Modeling and Control Strategies for Flight Service Vehicle Queue Behavior in Non Cooperative Games

From the perspective of the running process for service vehicles, based on the analysis of the benefits maximization of the special needs vehicles joining the queue as the scheduling goal, and aiming at the shortcomings of traditional task scheduling strategies, methods and models, a non-cooperative game behavior optimization scheduling model with security, economy and environmental protection as the optimization goals was proposed by using game theory. At the same time, the vehicle service capability, service times and import Angle are limited to ensure that the model is more in line with the actual operation scenario. In order to judge the deviation between the game result and the actual operation, the game efficiency matrix is constructed, and Nash equilibrium is used to prove that the model has an equilibrium point. On the basis of the game model, the strategy layer, the game layer and the optimization layer are divided into queue behavior control strategy methods. Finally, this paper verifies the feasibility of the model by using the actual parameter changes to determine the best parameter combination suitable for the model.


Introduction
As an important business scenario of airport production and operation, flight support is easy to cause low airport operation efficiency and poor passenger experience due to problems such as cross-operating units, diverse service links and tight security resources.Ensuring the timely arrival and effective use of vehicles will directly affect the quality and efficiency of airport services.According to statistics, the proportion of flight delays caused by support services accounted for 15. 45% of the total delay factors, among which the unreasonable deployment of support vehicles caused flight delays is one of the important delay factors.In order to improve the service quality steadily, it is imperative to optimize the controllability of vehicle queue behavior.
At present, there is a certain basis for the research on the operation of flight support vehicles at home and abroad, which mainly divides the operation of flight support vehicles into two categories:Wang Xinglong et al. [1] focused on the low efficiency of "first-come, first-serve" for deicing vehicles, proposed a cooperative scheduling model between deicing vehicles and aircraft, and solved the model by using improved genetic algorithm.Feng Xia et al. [2] established a collaborative scheduling model based on the operation service characteristics of refueling trucks and ferry trucks, and solved the model with multi-objective genetic algorithm to achieve the purpose of providing decision support for business departments.Zhu Xinping et al. [3] proposed a single parent genetic algorithm with progressive coding structure in order to solve the centralized scheduling problem of multi-class security vehicles serving multiple flights, which greatly improved the search ability of solution space and effectively reduced the number of flight delays.It can be seen that in the study of the guarantee vehicle scheduling algorithm, the problem is divided into: a single vehicle serves a single flight, different types of vehicles cooperate to serve the same flight, and multiple types of vehicles serve multiple flights.The other kind of research is the path planning of the support vehicle.In order to overcome the lack of experience in airport emergency drills, Shi Yongsheng et al. [4] built a multi-objective nonlinear mixed integer programming model by comprehensively considering factors such as path distance and path complexity, so as to achieve the purpose of obtaining the optimal path for emergency vehicles quickly and effectively.Liu Hongqing et al. [5] proposed to construct time-difference models and environment models respectively for vehicle path planning problems of different scales, and solved the above models by means of reinforcement learning.Lei Jinxian et al. [6] applied the improved ant colony genetic algorithm to the vehicle routing problem with time window, and found that the performance was significantly improved compared with the traditional algorithm.According to the above literature analysis, there are few studies on the behavior problems between vehicles during flight support operation.The random conflicts of time, start-up and operation between vehicles will also cause flight support service delay, and it is also necessary to pay attention to the research on the problem of lane transfer between vehicles.Zhang Ronghui et al. [7] established the vehicle position error model by studying the relationship between the inbound vehicle and the vehicle movement before and after the inbound point in the team, and the research results can provide references for vehicle collaborative driving.Lin Heng et al. [8] proposed a refined hybrid traffic flow numerical simulation model based on the Gipps safe distance model, which can effectively reduce vehicle reaction time and improve lane congestion.
Based on the behavior characteristics of vehicle entry queue, this paper fully considers the actual operation requirements and resource constraints of airport support vehicles, and proposes a rational noncooperative game model of flight support vehicle queue behavior.By combining the actual requirements of airport for safety and environmental protection, and combining the game utility matrix and Nash equilibrium theory, the deviation between the game theory effect and the practice and the existence of the game result are verified respectively.Finally, the best parameter combination is obtained by comparing the fluctuation of the game model under the change of each parameter.According to all feasible results, the equilibrium scheme area of this model is obtained, which is convenient to optimize the scheduling control of the actual operation process of the vehicle.In the process of flight support, it is assumed that the vehicle queue L travels at a safe distance d according to the established operating path R , as shown in Figure 1.Due to the needs of the temporary support plan, the support vehicles C need to be imported into the support team from the parallel state, that is, they need to be imported from the lane line 1 B to 2 B , and together with the team into the narrow road, reverse curve and other special sections to complete the support operation.At this time, the queue ( )

