A method for analyzing the flexibility of a military airport runway under air guidance countermeasure

Abstract

The application discloses a kind of military airport runway tenacity analysis methods under air guidance confrontation, creates the military facilities tenacity analysis framework considering the uncertainty of "red and blue both sides attack and defense strategy-weapon confrontation result-facility physical damage-facility function loss-facility function recovery " five links.Equipment runway function is used as evaluation object, and three-stage tenacity analysis method of "evaluation-optimization-analysis" is proposed.In the "evaluation" stage, the expression form and transmission process of the uncertainty of the last four links of military facilities tenacity analysis framework are described after establishing function index evaluation model, and the first link of "optimization" stage is described;Establish attack and defense strategy optimization model to describe the first link of military facilities tenacity analysis framework;The equilibrium strategy result obtained by attack and defense strategy optimization model is input into function index evaluation model, and the tenacity analysis result that can directly, quantitatively and reliably represent the function loss and recovery of military facilities is obtained.

Description

Technical Field

[0001] This invention relates to the field of airport damage assessment technology, specifically to a method for analyzing the toughness of military airport runways under air defense countermeasures. Background Technology

[0002] Military facilities primarily face the risk of deliberate enemy attacks. Assessing their functional loss and recovery time directly impacts the subsequent operational process and may even influence overall campaign decisions. Existing Damage Effect Evaluation (DEE) methods typically analyze the physical damage to military facilities after weapon attacks using numerical simulations, reliability-based damage trees, and artificial intelligence. They assess the facility's functionality by establishing a mapping between physical damage and functionality. However, current damage assessment methods do not consider the uncertainties arising from the strategies of both sides and the actual weapon-on-weapon confrontation, nor do they account for the uncertainties of the functional recovery process. Since opposing sides in combat often flexibly adapt their strategies based on the opponent's, the uncertainty surrounding the risks to military facilities is significant. Coupled with numerous uncertainties related to weapon performance and the operational environment, this presents a considerable challenge to the functional assessment of military facilities. The aforementioned damage effect evaluation results only represent the functional loss of facilities under a specific type of attack, resulting in a singular and one-sided assessment that cannot provide reliable data support for command and decision-making.

[0003] "Resilience" provides a theoretical framework for addressing the uncertainties in various aspects of system function under the influence of disasters. Resilience research related to natural disasters typically follows a sequence of steps: disaster risk analysis, physical damage to facilities, loss of facility function, and recovery of facility function. Each step is analyzed under the preconditions of the previous step, thus ensuring that the impact of each step and its uncertainties on functional loss and recovery is fully considered and transmitted. Current research on the resilience of military facilities mainly focuses on conceptual definition, indicator assessment, and practical application methods, lacking systematic research on the analysis and quantitative assessment of uncertainty transmission throughout the entire process. Summary of the Invention

[0004] In order to overcome the shortcomings of the above technologies, this invention provides a method for analyzing the toughness of military airport runways under air defense countermeasures.

[0005] Terminology Explanation:

[0006] 1. Air Defense and Missile Countermeasures (ADMC): This refers to the classic offensive and defensive confrontation between ballistic missiles and air defense missiles.

[0007] 2. AFRMI: Analysis Framework for the Resilience of Military Infrastructure.

[0008] 3. MOS: Minimum Operating Strip, representing the minimum runway operating plane required for fighter jet takeoff and landing.

[0009] 4. CEP: Circular Error Probable, which means that there is a 50% probability that the missile will fall into a circle with the aiming point as the center and CEP as the radius.

[0010] 5. SI: Situation.

[0011] 6. AC: Action.

[0012] 7. EF: Effect, result.

[0013] 8. DA: Damage.

[0014] 9. LO: Loss, loss.

[0015] 10. RE: Restoration.

[0016] 11. SSPK: Single-shot Probability of Kill, the probability of the blue team's interceptor missile intercepting a target with a single shot.

[0017] 12. TZMG: Two-player Zero-Sum Markov Games.

[0018] 13. MDP: Markov Decision Process.

[0019] 14. FMEM: Functional Metrics Evaluation Model.

[0020] 15. OMODS: Optimal Model of Offensive and Defensive Strategies.

[0021] The technical solution adopted by this invention to overcome its technical problems is:

[0022] A method for analyzing the resilience of military airport runways under air defense countermeasures includes the following steps:

[0023] S1. Based on the theory of resilience disaster prevention, a military facility resilience analysis framework, denoted as AFRMI, is created that considers the uncertainties of five stages: “offensive and defensive strategies of both sides - weapon confrontation results - physical damage to facilities - loss of facility function - recovery of facility function”.

[0024] S2. Taking the airport runway function under the background of air defense confrontation as the evaluation object, the resilience analysis of the airport runway is divided into three stages: "evaluation-optimization-analysis" based on AFRMI.

[0025] S3. Define and quantify the resilience index of the airport runway, determine the airport runway function curve based on the airport runway resilience index, and use the airport runway resilience index and airport runway function curve as the results of resilience analysis.

[0026] S4. In the "assessment" phase, the uncertainty of the four links of "weapon confrontation results - facility physical damage - facility function loss - facility function recovery" is analyzed, and a functional index assessment model is established, denoted as FMEM, to obtain the functional index of each airport runway under different offensive and defensive strategies.

[0027] S5. In the "optimization" phase, the "offensive and defensive strategies of the red and blue sides" are analyzed and an offensive and defensive strategy optimization model is established, denoted as OMODS. The offensive and defensive equilibrium strategy results obtained by solving the offensive and defensive strategy optimization model are input into the functional index evaluation model, thereby obtaining the resilience analysis results considering the offensive and defensive strategies of the red and blue sides during combat.

