An Airport Stand Assignment Problem considering the Passenger Boarding Distance

Te continued growth in the civil aviation industry leads to more trafc in the airport, resulting in a decline in operational efciency and the travel experience of passengers. Studying how to improve operational efciency and keep passenger satisfaction simultaneously is very signifcant. Tis study proposes to use total passenger boarding distance instead of total passenger walking distance to quantify passenger satisfaction and then model the airport stand assignment problem considering these two diferent objectives together with the gated percentage, respectively, and the NSGA-II algorithm is improved for a better solution speed. Tis study also performs a case study by applying a dataset of an airport in China. Te results of the case study prove that using the total passenger boarding distance can help the airport better balance operational efciency and passenger satisfaction, which can help provide theoretical support for airport management.


Introduction
Te civil aviation industry has maintained consistent growth over the past decade, resulting in more air trafc demand [1].Te growth challenges the whole industry, especially airports, which are essential parts of the air transportation system.Airports need to face the challenge of more fights and passengers as a result of more trafc demand.It is an undeniable fact that airports are becoming more and more crowded, leading to a steady decline in both operational efciency and the travel experience of passengers.As a result, eforts have been undertaken by both industry and academia to address these challenges.
Te most important stakeholders in the daily operation of an airport include the airport operator itself, the airlines, and the passengers [2].An important task for the airport operators is to assign each fight operated by diferent airlines to diferent stands and gates, and passengers go to these gates to board the fights on stands [3].During these procedures, the airport operators provide the service, while the airlines and passengers are the ones being served.Tis is called as airport gate assignment problem, AGAP for short.
During this procedure, passengers, airlines, and airport operators, all have their own interests and requirements [4].Passengers want more comfortable travel, such as shorter waiting times or boarding distances.Airlines always want their fights to be assigned to specifc gates.For example, if an airline uses a specifc terminal, the airline always prefers its fights to be assigned to gates in that terminal.Te airport operators have to ensure safety and operation efciency, as well as meet certain requirements of the local civil aviation administration.A general logistic procedure for a stand servicing fights is shown in Figure 1.
Academia has made great efort in the study of AGAP.Among those research studies, passenger satisfaction is one of the most focused research objectives.Passenger satisfaction is a metaphysical concept, so it is generally quantifed or characterized by using other characteristics.Te most common characteristics include the distance or walking time for the passengers to board their fights [5].Generally speaking, a shorter walking distance indicates better passenger satisfaction.But a shorter walking distance does not always make passengers satisfy.Te reason is, in an airport, there are always two styles of stands, which are remote stands and the stands with boarding gates, which will be called "gated stands" in the rest of this paper.Using gated stands can increase passenger satisfaction [6].If passengers board a fight at a gated stand, passengers can directly board the aircraft through the gate, which means the boarding distance can be considered as the walking distance.However, if the passengers board a fight at a remote gate, the passenger shall take a shuttle bus, making the total boarding distance to be walking distance together with the driving distance of the shuttle bus.Te driving distance of the shuttle bus can be rather long, especially in some hub airports, which can result in signifcantly reduced passenger satisfaction.Besides, satellite concourses are becoming very common in more and more hub airports.To access a satellite concourse, passengers need to travel with an APM (automated people mover) system [7], which will also increase the boarding distance of passengers and reduce passenger satisfaction.Te boarding procedure is shown in Figure 2.
Tus, we consider it would be interesting from both academic and industrial points of view to research would total boarding distance is a better objective.Motivated by the discussion above, we propose using the total passenger boarding distance instead of the passenger walking distance to quantify passenger satisfaction and compare the strategies under these two objectives.We also take the operational preferences of the airport into account and aim to assign more fights to gated stands.Te main work of this paper includes the following: (1) constructing a multiobjective stand assignment optimization model based on the passenger boarding distance, (2) improving the NSGA-II algorithm to solve this model, (3) validating this model by applying an airport operational dataset, and (4) comparing and discussing the optimal strategies under the minimal total boarding distance and total walking distance.Te paper is organized as follows: Section 2 presents some related literature review, Section 3 discusses the methodology of this research, Section 4 gives out the case study, and Section 5 concludes this research and gives out the further discussion.

