Autonomous Decision-Making for Air Gaming Based on Position Weight-Based Particle Swarm Optimization Algorithm

Abstract

As the complexity of air gaming scenarios continues to escalate, the demands for heightened decision-making efficiency and precision are becoming increasingly stringent. To further improve decision-making efficiency, a particle swarm optimization algorithm based on positional weights (PW-PSO) is proposed. First, important parameters, such as the aircraft in the scenario, are modeled and abstracted into a multi-objective optimization problem. Next, the problem is adapted into a single-objective optimization problem using hierarchical analysis and linear weighting. Finally, considering a problem where the convergence of the particle swarm optimization (PSO) is not enough to meet the demands of a particular scenario, the PW-PSO algorithm is proposed, introducing position weight information and optimizing the speed update strategy. To verify the effectiveness of the optimization, a 6v6 aircraft gaming simulation example is provided for comparison, and the experimental results show that the convergence speed of the optimized PW-PSO algorithm is 56.34% higher than that of the traditional PSO; therefore, the algorithm can improve the speed of decision-making while meeting the performance requirements.

1. Introduction

With the increasing complexity of modern air gaming scenarios, air gaming has gradually become a diversified challenge covering multi-domain applications [1,2]. The collaborative gaming strategy, due to its significant cost-effectiveness, high flexibility and adaptability, and powerful information fusion capability, has firmly occupied the forefront of international research and strategic deployment [3,4,5]. In this context, cluster countermeasures, as an emerging and highly promising research trend [6,7,8,9], are gradually transitioning from being the subject of theoretical explorations to real-world applications. In a coordinated multi-target decision-making environment, air gaming decision-making is not only directly related to victory or defeat on the battlefield but also involves complex problems such as massive data processing, real-time situational assessment, and dynamic target allocation [10,11,12,13]. This means that traditional algorithms face great challenges in dealing with such problems and need to rely on more advanced artificial intelligence, machine learning, big data analysis, and other modern technological means to realize deep mining, intelligent analysis, and rapid decision-making using information.

By reviewing the related literature, it can be observed that when solving air gaming decision-making problems, numerous studies tend to use a single algorithm or fuse multiple algorithms. However, air gaming problems are highly specific, requiring a precise grasp of our needs and complex judgments regarding the position of the aircrafts, which increases the complexity of the problem. In view of this, this paper carries out an in-depth analysis of specific scenarios in the process of air gaming and unifies the criteria regarding the parameters of the decision-making method and the evaluation system. Through simulating an air gaming problem, a way to improve the algorithm is obtained. A set of reasonable air gaming simulation experiments is designed, and an accurate evaluation of the decision-making method can be realized.

Achieving more efficient real-time decision-making in the highly dynamic and changing air gaming environment remains a challenge. Considering the complexity of air gaming systems and the extremely high demands regarding reaction speed, this paper proposes an innovative position weight-based particle swarm optimization algorithm (PW-PSO), which retains the high decision-making accuracy of the traditional PSO algorithm. Through the introduction of position weight information, the velocity update strategy can be optimized. This optimization strategy can intelligently and dynamically adjust the search speed of particles according to the distance between the current decision and the global optimal solution, thus accelerating the convergence of the algorithm while ensuring the accuracy of the decision-making results, which provides a more efficient and reliable solution for decision-making during air gaming.

The following sections are structured as follows: Section 2 describes the related works. Section 3 describes the specific air gaming problem and applies the AHP method to assign weights to the indicators at each level. Section 4 provides an in-depth introduction to the principles and implementations of the PW-PSO algorithm. Section 5 comprehensively verifies the effectiveness and operational efficiency of the proposed PW-PSO algorithm through simulation experiments. Section 6 summarizes the research.