[0028] Furthermore, step S1 specifically includes the following:

[0029] Based on the probability analysis framework of building structure function loss considering various uncertainties under natural disasters, the probability distribution of military facility function indicators under offensive and defensive confrontation is obtained, as shown in formula (1). Through the probability distribution of military facility function indicators, a military facility resilience analysis framework considering the uncertainties of five links is obtained: "offensive and defensive strategies of both sides - weapon confrontation results - physical damage to facilities - loss of facility function - recovery of facility function".

[0030] (1)

[0031] In formula (1), Given the current combat situation SI, the conditional probability density function for the offensive and defensive actions AC chosen by both the red and blue sides. To determine the conditional probability density function of the weapon confrontation outcome EF under offensive and defensive actions AC; To determine the conditional probability density function of the facility physical damage DA under the confrontation outcome EF; To determine the conditional probability density function of facility function loss LO under physical damage DA; To determine the cumulative distribution function of facility function recovery (RE) under functional loss (LO).

[0032] Furthermore, step S2 specifically includes the following:

[0033] In the analysis of airport runway resilience under the background of air defense confrontation, the functional indicators of each airport runway under all possible offensive and defensive strategies are first evaluated and the offensive and defensive benefit matrix is ​​constructed. Then, based on the offensive and defensive benefit matrix data, the offensive and defensive equilibrium strategies of the red and blue sides for each airport runway are optimized and solved. Finally, the offensive and defensive equilibrium strategies of the red and blue sides for each airport runway are analyzed to obtain the resilience analysis results considering offensive and defensive strategies.

[0034] Furthermore, step S3 specifically includes the following:

[0035] Airport runway resilience refers to the operational support capability of an airport runway after it has been damaged, as well as its ability to recover to a level of operational support required to exceed the damaged state.

[0036] The resilience of airport runways is quantified through two indicators. Specifically, under the offensive and defensive strategies of both sides, the probability distribution of the functional state of the airport runway at the initial moment of damage is defined as follows: After repair according to the repair strategy, the functional state changes from the initial functional state. Restore to the target functional state The total time was ,Will and As two indicators for quantifying the resilience of airport runways;

[0037] based on and The functional state probability distribution that changes with recovery time is obtained through the transition probability matrix. and will As an airport runway function curve, it is used to describe the change pattern of airport runway function from damage to recovery after offensive and defensive confrontation.

[0038] The resilience index of airport runways and and airport runway function curves These are the three results of the toughness analysis.

[0039] Furthermore, step S4 specifically includes the following:

[0040] S41. Air-to-air weapon confrontation simulation: The actions of the Red Force include at least the performance of the missile-launching weapon, the number of missiles, and the penetration method; the actions of the Blue Force include at least the performance of the interceptor weapon and the interception method. The weapon confrontation process is simulated through four stages: "missile launch detection and tracking - real and false warhead identification - interceptor missile launch - missile penetration and hit" to obtain the weapon confrontation result. The weapon confrontation result is represented by the probability of the combination of the number of missiles actually hit by the aiming point. The combination of the number of missiles actually hit by the aiming point is simply referred to as the hit combination.

[0041] S42. Random Destruction Scenario Simulation: The destruction scenario is defined as the damage to the runway surface by the submunitions after the Red Force missile completes its penetration. Due to the circular probability deviation of the mother missile and the bullet dispersion error, there are still random destruction scenarios under the same hit combination. The random destruction scenario modeling method is adopted to perform Monte Carlo simulation on a random destruction scenario under the hit combination to form a random destruction scenario sample set. Each sample in the random destruction scenario sample set contains the actual impact point coordinate information of all bullets.

[0042] S43. Construction of the mapping relationship between physical damage to airport runways and functional states: Based on the fact that airport runways support the take-off and landing of different types of fighter jets, functional states are divided. Assuming that the airport runways of the air force base need to support the take-off and landing of large, medium, and small fighter jets during the current operational phase, the functional states of the airport runways are divided into four types, as follows:

[0043] Fully functional state: Capable of supporting the takeoff and landing of large, medium, and small fighter jets. express;

[0044] Basic functional status: Cannot support the takeoff and landing of large fighter jets, but can support the takeoff and landing of medium and small fighter jets. express;

[0045] Minimum functional state: Cannot support the takeoff and landing of large and medium-sized fighter jets, only supports the takeoff and landing of small fighter jets. express;

[0046] Complete loss status: Unable to support the takeoff and landing of any type of fighter jet among the three categories: large, medium, and small. express;

[0047] For a sample set of random damage scenarios, a functional state classifier based on minimum operating plane MOS search is used to establish a mapping relationship between physical damage and functional state.

[0048] S44. Airport runway functional recovery analysis: Assuming that the airport runway repair mode includes two types, step-by-step repair and task-first repair, a functional recovery time analysis model based on discrete state Markov process is constructed to analyze the law of change of the airport runway functional state probability distribution function with repair time.