Literature Review
Some literatures which related to this study are reviewed in this section, focusing on airport stand assignment and associated modeling techniques and algorithms.
Te problem of airport stand assignment has long been a traditional area of research in daily airport management [8].As indicated in the introductory section, the stand assignment problem infuences a variety of stakeholders, with difering perspectives often leading to a broad set of objectives.Related studies need to fnd a balance between these divergent goals.Moreover, for the stand assignment problem, it is essential for the construction of math models following some universal constraints [9].For safety, the top-most priority in civil aviation procedures, any potential risks or conficts in aviation operations are unacceptable.In the context of stand assignment problems, this safety prerequisite is manifested in the following two non-negotiable constraints: uniqueness and exclusivity.Te constraint of uniqueness dictates that a single stand can be assigned to just one fight (or aircraft) at most at any moment.Te exclusivity constraint, on the other hand, ensures that the aircraft has a sole use of the stand in terms of time and space, that is, one stand can service only one fight (or aircraft) at a time.Tese two constraints form the basis for all stand assignment research.In addition, the stand assignment problem should generally adhere to a few more constraints, depending on the specifc operational rules of diferent airports.Common extra constraints include the stand-aircraft size constraint [10], the stand-fight characteristic constraint [11], the standairline constraint [12], and the time interval constraint [13].
To elaborate, stands can be categorized into various classes depending on the maximum size of the aircraft they can accommodate.Similarly, considering the types of fights they can service, stands can be classifed into domestic, international, or mixed categories.Some airports even have exclusive stands dedicated to one or a few specifc airlines.Hence, for airports with pertinent regulations and needs, stand assignments must observe the following rules: the size of the aircraft assigned to a stand cannot surpass the stand's capacity-smaller stands cannot accommodate larger aircraft, but smaller aircraft can be serviced by larger stands, demonstrating the stand-aircraft size constraint.Domestic fights cannot be assigned to international stands, nor can international stands be assigned to domestic fights, while mixed stands are not subject to these restrictions-this illustrates the stand-fight attribute constraint.For airports featuring exclusive stands, aircraft from diferent airlines cannot be assigned those stands, representing the standairline constraint.In addition, citing safety considerations, most airports prescribe a bufer time, typically between 10 and 50 minutes [14], for two consecutive aircraft using the same stand.Tis sensible bufer time is crucial for enhancing operational safety, averting stand conficts due to unforeseen delays, and strengthening the resilience of the stand assignment strategy [15].Particular problems should be analyzed and model constraints should be established in a manner that aligns with the specifc circumstances of the target airport.Furthermore, in stand assignment research, the model is often judiciously simplifed by establishing certain assumptions to enhance the efciency of the simulation solution.Common assumptions encompass the presumption that each stand operates independently of the others [16], that the aircraft that needs to be assigned stands all operate both an arrival and a departure fight [17], and that all stands are available for assignment at the beginning of assignment [18].
Stand assignments play a crucial role in determining operational efcacy and ensuring passenger satisfaction [19].A well-devised stand assignment strategy can curtail delays, enabling airlines to adhere more closely to their timetables [20] and elevating the overall efciency during an airport's ground turnaround [21].Despite multiple stakeholders being involved in stand assignment, the study of passenger satisfaction-based slot allocation is the oldest and most popular branch of research related to stand assignment, dating back to the 1970s [22].Earlier studies typically aimed to minimize passengers' total walking distance within the terminal to gauge their satisfaction [9].Subsequent research continued to spotlight this aspect, with current studies examining objectives such as the total distance passengers moving within the terminal [23], the total walking distance for transit passengers [24], or the average distance each passenger moves [25].For airline-oriented research in the context of stand assignments, common studies focus on minimizing taxi distances [26] and durations [27] and diminishing delays emanating from stand assignments [28].When considering airport-oriented stand assignment, the emphasis tends to be on assigning more fights to gated stands, maximizing the duration gated stands are occupied [29], or increasing the volume of passengers boarding via these gated stands [6].
Te study of multiobjective optimization of stand assignment problems is also becoming popular in recent years.Te research in this domain primarily addresses the diverse considerations related to passengers, airlines, and airport operators [30,31].Academics formulate mathematical models for multiobjective optimization of these problems and employ advanced optimization algorithms to decipher these models, resulting in a range of Pareto optimal outcomes.By doing so, scholars can balance the interests of all stakeholders and promote a benign game to achieve a multiwin situation.Te algorithms utilized for multiobjective optimization fall into the following two primary categories: exact algorithms and heuristic algorithms [32].While exact algorithms guarantee the procurement of Pareto optimal results, heuristic algorithms, although not ensuring exact optimal outcomes, are adept at handling vast problems and discrete models.Typical heuristic algorithms to tackle multiobjective optimization issues include evolutionary algorithms such as the nondominated sorting genetic algorithm-II (NSGA-II) and clustering algorithms such as the multiobjective particle swarm optimization (MOPSO).
Notably, heuristic algorithms tend to be slower, demand more computational resources, and exhibit lesser convergence compared to exact algorithms.However, in response to the limitations of both exact and heuristic methods, hybrid algorithms have been introduced recently as a means to address multiobjective optimization challenges more efciently [33].
Te abovementioned literature review underscores that the stand assignment challenge frequently encompasses multiple stakeholders, with passenger satisfaction often being a paramount concern.Given this, contemporary research on stand assignment predominantly leans towards a multiobjective optimization methodology.In addition, the viability of heuristic algorithms to tackle extensive stand assignment issues is acknowledged.Given this backdrop of existing research, this paper's primary contribution lies in introducing an innovative evaluation technique, which uses the total distance passengers need to board to measure their satisfaction with stand assignment.Leveraging these evaluation criteria, we construct a multiobjective stand assignment optimization model that considers the percentage of gates used.Utilizing the improved NSGA-II algorithm, this novel evaluation approach is then tested and contrasted using realworld airport operational data.Te detailed methodology of this investigation is elaborated further in Section 3.