2. Related Works

In the complex battlefield environment of large-scale air gaming, the efficient assignment of tasks among aircrafts is particularly important. This is not only related to whether each aircraft can fully utilize its performance advantages but also directly affects the overall effectiveness during gaming [14,15,16,17]. Nantogma, S, constructed a strategy-generating agent based on learning classifier-inspired coordination mechanism, which was inspired by an artificial immune system, skillfully fusing the strengths of a learning classifier system and an artificial immune algorithm to achieve intelligent learning for combat threat assessment and weapon allocation [18]. Liu, S, on the other hand, applied the partial triangular fuzzy number method to accurately define the relative distance and angle of the missile ships as key parameters of their formation, and then proposed an adaptive algorithm that integrates simulated annealing and particle swarm optimization (SA-PSO), which effectively solves the optimization problem of missile formation [19]. In addition, Liang, Z. et al. innovatively proposed a distributed self-organizing CISCS strategy for the challenges in multi-unmanned-aerial-vehicle (UAV) cooperative operations, which integrates finite-time formation control, the distributed ant colony optimization task assignment based on improved Q-Learning, and real-time obstacle avoidance path planning, significantly improving the efficiency of cooperative reconnaissance and operations [20]. Li et al. proposed a constrained strategy game approach to generate intelligent decisions for multiple UCAVs in a complex air combat environment by utilizing situational air combat information and time-sensitive information, and proposed a constrained strategy game-solving algorithm based on linear programming and linear inequalities (CSG-LL) [21].

In the process of multi-aircraft collaborative decision-making, the establishment of an appropriate air gaming performance indicator model will directly affect the effectiveness, accuracy, and real-time abilities of the decision-making system [22,23]. FANG et al., in response to the insufficient objectivity in air warfare threat assessment indicators, proposed an empowerment method combining the Analytical Hierarchy Process (AHP) method and the CRITIC method, realizing the combined empowerment of threat assessment indicators by using the AHP method to synthesize the expert’s judgment of the indicator’s weights and then determining the linkage between the objective data according to the CRITIC method [24]. Zhang et al. proposed an air attack target threat metrics ranking method based on the fusion of entropy and the AHP method to solve the problem of strong subjectivity and stereotyped logic of ranking in the evaluation of the target threat during traditional air defense operations [25].

PSO algorithm [26], as a stochastic optimization method based on group intelligence, is known for its rapid computation and excellent optimization ability, and is widely used in various scenarios of air combat decision-making [27]. Ding, Y. et al. constructed an effectiveness evaluation model based on the PSO-BP neural network by combining the fuzzy analysis method and BP neural network in the field of unmanned-surface-vehicle cooperative combat effectiveness evaluation [28]. Cheng, Z. et al. used agent-based modeling technology to deeply analyze the structure and function of the ballistic missile defense system and successfully applied the PSO algorithm to construct a multi-agent decision support system containing missile agents [29]. Wu et al. proposed an improved PSO algorithm based on the Zaslavskii chaos. The Zaslavskii chaotic mapping formulation is used to generate chaotic sequences, which improves the inertia weights and random variables of the particle swarm algorithm for solving the path planning problem of unmanned combat aerial vehicles (UCAVs) in threatened battlefields [30]. Zhao et al. proposed the SHOPSO algorithm, which combines a selfish population optimizer (SHO) and a PSO to maximize and minimize the cost of UAVs to avoid enemy radar reconnaissance and artillery attacks [31].

3. A Model of the Air Gaming Decision Problem

3.1. Description of the Problem

In the decision-making process of the air game, the platforms of the two sides of the aircraft and the decision-making results are not a simple one-to-one correspondence; locking accuracy, both sides of the position, attitude, distance and other factors will affect the final gaming results. However, as simulations involve more factors, their implementation becomes more challenging, so this paper simplifies the decision-making problem by reducing the gaming scenario from three dimensions to two. Next, the factors affecting the results are identified through AHP, which can be used to calculate the probability of the opposing equipment locking on to our platform and the probability of our equipment successfully locking on to the opposing aircraft.

Assuming both the other side and we have N aircrafts, in a period, an aircraft can only launch one. $P_o(i,j)$ is the success rate of the i-th aircraft lock on to the j-th opposing aircraft; the mission of the i-th aircraft is assigned to the j-th opposing aircraft, the probability of the j-th opposing aircraft surviving is $1-P_o(i,j)$. Correspondingly, the opposing aircraft can also use lock on us on our N aircrafts. $P_e(i,j)$ is the success rate of the opposant’s i-th aircraft destroying our j-th target. $J_o(i,j)$ represents the result of the decision of aircraft i against opposing aircraft j, where 0 means no action is taken and 1 means action is taken. Meanwhile, $J_e(i,j)$ denotes the decision result of opposing aircraft locking on our aircraft under our judgment.