[0049] Furthermore, in step S41, the air-to-air weapon countermeasure simulation specifically includes:

[0050] It has The first aiming point, the red direction on the airport runway. Missiles launched from a single aiming point At this point, the total number of combinations that hit the airport runway is... Assuming the penetration hit probability of each missile is independent, the first... The missile successfully penetrated the defenses and hit the target at the aiming point. The probability of sending Follows Bernoulli distribution Assuming the hit probability distribution for each aiming point is independent, then the hit combination... probability Represented as:

[0051] (2)

[0052] In formula (2), This indicates a combination of offensive and defensive actions. and These represent the attack actions of the red team and the defensive actions of the blue team, respectively. , This indicates the probability that a single missile fired by the Red Team will successfully penetrate the defenses. ;

[0053] Derived from the penetration probability model:

[0054] (3)

[0055] (4)

[0056] (5)

[0057] In formula (3), This indicates the probability that the detection and tracking accuracy of the missiles launched by the Red Force meets the launch requirements of the anti-missile system; This indicates the reliability of the interceptor missile platform launch, i.e., the probability of successfully launching an interceptor missile; This indicates the probability that a real warhead is correctly identified; This represents the probability of the three stages: "missile launch detection and tracking - real and false warhead identification - interceptor launch";

[0058] Formula (4) uses This represents the single-shot interception probability of the Blue Team's interceptor missiles against the target; assuming that each interceptor missile has the same single-shot interception probability against the target, then for each identified warhead, the Blue Team launches... Interceptor missiles were launched to intercept them. This indicates the probability that the missile was successfully intercepted;

[0059] Missiles only have two states: penetration and interception. Therefore, formula (5) represents the probability of a single missile penetrating the defense.

[0060] Furthermore, in step S42, the random destruction scenario simulation specifically includes:

[0061] Random destruction scenario sample set established by random destruction scenario modeling method Each sample in It contains the actual impact coordinates of all bullets;

[0062] Missile mother warhead There is a 50% probability that the target will land in a circle centered on the aiming point. Within a circle of radius , the mother missile carries the submunitions. The impact point coordinates are based on the actual impact point of the mother missile as the center, within a radius of... The projectiles are uniformly and randomly distributed within the circle. The expression for the bullet's impact point is shown in formula (6). The coordinates of the explosion point are expressed as in formula (7), and formulas (6) and (7) are as follows:

[0063] (6)

[0064] (7)

[0065] In formulas (6)-(7), the mother missile The coordinates of the aiming point are , The circular probability deviation (CEP) of a missile strike is represented by the number of missiles that strike it. and The random value representing the standard normal distribution in two dimensions, the mother missile The actual landing point coordinates are ;bullet Explosion point coordinates Therefore, the mother bomb Actual landing point coordinates Let be two independent, uniformly distributed random numbers originating from the origin. and Follows a uniform random distribution. Assume that the shape of each bullet crater is relative to the coordinates of the explosion point. Centered on a circle with radius equal to the bullet's damage radius. A standard circle.

[0066] Furthermore, in step S43, for the random damage scenario sample set, a functional state classifier based on minimum operating plane MOS search is used to establish the mapping relationship between physical damage and functional state, specifically including:

[0067] Indicates that the airport runway was initially damaged at the time of the incident. , , , The probability distribution of the four functional states; a sample set of random destruction scenarios. The samples are input into the functional state classifier and sorted according to the highest functional state. The corresponding MOS will sequentially judge and confirm the functional status of the sample;

[0068] Let the MOS of small, medium, and large fighter jets be represented as follows: , , , < < By optimizing the search algorithm, a sample set of random destruction scenarios was analyzed. Each sample in the dataset is evaluated to determine its functional status. Specifically, when no complete piece exists... At that time, continue to Perform a search; if it exists Then the functional state of the sample is determined to be... If there is no complete piece Then continue with Perform a search; if it exists Then the functional state of the sample is determined to be... If it does not exist Then the functional state of the sample is determined to be... This will allow for the creation of a random destruction scenario sample set. Classified as This corresponds to four functional states, thus completing the mapping from physical damage to functional states. The probability distribution of each functional state is determined by the number of samples from different functional states. Percentage of total sample size The proportion is expressed as follows:

[0069] (8)

[0070] right All possible combinations of hits were subjected to Monte Carlo simulation, resulting in:

[0071] (9)

[0072] Formula (9) incorporates uncertainties from three stages: "weapon confrontation outcome - facility physical damage - facility functional loss," and is used to obtain the resilience index. .

[0073] Furthermore, in step S44, the airport runway function recovery analysis specifically includes:

[0074] S441. Assume there are two runway repair modes: tiered repair and mission-priority repair. The runway repair mode is selected based on the urgency of the combat mission. Assume that crater repair time only considers the number of craters, and that all bullets are non-blocking bullets. Indicates the functional state Restore to functional state Repair required Furthermore, the repair time is directly proportional to the number of craters that need to be repaired, depending on the functional status. Restore to functional state Recovery time Represented as:

[0075] (10)

[0076] In formula (10), Indicates that repair is needed. Number of bullet holes This indicates the average time taken to repair one shell crater;

[0077] When the functional status of the airport runway is restored sequentially, the step-by-step repair mode is selected; when the functional status of the airport runway is restored in a skipped manner, the task priority mode is selected.