Methodology
Tis section introduces the stand assignment model and the corresponding solution algorithm.Te model is for optimizing an airport stand assignment problem with two optimization objectives as follows: minimizing the passenger boarding distance or walking distance and maximizing the proportion of fights assigned to gated stands.

Defnition of the Mathematical Elements.
In this subsection, we will frst defne all the mathematical elements necessary for the modeling in Table 1.
Regarding the sets, we denote the set of all outbound fights as P O � p o 1 , . . ., p o p   and the set of all available airport stands as Q � q 1 , . . ., q q  .Regarding the parameters, for fight ∈ P O , a f i and d f i are defned start and end time at a stand, as given in the data.n i stands for the number of passengers who take fight i ∈ P O , which is calculated by c i , the passenger load factor for fight i ∈ P O and s i , the number of seats for fight i ∈ P O .D ie is the boarding distance for each passenger when fight i ∈ P O is using the stand e ∈ Q while d ie is the walking distance for each passenger when fight i ∈ P O is using the stand e ∈ Q.Also, t s signifes the necessary gap in time between two consecutive fights at the same stand for safety reasons.Regarding the variables, g e i serves as an indicator of stand assignment, which equals 1 if the fight i ∈ P O is assigned to the stand e ∈ Q; otherwise, 0. The variable c i a categorizes the aircraft type for i ∈ P O .flight A value of 2 indicates a double −aisle aircraft, 1 signifies a singleflight − aisle aircraft, and 0 marks a regional aircraft.Te stand size is denoted by c e b .It assumes a value of 2 if the stand e ∈ Q can accommodate a double-aisle aircraft and 1 otherwise.Te presence of a boarding bridge at the stand e ∈ Q is indicated by h e , with 1 meaning yes and 0 meaning no.Lastly, the variable y e ij is set to 1 if the fight j ∈ P O follows fight i ∈ P O at the stand e ∈ Q; otherwise, it remains 0.

