An airport cluster flight schedule multi-agent configuration method for operation resource configuration

Abstract

The application discloses an airport cluster flight schedule multi-agent configuration method for operation resource configuration, and belongs to the technical field of aviation transportation. The method firstly constructs a schedule resource fitness evaluation standard, evaluates flights by using an entropy weight method, and selects to-be-optimized flights; then constructs an airport cluster cooperative game flight schedule optimization model, and solves the model by using a whale optimization algorithm; further constructs a dynamic game model covering the interests of airlines, airports, passengers and the like, solves the model by using an improved particle swarm optimization algorithm, and finally adjusts flights according to the optimal solution obtained, so as to realize overall optimized configuration of airport cluster schedule resources, solve the problem that the existing method does not consider the reasonable configuration and optimized utilization of resources when arranging flight schedules between different airports, and improve resource utilization efficiency and system fairness.

Description

Technical Field

[0001] This application relates to the field of aviation transportation technology, and in particular to a multi-entity method for allocating flight slots for airport clusters in order to allocate operational resources. Background Technology

[0002] Currently, airport clusters are playing an increasingly important role as regional transportation hubs in air transport networks. In recent years, the government has increased its investment in airport cluster construction, and their scale has maintained steady growth. However, this growth has also brought about a series of unbalanced and insufficient development problems—primary hub airports within airport clusters are allocated too many slots, easily exceeding their capacity, while secondary and tertiary hub airports typically face low utilization rates. How to rationally allocate flight slot resources within airport clusters to both meet the market positioning of individual airports and improve the overall operational efficiency of the airport cluster is becoming an increasingly important research focus.

[0003] Flight schedule optimization is a systematic management process that scientifically adjusts flight takeoff and landing times and coordinates the allocation of airspace and ground resources to improve air transport efficiency, reduce delays, and maximize resource utilization. Its core lies in balancing supply and demand, addressing the conflict between airport capacity limitations and flight volume growth, and thus playing a crucial role in optimizing aviation resource allocation, improving airline operational efficiency, and achieving equitable development among airports. The problem of optimizing flight schedules within airport clusters must consider not only the rational allocation of flight slots within the cluster but also the interests of other stakeholders in the air traffic system. Currently, airport cluster flight schedule optimization is typically approached from the perspectives of air traffic control, airports, and airlines, using total slot adjustments, delay times, and system punctuality rates as objective functions, but often neglecting passenger needs and experience. Passenger experience is crucial for improving the overall service quality of the civil aviation system, enhancing satisfaction among all parties, and ultimately promoting the healthy development of the air transport system. Furthermore, game theory in the civil aviation field often focuses on how to rationally determine ticket prices from the perspective of airlines or governments; few scholars analyze the game relationships between airports within an airport cluster to optimize flight schedules. Summary of the Invention

[0004] To address the aforementioned shortcomings in the existing technology, this application provides a multi-subject allocation method for airport cluster flight slots oriented towards operational resource allocation, which solves the problem that existing methods do not comprehensively consider the rational allocation and optimized utilization of resources when scheduling flight slots between different airports.

[0005] To achieve the aforementioned objectives, the technical solution adopted in this application is as follows: This application provides a multi-agent method for configuring flight slots in airport clusters based on operational resource allocation, including: S1: Construct a time slot resource fit evaluation standard with the goal of meeting the airport's functional positioning, and use the entropy weight method to determine the evaluation value of the flight, and obtain the flight to be optimized based on the evaluation value; S2: Construct a cooperative game-theoretic flight schedule optimization model for airport clusters, and use the whale optimization algorithm to solve the cooperative game-theoretic flight schedule optimization model for airport clusters to obtain the optimal solution of the cooperative game-theoretic flight schedule optimization model for airport clusters. S3: Based on the needs of air traffic control, airlines, airports and passengers, the multi-objective optimization problem is transformed into a dynamic game problem. A dynamic game theory model of the multi-objective optimization problem is constructed, and an improved particle swarm optimization algorithm with the introduction of Logistic mapping is used to solve the optimal solution of the dynamic game theory model. S4: Based on the optimal solution of the airport cluster cooperative game flight schedule optimization model and the optimal solution of the dynamic game theory model, the flights to be optimized are adjusted to achieve the optimal airport cluster flight schedule configuration.

[0006] Further, S1 includes: S101: Based on the principles of objective evaluation, subjective evaluation, typicality, and operability, and utilizing the airport's own market positioning and operational efficiency, establish multiple indicators for the evaluation system of hub airport slot resources. These indicators include destination categories, flight types, transfer efficiency, flight punctuality rate, and load factor. S102: Standardize each indicator. The standardization formula is as follows:

[0007] In the formula, For standardized sequences, The original sequence, For the first The last sample value of each indicator. For the first One flight review, For the first One indicator, The total number of indicators; S103: Based on the standardized indicators, calculate the information entropy of each indicator. The calculation formula is as follows:

[0008]

[0009] In the formula, For the first Information entropy of each indicator For entropy scaling parameters, For the first The first evaluated flight was on The proportion of each indicator The total number of evaluation objects; S104: Calculate the information entropy redundancy of each indicator. The calculation formula is as follows:

[0010] In the formula, For the first Information entropy redundancy of each indicator; S105: Calculate the weight of each indicator based on the information entropy redundancy. The calculation formula is as follows:

[0011] In the formula, For the first The weight of each indicator; S106: Based on the weights of each indicator, calculate the evaluation value of the flight. The calculation formula is as follows:

[0012] In the formula, For the first The rating value of each flight; S107: Based on the evaluation value of the evaluated flights, evaluate the fit between the flight attributes and the hub airport positioning to obtain the flights to be optimized.

[0013] Further, S2 includes: S201: Treating the airports in the airport cluster as game participants, calculate the marginal contribution of each airport to different cooperation combinations, and allocate benefits based on the marginal contribution. The calculation formula is as follows:

[0014] In the formula, For the first The profit distribution value of each airport. For characteristic function, The number of airports among the participating parties in the alliance. yes Marginal contribution to the alliance, This represents the total number of participants within the airport. To coordinate revenue for the overall operation of the cooperative game model, For the alliance remove Post-cooperative alliance operational coordination revenue; S202: Based on benefit distribution, construct an objective function that minimizes flight schedule adjustments, maximizes flight function positioning indicators, and minimizes the Gini coefficient. The expression for the objective function is:

[0015]

[0016]

[0017] In the formula, For flight schedule adjustments, Flights in the timetable The actual time allocated, Flights in the timetable When to apply As decision variables, For the number of flights, For a set of time periods, As a functional positioning indicator for flights, Indicates the aircraft that took off from Airport Time arrives Do the airport's flights align with its strategic positioning? Gini coefficient, This is a collection of airports within an airport cluster. To adjust the average value at any time, For the first The actual time slots allocated to each airport For the first The number of flights allocated to each airport. Time allocated to the plan; S203: The constraints are flight uniqueness, airport capacity, flight continuity, maximum allowable adjustment, and shared waypoint capacity. The expression for the constraints is as follows:

[0018]

[0019]

[0020]

[0021]

[0022] In the formula, For the airport Airport capacity within a specified time period For time pool, In order to be in Flights departing at specific times In order to be in Flights landing at specific times It is a binary variable. Minimum transfer time, For the maximum transit time, Traffic constraints for shared waypoints in an airport cluster; S204: Constructing a cooperative game-theoretic flight schedule optimization model for airport clusters based on the objective function and constraints. :

[0023] S205: The optimal solution is obtained by using the whale optimization algorithm to solve the airport group cooperative game flight schedule optimization model.