[0078] S442, The blue team's goal is to find the shortest... That is, to find the one with the least of , recorded as The sample set was obtained through Monte Carlo simulation. Sample set This represents the sample set under all possible hit combinations. Divided into different subsets , subset Samples in Search and calculate the minimum number of craters to repair for each sample. Thus, the total time taken to repair the sample is obtained;

[0079] Assume the initial functional state of the airport runway after the attack. Set the target functional state That is, the final repair goal. Let the functional state between the initial functional state and the target functional state be... ;

[0080] The total time taken to restore from the initial functional state to the target functional state in the step-by-step repair mode is calculated as follows:

[0081] (11-1)

[0082] In the task-priority mode, the total time to recover from the initial functional state to the target functional state is:

[0083] (11-2)

[0084] Statistical sample data, to obtain The probability density function and cumulative distribution function of the step-by-step repair mode and the task-priority mode are calculated using the same methods. The probability density function and cumulative distribution function are shown in Equation (12) and Equation (13) respectively, as follows:

[0085] (12)

[0086] (13)

[0087] Formulas (12) and (13) are based on the hit combination As a prerequisite, formulas (12) and (13) are weighted and summed, with the weights being the combination of offensive and defensive actions. Lower Hit Combination The probability of is expressed as follows:

[0088] (14)

[0089] (15)

[0090] S443. Construct a functional recovery time analysis model based on discrete-state Markov processes to analyze the variation of the functional state probability distribution function of an airport runway with repair time. Specifically, the functional state probability distribution is expressed as a function of repair time, as follows:

[0091] (16)

[0092] In formula (16), Indicates the initial moment of repair. The probability distribution of the functional state at that time; This represents the transition probability matrix, due to the one-way nature of the repair. An upper triangular matrix is ​​represented as follows:

[0093] (17)

[0094] In formula (17), Indicates the initial time. Functional status is Under the conditions, The status of the time function is changed by Turn to The probability, , ;

[0095] Represented as:

[0096] (18)

[0097] In formula (18), Indicates the functional state A higher level of functional status, Indicates functional status Restore to functional state The cumulative distribution function of total time.

[0098] Furthermore, step S5 specifically includes the following:

[0099] The attack and defense strategy optimization model is configured to solve the problem of a zero-sum Markov game between red and blue sides, denoted as TZMG, and represented by the following 8-tuple:

[0100] (1) The operational situation space should include at least the remaining weapon types and quantities of the Red Force, the remaining weapon types and quantities of the Blue Force, the dimensions of the airport runway in the current offensive and defensive confrontation, and the operational support tasks.

[0101] (2) This represents the red side's action space, and each action... Indicates the type and quantity of missiles used;

[0102] (3) This represents the action space of the blue team, and each action... Indicates the type of interceptor missile used and the method of interception;

[0103] (4) Representing state Red takes action The probability of;

[0104] (5) Representing state The blue team takes action The probability of;

[0105] (6) If the red team's reward set is represented, then the blue team's reward is a negative number of the red team's reward. The reward setting is based on the output of the functional indicator evaluation model.

[0106] (7) Represents the state transition probability, indicating that in At that moment, the red team took action. Take action with the blue team Post-state Transition to the next state The probability of;

[0107] (8) This is a discount factor, indicating the importance of future rewards. This illustrates that the outcome of each round of competition is equally important. This indicates that the results of the current round are given more importance;

[0108] The analysis process of air defense and air-guided attack and defense strategies for multiple airport runways is as follows: First, a Markov decision process simulation of attack and defense is performed, with the initial state... It consists of the initial resource inputs of both the red and blue teams and any unassigned airport runway; both sides use strategies... and Select initial state Offensive and defensive actions offensive and defensive actions and state The input is fed into the functional performance evaluation model, and the output of the functional performance evaluation model is selected as the reward for the first round of competition. The status has been updated to Repeat the above process until the Red team's resources are exhausted or all airport runways are used up, at which point the process terminates. When the entire confrontation process ends and a trajectory is formed, it is represented as... The outcome of the entire confrontation process is represented by a cumulative discounted reward, also known as a "reward," as follows:

[0109] (19)

[0110] In formula (19), Indicates the initial state According to the strategy and The reward of a round of simulated combat; due to the uncertainty of offensive and defensive strategies, both sides focus on the expected reward, initial state. The expected return is:

[0111] (20)

[0112] In formula (20), Indicating in strategy and Below Seeking expectations, Represents the initial state According to the strategy and The expected return of engaging in confrontation;

[0113] The goal of the Red team is to optimize their offensive strategy. Make The goal is to maximize the objective, while the blue team's objective is to optimize their defensive strategy. Make Minimum; exists a Nash equilibrium joint strategy This makes formula (21) true, where and These represent the optimal offensive strategy for the red side and the optimal defensive strategy for the blue side, respectively:

[0114] (twenty one)

[0115] In formula (21), This represents the expected return for the Red team when it adopts the optimal offensive strategy and the Blue team when it does not adopt the optimal defensive strategy; This represents the expected reward when both the red and blue teams adopt the optimal strategy. This represents the expected return for the Red team not adopting the optimal offensive strategy and the Blue team adopting the optimal defensive strategy;

[0116] The expected cumulative reward for multiple airport runways under the Nash equilibrium joint strategy is simplified as formula (22):

[0117] (twenty two)

[0118] The Q-learning algorithm is used to solve the attack and defense strategy optimization model represented by formula (22) to obtain the Nash equilibrium joint strategy. The specific form.