Scenario description of game problems
1 : I L l l occupies part of the lane space, which easily leads to the failure of support vehicle C to complete the import and merge process.Some vehicles in the queue can obtain the import demand of parallel support vehicles C .This phenomenon is more common in airport support vehicles, there is a game problem, the result of which is limited by the cost function of both sides of the game, and the cost function is affected by the acceleration of the support vehicle.When the support vehicle C joins the lane 2 B , due to the need to follow the lane 2 B support vehicle, the speed of the support vehicle C is limited by the speed of the vehicle queue, and it needs to slow down or speed up to the same speed as the queue.To sum up, when the above-mentioned right-of-way conflicts occur, there is competition and cooperation between all the security vehicles, and each security vehicle aims to maximize its own security benefits.At the same time, the indexes of safety, economy and environmental protection are regarded as cost functions, and the optimal equilibrium solution of all guaranteed vehicles is determined, so that all guaranteed vehicles can drive out of the disputed area "safely, fairly and reasonably".

The basic assumption of the game
1) Ignore the effects of vertical movement and longitudinal and transverse aerodynamics on vehicle movement; 2) Assuming that all the vehicles involved in the game are rational participants, their final decision returns are greater than other decision returns.
3) It is assumed that the special vehicles can be divided into three types: radical type, ordinary type and conservative type.The radical type will have a high probability of merging, while the small probability will maintain the status quo and wait for the opportunity to change lanes.Conservative high probability will wait for the opportunity, small probability will force the lane change.

Symbols and variables
In order to facilitate the study of the game model, the symbolic parameters used in the subsequent model are described as follows: d is the safe distance between the vehicles in the process of ensuring the formation of vehicles; δ is the allowable deviation coefficient, ( ) Q is the cost and benefit of each guarantee vehicle, among C is a special need vehicle that needs to be incorporated into the security vehicle queue, , Where j C is the total number of special vehicles outside the queue; L is to ensure the vehicle queue set, , Where i L is the total number of vehicles included in the guaranteed vehicle queue; B is the set of lanes in a given path, , Where n B is the total number of parallel lanes in the given path; T refers to the time required for special vehicles to join the security vehicle queue; H is the displacement of the lateral movement of each guarantee vehicle.Where C H is the displacement of the lateral movement of the special vehicle.
, L i H is the displacement of the lateral movement required by the conflicting vehicle i in the queue when the special vehicles move horizontally; Z is the displacement of the longitudinal movement of each guarantee vehicle.Where C Z is the displacement of the longitudinal movement of the special vehicle., Z is the displacement of the longitudinal movement required by the conflicting vehicle i in the queue when the special vehicles move longitudinally;

Objective Function
The objective function comprehensively considers the safety, economy and environmental protection of airport support vehicles in the process of integration game., , , The , , formula represents the safety cost, the economic cost, and the environmental saving cost respectively.In order to apply the objective function to airport operation requirements and related safety and environmental protection policies, it is necessary to refine the safety, economy and environmental protection cost functions of the game model.