[0024] Furthermore, the step of using the whale optimization algorithm to solve the airport cluster cooperative game flight schedule optimization model to obtain the optimal solution includes: A1: For each flight, a time window is randomly generated within the time window between the earliest and latest times to obtain the initial flight time scheme, and the population size is designed based on the number of flights in the airport cluster; A2: Calculate the performance indicators of flight time adjustment, flight function positioning index, and Gini coefficient for each flight time plan; A3: Compare all initial flight time schemes, find the scheme that is not surpassed by other solutions in terms of performance indicators, and store the found scheme in an external archive as the current set of optimal solutions; A4: Simulate the behavior of whales surrounding prey, capturing prey with bubble nets, and searching for prey. Update the optimal solution set and repair the flight schedule schemes in the newly generated optimal solution set. Recalculate the fitness of the repaired new timetable schemes and add the newly generated high-quality solutions to the external archive. Select the solution in the external archive that is closest to the current flight schedule scheme as a reference. A5: Determine if the current iteration count has reached the preset number. If yes, obtain the optimal solution; otherwise, proceed to A4 for iteration.

[0025] Further, S3 includes: S301: Based on the needs of air traffic control, airlines, airports, and passengers, an objective function is constructed to represent the interests of each party, with the objective function being: minimizing the variance of the total number of time slot adjustments, minimizing the fairness differences among airlines, minimizing the fairness differences among airports, and minimizing passenger lost time. The expression for the objective function is as follows:

[0026]

[0027]

[0028]

[0029] In the formula, To minimize the variance of the total number of adjustments at any given time, To minimize fairness differences among airlines, To minimize fairness differences at airports, To minimize passenger loss time, For the first The number of times a flight is adjusted. This is the average number of flight adjustments across all flights. A collection of all airlines. This refers to the aggregation of flights from all airlines within the airport cluster. This refers to the collection of flights from all airports within the airport cluster. For the airline's fairness bias, For the airport's fairness bias, For airlines Flight offset, This refers to the flight offset at the airport. S302: Air traffic control, airlines, airports, and passengers are treated as the optimization agents in the objective function and as players in a game. The set of decision variables represents the players' strategy set, the objective function serves as the payoff function in dynamic game theory, and the constraints can be used as dynamic game constraints, resulting in a dynamic game theory model for a multi-objective optimization problem. :

[0030] In the formula, Represents the players in a dynamic game. A set of strategies representing the players in the game. Represents the payoff function for the players. For the first dynamic game One of the people involved in the game, For the first A collection of strategies for each player in the game. For the first The payoff function of each player in the game The number of people in the game; S303: Solve the optimal solution of a dynamic game theory model using an improved particle swarm optimization algorithm that incorporates the Logistic mapping.

[0031] Furthermore, the improved particle swarm optimization algorithm includes: introducing a Logistic mapping based on the particle swarm optimization algorithm, improving the inertia weight and learning factor of the particle swarm optimization algorithm, and selecting the reward function as the fitness function to obtain the improved particle swarm optimization algorithm. The expression for the Logistic mapping is:

[0032] In the formula, For state variables, For independent parameters, and For the first The second iteration and the first The next iteration; The improved expressions for inertia weight and learning factor are:

[0033]

[0034]

[0035] In the formula, For the improved inertia weight, and For the improved learning factor, and The boundary value of the dynamic inertia weight. The maximum number of iterations. This represents the current iteration number. This is the initial value of the individual learning factor. This represents the final value of the individual learning factor. This is the initial value for the social learning factor. This represents the final value of the social learning factor. The expression for the fitness function is:

[0036] In the formula, For the fitness function, As a penalty item, It is a binary variable. For an optimization model that includes an objective function, For the first One objective function.

[0037] Furthermore, the method of using the improved particle swarm optimization algorithm incorporating the Logistic mapping to solve the optimal solution of the dynamic game theory model includes: B1: Select data from the flight schedule as the initial particle swarm, use the flight departure time as the initial dimension, initialize the dimension and the maximum number of iterations, and determine the type of chaotic mapping; B2: Determine the specific position and movement speed of the initial particles, where the specific position is the takeoff time of each flight; B3: Calculate the fitness of the particles based on the fitness function; B4: If the fitness of the current particle exceeds the best state in its history, update the optimal solution for the current particle; if the fitness of the current particle exceeds the global best solution in its history, update the global best solution. B5: Select the top-ranked particles by fitness and perform Logistic mapping. Calculate the particle fitness based on the fitness function and update the particles according to their fitness until each particle converges to its corresponding weighted average point. Output the optimal solution.

[0038] The beneficial effects of this application are: This application provides a multi-stakeholder allocation method for airport cluster flight slots oriented towards operational resource allocation. It utilizes a cooperative game-theoretic flight slot optimization model for airport clusters to readjust flight slots that do not conform to the positioning, thereby improving the fairness of flight slot resource allocation within the airport cluster. Furthermore, from the perspective of the entire civil aviation operation system, it establishes a multi-scale, multi-objective optimization model for airport cluster flight slots that considers the needs of all four parties, thereby increasing the capacity utilization efficiency of each airport, reducing flight delays, and improving the overall operational efficiency of the airport cluster. Attached Figure Description

[0039] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other embodiments can be obtained based on these drawings.

[0040] Figure 1 This is a flowchart illustrating a multi-subject configuration method for airport cluster flight schedules oriented towards operational resource allocation, provided in an embodiment of this application.

[0041] Figure 2 This is a schematic diagram of the fitness change curve of a whale optimization algorithm provided in an embodiment of this application.

[0042] Figure 3 A graph showing the relationship between the Gini coefficient and flight deviation is provided for an embodiment of this application.

[0043] Figure 4 This is a schematic diagram illustrating the changes in flight slot resources at the three airports in Beijing and Tianjin before and after optimization, as provided in an embodiment of this application.

[0044] Figure 5 This is a schematic diagram illustrating the distribution and changes in the number of departing flights per unit time at each airport before and after optimization, as provided in an embodiment of this application.

[0045] Figure 6 This is a schematic diagram illustrating the iterative process of the values ​​of four objective functions provided in an embodiment of this application.

[0046] Figure 7 This is a schematic diagram showing the flight schedule distribution of three airports before and after optimization, provided as an embodiment of this application. Detailed Implementation

[0047] The technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments.

[0048] Example 1: This application provides a multi-entity configuration method for flight slots in airport clusters, oriented towards operational resource allocation. Taking the Beijing-Tianjin airport cluster as an example, it mainly focuses on Beijing Capital International Airport (ZBAA), Beijing Daxing International Airport (ZBAD), and Tianjin Binhai International Airport (ZBTJ). This method can be found in [reference needed]. Figure 1 ,include: S1: Construct a time slot resource fit evaluation standard with the goal of meeting the airport's functional positioning, and use the entropy weight method to determine the evaluation value of flights, and obtain the flights to be optimized based on the evaluation value.

[0049] In one embodiment of this application, Beijing Capital International Airport, Beijing Daxing International Airport, and Tianjin Binhai International Airport are important aviation hubs in the Beijing-Tianjin-Hebei region and even in northern China. When optimizing flight schedules for the three airports in Beijing and Tianjin, it is necessary to consider the balanced allocation of resources and flight schedules among different airports to ensure that each airport can operate under relatively equal conditions, thereby supporting the balanced development of the aviation network of the city. This application defines it from the following two aspects: (1) Market positioning: Considering the market positioning of each airport is to ensure that each airport can operate effectively according to its specific functions and target markets, while promoting the coordinated development and efficient operation of the entire regional aviation network. In this embodiment of the application, by evaluating the fit between flight attributes and hub airport positioning, flights within each airport that do not conform to their own positioning are identified as subsequent optimization targets. In this way, unreasonable flight schedules are adjusted from the perspective of market positioning to avoid blind allocation of time slot resources and improve the utilization efficiency and fairness of time slot resources. (2) Airport operational efficiency: The operational efficiency of multiple airports involves ensuring that each airport allocates operational resources and slots fairly according to its capacity and demand in order to improve the efficiency and fairness of the overall aviation network. Such considerations of fairness not only help to achieve optimal resource allocation, but also relate to the healthy development of the aviation industry and the public interest.