[0119] The beneficial effects of this invention are as follows: 1. This invention proposes a military facility resilience analysis framework, denoted as AFRMI, that considers the uncertainties of five stages: "offensive and defensive strategies of both sides, weapon confrontation results, physical damage to facilities, loss of facility function, and recovery of facility function." Taking the assessment of airport runway function under air-to-air missile confrontation as an example, the application method and advantages of AFRMI are introduced. Compared with existing military facility damage effect assessments, AFRMI can not only obtain an offensive and defensive equilibrium strategy that conforms to the current combat situation and considers the resource constraints of both sides, but also obtain the impact of weapon confrontation results on the function of military facilities under this equilibrium strategy. Furthermore, it uses function curves based on probability distribution to characterize the loss and recovery of military facility function, providing quantitative and comprehensive resilience analysis results. Simultaneously, it can also serve as an analytical tool for higher-level strategic decisions such as optimizing operational objectives, weapon resource investment, and repair force allocation, providing more accurate, effective, and advantageous decision support for combat command in battlefield environments. 2. This invention takes airport runways under air-to-air missile countermeasures as the resilience analysis object and constructs a functional index evaluation model (FMEM) for airport runways under known offensive and defensive strategies. FMEM consists of four modules: "weapon confrontation simulation," "random destruction scenario simulation," "construction of the mapping relationship between airport runway physical damage and functional state," and "airport runway functional recovery analysis." Specifically, for the uncertainty in the "weapon confrontation simulation" stage, analytical formulas are used, and the probability of hit combinations can intuitively and clearly represent the uncertainty of the weapon confrontation outcome. For the uncertainty in the "random destruction scenario simulation," the mother missile impact point deviation and bullet dispersion error are used for description. Random samples are generated through Monte Carlo simulation to ensure the reliability of the random samples. For the uncertainty in the "mapping relationship between airport runway physical damage and functional state," an optimized search algorithm is used to classify random destruction scenario samples, and the probability of functional state is expressed by frequency, concisely and efficiently reflecting the uncertainty of functional state. For the uncertainty in the "airport runway functional recovery analysis," it is indirectly reflected by the uncertainty of the number of craters to be repaired, and two methods, namely step-by-step repair and task-priority, are proposed, with statistical analysis of the probability distribution of repair time for each method. 3. This invention studies the uncertainty propagation in five stages: "offensive and defensive strategies of both sides - weapon confrontation results - physical damage to facilities - loss of facility function - recovery of facility function." It uses multiple conditional probability integrals to represent the probability distribution of military facility function indicators to describe and handle the uncertainty propagation process in these four stages. 4. This invention analyzes offensive and defensive strategies, reducing the problem to a zero-sum Markov game between the red and blue sides. It establishes an optimization model for these strategies and uses the Q-learning algorithm to solve for the equilibrium strategy. Attached Figure Description

[0120] Figure 1This is a flowchart illustrating the method for analyzing the toughness of military airport runways under air defense countermeasures, as described in an embodiment of the present invention.

[0121] Figure 2 This is a schematic diagram of the method for analyzing the toughness of military airport runways under air defense countermeasures, as described in an embodiment of the present invention.

[0122] Figure 3 This is a schematic diagram of the military facility resilience analysis framework described in an embodiment of the present invention.

[0123] Figure 4 This is a schematic diagram illustrating the simulation process of a random runway destruction scenario as described in an embodiment of the present invention.

[0124] Figure 5 This is a schematic diagram illustrating the mapping relationship between physical damage to functional status of an airport runway as described in an embodiment of the present invention.

[0125] Figure 6 This is a schematic diagram of the airport runway function recovery analysis process described in an embodiment of the present invention.

[0126] Figure 7 This is a functional curve diagram illustrating the step-by-step repair mode for airport runways as described in an embodiment of the present invention.

[0127] Figure 8 This is a functional curve diagram illustrating the task-priority repair mode for airport runways as described in an embodiment of the present invention.

[0128] Figure 9 This is a schematic diagram of the attack and defense strategy analysis process described in an embodiment of the present invention.

[0129] Figure 10 This is a schematic diagram of the 9×3 attack and defense benefit matrix of airport runway 1 obtained by using the functional index evaluation model described in this embodiment of the invention.

[0130] Figure 11 This is a schematic diagram of the 9×3 attack and defense benefit matrix of airport runway 2 obtained by using the functional index evaluation model described in this embodiment of the invention.

[0131] Figure 12 This is a schematic diagram of the 9×3 attack and defense benefit matrix of airport runway 3 obtained by using the functional index evaluation model described in this embodiment of the invention.

[0132] Figure 13 This is a schematic diagram of the 9×3 attack and defense benefit matrix of airport runway 4 obtained by using the functional index evaluation model described in this embodiment of the invention.

[0133] Figure 14 This is a schematic diagram comparing the cumulative returns of the random attack strategy and the balanced attack and defense strategy described in an embodiment of the present invention.

[0134] Figure 15 This is a schematic diagram of the cumulative distribution function of the functional state transitions of the airport runway 2 according to an embodiment of the present invention.

[0135] Figure 16 This is a schematic diagram of the functional curves of airport runway 2 under the offensive and defensive balance strategy described in this embodiment of the invention. Detailed Implementation

[0136] To facilitate a better understanding of the present invention by those skilled in the art, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. The following are merely exemplary and do not limit the scope of protection of the present invention.

[0137] This invention discloses a method for analyzing the toughness of military airport runways under air defense countermeasures, such as... Figure 1 and Figure 2 As shown, it includes the following steps:

[0138] S1. Based on the theory of resilience disaster prevention, a military facility resilience analysis framework, denoted as AFRMI, is created, considering the uncertainties of five stages: "offensive and defensive strategies of both sides - weapon confrontation results - physical damage to facilities - loss of facility function - recovery of facility function." Figure 3 As shown.

[0139] In this embodiment, step S1 specifically includes: based on the probability analysis framework for the functional loss of building structures under natural disasters, considering the transmission of various uncertainties, the probability distribution of military facility functional indicators under offensive and defensive confrontation is obtained, as shown in formula (1). Through the probability distribution of military facility functional indicators, a military facility resilience analysis framework considering the uncertainties of five links are obtained: “offensive and defensive strategies of both sides - weapon confrontation results - physical damage to facilities - functional loss of facilities - functional recovery of facilities”.