Modeling the Stand Assignment Problem.
When tackling the stand assignment problem, a systematic approach is essential.Tis entails designating each fight to a specifc stand, refecting its unique features while adhering to particular optimization goals.Within the scope of this study, the foundational premises and limitations concerning stand assignment are as follows: (1) An individual aircraft is restricted to a single stand; simultaneously, a stand cannot cater to multiple aircrafts.
(2) Te compatibility between the dimensions of the aircraft and the stand is crucial.Tus, only larger stands can accommodate bigger aircraft but smaller aircraft have the fexibility to occupy any stand.(3) For safety protocols, when two consecutive fights are designated to the same stand, a sufcient time interval must be ensured between them.
Furthermore, it is assumed that all the stands in the dataset are available for assignment throughout the entire period under consideration.Initially, no aircraft is present at any of the stands in the dataset.A series of Pareto optimal solutions can be achieved by adhering to these constraints and focusing on the objective functions.
Based on the earlier research motivation and the outlined problem description, the objective functions can be written as follows: Equation (1a) represents the minimum total boarding distance, while equation (1b) represents the minimum total walking distance.Equation (2) represents maximizing the gated percentage.Te objective functions should follow the constraints as follows: (a Equation ( 3) indicates the method to calculate the number of passengers with the passenger load factor and the seat number on fights.Equation (4) limits the relation among y e ij , g e i , and g e j for logical reasons.Equations ( 5) and ( 6) mean an individual aircraft is restricted to a single stand and a stand cannot cater to multiple aircrafts.Equation ( 7) means only larger stands can accommodate bigger aircraft but smaller aircraft have the fexibility to occupy any stand.Equation ( 8) limits that when two consecutive fights are designated to the same stand, a sufcient time interval must be ensured between them.

Algorithms.
In our research, we employ the NSGA-II algorithm, a well-established heuristic technique designed for tackling multiobjective optimization challenges [34].For a multiobjective optimization problem, a set of solutions usually exists which is not comparable in merit among them.Te set of solutions is called Pareto optimal, in which none of the objective functions can be improved without degrading some of the other objective values [35].If a solution is inferior to the Pareto solution set in each objective function, it is called a solution dominated by the Pareto solution set.Te NSGA-II algorithm fnds the Pareto solution set by a fast nondominated ranking and then selects the better individuals in the iteration according to the ranking to participate in the crossover variation to fnd the Pareto optimal solution to the solution problem.Te algorithm applied in this research is an improved NSGA-II algorithm based on a previous research [36].To explain this algorithm in more detail, the individual chromosomes are coded in natural numbers, and the assignment scheme represented by each individual is a feasible solution to the problem.Te length of each chromosome equals the number of fights to be assigned, and the gene at each gene locus is the aircraft position assigned to that fight.For example, if the total number of fights to be assigned is 10 and the available stands are numbered 1, 2, 3, and 4, the chromosome length of each individual is 10, and the chromosome coded as shown in Figure 3 would indicate that fight 1 is assigned to stand 3, fight 2 is assigned to stand 1, fight 3 is assigned to stand 2, and so on.However, the NSGA-II algorithm intrinsically exhibits randomness in its initialization and cross-variance procedures.Concurrently, the stand assignment problem is a quintessential NP-hard issue.As a result, performing random initialization or cross-variance can produce numerous nonviable solutions, hampering the efciency in identifying optimal solutions.Consequently, we have refned the initialization and cross-variance techniques, detailing the modifcations in the pseudocode provided subsequently in Algorithm 1.
To go into more detail, for the initialization, a series of available stands for the frst fight is generated.Ten, a stand is randomly selected from the set of available stands as the initial stand for the frst fight.After this fight is assigned, the occupancy schedule for this stand is updated.Repeat the abovementioned operation to initialize the next fight in the order of the fights until all fights are assigned a stand.Te crossover procedures frst set the number of chromosome crossovers as w and randomly generate the crossover point q, so the crossover position is from q to q + w − 1. Genes from the paternal chromosome outside the crossover position are passed directly to the ofspring.Ten, update the occupancy schedule for all stands.Starting from the gene q of the maternal chromosome, we sequentially determine whether the inheritance of the maternal chromosome to the ofspring will produce a confict.If so, a series of available stands for fight q is generated and then a stand for fight q is randomly selected from this set.If not, the gene q of the maternal chromosome is passed directly to the ofspring.Ten, we update the occupancy schedule for all stands and repeat the abovementioned operation.Te next gene of the maternal chromosome is judged sequentially until the crossover termination point is reached.Te mutation process frst randomly selects a gene r on the chromosome as the point of variation.Ten, a series of available stands for fight r is generated and randomly selected a stand from the set as the stand for fight r.Upon deriving the optimized strategy, it will be presented as a feasible result.