Security cost function
In order to reasonably describe the safety between the special vehicles and the guaranteed vehicle queue, this paper proposes an evaluation index of the safety distance between vehicles.The specific formula is shown in (2).The relationship between safety distance and cost is that the smaller the safety distance, the higher the cost.The greater the safety distance, the lower the cost.

Economic cost function
For special vehicles, if the i car in the queue changes, the subsequent I i − cars will take the same policy action.The economic cost function of special demand vehicles and queue vehicles is shown in equation (3) below.
1st International Conference on Applied Physics and Mathematics

Environmental cost function
In order to reasonably describe the impact of security vehicles on the airport environment, this paper uses the carbon emission rate estimation function, as shown in Formula (4) below.The greater the acceleration of the vehicle, the greater the carbon emissions and the higher the cost.On the other hand, the smaller the speed, the less carbon emissions, the smaller the cost.
Where: γ indicates the load rate; a represents the acceleration of each support vehicle; , , , , , β β β β β β is the defined parameter, and the value of the support vehicle varies according to the vehicle type.

Constraint model
In order to make the model applicable to airport operation requirements and related safety and environmental protection policies, it is necessary to restrict the safety, economy and environmental protection of the game model.1) Safety: In order to avoid collision during the process of importing a special vehicle into the vehicle queue, the relative position relationship between the vehicles before and after the importing position should be considered.
Where: C W and C G represent the width and length of the special vehicle, respectively.θ is the Angle between the front wheel Angle and the forward direction of the special vehicle.
2) Economy: In order to accurately describe the operation characteristics of flight support vehicles, it is necessary to limit the service times and service capabilities of various types of support vehicles, and ensure that each flight can only have one type of support vehicles for support services and only one service.
Where: , x y respectively represents the starting departure position of the guaranteed vehicle and the planned parking position of the flight served.
3) Environmental protection: the load of various types of security vehicles will affect the carbon emissions of the airport area, and the overall matching of the load of security vehicles required for each flight will effectively improve the ecological environment around the airport and enhance the service efficiency of the airport.

Nash equilibrium analysis
In non-cooperative games, when there is a strategy that makes all players have a strict advantage, this absolute favorable strategy is the one that all players are willing to choose and represents the most stable outcome of the game.However, in practical engineering problems, due to the utility conflict between the two game parties, it is difficult to realize the ideal strategy that all game parties achieve strict advantages.Non-cooperative Nash equilibrium is a solution based on such problems.Different from multi-objective programming, which coordinates the joint actions of all optimization objectives, in noncooperative Nash equilibrium, each player only makes absolute rational adjustments for his own benefits.When all players choose the best strategy compared with other strategies, the whole game system reaches Nash equilibrium.So when the players satisfy equation ( 8), the game reaches Nash equilibrium.
As the cost function is adopted in this paper, the purpose of control is to minimize the cost function, and the action strategies of each game party are bounded by constraints, which need to satisfy [ ] min max , a a a ∈ .To sum up, in order to ensure that the game problem between flights and vehicles always has a Nash equilibrium solution, the sufficient condition is: the cost function is quasi-convex, in which case the cost function has a minimum value.
The formula (9) is obtained by taking the partial derivative of the acceleration of the cost function of each vehicle in the special need vehicle and support vehicle queue and setting it equal to 0. There are 1 i + equations and 1 i + unknowns: acceleration.So we can find the Nash equilibrium solution. , At this time, if each player makes any change, it will only lead to its own income decline, and the system will become a stable state that no player can change.