[0140] (1)

[0141] In formula (1), Given the current combat situation SI, the conditional probability density function for the offensive and defensive actions AC chosen by both the red and blue sides. To determine the conditional probability density function of the weapon confrontation outcome EF under offensive and defensive actions AC; To determine the conditional probability density function of the facility physical damage DA under the confrontation outcome EF; To determine the conditional probability density function of facility function loss LO under physical damage DA; To determine the cumulative distribution function of facility function recovery (RE) under functional loss (LO).

[0142] Formula (1) is a multiple integral formula, which means that the analysis of the functional recovery of military facilities under the current combat situation needs to be carried out in accordance with the steps of "1. Offensive and defensive strategy analysis - 2. Confrontation result analysis - 3. Physical damage analysis - 4. Functional loss analysis - 5. Functional recovery analysis". Each analysis step is carried out on the basis of the previous analysis step, so as to ensure that the uncertainty of the intermediate links in the analysis of the functional recovery of military facilities under the known current combat situation is fully considered and transmitted.

[0143] S2. Taking the airport runway function under the background of air defense as the evaluation object, the resilience analysis of the airport runway is divided into three stages: "evaluation-optimization-analysis" based on AFRMI.

[0144] In this embodiment, step S2 specifically includes the following:

[0145] When using AFRMI, the uncertainty in the analysis of the four steps of "weapon confrontation outcome - facility physical damage - facility functional loss - facility functional recovery" includes objective factors such as weapon performance, facility structure, and environmental conditions. The uncertainty in the analysis of the "red and blue sides' offensive and defensive strategies" step comes from the decisions of the commanders on both sides, and the decision-making is based on the evaluation results of the four steps of "weapon confrontation outcome - facility physical damage - facility functional loss - facility functional recovery". The analysis of the "red and blue sides' offensive and defensive strategies" is supported by the analysis and evaluation results of the four steps of "weapon confrontation outcome - facility physical damage - facility functional loss - facility functional recovery", optimizes the final form of the strategies of both sides, and then uses the results of the offensive and defensive strategy analysis as input to obtain the resilience analysis results considering the offensive and defensive strategies. Therefore, in the analysis of airport runway resilience under the background of air defense confrontation, the functional indicators of each airport runway under all possible offensive and defensive strategies are first evaluated and the offensive and defensive benefit matrix is ​​constructed. Then, based on the offensive and defensive benefit matrix data, the offensive and defensive equilibrium strategies of the red and blue sides for each airport runway are optimized and solved. Finally, the offensive and defensive equilibrium strategies of the red and blue sides for each airport runway are analyzed to obtain the resilience analysis results considering offensive and defensive strategies.

[0146] S3. Define and quantify the resilience index of the airport runway, determine the airport runway function curve based on the airport runway resilience index, and use the airport runway resilience index and airport runway function curve as the results of resilience analysis.

[0147] In this embodiment, step S3 specifically includes the following:

[0148] Airport runway resilience refers to the operational support capability of an airport runway after it has been damaged, as well as its ability to recover to a level of operational support required to exceed the damaged state.

[0149] The resilience of airport runways is quantified through two indicators. Specifically, under the offensive and defensive strategies of both sides, the probability distribution of the functional state of the airport runway at the initial moment of damage is defined as follows: After repair according to the repair strategy, the functional state changes from the initial functional state. Restore to the target functional state The total time was ,Will and These are two indicators used to quantify the resilience of airport runways.

[0150] based on and The functional state probability distribution that changes with recovery time is obtained through the transition probability matrix. and will As an airport runway function curve, it is used to describe the change pattern of airport runway function from damage to recovery after offensive and defensive confrontation.

[0151] The resilience index of airport runways and and airport runway function curves These are the three results of the toughness analysis.

[0152] S4. In the "Assessment" phase, the uncertainties of the four stages of "weapon confrontation results - facility physical damage - facility function loss - facility function recovery" are analyzed, and a functional index assessment model, denoted as FMEM, is established to obtain the functional index of each airport runway under different offensive and defensive strategies.

[0153] In this embodiment, step S4 specifically includes the following:

[0154] S41. Simulation of missile-guided air defense weapon confrontation: The actions of the Red Force include at least the performance of the missile-launching weapon, the number of missiles, and the penetration method; the actions of the Blue Force include at least the performance of the interceptor weapon and the interception method. The weapon confrontation process is simulated through four stages: "missile launch detection and tracking - real and false warhead identification - interceptor missile launch - missile penetration and hit". The weapon confrontation result is represented by the probability of the combination of missiles actually hit by the aiming point. The combination of missiles actually hit by the aiming point is simply referred to as the hit combination.

[0155] In this embodiment, the simulation of air-to-air weapon countermeasures specifically includes:

[0156] It has The first aiming point, the red direction on the airport runway. Missiles launched from a single aiming point At this point, the total number of combinations that hit the airport runway is... Assuming the penetration hit probability of each missile is independent, the first... The missile successfully penetrated the defenses and hit the target at the aiming point. The probability of sending Follows Bernoulli distribution Assuming the hit probability distribution for each aiming point is independent, then the hit combination... probability Represented as:

[0157] (2)

[0158] In formula (2), This indicates a combination of offensive and defensive actions. and These represent the attack actions of the red team and the defensive actions of the blue team, respectively. , This indicates the probability that a single missile fired by the Red Team will successfully penetrate the defenses. ;

[0159] Derived from the penetration probability model:

[0160] (3)

[0161] (4)

[0162] (5)

[0163] In formula (3), This indicates the probability that the detection and tracking accuracy of the missiles launched by the Red Force meets the launch requirements of the anti-missile system; This indicates the reliability of the interceptor missile platform launch, i.e., the probability of successfully launching an interceptor missile; This indicates the probability that a real warhead is correctly identified; This represents the probability of the three stages: "missile launch detection and tracking - identification of real and false warheads - interceptor launch".