[0050] This application establishes a hub airport slot resource assessment standard. Considering the hub airport's market positioning, it assesses whether multiple flight attributes align with the airport's market positioning. If a flight's time and service match the airport's market positioning, it indicates that the flight schedule is consistent with the airport's functional objectives. Conversely, if they do not, this mismatch indicates that the flight's time arrangement is inconsistent with the airport's strategic objectives, and therefore, such flights should be considered key targets for airport slot resource optimization and adjustment. This assessment method adjusts flights that do not conform to the airport's market positioning, ensuring the fairness, rationality, and efficiency of slot resource allocation among airports within the airport cluster, further promoting balanced development among airports in the region, thereby driving economic growth in their respective areas.

[0051] This application embodiment starts from the international hub function of a first-level hub airport and the domestic hub function of a third-level hub airport, and constructs a time slot resource function fit assessment standard. The selection of assessment indicators should be scientific and reasonable, with clear levels, and be able to clearly screen out flights that do not conform to the positioning of the hub airport, so as to facilitate subsequent optimization of flight slots. When constructing the fit assessment indicator standard, the following principles need to be followed to ensure the comprehensiveness, scientificity and operability of the assessment: (1) Combining objective assessment with subjective assessment: It is necessary to assess both objective facts, such as the economic benefits brought by the optimization of flight slots, and subjective feelings, such as passenger satisfaction; (2) Typicality principle: Select representative indicators to truly reflect the market positioning and operational efficiency of each airport; (3) Operability principle: The selected assessment indicators should have clear meanings, be easy to understand, and the data sources should be true and reliable, with a basis, so as to be easy to apply to practice and provide real help for future optimization.

[0052] From the perspective of the airport's own market positioning, and while ensuring the airport's operational efficiency, adhering to the principles of scientificity, comparability, and fairness of indicators, five indicators of the hub airport slot resource assessment system were established: (1) Destination category: Destination category refers to the classification of airports in the route network. This classification is based on multiple dimensions such as the airport's service type, facility scale, and flight density. These classifications help the authorities, airlines, and passengers to better understand the functions and service scope of each airport or destination; (2) Flight category: Flight categories are usually divided into international flights and domestic flights. First-level hub airports tend to serve more international flights, while third-level hub airports focus on the operation of domestic flights; (3) Transfer efficiency: Transfer efficiency of flights within the airport refers to the flow of passengers transferring from one flight to another within the airport. (3) Smoothness and time efficiency. First-tier hub airports usually pay more attention to transfer efficiency because they rely on efficient transfer processes to maintain their status as core nodes of the international aviation network, maintain competitiveness with other global hubs, and provide better service experience for international passengers; (4) Flight punctuality rate: a key indicator for measuring the accuracy of flight departure and arrival on schedule. This indicator reflects the consistency between the actual departure and arrival time of the flight and its scheduled time. If the flight punctuality rate is low, it means that there are many delays at the airport and the airport utilization rate is high; (5) Flight load factor: an important indicator for measuring the economic benefits of flights. A high load factor may indicate a competitive advantage and strong market attractiveness. If the load factor of a hub airport is low, it will waste flight resources. In this case, the time slots should be adjusted to avoid the waste of hub airport resources.

[0053] Entropy weighting is an objective weighting method commonly used in multi-indicator comprehensive evaluation to determine the weight of each evaluation indicator. Its core idea is that the lower the information entropy of an indicator, the greater its role in the overall evaluation, and therefore it should be assigned a higher weight. Based on this, this application introduces information entropy for evaluation: (1) Standardize each indicator. The standardization formula is as follows:

[0054] In the formula, For a standardized sequence, if an element becomes 0 after standardization, it is replaced with a very small number. To avoid meaningless cases when taking the natural logarithm, The original sequence, For the first The last sample value of each indicator. For the first One flight review, For the first One indicator, This represents the total number of indicators.

[0055] (2) Calculate the information entropy of each indicator. The calculation formula is as follows:

[0056]

[0057] In the formula, For the first Information entropy of each indicator For entropy scaling parameters, For the first The first evaluated flight was on The proportion of each indicator The total number of objects being evaluated.

[0058] (3) Calculate the information entropy redundancy of each indicator. The information entropy redundancy reflects the effective information content of the indicator. The calculation formula is:

[0059] In the formula, For the first The information entropy redundancy of an indicator is as follows: the smaller the information entropy of an indicator, the greater its redundancy, which means that the indicator carries more effective information in the evaluation.

[0060] (4) Calculate the weight of each indicator based on the information entropy redundancy. The calculation formula is as follows:

[0061] In the formula, For the first The weight of each indicator.

[0062] (5) Calculate the overall score, i.e. the evaluation value of the flight, based on the weight of each indicator. The calculation formula is as follows:

[0063] In the formula, For the first The evaluation value of each flight is used to evaluate the flight attributes and their alignment with the hub airport's positioning, thus identifying flights to be optimized.

[0064] This application example describes the situation with Beijing Capital International Airport and Beijing Daxing International Airport positioned as Tier 1 hub airports, and Tianjin Binhai International Airport positioned as a Tier 3 hub airport. A total of 121 flights departing from the three airports between 7:00 and 8:00 on April 30, 2024 were selected, including 60 flights departing from Beijing Capital International Airport, 39 flights departing from Beijing Daxing International Airport, and 22 flights departing from Tianjin Binhai International Airport. Flight transfer efficiency and flight schedules were obtained from OAG data, and flight punctuality rate and load factor were obtained from FlightAware. Beijing Capital International Airport is shown in Table 1.

[0065] Table 1 Initial Data Table for Beijing Capital International Airport

[0066] The weight coefficients of each indicator can be obtained, as shown in Table 2.

[0067] Table 2 Weighting Coefficients of Various Indicators for Beijing Capital International Airport

[0068] As the table shows, Beijing Capital International Airport, as a Tier 1 hub airport, has a relatively large weighting for destination type, transfer efficiency, and flight type. This is because hub airports need to connect multiple international and domestic destinations, building a vast route network, making the airport's radiation capacity and influence particularly important. Meanwhile, efficient transfer services can attract more airlines and passengers to choose the airport as a transit point, enhancing the airport's competitiveness in the global aviation network. Flight type diversification indicates that hub airports need to provide diversified flight services, including short-haul and long-haul, domestic and international flights, to meet the needs of different passenger groups, thereby improving the airport's overall profitability. Tianjin Binhai International Airport, as a Tier 3 hub airport, has a relatively large weighting for destination type and load factor. The main function of a Tier 3 hub airport is to improve regional air connectivity. By increasing destinations, a Tier 3 hub airport can cover more small and medium-sized cities and local airports, meeting regional aviation demand. Maintaining a high load factor ensures sustainable flight operations, avoids flight cancellations due to low load factors, thus maintaining the stability and reliability of flight schedules and improving airport facility utilization. In summary, the objective weights calculated using the entropy weight method can effectively reflect Beijing Airport's market positioning as a large international hub airport and a comprehensive transportation hub in the Beijing-Tianjin region, while Tianjin Binhai International Airport is a large regional aviation hub. This provides a theoretical basis for clarifying the functions of each airport and for subsequent assessment of slot resources within the airport cluster. The final flight assessment results are shown in Table 3.

[0069] Table 3. Flight Evaluation Results for Beijing Capital International Airport (Top Five and Bottom Five)

[0070] The table above shows the comprehensive evaluation results for each flight between 7:00 and 8:00 on April 30, 2024. The evaluation results show that international flights and flights to Tier 1 and Tier 2 hub airports at Beijing Airport received higher comprehensive scores, while flights to non-Tier 1 and Tier 2 hub airports received lower scores. This is mainly because Beijing Capital International Airport's primary market positioning is as a large international hub airport. Tianjin Binhai International Airport received lower scores for international flights and higher scores for flights to Tier 3 hub airports. This is mainly because Tianjin Binhai International Airport is positioned as a regional hub airport, absorbing the spillover demand from Beijing Airport.

