Soolaki
13 of 19 • Scenario 3: Optimized solution from mathematical modelling and GA application (indicated inFig. 7). • Scenario 4: Optimized solution considering increased seat load of 66% (115 passengers). • Scenario 5: Optimized solution considering for maximum number of passengers (174).The corresponding solutions for the scenarios are illustrated in Fig. 9. In the scenarios 1 to 3, thenumber of passengers is fixed to 87 passengers. The values of the objective function (O.F.) of these threescenarios exhibit that our approach (scenario 3) for an optimized seat layout results in a significantlyreduction of the transmission risk: a reduction of 94% compared to scenario 1 (seats are assigned randomlyto passengers with a maximum distance), and a reduction of 90% compared to scenario 2 (seats areassigned randomly, group members in the same zone). In addition, the optimization method uses theavailable space in the best way, so airlines could benefit from our approach. For example, although wewere increasing the number of passengers by 33% and created the scenario 4, the objective function of thatscenario is still lower than the second case. Finally, the consideration of a seat load of 100% indicates anupper boundary (scenario 5) for the objective function.
Scenario 1: Pass. No.= 87O.F.= 154.155 Scenario 2: Pass. No.= 87O.F.= 89.099 Scenario 3: Pass. No.= 87O.F.= 9.191 Scenario 4: Pass. No.= 115O.F.= 83.514 Scenario 5: Pass. No.= 174O.F.= 431.051
Scenarios 1,2 and 3 Scenario 4 Scenario 5
Figure 9.
Five different solutions for optimized seat allocation in the aircraft cabin considering 87, 115, and174 passengers assigned to different groups.
If the seat load increases over 50% (87 passengers) the values of the objective function (transmissionrisk) progressively increases as shown in Fig.10. Assuming an average seat load of 85% (147 passengers)airlines could significantly reduce the transmission risk by two third by implementing our group approach reprint - submitted to journal - Michael Schultz and Majid Soolaki
14 of 19 and a reduced seat load of 66% (115 passengers). To show the general behaviour of the objective function,we use a power law function y = ax b with a = × − and b = The number of passengers F i t n e ss F un c t i o n V a l u e Best Fitness Function ValuePower Law Function
Figure 10.
Progressive increase of transmission risk over the number of seated passengers.
The implementation of the mixed-integer linear programming model and the genetic algorithm resultin an optimized layout for the passengers to be seated in the aircraft cabin. This layout will be usedas input for the passenger boarding model, which was extended by a transmission module to evaluatetransmission risk during aircraft boarding, to derive an optimum sequence to board the passengers. In ourcontribution, we will not provide an optimization of the deboarding process.Analyses in the context of appropriate boarding sequence accompanied by the introduction ofinfrastructural changes showed that an optimized sequence comprises a mix of boarding per seat (fromwindow to aisle) and per seat row (from the rear to the front) [52]. First and foremost, per-seat boarding(window seats first) is the most important rule to ensure seating without additional interaction in theseat rows. Starting with an outer seat in the last row, the number of group members and the necessaryphysical distance between passengers (1.6 m) defines the subsequently following seat row, which couldbe used in parallel (e.g. 6 passengers with seat row 29 will block the aisle until seat row 27 (waiting), thephysical distance requires to block row 26 and 25, the next group must have seats in front of row 25). Thisprocess of seat and row selection is repeated until the front of the aircraft is reached and is repeated untilall passengers are seated. We further assume that the passengers in each group will organize themselvesappropriately to minimize local interactions. In Fig. 11, the result of this sequencing algorithm is exemplaryillustrated. individual sequence (from back to front)
Figure 11.
Optimized individual boarding sequence considering a physical distance of 1.6 m betweenpassengers (scenario 1).
If the sequencing algorithm is applied to the optimized seat layout from scenario 3, the passengergroups are boarded in five segments. Inside each group, the distance between passengers is not restricted reprint - submitted to journal - Michael Schultz and Majid Soolaki
15 of 19 but between groups it is constrained by 1.6 m (last member of the first group and the first member ofthe following group). The first segment starts with group no. 31 and the last segment with group no. 14(see Fig. 12). As an example, the passengers inside group no. 31 (yellow) are organized by the followingsequence of seats, which results in a minimum of individual seat and row interactions: 29A, 29B, 28A, 28B,27A, 27B, and 27C. Considering distances between groups, the best candidate will be group no. 27 (red)with the seats 23F, 23E, 22F, 22E, and 22D. This sequence allows both groups to start the seating process inparallel, without waiting time due to a too small distance between the seat rows.
Figure 12.
Optimized individual boarding sequence considering a physical distance of 1.6 m betweenpassengers (scenario 3).