[0164] Formula (4) uses This represents the single-shot interception probability of the Blue Team's interceptor missiles against the target; assuming that each interceptor missile has the same single-shot interception probability against the target, then for each identified warhead, the Blue Team launches... Interceptor missiles were launched to intercept them. This indicates the probability that the missile is successfully intercepted.

[0165] Missiles only have two states: penetration and interception. Therefore, formula (5) represents the probability of a single missile penetrating the defense.

[0166] S42. Random Destruction Scenario Simulation: The destruction scenario is defined as the damage to the runway surface by the submunitions after the Red Force missile completes its penetration. Due to the circular probability deviation of the mother missile and the dispersion error of the submunitions, there are still random destruction scenarios under the same hit combination. The random destruction scenario modeling method is adopted to perform Monte Carlo simulation on a random destruction scenario under the hit combination to form a random destruction scenario sample set. Each sample in the random destruction scenario sample set contains the actual impact point coordinate information of all submunitions.

[0167] In this embodiment, the blue team's runway is 2800m × 30m. The red team launches two missiles at three aiming points (700, 15), (1400, 15), and (2100, 15), and both missiles successfully penetrate the defenses. This is a random destruction scenario under this hit combination. like Figure 4 As shown, each hit combination obtained from the simulation of air-to-air weapon countermeasures Monte Carlo simulations were conducted to generate a sample set of random destruction scenarios. Each of the samples It contains the actual impact coordinates of all bullets. The actual impact coordinates of each missile's bullets are as follows: Figure 4 As shown in the black dashed box, specifically, the missile's mother missile. There is a 50% probability that the target will land in a circle centered on the aiming point. Within a circle of radius , as shown by the blue dashed box, are the submunitions carried by the mother missile. The impact point coordinates are based on the actual impact point of the mother missile as the center, within a radius of... The components are uniformly and randomly distributed within the circle, as shown in the red dashed box. (The parent missile...) The expression for the bullet's impact point is shown in formula (6). The coordinates of the explosion point are expressed as in formula (7), and formulas (6) and (7) are as follows:

[0168] (6)

[0169] (7)

[0170] In formulas (6)-(7), the mother missile The coordinates of the aiming point are , The circular probability deviation (CEP) of a missile strike is represented by the number of missiles that strike it. and The random value representing the standard normal distribution in two dimensions, the mother missile The actual landing point coordinates are ;bullet Explosion point coordinates Therefore, the mother bomb Actual landing point coordinates Let be two independent, uniformly distributed random numbers originating from the origin. and Follows a uniform random distribution. Assume that the shape of each bullet crater is relative to the coordinates of the explosion point. Centered on a circle with radius equal to the bullet's damage radius. A standard circle.

[0171] S43. Construction of the mapping relationship between physical damage to airport runways and functional states: Based on the fact that airport runways support the take-off and landing of different types of fighter jets, functional states are divided. Assuming that the airport runways of the air force base need to support the take-off and landing of large, medium, and small fighter jets during the current operational phase, the functional states of the airport runways are divided into four types, as follows:

[0172] Fully functional state: Capable of supporting the takeoff and landing of large, medium, and small fighter jets. express.

[0173] Basic functional status: Cannot support the takeoff and landing of large fighter jets, but can support the takeoff and landing of medium and small fighter jets. express.

[0174] Minimum functional state: Cannot support the takeoff and landing of large and medium-sized fighter jets, only supports the takeoff and landing of small fighter jets. express.

[0175] Complete loss status: Unable to support the takeoff and landing of any type of fighter jet among the three categories: large, medium, and small. express.

[0176] For a sample set of random damage scenarios, this embodiment proposes to use a functional state classifier based on minimum operating plane MOS search to establish a mapping relationship between physical damage and functional state, such as... Figure 5 As shown, it specifically includes:

[0177] Indicates that the airport runway was initially damaged at the time of the incident. , , , The probability distribution of the four functional states; a sample set of random destruction scenarios. The samples are input into the functional state classifier and sorted according to the highest functional state. The corresponding MOS (Method in Module) is sequentially judged and the sample functional status is confirmed. Let the MOS for small, medium, and large fighter jets be represented as follows: , , , < < By optimizing the search algorithm, a sample set of random destruction scenarios was analyzed. Each sample in the dataset is evaluated to determine its functional status. Specifically, when no complete piece exists... At that time, continue to Perform a search; if it exists Then the functional state of the sample is determined to be... If there is no complete piece Then continue with Perform a search; if it exists Then the functional state of the sample is determined to be... If it does not exist Then the functional state of the sample is determined to be... This will allow for the creation of a random destruction scenario sample set. Classified as This corresponds to four functional states, thus completing the mapping from physical damage to functional states. The probability distribution of each functional state is determined by the number of samples from different functional states. Percentage of total sample size The proportion is expressed as follows:

[0178] (8)

[0179] right All possible combinations of hits were subjected to Monte Carlo simulation, resulting in:

[0180] (9)

[0181] Formula (9) incorporates uncertainties from three stages: "weapon confrontation outcome - facility physical damage - facility functional loss," and is used to obtain the resilience index. .