[0071] S2: Using the objective functions of minimizing flight schedule adjustments, maximizing flight function positioning indicators, and minimizing the Gini coefficient, a cooperative game-theoretic flight schedule optimization model for airport clusters is constructed. The whale optimization algorithm is used to solve the cooperative game-theoretic flight schedule optimization model for airport clusters, resulting in the optimized flight schedule.

[0072] The core of optimizing flight schedules within an airport cluster lies in improving flight efficiency, reducing delays, and enhancing passenger travel experience by adjusting the time slots allocated to each airport, while ensuring the optimization scheme meets actual operational requirements. This application's optimization approach focuses on alleviating the burden on primary and tertiary hub airports within the airport cluster. Flights exceeding capacity or not conforming to airport positioning at hub airports are reassigned to departure airports and departure times according to airport function. The optimization model in this application considers only the perspective of each airport within the airport cluster. By analyzing the relationships between participants in a cooperative game model, it is found that each airport within the cluster can be viewed as a cooperative player. Time slot resources are optimized and reorganized to ensure a fairer allocation of adjustments among the airports. Constraints are set on flight uniqueness, maximum flight adjustment, flight continuity, and airport capacity to guarantee the model's scientific validity and effectiveness. The optimized flight schedule is obtained using the whale optimization algorithm. Data analysis shows that the optimized schedule not only keeps flights within the capacity range of each airport but also improves the operational efficiency of each airport and reduces flight delays.

[0073] The assessment results regarding the functional fit of time slot resources are used to rank the hourly flight evaluation values ​​of each airport. Flights exceeding the airport's hourly capacity limit and ranking relatively low are considered for optimization. In short, time slot adjustments and optimizations are made for overloaded and relatively inefficient flights. Cooperative game theory studies how participants can engage in games based on principles of fairness, impartiality, and utility through collaboration. It emphasizes group rationality and the pursuit of overall benefit effectiveness. Unlike non-cooperative games, in cooperative games, participants can form alliances and improve overall returns through collaborative decision-making and resource sharing. Cooperative games emphasize collective action and common interests, achieving Pareto solutions that satisfy the comprehensive interests of all parties through collective rational game theory. It is typically used to solve complex problems involving multiple stakeholders. Through a reasonable benefit distribution mechanism, it incentivizes all parties to actively participate in cooperation. The outcome should increase the interests of all decision-makers involved in the game, or at least increase the interests of at least one decision-maker without harming the interests of others, achieving a win-win or multi-win situation. The problem of optimizing flight schedules within an airport cluster involves numerous constraints and optimization indicators, requiring collaborative decision-making among multiple airports to fully utilize spatiotemporal resources for real-time flight scheduling. Cooperative game theory studies how multiple decision-makers participate in decision-making through alliances and other means, addressing issues of overall optimal benefit and equilibrium. Constructing a multi-airport flight schedule optimization model based on the collaborative efforts of airports within two airport clusters to maximize the interests of all parties can serve as a valuable tool for describing and solving the problem of efficient allocation of flight schedules across multiple airports. This application's embodiment selects Beijing and Tianjin airports as the cooperative game players. Through negotiation, each airport shares real-time information such as flight dynamics, weather conditions, and operational status to improve overall emergency response capabilities. Under constraints such as satisfying the interests of all parties and the capacity limitations of each airport and airport cluster, a flight schedule is jointly formulated to avoid flight schedule conflicts and reduce flight delays. The Shapley value considers the marginal contribution of each game player (airport) to different cooperative combinations (subsets). Specifically, for each player, the incremental contribution (i.e., marginal contribution) brought about by introducing that player into all possible player subsets is calculated using the following formula:

[0074] In the formula, For the first The profit distribution value of each airport. For characteristic function, The number of airports among the participating parties in the alliance. yes Marginal contribution to the alliance, This represents the total number of participants within the airport. To coordinate revenue for the overall operation of the cooperative game model, For the alliance remove Revenue from post-cooperative alliance operations coordination.

[0075] After using the Shapley cooperative game model for benefit allocation, an objective function is constructed to minimize flight schedule adjustments, maximize flight function positioning indicators, and minimize the Gini coefficient. Further multi-objective optimization is performed on the objective values ​​to balance the interests of each airport in the airport cluster, thereby improving the robustness of the overall solution. Specifically, the total schedule adjustment within the three airports is minimized.

[0076] In the formula, For flight schedule adjustments, Flights in the timetable The actual time allocated, Flights in the timetable When to apply As decision variables, For a set of time periods, This refers to the number of flights.

[0077] To fulfill the airport's maximum functional positioning:

[0078] In the formula, As a functional positioning indicator for flights, Indicates the aircraft that took off from Airport Time arrives Do the airport's flights align with its strategic positioning?

[0079] This application introduces the fairness indicator—the Gini coefficient—and minimizes it to ensure that the total flight slot adjustment is borne relatively fairly by each airport, thereby guaranteeing the fairness of slot allocation among airports within the airport cluster and improving the utilization efficiency of each airport within the cluster. The formula for minimizing the Gini coefficient is:

[0080] In the formula, Gini coefficient, This is a collection of airports within an airport cluster. To adjust the average value at any time, For the first The actual time slots allocated to each airport For the first The number of flights allocated to each airport. The time allocated for the plan.

[0081] In summary, a cooperative game theory flight schedule optimization model for airport clusters is established. : .

[0082] For the alliance In terms of its cooperative game model for: .

[0083] The constraints satisfied by the airport cluster cooperative game-theoretic flight schedule optimization model include: Flight uniqueness constraint: Within the considered time frame, each flight in the airport cluster is only allowed to perform a takeoff mission once at the same airport, that is, it is only assigned to a specific time period, expressed as: .

[0084] Airport capacity constraints refer to various factors that limit airport operational efficiency and throughput. These factors determine the number of flights and passengers an airport can safely and efficiently handle within a specific timeframe. Adhering to airport capacity constraints helps to rationally schedule flight times, reduce the probability of flight delays and cancellations, improve flight punctuality, and thus enhance overall operational efficiency and passenger satisfaction. It also ensures that flight schedules do not exceed the airport's physical handling capacity. This application's embodiments consider the airports in the Beijing-Tianjin-Hebei region as a whole for study; therefore, when considering the capacity of individual airports, it is also necessary to ensure that the entire airport group does not exceed its capacity. This application sets the unit time to 60 minutes to avoid flight schedules exceeding the airport's published time slot capacity limits and to achieve a uniform distribution of flight slots. Airport capacity constraints:

[0085] In the formula, For the airport Airport capacity within a specified time period For time pool.

[0086] Flight continuity constraint: For passengers requiring transfers, sufficient transfer time must be guaranteed, including considering connection times between different flights (minimum and maximum transfer times), expressed as:

[0087] In the formula, In order to be in Flights departing at specific times In order to be in Flights landing at specific times It is a binary variable. Minimum transfer time, This is the maximum transit time.

[0088] When establishing a flight schedule optimization model, it is crucial to meet the maximum permissible adjustment constraint. This constraint ensures the feasibility and practicality of flight schedule adjustments, contributing to the maintenance of operational stability for airlines and airports. The maximum permissible adjustment constraint is as follows:

[0089] The flow of departing flights within an airport cannot exceed the flow limit of each shared waypoint when passing through it. This essentially reflects the capacity constraints of an airport cluster caused by airspace resource limitations.

[0090] In the formula, Traffic constraints for shared waypoints in an airport cluster.

[0091] The whale optimization algorithm is used to solve the airport group cooperative game flight schedule optimization model to obtain the optimal solution, including: A1: Initialize the whale population (generate initial flight schedule scheme): Each individual whale represents a complete flight schedule scheme. If there are M flights in the system, then each individual is a vector of length M, where each element represents the planned departure time of the corresponding flight. For each flight, the initialization method is as follows: randomly generate a time within its allowed time window (between the earliest and latest times) to obtain the initial flight schedule scheme, and design the population size according to the complexity of the problem and the number of flights in the airport group.