Establishment of non-cooperative game effectiveness matrix
In order to obtain the effective results of the game model, according to the mathematical characteristics of the cost function as quasi-convex function and the utility characteristics of the actual engineering situation, the utility matrix of non-cooperative game under different strategies is constructed.The matrix is a combination of Hessian matrix for solving quasi-convex function and fuzzy clustering method to achieve flexible clustering of variables.The process is described as follows.
1) Efficiency dimension processing.Since the utility dimensions of the two sides of the game are different, dimensional processing is needed to eliminate the influence, as shown in equation (10).
2) Establish a cooperative cost function (CCF).This function presets the non-cooperative problem as each player participates in the game with the concept of cooperation, which can be expressed as: 3) Construct the Hessian matrix.According to the canonical form of Hessian matrix, the extreme value of the cost function is judged by judging the properties of the matrix (positive definite matrix, negative definite matrix, semi-positive definite matrix, semi-negative definite matrix).The formula is expressed as follows: When matrix * H is positive definite, the cooperation cost function has a minimum value.When matrix * H is negative definite, the cooperation cost function has a maximum value.When matrix * H is a semipositive definite or semi-negative definite matrix, there are extreme points to be measured, and other methods need to be used to assist the judgment.
4) The fuzzy similar matrix of Hessian matrix is constructed.Fuzzy similarity matrix ( ) established.The similar matrix of Hessian matrix is obtained according to the similar property.At the same time, ij s also needs to take into account the similarity of ik ϕ′′ and jk ϕ′′ , and finally calculate through the absolute distance method, as shown in formula ( 14).
Where: ε is the similarity correction coefficient, the value must be 5) The fuzzy equivalent matrix is established.The truncated number χ is selected according to the clustering demand to classify the matrix.
to get different cut sets, Finally, the same truncated set elements are clustered to realize the strategy aggregation mapping of various behavior situations to different game parties.
The vehicle collaborative modeling problem studied in this article belongs to cooperative games, where multiple variables are assigned to the same variable for control, such as acceleration.So, the examples in the following text mainly optimize the correlation coefficients and variables to achieve the goal of collaborative operation of vehicles in real scenarios.

Queue behavior policy control method
When a special need vehicle joins the support vehicle queue, the front vehicle needs to do uniform acceleration and the rear vehicle needs to do uniform deceleration.Therefore, in the process of vehicle operation, it is necessary to pay attention to the load of each vehicle, acceleration and the safety distance between the front and back of the vehicle.
When the non-cooperative game model is used to optimize the behavior of the vehicle queue, the strategy layer is established by analyzing the special vehicles and the vehicles in the queue.Then, in the game layer, fuzzy clustering is used to assign all kinds of behaviors between the special vehicles and the vehicles in the queue, and the cost function is constructed to form the game utility matrix with the goal of safety, economy and environmental protection.Finally, at the optimization level, the optimal control strategy is obtained by using the game utility matrix and Nash equilibrium analysis of queue behavior.The whole control method is shown in Figure 2.

Policy layer queue Behavior Verification
There are differences in the cognitive importance of drivers of different types of vehicles to their driving safety benefits, economic benefits and environmental benefits.For aggressive drivers, they will join the queue and pay more attention to economic benefits.For transport fuel powered vehicles, they are conservative and pay more attention to driving safety, generally take to move on, and their safety benefits are higher.The fuzzy weights of different types of drivers for each income are shown in Table 1.The membership function of conflict income is shown in Figure 3.
Table 1 As can be seen from Figure 3, due to the differences in the preferences of various drivers and the obvious conflicts and antagonisms in the interests and preference structures among them, this paper divides the behavior of vehicle queuing into the strategy layer, the game layer and the optimization layer.The hybrid fuzzy multi-objective multi-person non-cooperative game theory considering the driver's goal weight preference is used to describe the driver's game behavior.1) In Figure 4-a, as C λ increases, the safety of the vehicle gradually increases.The regulation of the lateral and longitudinal displacement of the vehicle is more careful, and the overshoot of the lateral movement is gradually slowed down, and the error in the process of importing the special vehicles is greatly reduced.
2) As shown in Figure 4-b, while taking into account the safety and economy, it is bound to produce a game between different main body guarantee vehicles.With the gradual change of C µ , the overshoot of the curve gradually increases.The main reason for this phenomenon is that the change of safety weight coefficient promotes the increase or decrease of vehicle spacing, which further affects the economic benefits of vehicle import behavior.
3)In Figure 4-c, With the C η increase, the lateral movement gradually slows down, the convergence speed and overshoot are significantly improved, but when the environmental protection is overcontrolled, it will inevitably lead to the widening of vehicle spacing and serious economic losses.Therefore, the effective control of vehicle spacing is the key factor to realize the environmental control optimization of vehicle import behavior.2) The vehicle behavior attributes are further divided into aggressive type, ordinary type and conservative type.According to various behavior attributes, corresponding control strategies are developed to form a policy layer.The game model is used to balance the actual demands of all parties and form a game layer.Finally, Nash equilibrium is used to achieve reasonable optimization of various demand objectives, and an optimization layer is constructed.Based on the strategy layer, game layer and optimization layer, an NCG-based behavioral strategy optimization control method is formed.
3) According to the above theories, methods and models, the actual operation cases of airports are applied for simulation.Finally, the numerical values of various weight coefficients and influence parameters that meet the actual requirements of safety, economy and environmental protection can be obtained by simulation, which provides a basis for model control.Moreover, the optimal feasible solution set and equilibrium point set of the model are further summarized.It is convenient to realize multi-objective and multi-attribute full strategy optimization control on the inbound queue behavior of guaranteed vehicles.