The success probability of all N aircrafts’ actions in concert for the opposant:

$$P_{og}(j)={\displaystyle {\sum }_{i=1}^N{P}_o(i,j)\cdot J_o(i,j)}.$$

(1)

Correspondingly, in the case of an opposant’s action, the success probability of escape for our aircraft j is

$$P_{eg}(j)=1-{\displaystyle {\sum }_{i=1}^N{P}_e(i,j)\cdot J_e(i,j)}.$$

(2)

Set the number of our aircrafts at 6 and the number of opposing aircrafts at 6. In this paper, we randomly generated the position and attitude of 12 air aircrafts within a 1000 × 1000 grid.

3.2. AHP-Based Constraint Establishment

Multi-aircraft air gaming has various attributes, a complex structure, and difficulties in quantitatively assessing the evaluation of the object. The aircraft launching an action must consider multiple factors, requiring a more comprehensive and specific approach to account for various elements. This ensures the authenticity of the experiment during the simulation process. Therefore, this paper chooses the AHP method to evaluate the factors that need to be considered for launching an action, determining the multi-level and multi-element constraints on the aircraft’s action evaluation.

The basic idea of the AHP method is to decompose the complex problem into various constituent factors, and then group the factors according to their dominant relationship to form a hierarchical structure. The relative importance of each factor in the hierarchy is determined by pairwise comparison, combining quantitative and qualitative analysis, using the experience of the decision-maker to judge the relative importance of each criterion for measuring whether the goal can be achieved, and reasonably giving the weight of each criterion for each decision-making scheme, and using the weights to find out the order of the scheme’s advantages and disadvantages.

3.2.1. Hierarchical Modeling

The multi-level structural model is constructed as shown in Figure 1. Among them, the target layer is the air gaming capability index, the criterion layer includes four items of the environment, aircraft flight performance, weapon system, and aircraft hardware performance, and the indicator layer covers three items of aircraft attack range D, aircraft detection range R and aircraft attack angle A. The action capability of the aircraft is also reflected through the above three indicators.

3.2.2. Judgment Matrix Construction

Two-by-two comparisons were made for each factor at the same level to assess their importance relative to specific criteria at the previous level. This process uses a 1–9 scale to quantitatively express the relative importance between factors, and a judgment matrix is constructed accordingly. The generation of the judgment matrix is based on expert evaluation, and sub-indicators are obtained for each assessment indicator, resulting in a matrix of assessment indicator values A. It is worth noting that the matrix A exhibits positive reciprocity, i.e., the elements in the matrix satisfy the following:

$$\left\{\begin{array}{l}0<a_{ij}\le 9\\ a_{ij}={\displaystyle \frac{1}{a_{ji}}}(i\ne j)\\ a_{ii}=1\end{array}\right..$$

(3)

This characteristic ensures the consistency and rationality of the judgment matrix. The judgment matrices obtained in this paper are shown in

Table A2. A-B and B-C two-level judgment matrices and processing.