[0092] A2: Calculate the multi-objective function value for each solution: For each flight time slot scheme, calculate multiple performance indicators, including flight time slot adjustment (calculate the sum of the absolute deviations between the actual and planned times of all flights to reflect the overall operational efficiency), flight function positioning index (calculate whether the time slot adjustment meets the airport's strategic positioning standards, take the maximum value, and ensure that the time slot adjustment meets the needs of each airport), and Gini coefficient (statistically count the number of flight slots allocated to each airport and calculate the Gini coefficient accordingly).

[0093] A3: Initialize external archive (establish initial non-dominated solution set): (1) Non-dominated solution screening: compare all initial flight time schemes and find the schemes that are not surpassed by other solutions in terms of performance indicators; (2) Archive establishment: store these non-dominated solutions in the external archive as the current optimal solution set.

[0094] A4: Main optimization loop (iterative improvement): (1) Whale individual update process: Select a guiding solution from the external archive as the "leader" and adopt a roulette wheel selection based on crowding distance: the probability of sparsely distributed solutions in the target space being selected is higher, which helps to explore different areas. Three update strategies: Surround the prey (development): When the parameter |A|<1, the whale moves closer to the leader solution, which is reflected in the flight time adjustment, that is, to move its own flight time closer to the leader's time; Random search (exploration): When |A|≥1, randomly select an individual as a reference for exploration, which helps to escape local optima; Bubble attack: Update the position in a spiral manner, combining local development and global exploration, and make nonlinear adjustments to the flight time. (2) The newly generated flight schedule may violate actual operational constraints and needs to be repaired: Time window constraint repair ensures that the time of each flight is within its earliest and latest time range, and if it exceeds the range, it is adjusted to the boundary value; Safety interval repair checks whether the time interval of consecutive flights using the same runway meets the minimum safety requirements, and if it does not meet the minimum requirements, it is adjusted to the later time; Capacity constraint repair detects peak periods and adjusts some flights to adjacent periods with lower utilization; Aircraft turnaround repair ensures that there is enough turnaround time for cleaning, refueling, passenger pick-up and drop-off for consecutive flights of the same aircraft. (3) For the repaired new timetable, recalculate all objective function values ​​and evaluate its overall performance. (4) Add the newly generated high-quality solution to the external archive, compare the dominance relationship between the new solution and all solutions in the archive, and if the new solution is not dominated by any archive solution, add it to the archive, and remove those old solutions dominated by the new solution to ensure that the archive contains solutions with different trade-off tendencies. (5) When the archive size exceeds the preset limit, pruning is performed to maintain diversity; for each solution, the distance between it and its neighboring solutions in the target space is calculated; solutions with large crowding distances are retained, that is, solutions that are sparsely distributed in the target space and represent unique trade-offs. (6) Individual optimal update: In a multi-objective environment, each whale individual also needs to remember its own historical optimal state. Since there are multiple objectives, the individual's "historical optimal" may be a set of non-dominated solutions, or the solution in the archive that is closest to the current individual may be selected as a reference.

[0095] A5: Determine if the current iteration count has reached the preset number. If yes, obtain the optimal solution; otherwise, repeat step A4. After the algorithm terminates, the solution set in the external archive constitutes the Pareto front, representing the optimal trade-off between different objectives.

[0096] Since the three airports in Beijing and Tianjin have a total of 1000 flight slots on that day, the data volume is large and the model is complex. Therefore, this application embodiment first selects a small range of data for model verification, as shown in Table 4.

[0097] Table 4. Departure flight schedules for the Beijing-Tianjin airport cluster from 7:00 to 8:00 (partial)

[0098] The whale optimization algorithm was solved using MATLAB, and the fitness change curve of the whale optimization algorithm is shown below. Figure 2 As shown. When the iterations reach approximately 100, the fitness value remains essentially unchanged, the whale optimization algorithm converges, and the corresponding value can be considered the Pareto optimal solution. Through calculation and analysis, the fairness deviation of time allocation in the initial timetable allocation scheme decreased from 0.58271 to 0.05471, with a total time adjustment of 230 minutes. Figure 3 As shown, before the flight schedule optimization, the Gini coefficient was 0.601. According to the definition of the Gini coefficient, flight delays varied significantly between airports. After optimization, the Gini coefficient decreased to 0.059, and the flight deviation was 220 minutes. Compared with before optimization, fairness among airports was significantly improved. Based on the optimization results, 14 out of 121 flights operating between 7:00 and 8:00 AM at the Beijing and Tianjin airports were adjusted. The optimization results of these 14 flights are shown in Table 5.

[0099] Table 5. Adjusted and Optimized Flight Schedule

[0100] Continued table

[0101] Table 6 shows a comparison of the fit evaluation results for the 14 flights that were adjusted as described above. Based on the comparative analysis of data before and after the flight schedule adjustment, the 14 flights with optimized scheduling showed a significant advantage in the overall fit evaluation value, with an average fit of 0.068, which is about 3 times higher than the 0.022 of the unadjusted flights.

[0102] Table 6 Comparison of data before and after flight schedule adjustment

[0103] like Figure 4 The figure shows the changes in flight slot resources at the three airports before and after optimization. The results indicate that Beijing Capital International Airport saw the largest adjustment. International flights and flights to which the destination airport is a primary hub were relocated to Beijing Daxing International Airport, aligning with Beijing Daxing's market positioning. Flights to which the destination airport is a tertiary hub were relocated to Tianjin Binhai International Airport, leveraging Tianjin Binhai's function as a tertiary regional hub. Relocating international flights and flights to which the destination airport is a primary hub from Tianjin Binhai International Airport to Beijing Daxing International Airport alleviates operational pressure on Tianjin Binhai International Airport and improves the capacity utilization rate of Beijing Daxing International Airport, meeting the expected optimization goals.

[0104] Figure 5Images (a), (b), and (c) show the distribution and changes in the number of departing flights per unit time at each airport before and after optimization. (a) represents Beijing Capital International Airport, (b) Beijing Daxing International Airport, and (c) Tianjin Binhai International Airport. Before optimization, Beijing Capital International Airport experienced significant flight fluctuations. After optimization, the number of takeoffs and landings became relatively stable, with a more even distribution of flight times, and all takeoffs and landings remained within the airport's capacity. Beijing Daxing International Airport saw an increase in the number of flights after optimization, with flights more evenly distributed across different times. Tianjin Binhai International Airport's data still showed some fluctuations after optimization, but the curve was more stable than before, improving the stability of flight times to some extent. However, the above model only used the one-hour timeframe from 7:00 AM to 8:00 AM on April 30, 2024, resulting in a small dataset that could not accurately reflect the flight times for the entire day. Therefore, a total of 1236 flights for the entire day were selected for optimization. Using the Whale Optimization Algorithm, the total number of flights adjusted in the Beijing-Tianjin airport cluster was 108, with 45 at Beijing Capital International Airport, 38 at Beijing Daxing International Airport, and 25 at Tianjin Binhai International Airport. The specific airport flight schedule optimization plan before and after the 24-hour optimization is as follows: Figure 5 As shown in (d), (e), and (f), (d) is a comparison chart of the optimized flight schedules at Beijing Capital International Airport over 24 hours. After optimization, the flight schedules at the airport are more evenly distributed, and the number of flights during peak hours has been reduced, keeping the number of flights within the airport's capacity. Figure 5 (e) is a comparison chart of the optimized flight schedules at Beijing Daxing International Airport over 24 hours. It can be seen that after the optimization, the number of flights at the airport has increased, taking on a portion of the flight volume of Beijing Capital International Airport. This not only improves the utilization rate of the airport itself but also reduces the operational pressure on Beijing Capital International Airport. Figure 5 (f) is a comparison chart of the optimized flight schedules at Tianjin Binhai International Airport over 24 hours. After optimization, the 7:00-8:00 time slots no longer exceed the airport's capacity, and the overall flight distribution is more stable.