In the first three scenarios 87 passengers are boarded with different strategies (see Fig. 9): individualpassengers in a regular pattern (scenario 1), groups in a regular pattern (scenario 2), and groups in anoptimized seat layout (scenario 3). Scenario 1 is used as reference case to evaluate the performance(boarding time) and the transmission risk of Scenario 2 and 3 (Tab. 2a). Therefore, the passenger sequenceis established for both random and individual boarding strategy (optimized, Fig. 11). The boarding timefor the random strategy is set to 100%. As shown in Tab. 2a, the implementation of the individual strategywill reduce the boarding time to 45.2% at a minimum of transmission risk. The consideration of groups(scenario 2 and 3) using the random strategy already reduces the boarding time by about a third at acomparable level of transmission risk. If the optimized seat allocation is used together with the individual(group) boarding the boarding time could be further reduced to 41.1% at a low transmission risk of 0.09new infected passengers at average (85% reduction). Tab. 2b emphasizes the portability of the resultsachieved by the evaluation of scenarios 4 and 5. A corresponding baseline was calculated for each scenario(random boarding of individual passengers). The boarding of groups reduces the boarding time andthe transmission risk, and the optimization of the boarding sequence additionally leads to a significantreduction of the transmission risk of about 65%.
Table 2.
Evaluation of average aircraft boarding times and transmission risk for one randomly seatedinfectious passenger. (a)
Scenario 1,2, 3 with 87 passengers.Sce- Strategy Boarding Transmis-nario Time (%) sion risk1 random 100.0 0.58individual 45.2 0.002 groups, random 68.0 0.62groups, individual 51.9 0.203 groups, random 69.0 0.57groups, individual 41.1 0.09 (b)
Scenario 4 (115 passengers) and 5 (174 passengers)Sce- Strategy Boarding Transmis-nario Time (%) sion risk4 random 100.0 1.11groups, random 60.5 0.94groups, individual 38.1 0.315 random 100.0 2.09groups, random 65.1 1.96groups, individual 34.4 0.66 reprint - submitted to journal - Michael Schultz and Majid Soolaki
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Finally, our results show that optimized group boarding of 174 passenger (scenario 5) possesses atransmission risk of 0.66, which is close to the random strategy in scenario 1 (0.58). Furthermore, thisparticular scenario 5 performs about 20% faster than the random strategy in scenario 1 (87 passengers) andreaches pre-pandemic boarding times.
5. Discussion and outlook
Along the passenger journey, the processes in the aircraft cabin require sharing a confined environmentwith other passengers during boarding and flight. These processes have the risk of virus transmissionbetween passengers and require appropriate seat configuration and risk mitigation strategies. A physicaldistance between passengers during boarding and staggered seat configurations are part of the riskmitigation strategy. However, the side effect from an operational point of view is a doubled boarding timecompared to the situation before the coronavirus pandemic situation.In our contribution, we consider passenger groups as an important factor for the operational efficiency.The main idea behind our approach is that members of one group are allowed to be close to each other, asthey already are before boarding, while different groups should be as far apart as necessary. We providea customer-oriented solution for both airlines and passengers, which enables a situative approach toestablish appropriate seat allocation and aircraft entry sequences considering a minimum transmissionrisk between groups of passengers. Thus, we developed a new mathematical model, which providesan optimized seat allocation, while minimizing the sum of shedding rates that an infected passengercan cause. The developed model was used to evaluate the transmission risk of a seat allocation schemeand to solve this optimization problem with a genetic algorithm for three different scenarios of groupedpassengers (87, 115, 174). The optimization of a standard scenario with a seat load of 50% (87 passengers)shows that with the consideration of groups the value for the objective function was reduced from 154 to 9,which means a significant reduction of the transmission risk induced by the new seat allocation.Five seat and group configurations were used as input for the boarding simulation (stochastic cellularautomata), which evaluates the transmission risk during the passenger movements in the cabin (walkthe aisle, store luggage, take the seat). Therefore, the sequence of groups were optimized to keep theboarding time as low as possible. Our simulation results exhibit that the optimized seat allocation forgroups (scenario 3) performs best for the boarding time (41.1% in relation to random boarding with nogroups) at a low level of transmission risk (0.09, while random boarding without groups leads to a risk of0.58). We could also demonstrate that the effective consideration of passenger groups is a major impactfactor for fast and safe passenger boarding (e.g., board more passengers at the same level of transmissionrisk). In the context of aircraft ground operations (turnaround), shorten boarding times could compensatethe extended ground times caused by additional disinfection procedures in the aircraft cabin.
Conflicts of Interest:
The authors declare no conflict of interest.
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