A	B1	B2	B3	B4	${\mathit{W}}_{\mathit{i}}$	${\mathit{W}}_{\mathit{i}}^{\mathbf{0}}$	${\mathit{\lambda }}_{\mathit{m}\mathit{i}}$	$\begin{array}{l}{\lambda }_{\max }\approx 4.153\\ C.I.=0.018\\ R.I.=0.89\\ C.R.\approx 0.057<0.1\end{array}$
B1	1	1/2	1/3	1/4	0.452	0.097	4.104
B2	2	1	1/2	1/3	0.759	0.164	4.078
B3	3	2	1	2	1.861	0.401	4.226
B4	4	3	1/2	1	1.565	0.338	4.205
B1	C1	C2	C3		${\mathit{W}}_{\mathit{i}}$	${\mathit{W}}_{\mathit{i}}^{\mathbf{0}}$	${\mathit{\lambda }}_{\mathit{m}\mathit{i}}$	$\begin{array}{l}{\lambda }_{\max }\approx 3.004\\ C.I.=0.002\\ R.I.=0.52\\ C.R.\approx 0.004<0.1\end{array}$
C1	1	1/3	2		0.874	0.230	3.002
C2	3	1	5		2.466	0.648	3.004
C3	1/2	1/5	1		0.464	0.122	3.005
B2	C1	C2	C3		${\mathit{W}}_{\mathit{i}}$	${\mathit{W}}_{\mathit{i}}^{\mathbf{0}}$	${\mathit{\lambda }}_{\mathit{m}\mathit{i}}$	$\begin{array}{l}{\lambda }_{\max }\approx 3.039\\ C.I.=0.020\\ R.I.=0.52\\ C.R.\approx 0.038<0.1\end{array}$
C1	1	1/3	1/5		0.406	0.105	3.036
C2	3	1	1/3		1.000	0.258	3.040
C3	5	3	1		2.466	0.637	3.040
B3	C1	C2	C3		${\mathit{W}}_{\mathit{i}}$	${\mathit{W}}_{\mathit{i}}^{\mathbf{0}}$	${\mathit{\lambda }}_{\mathit{m}\mathit{i}}$	$\begin{array}{l}{\lambda }_{\max }\approx 3.003\\ C.I.=0.0015\\ R.I.=0.52\\ C.R.\approx 0.003<0.1\end{array}$
C1	1	7	3		2.759	0.682	3.003
C2	1/7	1	1/2		0.415	0.103	3.003
C3	1/3	2	1		0.874	0.215	3.002
B4	C1	C2	C3		${\mathit{W}}_{\mathit{i}}$	${\mathit{W}}_{\mathit{i}}^{\mathbf{0}}$	${\mathit{\lambda }}_{\mathit{m}\mathit{i}}$	$\begin{array}{l}{\lambda }_{\max }\approx 3.0099\\ C.I.=0.0045\\ R.I.=0.52\\ C.R.\approx 0.009<0.1\end{array}$
C1	1	3	2		1.817	0.540	3.009
C2	1/3	1	1/2		0.550	0.163	3.010
C3	1/2	2	1		1.000	0.297	3.008

in Appendix A.

3.2.3. Calculation of Relative Weights of Elements

The square root method is a technical tool to compute the normalized relative importance vector $W_i^0$ of each element associated with a particular element in its upper layer. Specifically, the calculations are based on the following formula:

$$W_i={({\displaystyle {\prod }_{j=1}^n{a}_{ij}})}^{{\displaystyle \frac{1}{n}}},$$

(4)

$$W_i^0={\displaystyle \frac{W_i}{{\displaystyle {\sum }_i{W}_i}}}.$$

(5)

By applying this formula to the five judgment matrices constructed in Section 3.2.2, we can calculate the relative importance vectors corresponding to each judgment matrix. The results of the calculations are exhaustively organized and presented in Table A2 in Appendix A.

3.2.4. Consistency Test

In hierarchical analysis, consistency tests must be performed to verify the reasonableness and feasibility of the judgment matrix obtained. Two key indicators commonly used in this process are the consistency indicator (C.I.) and the consistency ratio (C.R.). The specific formulas for calculating these two indicators are as follows:

$$C.I.={\displaystyle \frac{{\lambda }_{\max }-n}{n-1}},$$

(6)

$${\lambda }_{\max }\approx {\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\frac{{(AW)}_i}{W_i}=\frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\frac{{\displaystyle {\sum }_{j=1}^n{a}_{ij}{W}_j}}{W_i}},$$

(7)

where ${(AW)}_i$ denotes the i-th component of the vector AW.

$$C.R.={\displaystyle \frac{C.I.}{R.I.}}<0.1.$$

(8)

In the hierarchical analysis method, the Random Index (R.I.) is used as the average random consistency indicator, and its value is given through

Table A1. Average random consistency indicator.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
R.I.	0	0	0.52	0.89	1.12	1.26	1.36	1.41	1.46	1.49	1.52	1.54	1.56	1.58

in Appendix A, which shows the average random consistency indicator from 1000 calculations based on the positive inverse matrices of order 1 to 14. Since R.I. represents the average level of the consistency indicator of the same-order stochastic judgment matrices, its introduction effectively mitigates the problem that the consistency judgment indicator may significantly increase as the matrix order n increases.