[0105] S3: Based on the needs of air traffic control, airlines, airports and passengers, this paper constructs an objective function to minimize the variance of the total number of time slot adjustments, minimize the fairness differences among airlines, minimize the fairness differences among airports and minimize the passenger loss time. The multi-objective optimization problem is transformed into a dynamic game problem. A dynamic game theory model of the multi-objective optimization problem is constructed, and an improved particle swarm optimization algorithm is used to solve for the optimal solution.

[0106] An airport cluster refers to an air transport system comprised of multiple airports within the same city or city cluster. In such systems, flight slot optimization involves balancing and coordinating the interests of all members. Air traffic control, airlines, airports, and passengers are the main stakeholders in slot and airspace resource utilization. They need to jointly participate in the allocation of flight slots within the airport cluster. Based on the needs of these four parties—air traffic control, airlines, airports, and passengers—an objective function is constructed to minimize the variance of the total number of slot adjustments, minimize fairness differences among airlines, minimize fairness differences among airports, and minimize lost passenger time.

[0107] Air traffic control angle: Minimize the variance of the total number of time adjustments, reducing the number of times controllers need to adjust flights. This reduces the workload of controllers and ensures that the number of adjustments for each flight is as even as possible, guaranteeing fairness in adjustments. The expression is:

[0108] In the formula, To minimize the variance of the total number of adjustments at any given time, No. The number of times a flight is adjusted. This is the average number of adjustments made to all flights.

[0109] From the airline's perspective: Minimize airline fairness deviation. The airline fairness index is defined as: considering the ratio of a particular airline's flight offset to the total flight offset and the number of slots requested by that airline to the total number of slots requested by all airlines, and then comparing the fairness of any airline with the mean fairness of all airlines, minimizing the maximum deviation, ensuring that the deviation between the most unfair situation and the average fairness is as small as possible. The expression is:

[0110]

[0111] In the formula, To minimize fairness differences among airlines, For the airline's fairness bias, A collection of all airlines. For airlines Flight offset, This refers to the collection of flights from all airlines within the airport cluster.

[0112] From the airport's perspective: Minimizing airport fairness deviation. Based on the objective function established by the airlines, the minimum fairness deviation for an airport is similarly derived: the ratio of a certain airport's flight offset to the total flight offsets and the ratio of the number of time slots requested by that airport to the total number of time slots requested by all airports. The expression is:

[0113]

[0114] In the formula, To minimize fairness differences at airports, For the airport's fairness bias, This represents the flight offset at the airport. This refers to the collection of flights from all airports within the airport cluster.

[0115] From the passenger's perspective: Minimize passenger lost time. Passenger lost time mainly refers to the extra time passengers spend at the airport or during their journey due to flight delays or cancellations. This includes time spent waiting to board, being stranded at the airport, transferring to other means of transportation, and flight delay time (the length of time the actual departure or arrival time of the flight is later than the scheduled time). This application embodiment, in order to maximize the satisfaction of passenger travel needs, only considers the impact of flight delay time on the passenger travel experience. Therefore, passenger lost time is set as the deviation between the departure (or arrival) flight time and the original flight time adjustment, aiming to minimize the disturbance to existing airport flight time resources. The expression is:

[0116] In the formula, To minimize passenger loss time.

[0117] Game theory provides a framework for understanding and handling complex decision-making situations involving strategic interactions among multiple participants. Flight slot optimization requires the joint participation of air traffic control, airlines, airports, and passengers in allocating flight slots within an airport cluster. Air traffic control determines the total number of flights and flow control strategies for the day based on airspace capacity and safety requirements; this decision influences the planning of airlines and airports. Airlines, under the capacity constraints determined by air traffic control, submit flight slot applications, selecting the optimal flight slots to maximize their revenue and market demand. Airports allocate runway, taxiway, and gate resources based on airline slot applications and their own resource constraints to ensure smooth flight takeoffs and landings. Passengers choose flights based on airport slots, prices, and services, influencing airline decisions through feedback and market reactions. Thus, a time-sequential decision-making process exists among the various participants; therefore, dynamic game theory models can be chosen to analyze the multi-agent airport cluster flight slot optimization problem. Dynamic game theory models and multi-objective optimization problems share many theoretical similarities, especially when dealing with the different objectives of multiple stakeholders. Both aim to find a balance point among multiple objectives. Therefore, by applying dynamic game theory models, multi-objective optimization problems can be solved, thereby improving overall efficiency. The dynamic game model mainly includes three main factors: players, game strategies, and payoffs. It transforms the multi-objective optimization model into a dynamic game model problem, constructing a multi-objective optimization dynamic game model, specifically described as follows: (1) Air traffic control, airlines, airports, and passengers are the optimization agents in the objective function, i.e., they can be considered as players in the game; (2) The set of decision variables... It can serve as a set of game strategies for players; (3) the objective function in multi-objective optimization. (3) It can be used as the payoff function in dynamic game theory; (4) The constraints of multi-objective optimization can be used as the constraints of dynamic games, and the dynamic game model of multi-objective optimization problems. The formula is as follows:

[0118] In the formula, Represents the players in a dynamic game. A set of strategies representing the players in the game. Describes the payoff function for the players. For the first dynamic game One of the people involved in the game, For the first A collection of strategies for each player in the game. For the first The payoff function of each player in the game The number of people involved.

[0119] Subgame perfect Nash equilibrium is considered a core concept in dynamic game theory. This idea emphasizes that in dynamic games, the strategy combinations of all participants must not only reach Nash equilibrium throughout the entire game but also in all possible subgame scenarios. This means that at any stage of the dynamic game, the participants' strategy choices are optimal, and these strategies remain optimal in every subgame. Pareto optimality, however, is a broader concept of equilibrium, allowing for untrustworthy threats or commitments in certain situations—that is, strategies that may not actually be implemented. Compared to Pareto optimality, subgame perfect Nash equilibrium eliminates these untrustworthy behaviors, ensuring that all strategies are trustworthy, providing a more stringent equilibrium standard. Therefore, perfect Nash equilibrium strategies in subgames can effectively eliminate Nash equilibria built on unstable threats or commitments, maintaining only truly stable strategy combinations. Solving airport cluster flight schedule optimization models based on dynamic game theory requires constructing a dynamic game theory model for airport cluster flight schedule optimization. The four optimization objectives in this application act as players in a dynamic game, making decisions in a specific order, each affecting subsequent decisions and the final outcome. The payoff function of the first player is to minimize the variance of the number of adjustments at each time step (…). The payoff function for the second player is to minimize the airline fairness bias. The third player aims to minimize airport fairness bias. The fourth player's payoff function is to minimize the passenger's lost time. The dynamic game model can then be described as follows:

[0120] The perfect Nash equilibrium solution of the subgame in this model is actually the optimal output of the airport cluster flight schedule optimization model. In multi-objective games, Nash equilibrium means that each optimization objective, after considering the decisions of other objectives, chooses its own optimal strategy, and no single objective can dominate the decision-making process alone. The main purpose of the Nash equilibrium solution is to find a strategy combination in the game process such that each participant, after considering the strategies of other participants, chooses the optimal response without changing its own strategy. In this equilibrium state, no participant can gain a greater benefit by unilaterally changing its strategy, thus achieving a balance and stability of interests among all parties. This ensures that each optimization objective gets as close as possible to its own optimal solution without harming the interests of other optimization objectives. The calculation formula is as follows:

[0121] In the formula, It is the first Criteria for evaluating the Nash equilibrium solution of an individual. The percentage of deviation from the target benchmark. Indicates the first The first individual One target value, Indicates the first The optimal value for each objective.

[0122] Particle Swarm Optimization (PSO) is an ideal choice for finding Nash equilibrium solutions. Although PSO performs well on many optimization problems, it still has some limitations, such as being prone to premature convergence and global optima. To address these limitations, the Logistic mapping is introduced. Leveraging its nonlinear and non-periodic characteristics, as well as its ability to generate pseudo-random number sequences, the global search capability of PSO is enhanced, and the algorithm's versatility is improved. The expression for the Logistic mapping is:

[0123] In the formula, For state variables, For independent parameters, and For the first The second iteration and the first iteration This indicates that the particle has entered a state of complete chaos.