Case Study
4.1.General Information.An operational dataset for one day in April 2023 of Beijing Capital Airport (ICAO: ZBAA, IATA: PEK), which is one of the hub airports in China, is applied for this case study.Te dataset includes all domestic fights operated in terminal 3. Terminal 3 mainly service Air China (ICAO: CCA, IATA: CA) and some other airlines, and the layout of terminal 3 is presented in Figure 4. Te domestic fights are operated in the area of T3C and T3D.Te T3C area has 37 gated stands and the T3D area has 10 gated stands.If passengers need to board a fight at the T3D area, they need to take the APM.Te sample of stands information is presented in Table 2.
Te dataset for the case study included 252 fights using 93 stands.Te information including fight no., type of aircraft, and the number of seats on the aircraft are known in the dataset.Te start and ending times for an aircraft using a stand, also called a f i and d f i , are also known.Te sample of the dataset is presented in Table 3.
Te next step is to determine the value of the mathematical elements.Te elements including and h e can be directly known from the stands information and the dataset.For other elements, n i is calculated by s i and c i from equation (3), and the value of c i can be found in the operation report of airlines and CAAC (Civil Aviation Administration of China).In this case study, the c i for Air China is 73.7\%, and for other airlines, c i is 77.6%.Te D ie and d ie can be valued via the passenger boarding procedure as shown in Figure 2, the distance for walking, APM travel distance and ferry bus travel distance are known from the airport and OSM (open street map), a geographic information system.Considering the practical operation, the safety interval time t s is 10 minutes in this case study.