Eventually, the calculated C.R. values, are shown in Table A2 in Appendix A, all the listed C.R. values satisfy the criterion of less than 0.1. This result shows that the degree of inconsistency of the above judgment matrix is controlled within acceptable limits, thus validating the reasonableness and effectiveness of the judgment matrix.

Table 1. Table of indicator weights.

	B1	B2	B3	B4	W
	0.097	0.164	0.401	0.338
C1	0.230	0.105	0.682	0.540	0.496
C2	0.648	0.258	0.103	0.163	0.202
C3	0.122	0.637	0.215	0.297	0.302

summarizes the results of the stratified analysis of the overall importance of the indicators:

Based on the table above, the importance weights for each of the three indicators in the indicator layer can be determined. Subsequently, when constructing the objective function, the weights assigned to these indicators will directly refer to the corresponding data in Table 1.

By using the AHP method, the factors influencing the success or failure of an aircraft action and the corresponding weights are determined, in which the relationship between the parameters of the aircraft attack range D, the angle of the opposing aircraft relative to us A, and the detection range R of the aircraft is shown in Figure 2.

In the above figure, orange aircraft indicates our aircraft, blue indicates opposing aircraft, and yellow areas indicate the attackable range of our aircraft. $\overrightarrow{VO}$ is the direction vector of our aircraft, $\overrightarrow{OE}$ is the vector of the opposing aircraft relative to our aircraft, D is the distance of the opposing aircraft relative to our aircraft, and R is the detection range of our aircraft. The angle at which our aircraft can action includes only 180° in front of the aircraft. Only when the distance of the enemy aircraft relative to our aircraft is less than the detection range R of our aircraft and the angle of the opposing aircraft relative to our aircraft is within the detection range of our aircraft:

$$\left\{\begin{array}{l}D\le R\\ \overrightarrow{VO}\cdot \overrightarrow{OE}>0\end{array}\right..$$

(9)

In addition, since an aircraft can select only one opposing aircraft in an operation, no more than N actions are made by all the aircraft of our team in an operation and no more than 1 action is made by one aircraft:

$$\left\{\begin{array}{l}{\displaystyle {\sum }_{i=1}^N{\displaystyle {\sum }_{j=1}^N{J}_o(i,j)\le N}}\\ {\displaystyle {\sum }_{j=1}^N{J}_o(i,j)\le 1}\end{array}\right..$$

(10)

The constraints generated are based on a combination of the aircraft’s current relevant information and the action constraints set by the weapon system, as described in the previous section.

3.3. Objective Function

In the air gaming problem, the main task is to eliminate the maximum loss to us from the target, i.e., to ensure the maximum damage to the opposant while minimizing the loss to ourselves. Firstly, in terms of ensuring the maximum loss to the opposant from our side, the assignment is expressed by the formula:

$$\max P_o={\alpha }_1\cdot (1-D/R)+{\alpha }_2\cdot J_o.$$

(11)

And in terms of ensuring that we minimize the loss incurred to our side, the assignment is expressed as follows:

$$\min P_e={\beta }_1\cdot (1-D/R)+{\beta }_2\cdot J_e.$$

(12)

The main point of importance is to decide which opposing aircraft to attack by judging their loss value to us. If the opposant’s aircraft is not within our action range, then $P_e$ takes the maximum value of the opposant’s threat to our aircraft.

The air gaming decision-making problem is a multi-objective function problem, which is simplified and solved using a particle swarm algorithm. In this paper, we use the linear weighted sum method to simplify the two objective functions, set the weight coefficients according to the importance of the two objective functions, and set the weight coefficient of the objective function (11) to be ${\omega }_1$, and the weight coefficient of the objective function (12) to be ${\omega }_2$. Then, the objective function can be simplified into

$$Q={\omega }_1\cdot P_o-{\omega }_2\cdot P_e.$$

(13)

The primary purpose of the air gaming decision is to inflict damage on the enemy; therefore, the initialization sets the weight coefficients ${\omega }_1=0.7$ and ${\omega }_2=0.3$.