[0124] By incorporating the Logistic mapping into the particle swarm optimization algorithm, this method performs particularly well when dealing with complex or multimodal optimization problems. The chaotic mapping operates as follows:

[0125]

[0126] In the formula, For the first The value of the chaotic iteration. For the first The value of the chaotic iteration. For the original multidimensional data, For the first One particle, The number of chaotic iterations. As a dimension, , For the first Search boundary of the dimensionless variable. When performing chaotic mapping, applying Logistic mapping to all particles may increase the computational burden of the algorithm and may cause the overall performance of the particle swarm to decline due to overexploration. Therefore, selecting the top 20% of excellent particles for mapping is a trade-off between efficiency and effectiveness, which can improve algorithm performance while controlling computational complexity. The specific operation is as follows:

[0127] in, For the top 20% of particles in terms of fitness, The optimal particle position and the overall optimal position of the particle swarm were recalculated. Through chaotic mapping, population diversity was enhanced, thereby improving the algorithm's performance in global search and avoiding the risk of falling into local optima. This resulted in better performance when handling high-dimensional problems. However, while chaotic mapping successfully solved some problems of traditional particle swarm optimization algorithms, it may also reduce the algorithm's convergence speed. Therefore, this problem is addressed by optimizing the inertia weight and learning factor, thereby improving the algorithm's execution performance and convergence speed. In the PSO algorithm, the inertia weight and learning factor are usually fixed constants, which leads to relatively slow search and convergence speeds in solving complex problems. Therefore, this application improves its inertia weight and learning factor, with the specific formula as follows:

[0128]

[0129]

[0130] In the formula, For the improved inertia weight, and For the improved learning factor, and These are the boundary values ​​for the dynamic inertia weight. The maximum number of iterations. This represents the current iteration number. This is the initial value of the individual learning factor. This represents the final value of the individual learning factor. This is the initial value for the social learning factor. This represents the final value of the social learning factor.

[0131] The fitness function plays a crucial role in particle swarm optimization (PSO) algorithms, evaluating the positional quality of each particle in the solution space and guiding its search direction. In dynamic game optimization, it's necessary to balance the relationships between multiple players. Therefore, this embodiment selects the payoff function from the model as the fitness function. Since different payoff functions have different units of measurement, they are first standardized to remove dimensions before calculating the fitness function. The specific formula is as follows:

[0132] in The standardized return function, For the original payoff function, This is the average value. The standard deviation is used. A large penalty term is added to the fitness function. This method penalizes particles that do not meet the constraints, ensuring that the particle swarm optimization algorithm keeps the number of particles within a feasible range during the search process. Simulation results show that this method effectively improves the convergence speed and solution accuracy, and each particle's fitness function has its own specific mathematical formula:

[0133] In the formula, For the fitness function, For an optimization model that includes an objective function, For the first One objective function; ; It is a 0-1 variable; when the particle meets the model constraints... When not in compliance .

[0134] Solving for optimal solutions to dynamic game theory models using an improved particle swarm optimization algorithm incorporating a logistic mapping includes: B1: Select data from the flight schedule as the initial particle swarm, initialize the dimensions (each dimension corresponds to the departure time of a flight), the maximum number of iterations, and determine the type of chaotic mapping; B2: Determine the specific position and movement speed of the initial particles, where the specific position is the takeoff time of each flight; B3: Calculate the fitness of the particles based on the fitness function; B4: If the fitness of the current particle exceeds the best state in its history, update the optimal solution for the current particle; if the fitness of the current particle exceeds the global best solution in its history, update the global best solution. B5: Select the top-ranked particles by fitness and perform Logistic mapping. Calculate the particle fitness based on the fitness function and update the particles according to their fitness until each particle converges to its corresponding weighted average point. Output the optimal solution.

[0135] In one embodiment of this application, after running the model in the MATLAB software environment, the values ​​of the four objective functions gradually converge during the iteration process, eventually reaching a stable state, as shown below. Figure 6 As shown in the figure, the first curve represents the convergence curve from the air traffic control perspective, the second from the airline's perspective, the third from the airport's perspective, and the fourth from the passenger's perspective. This process demonstrates that the model continuously optimizes during iteration, ultimately achieving the predetermined objective function value. The figure shows that all four objective functions converge to their minimum values ​​after approximately 100 iterations. Specifically, the variance of the total number of time adjustments by air traffic controllers decreased from 546 to 360, the fairness bias of airlines decreased from 0.42 to 0.15, the fairness bias of airports decreased from 0.34 to 0.14, and the lost time for passengers decreased from 1700 to 900. All four objectives were well optimized. Therefore, this dynamic game model reduces the workload of air traffic controllers, preventing some airlines from experiencing operational disruptions due to excessive time adjustments, while also avoiding resource idleness caused by some airlines having too few time adjustments.

[0136] S4: Based on the optimal solution of the airport cluster cooperative game flight schedule optimization model and the optimal solution of the dynamic game theory model, the flights to be optimized are adjusted to achieve the optimal airport cluster flight schedule configuration.

[0137] The model established in this application not only meets the needs of all parties but also improves the overall efficiency of various community members within the civil aviation system. The flight schedule distribution of the three airports before and after optimization is as follows: Figure 7 As shown in the figure, the first airport is Beijing Capital International Airport, the second is Beijing Daxing International Airport, and the third is Tianjin Binhai International Airport. The figure shows that the optimized airports all meet their maximum departure capacity, and flights exceeding capacity limits during peak hours are moved to before or after low-traffic periods. Therefore, the distribution of flight numbers in each time period fluctuates around capacity as much as possible. After the flight schedule adjustments, the overall time distribution at Beijing Capital International Airport and Beijing Daxing International Airport has become more balanced. This adjustment helps optimize resource allocation, improve operational efficiency, and provide more convenient services for passengers. It can be seen that the above model is of great significance for improving the capacity utilization efficiency of each airport and promoting the balanced development of airports within an airport cluster.

[0138] The method provided in this application utilizes a cooperative game-theoretic flight schedule optimization model for airport clusters to readjust flight schedules that do not conform to the positioning, thereby improving the fairness of flight schedule resource allocation within the airport cluster. Furthermore, starting from the entire air traffic control system, and balancing the interests of controllers, airlines, airports, and passengers, a multi-scale, multi-objective optimization model for airport cluster flight schedules that considers the needs of all four parties is established, which increases the capacity utilization efficiency of each airport, reduces flight delays, and improves the overall operational efficiency of the airport cluster.

[0139] It should be noted that the embodiments described herein are for the purpose of helping readers understand the principles of this application, and should be understood as not limiting the scope of protection of this application to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this application without departing from the essence of this application, and these modifications and combinations are still within the scope of protection of this application.

Claims

1. A multi-entity method for allocating flight slots in airport clusters based on operational resource allocation, characterized in that, include: S1: Construct a time slot resource fit evaluation standard with the goal of meeting the airport's functional positioning, and use the entropy weight method to determine the evaluation value of the flight, and obtain the flight to be optimized based on the evaluation value; S2: Construct a cooperative game-theoretic flight schedule optimization model for airport clusters, and use the whale optimization algorithm to solve the cooperative game-theoretic flight schedule optimization model for airport clusters to obtain the optimal solution of the cooperative game-theoretic flight schedule optimization model for airport clusters. S3: Based on the needs of air traffic control, airlines, airports and passengers, the multi-objective optimization problem is transformed into a dynamic game problem. A dynamic game theory model of the multi-objective optimization problem is constructed, and an improved particle swarm optimization algorithm with the introduction of Logistic mapping is used to solve the optimal solution of the dynamic game theory model. S4: Based on the optimal solution of the airport cluster cooperative game flight schedule optimization model and the optimal solution of the dynamic game theory model, the flights to be optimized are adjusted to achieve the optimal airport cluster flight schedule configuration.