Findings and Discussion
. Te algorithm is implemented using MATLAB 2020a.Initially, objective functions aiming to minimize the total boarding distance are addressed, setting the population size at 100 and capping the genetic iterations at 300.Notably, convergence is attained after roughly 100 generational iterations.Due to the scale of the dataset being not large enough, there are only 3 sets of solutions on the Pareto front surface.We selected the one with the minimum boarding distance, and the result is shown in Figure 5. Te yellow lines represent the gated percentage, while the blue lines represent the boarding distance.Te solid lines represent the optimized results and the dotted lines represent the result calculated with the dataset.Te total boarding distance after optimization is 39,693,900 meters and the gated percentage is 70.24%.In comparison, the boarding distance calculated from the dataset is 44,956,020 meters and the gated percentage is 75.79%.
Similarly, the objective functions with minimized total walking distance are also processed.Te population size is also 100 and the maximum number of genetics is also 300.Te algorithm achieved a convergence after about 200 generations of iterations and there are 2 sets of solutions on the Pareto front surface.Here, the result with the minimum Journal of Advanced Transportation walking distance is shown in Figure 6.Same with Figure 5, the yellow lines represent the gated percentage, while the blue lines represent the boarding distance, and the solid lines represent the optimized results while the dotted lines represent the result calculated with the dataset.Te total walking distance after optimization is 14,563,760 meters and the walking distance calculated from the dataset is 16,349,850 meters.Te gated percentage is 68.26%.It can be found after the optimization that the total boarding distance decreased by 11.71%, while the gated rate also decreased by 5.55%.When considering the walking distance, the total walking distance decreased by 10.92%, and the gated rate decreased by 7.53%.Tis indicates a large range of optimization in terms of both boarding distance and walking distance.Compared to walking distance, the boarding distance can be improved with less gated percentage reduction.We also calculate the walking distance under the minimized boarding distance and the boarding distance under the minimized walking distance, respectively, are 16,969,250 meters and 53,214,460 meters.Both distances have some increases compared with those calculated from the dataset, and it is reasonable for other objectives to get    6 Journal of Advanced Transportation worse when one objective is optimized.Te passenger walking distance only increases by 3.79% when achieving the minimized passenger boarding distance, while the boarding distance increases by 18.37% when achieving the minimized walking distance.Tis means that the optimizing boarding distance can be achieved without signifcantly increasing the passenger walking distance and better accommodating the need of the airport for the gated percentage.
To compare, we also process a single-objective optimization only considering the gated percentage, and the result is presented in Figure 7. Te maximized gated percentage is 76.19%, while the gated percentage calculated from the dataset is 75.79%.Tis indicates the stand assignment strategy in practice is very near the strategy with the maximized gated percentage.Tis means the airport considers more about the gated percentage more than any other objectives.Te result also illustrates the decrease in gated percentage during the optimization of the boarding distance and walking distance.Improving other objectives will inevitably lead to a decrease in the gated percentage.From this perspective, it may also be a good idea to consider the boarding distance than the walking distance.As this may help the airport sacrifces less on the gated percentage but still ensure the passenger satisfaction.Studying a triobjective optimization considering boarding distance, walking distance, and gated rate may also be interesting.

Conclusion
In this study, we discuss using the boarding distance to evaluate passenger satisfaction instead of using the passenger total walking distance in an airport stand assignment problem.Te main work includes modeling and improving the NSGA-II algorithm for a better solution speed.We also conduct a case study using operational data from a Chinese airport.Tis research can assist airports in enhancing their operational management and shaping strategies from a theoretical perspective.
In the coming works, we are going to consider more realistic scenarios and consider the interests of more stakeholders.It would also be interesting to consider the robustness of the stand assignment strategy.Finally, we believe it would be inspiring to connect this work together with the total operational management of airports and datadriven intelligent airport operation.Journal of Advanced Transportation

Figure 3 :
Figure 3: Sample of the chromosome.
a , c e b , Dis i , dis i , Stand, flight, Fn 1 , Generations U, N U (1) Get totalnum 1 by Fn 1 + flight (2) Initialize the population (N U •totalnum 1 ) (3) for i ← 1 to N U do (4) Selection cells in mating pool (5) Crossover and mutate of cells (6) Ofspring cells into new population (7) if stand assignment not satisfy constraints then (8) Regenerate and replace set of fights' stands (9) end if (10) Calculate objective functions result (11) Nondominate sort the cells (12) Calculate the crowd distance (13) Sort population in descending order (14) end for Output: Best solution for stand assignment ALGORITHM 1: Te NSGA-II algorithm for stand assignment pseudocode.

Figure 5 :
Figure 5: Process of fnding optimized boarding distance and gated percentage.

Figure 6 :Figure 7 :
Figure 6: Process of fnding optimized walking distance and gated percentage.

Table 1 :
Mathematical elements for modeling.

Table 2 :
Data sample for the case study sample of stands information.

Table 3 :
Data sample for the case study sample of the dataset.