Figure 1 .
Figure 1.Schematic diagram of vehicle game scenario to guarantee the vehicle cost function in the vehicle queue; , , λ µ η is the target weight coefficient of the cost function, among , , C C C λ µ η is the weight coefficient of the cost function of special vehicles for lane change, , , L L L λ µ η is the weight coefficient of the vehicle queue cost function;

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International Conference on Applied Physics and Mathematics Journal of Physics: Conference Series 2729 (2024) 012023 IOP Publishing doi:10.1088/1742-6596/2729/1/012023represents the acceleration of each support vehicle, Where , L i a represents the acceleration of the Ith vehicle in the queue, , C j a represents the acceleration of the special vehicle j outside the queue; formula, the range transformation is shown in equation (11).

Figure 2 .
Figure 2. Optimization Control Method for Vehicle Queue Behavior Strategy Based on NCG

Figure 3 .Figure 4 .
Figure 3. Membership function of queue behavior returns5.2.Comparison and verification of control methods at the game level of queue behaviorAfter the partial derivation of the acceleration velocity of the game model, it can be concluded that the main factors affecting the acceleration include each weight coefficient, load coefficient and adjustment time.In order to analyze the influence of different parameters on import behavior, this part simulates and compares the effect of different influencing factors under different values.

4 )
As shown in Figure4-d, the adjustment time is the intermediate variable of the game model.Reasonable control of the adjustment time will fully improve the vehicle spacing, enhance the safety and economy of the vehicle queue, and the convergence rate of vehicle errors will gradually slow down with the increase of time.5)InFigure4-e, the load of the guarantee vehicle affects the running speed of the vehicle.With the gradual increase of the load, the overshoot of the lateral running of the vehicle increases greatly, which affects the safety and environmental protection of the vehicle importing behavior.Proper control of vehicle load protection will effectively optimize the overall performance of vehicle queue behavior.

Figure 5 .
Figure 5. Schematic Diagram of Game Returns Set and Nash Equilibrium ReturnsThe results of various cases are solved through calculation, and the feasible set of the game and the Nash equilibrium return calculation are drawn in Figure5.The yellow region is the solution set composed of all game behavior return points; The blue area is Nash equilibrium income.When the guaranteed vehicle behavior income is in the blue area, both sides of the game can satisfy the Nash equilibrium state.

1 )
Under the actual operation conditions of the airport, the corresponding requirements of safety, economy and environmental protection in the process of importing special vehicles into the security vehicle queue are analyzed, and factors such as the service capability, carbon emission and the Angle of importing vehicles are restricted to form a non-cooperative game model among the security vehicles.
. Fuzzy Weights of Different Types of Drivers for Each Cost Benefit

Table 2 .
Table of Weight Coefficients for Game Model of Ensuring Vehicle Queue Behavior