4. Particle Swarm Algorithm Based on Position Weight Velocity Update Strategy

4.1. Particle Swarm Algorithm (PSO)

The PSO algorithm is an intelligent optimization algorithm inspired by the foraging behavior of bird flocks. In this algorithm, a population of randomly distributed particles is first initialized, with each particle representing a potential solution to the problem. Subsequently, each particle dynamically adjusts its flight speed and position based on its own historical best position (the individual optimal solution) as well as the global best position (the global optimal solution) discovered by the entire flock so far. This adjustment mechanism enables the particles to efficiently search for and approximate the optimal solution in the solution space.

The updating of position and velocity in particle swarm algorithms usually follows the following formula:

$$v_i(t+1)=\omega v_i(t)+c_1(P_{best}(t)-x_i(t))+c_2(G_{best}(t)-x_i(t)),$$

(14)

$$x_i(t+1)=x_i(t)+v_i(t+1),$$

(15)

where $v_i(t)$ is the velocity of particle i at the t-th iteration on dimension d; $\omega$ is the inertia weight, which is used to control the degree of inheritance of the particle’s velocity; $c_1$ and $c_2$ are the learning factors, that regulate the step size of the particle’s leaning towards the individual optimum and global optimum; $P_{best}$ is the individual optimal position; $G_{best}$ is the value of the global optimal position on dimension d; $x_i(t)$ is the position of particle i at the t-th iteration on dimension d.

By iteratively performing the above velocity and position updating process, the particle swarm algorithm can induce the entire population of particles to gradually converge to or approach the optimal solution region of the problem.

When dealing with large-scale complex problems, PSO often experiences high consumption of computational resources and takes a long time to explore and converge to the optimal solution. However, a common challenge in the process of approximating the optimal solution is the rapid slowdown of the particles or the limitation of the updating mechanism, which may weaken the fineness of the search and make it difficult for the algorithm to pinpoint the globally optimal solution accurately. To overcome this challenge, this paper innovatively proposes a velocity updating strategy based on position weights, aiming to improve the search precision and efficiency of the algorithm.

4.2. Speed Update Strategy Based on Positional Weights (PW-PSO)

To achieve a more rapid and accurate judgment of strategies in an intense air gaming environment, we have targeted and optimized the velocity update mechanism in the particle swarm algorithm. By incorporating a position weight parameter into the velocity update formula, we provide the algorithm with a kind of intelligent adjustment capability: when the decision judged by a particle (potential strategy) is at a large distance from the optimal decision, the parameter will automatically increase, prompting the particle to accelerate in the optimal direction to quickly narrow the gap; when the decision is close to the optimal solution, the weight will be reduced accordingly, slowing down the speed of the particle, making the search process more detailed and avoiding missing potential better solutions near the optimal solution.

According to the idea of the dynamic adjustment of the speed of the decision and the distance from the optimal solution, this design, in which the position weight parameter is introduced, adds the parameter r and weight, and updates the speed formula, which will be adjusted as follows:

$$v_i(t+1)=\omega v_i(t)+(c_1(P_{best}(t)-x_i(t))+c_2(G_{best}(t)-x_i(t)))(G_{best}(t)-x_i(t)).$$

(16)

The main steps of the proposed PW-PSO method are as Algorithm 1, where eval is the number of times the function is evaluated and MaxEval is the maximum number of times the function is evaluated.

Algorithm 1: PW-PSO-based air gaming decision-making

Begin

for each particle i do:

Randomly initialize the position $x_i$;

Randomly initialize the velocity $v_i$;

Calculate fitness value EFFECT($x_i$);

Set individual optimal position $P_{best}(i)=x_i$;

Set the global optimal position $G_{best}$ as the position of the particle with the best fitness value among all $P_{best}(i)$;

while eval ≤ MaxEval do

if r < 0.6 then

for each particle i do

Update the velocity according Equation (16);

Update position according to Equation (15);

Calculate the fitness value;

eval++;

end for

else

for each particle i do

Update the velocity according Equation (14);