2. The multi-entity configuration method for airport cluster flight slots based on operational resource allocation as described in claim 1, characterized in that, S1 includes: S101: Based on the principles of objective evaluation, subjective evaluation, typicality, and operability, and utilizing the airport's own market positioning and operational efficiency, establish multiple indicators for the evaluation system of hub airport slot resources. These indicators include destination categories, flight types, transfer efficiency, flight punctuality rate, and load factor. S102: Standardize each indicator. The standardization formula is as follows: In the formula, For standardized sequences, The original sequence, For the first The last sample value of each indicator. For the first One flight review, For the first One indicator, The total number of indicators; S103: Based on the standardized indicators, calculate the information entropy of each indicator. The calculation formula is as follows: In the formula, For the first Information entropy of each indicator For entropy scaling parameters, For the first The first evaluated flight was on The proportion of each indicator The total number of evaluation objects; S104: Calculate the information entropy redundancy of each indicator. The calculation formula is as follows: In the formula, For the first Information entropy redundancy of each indicator; S105: Calculate the weight of each indicator based on the information entropy redundancy. The calculation formula is as follows: In the formula, For the first The weight of each indicator; S106: Based on the weights of each indicator, calculate the evaluation value of the flight. The calculation formula is as follows: In the formula, For the first The rating value of each evaluated flight; S107: Based on the evaluation value of the evaluated flights, evaluate the fit between the flight attributes and the hub airport positioning to obtain the flights to be optimized.

3. The multi-entity configuration method for airport cluster flight slots based on operational resource allocation as described in claim 1, characterized in that, S2 includes: S201: Treating the airports in the airport cluster as game participants, calculate the marginal contribution of each airport to different cooperation combinations, and allocate benefits based on the marginal contribution. The calculation formula is as follows: In the formula, For the first The profit distribution value of each airport. For characteristic function, The number of airports among the participating parties in the alliance. yes Marginal contribution to the alliance, This represents the total number of participants within the airport. To coordinate revenue for the overall operation of the cooperative game model, For the alliance remove Post-cooperative alliance operational coordination revenue; S202: Based on benefit distribution, construct an objective function that minimizes flight schedule adjustments, maximizes flight function positioning indicators, and minimizes the Gini coefficient. The expression for the objective function is: In the formula, For flight schedule adjustments, Flights in the timetable The actual time allocated, Flights in the timetable When to apply As decision variables, For the number of flights, For a set of time periods, As a functional positioning indicator for flights, Indicates the aircraft that took off from Airport Time arrives Do the airport's flights align with its strategic positioning? Gini coefficient, This is a collection of airports within an airport cluster. To adjust the average value at any time, For the first The actual time slots allocated to each airport For the first The number of flights allocated to each airport. Time allocated to the plan; S203: The constraints are flight uniqueness, airport capacity, flight continuity, maximum allowable adjustment, and shared waypoint capacity. The expression for the constraints is as follows: In the formula, For the airport Airport capacity within a specified time period For time pool, In order to be in Flights departing at specific times In order to be in Flights landing at specific times It is a binary variable. Minimum transfer time, Maximum transit time, Traffic constraints for shared waypoints in an airport cluster; S204: Constructing a cooperative game-theoretic flight schedule optimization model for airport clusters based on the objective function and constraints. : S205: The optimal solution is obtained by using the whale optimization algorithm to solve the airport group cooperative game flight schedule optimization model.

4. The multi-entity configuration method for airport cluster flight slots based on operational resource allocation according to claim 3, characterized in that, The process of using the whale optimization algorithm to solve the airport cluster cooperative game flight schedule optimization model to obtain the optimal solution includes: A1: For each flight, a time window is randomly generated within the time window between the earliest and latest times to obtain the initial flight time scheme, and the population size is designed based on the number of flights in the airport cluster; A2: Calculate the performance indicators of flight time adjustment, flight function positioning index, and Gini coefficient for each flight time plan; A3: Compare all initial flight time schemes, find the scheme that is not surpassed by other solutions in terms of performance indicators, and store the found scheme in an external archive as the current set of optimal solutions; A4: Simulate the behavior of whales surrounding prey, capturing prey with bubble nets, and searching for prey. Update the optimal solution set and repair the flight schedule schemes in the newly generated optimal solution set. Recalculate the fitness of the repaired new timetable schemes and add the newly generated high-quality solutions to the external archive. Select the solution in the external archive that is closest to the current flight schedule scheme as a reference. A5: Determine if the current iteration count has reached the preset number. If yes, obtain the optimal solution; otherwise, proceed to A4 for iteration.

5. The multi-subject configuration method for airport cluster flight slots based on operational resource allocation according to claim 4, characterized in that, S3 includes: S301: Based on the needs of air traffic control, airlines, airports, and passengers, an objective function is constructed to represent the interests of each party, with the objective function being: minimizing the variance of the total number of time slot adjustments, minimizing the fairness differences among airlines, minimizing the fairness differences among airports, and minimizing passenger lost time. The expression for the objective function is as follows: In the formula, To minimize the variance of the total number of adjustments at any given time, To minimize fairness differences among airlines, To minimize fairness differences at airports, To minimize passenger loss time, For the first The number of times a flight is adjusted. This is the average number of flight adjustments across all flights. A collection of all airlines. This refers to the aggregation of flights from all airlines within the airport cluster. This refers to the collection of flights from all airports within the airport cluster. For the airline's fairness bias, For the airport's fairness bias, For airlines Flight offset, This refers to the flight offset at the airport. S302: By treating air traffic control, airlines, airports, and passengers as the optimization agents in the objective function and as players in a game, the set of decision variables as the players' strategy set, the objective function as the payoff function in dynamic game theory, and the constraints as the dynamic game's constraints, a dynamic game theory model for a multi-objective optimization problem is obtained. : In the formula, Represents the players in a dynamic game. A set of strategies representing the players in the game. Represents the payoff function for the players. For the first dynamic game One of the people involved in the game, For the first A collection of strategies for each player in the game. For the first The payoff function of each player in the game The number of people in the game; S303: Solve the optimal solution of a dynamic game theory model using an improved particle swarm optimization algorithm that incorporates the Logistic mapping.

6. The multi-entity configuration method for airport cluster flight slots based on operational resource allocation according to claim 5, characterized in that, The improved particle swarm optimization algorithm includes: introducing a Logistic mapping based on the particle swarm optimization algorithm, improving the inertia weight and learning factor of the particle swarm optimization algorithm, and selecting the reward function as the fitness function to obtain the improved particle swarm optimization algorithm. The expression for the Logistic mapping is: In the formula, For state variables, For independent parameters, and For the first The second iteration and the first The next iteration; The improved expressions for inertia weight and learning factor are: In the formula, For the improved inertia weight, and For the improved learning factor, and The boundary value of the dynamic inertia weight. The maximum number of iterations. This represents the current iteration number. This is the initial value of the individual learning factor. This represents the final value of the individual learning factor. This is the initial value for the social learning factor. This represents the final value of the social learning factor. The expression for the fitness function is: In the formula, For the fitness function, As a penalty item, It is a binary variable. For an optimization model that includes an objective function, For the first One objective function.

7. The multi-subject configuration method for airport cluster flight slots based on operational resource allocation according to claim 6, characterized in that, The method for finding the optimal solution to a dynamic game theory model using an improved particle swarm optimization algorithm incorporating a Logistic mapping includes: B1: Select data from the flight schedule as the initial particle swarm, use the flight departure time as the initial dimension, initialize the dimension and the maximum number of iterations, and determine the type of chaotic mapping; B2: Determine the specific position and movement speed of the initial particles, where the specific position is the takeoff time of each flight; B3: Calculate the fitness of the particles based on the fitness function; B4: If the fitness of the current particle exceeds the best state in its history, update the optimal solution for the current particle; if the fitness of the current particle exceeds the global best solution in its history, update the global best solution. B5: Select the top-ranked particles by fitness and perform Logistic mapping. Calculate the particle fitness based on the fitness function and update the particles according to their fitness until each particle converges to its corresponding weighted average point. Output the optimal solution.

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