Update position according to Equation (15);

Calculate the fitness value;

eval++;

end for

end if

Update the $P_{best}$ and $G_{best}$ in the population;

end while

End

5. Simulation Analysis

This paper takes the two-dimensional static multi-objective air gaming decision-making problem as the research object, establishes a mathematical model, and formulates an objective function according to the two directions of maximizing the action effect of our side and minimizing the loss level of the opposant; and applies the linear weight method to simplify the multi-objective function into a single objective function. According to the constraints, the objective function is corrected in the process of randomly generating the initial solution to make it conform to the constraints, and the PW-PSO is utilized for the solution calculation. Moreover, under the same conditions, the problem is solved using the original PSO algorithm, and the two are compared and analyzed to observe the capability enhancement effect of the PW-PSO algorithm. The computer configuration chosen for the experiment was equipped with an Intel(R) Core(TM) i5-10400F CPU running Windows 10 operating system, and MATLAB 2022a was used as the experimental software.

5.1. Randomized Parameter Setting

Within the framework of the simulation experiments in this paper, the pre-positional attitude information is obtained by random generation. For the position information part, the x and y coordinates are randomly selected within a wide range of [0, 1000] to simulate different spatial positions. The angle A, as the angle between the direction towards which the aircraft is facing and the x-axis, has a value randomly determined within a complete circle of [0, 360] degrees, as shown in the representation below. In the experiment, we recorded the specific data generated randomly once, as shown in

Table 2. Aircraft position attitude numeric.

	${\mathbf{Position}}_{\mathbf{o}}$						${\mathbf{Position}}_{\mathbf{e}}$
x	930.85	416.03	842.14	852.24	375.65	821.45	787.02	326.78	810.63	374.63	183.52	68.71
y	655.17	771.38	45.47	976.99	566.90	527.89	52.94	646.00	743.36	442.05	862.94	902.56
A	20.53	35.95	58.41	216.17	129.67	265.52	54.24	165.82	211.89	77.61	141.74	171.79

.

Converting the data in Table 2 into a graphical representation, we can visualize the 2D planar position map of the aircraft, as shown in Figure 3. In this figure, the orange markers represent the position and attitude of our aircraft, while the blue markers indicate position and orientation of the opposing aircraft.

5.2. Results and Analysis

Based on the position and attitude information obtained in the previous section, combined with the constraints defined in Section 3 and the designed fitness function, four optimization algorithms, namely genetic algorithm (GA) [32], PSO algorithm, adaptive particle swarm algorithm (Adaptive-PSO) [33], and PW-PSO, were selected for the experiments in this study. All the algorithms reached the same maximum value of 1.10301 in the experiments. To further evaluate the accuracy and stability of these algorithms, each algorithm independently conducted 100 repetitions of the experiments for the purpose of counting the number of times that each of them reached the known maximum value. Based on the results of these experiments, the accuracy assessment of each algorithm is shown in Figure 4.

As can be obtained from Figure 4, the accuracy of experiments using PSO and its two improved versions (Adaptive-PSO and PW-PSO) both exceeded 80%, demonstrating the superior performance of the PSO and its variants in dealing with the specific problem. In contrast, the GA exhibited a lower accuracy of 30%, indicating that the GA may not be the best choice in this application scenario. In light of these results, the following section focuses on more in-depth experiments with these three high-performing PSO to explore and compare their convergence rates.

Comparative experiments of the three PSO algorithms were conducted to obtain Figure 5 and Figure 6. Figure 5 shows the fitness function curves of the three algorithms during one of the experiments. Figure 6 shows a box plot of the number of generations of convergence of the three algorithms after 100 experiments on the three algorithms.

From Figure 5 and Figure 6, it is obvious that the distribution of convergence generations of the PSO algorithm is broader, mainly focusing on the range of 0 to 300 iterations, showing greater volatility, and the average number of convergent generations of the algorithm is about 142 times. In contrast, the Adaptive-PSO algorithm shows improvement in convergence speed, and its average number of convergent generations is reduced to about 124, indicating that the adaptive mechanism helps to accelerate the convergence process of the algorithm.

The PW-PSO algorithm shows a significant improvement in convergence speed, and most of its convergent generations are concentrated in the smaller range of 0 to 100 iterations, with an average number of convergent generations of 62 approximately. This result indicates that the convergence speed of the PSO algorithm is improved by introducing the position weighting mechanism to optimize the PSO algorithm, which can significantly improve the convergence performance and stability.

In addition, to comprehensively evaluate the performance of the three algorithms, the key indexes of the algorithms, such as the success rate, the average convergence value, and the average number of convergence generations, are counted and summarized in

Table 3. Summary of experimental results data.

	Number of Experiments	Success Rate (%)	Optimal Convergence Value	Average Convergence Value	Average Convergent Algebra	Convergence Value Variance
PSO	100	87	1.10301	1.07665	142	0.00519
PW-PSO	100	99	1.10301	1.10152	62	0.00022
Adpative-PSO	100	84	1.10301	1.07195	124	0.00578

.

The PW-PSO algorithm performs well in 100 independent experiments, with 80 experiments converging significantly faster than the PSO algorithm, which proves the significant improvement of PW-PSO in convergence efficiency. In addition, the experimental success rate of PW-PSO is as high as 98%, which is not only much higher than the 87% of PSO but also higher than the 84% of Adaptive-PSO, which demonstrates its high stability and reliability.

Further analysis of the average convergence value and the variance of the convergence value reveals that PW-PSO also shows advantages in precision and accuracy, outperforming both PSO and Adaptive-PSO. This indicates that PW-PSO can not only converge quickly in the process of searching for the optimal solution, but also approach the true optimal value with higher precision, reducing the volatility and error of the results.

In particular, the average number of convergence generations of PW-PSO is only 62, which is a significant reduction compared to 142 generations of PSO and 124 generations of Adaptive-PSO. The 56.34% improvement in the number of convergence generations for PW-PSO compared to PSO is a figure that intuitively reflects the dramatic improvement in the convergence speed of the algorithm.

In summary, the PW-PSO algorithm outperforms the traditional PSO and Adaptive-PSO algorithms in several aspects such as convergence speed, success rate, precision, and accuracy.

6. Conclusions

With the increasing complexity of air gaming scenarios, more stringent requirements are imposed on the efficiency and accuracy of the decision-making process. To effectively deal with this challenge, this paper innovatively proposes PW-PSO. The algorithm first extracts and models the key parameters of the aircraft and environment in air gaming, and transforms them into a complex multi-objective optimization problem. Subsequently, the multi-objective problem is successfully transformed into a more manageable single-objective optimization problem by skillfully applying hierarchical analysis and linear weighting techniques. To address the shortcoming of the traditional PSO algorithm’s insufficient convergence speed in air gaming scenarios, the PW-PSO algorithm achieves a significant improvement in the performance through the introduction of the concept of positional weights and the optimization speed update mechanism. To verify this optimization result, we designed a 6v6 air gaming simulation scenario for comparison experiments. The experimental data clearly show that, compared to the traditional PSO algorithm, the convergence speed of the PW-PSO algorithm is improved by 56.34%, and at the same time, the decision-making process is significantly accelerated under the premise of satisfying the performance constraints, which lays a solid foundation for the efficient execution of air gaming decisions.

Although the PW-PSO algorithm shows significant advantages in convergence speed and stability, it may still fall into a local optimal solution under specific situations. Furthermore, when applying the PW-PSO algorithm to actual air gaming decision-making scenarios, it is necessary to consider numerous practical variables, including the dynamic changes in the environment and the differences in the performance of equipment, etc., which may limit the application of the algorithm to a certain extent. The limitations in the above two areas are also directions for further improvement of the algorithm.

For the limitations of the PW-PSO algorithm, the algorithm can be further optimized in the future to improve its efficiency and accuracy in dealing with complex problems. For the problem of falling into local optimal solutions, more advanced parameter tuning methods can be explored to reduce the dependence on parameter settings and improve the global search capability of the algorithm. To overcome the challenge of limited complex scenarios, more complex simulation scenarios can be designed to consider more practical factors in order to evaluate the performance of the algorithm more comprehensively.