[0182] S44. Airport runway functional recovery analysis: Assuming that the airport runway repair mode includes two types, step-by-step repair and task-first repair, a functional recovery time analysis model based on discrete state Markov process is constructed to analyze the law of change of the airport runway functional state probability distribution function with repair time.

[0183] In this embodiment, the airport runway function recovery analysis specifically includes:

[0184] S441. Assume there are two runway repair modes: tiered repair and mission-priority repair. The runway repair mode is selected based on the urgency of the combat mission. Assume that crater repair time only considers the number of craters, and that all bullets are non-blocking bullets. Indicates the functional state Restore to functional state Repair required Furthermore, the repair time is directly proportional to the number of craters that need to be repaired, depending on the functional status. Restore to functional state Recovery time Represented as:

[0185] (10)

[0186] In formula (10), Indicates that repair is needed. Number of bullet holes This indicates the average time taken to repair one crater.

[0187] When the functional status of the airport runway is restored sequentially, the step-by-step repair mode is selected; when the functional status of the airport runway is restored in a skipped manner, the task priority mode is selected.

[0188] S442, The blue team's goal is to find the shortest... That is, to find the one with the least of , recorded as The sample set was obtained through Monte Carlo simulation. Sample set This represents the sample set under all possible hit combinations. Divided into different subsets , subset Samples in Search and calculate the minimum number of craters to repair for each sample. Thus, the total time taken to repair the sample is obtained.

[0189] Assume the initial functional state of the airport runway after the attack. Set the target functional state That is, the final repair goal. Let the functional state between the initial functional state and the target functional state be... .

[0190] The total time taken to restore from the initial functional state to the target functional state in the step-by-step repair mode is calculated as follows:

[0191] (11-1)

[0192] In the task-priority mode, the total time to recover from the initial functional state to the target functional state is:

[0193] (11-2)

[0194] Statistical sample data, to obtain The probability density function and cumulative distribution function of the step-by-step repair mode and the task-priority mode are calculated using the same methods. The probability density function and cumulative distribution function are shown in Equation (12) and Equation (13) respectively, as follows:

[0195] (12)

[0196] (13)

[0197] Formulas (12) and (13) are based on the hit combination As a prerequisite, formulas (12) and (13) are weighted and summed, with the weights being the combination of offensive and defensive actions. Lower Hit Combination The probability of is expressed as follows:

[0198] (14)

[0199] (15)

[0200] Due to the inherent uncertainties, the steps of runway function recovery analysis are as follows: Figure 6 As shown, if analysis is required Recovery time, ① Sample set after functional state classification Select sample Search using optimization algorithms Location (e.g.) Figure 6 (The red rectangle in the middle) is represented by a triangle. The origin ( The bottom left corner, with coordinates (1400, 10), is used to search for the sample. 5; ② For Each sample All were optimized using algorithm search. The spatial distribution of the origin is as follows Figure 6 The triangular distribution of the runway surface is shown, and statistics can be obtained. middle The probability distribution; ③ medium sample Substituting into formula (10) allows us to calculate the sample. Recovery time and to All samples Statistical analysis yields the probability density function. .

[0201] However, even for the same sample with the same ultimate repair goal, the recovery time can vary under different repair strategies. For example, assuming the runway's functional state after damage is... The ultimate goal of the repair is Based on the urgency of the current combat mission, the takeoff and landing priorities are small, medium, and large fighter jets, respectively. Therefore, the repair targets are, in order: , , If the current mission requires the emergency takeoff and landing of medium-sized fighter jets, the primary target to repair is... , and then for In this embodiment, the functional states will be restored sequentially in order (e.g.: The repair strategy is defined as a step-by-step repair mode, and the function curve using the step-by-step repair mode is as follows: Figure 7 As shown; restore the function state across levels (e.g.: The repair strategy is defined as task-first mode, and the function curve for task-first mode is as follows: Figure 8 As shown.

[0202] S443. Construct a functional recovery time analysis model based on discrete-state Markov processes to analyze the variation of the functional state probability distribution function of an airport runway with repair time. Specifically, the functional state probability distribution is expressed as a function of repair time, as follows:

[0203] (16)

[0204] In formula (16), Indicates the initial moment of repair. The probability distribution of the functional state at that time; This represents the transition probability matrix, due to the one-way nature of the repair. An upper triangular matrix is ​​represented as follows:

[0205] (17)

[0206] In formula (17), Indicates the initial time. Functional status is Under the conditions, The status of the time function is changed by Turn to The probability, , ;

[0207] Represented as:

[0208] (18)

[0209] In formula (18), Indicates the functional state A higher level of functional status, Indicates functional status Restore to functional state The cumulative distribution function of total time.

[0210] S5. In the "optimization" phase, the "offensive and defensive strategies of the red and blue sides" are analyzed and an offensive and defensive strategy optimization model is established, denoted as OMODS. The offensive and defensive equilibrium strategy results obtained by solving the offensive and defensive strategy optimization model are input into the functional index evaluation model, thereby obtaining the resilience analysis results considering the offensive and defensive strategies of the red and blue sides during combat.

[0211] Furthermore, step S5 specifically includes the following:

[0212] The attack and defense strategy optimization model is configured to solve the problem of a zero-sum Markov game between red and blue sides, denoted as TZMG, and represented by the following 8-tuple:

[0213] (1) The operational situation space should include at least the remaining weapon types and quantities of the Red Force, the remaining weapon types and quantities of the Blue Force, the dimensions of the airport runway in the current offensive and defensive confrontation, and the operational support tasks.

[0214] (2) This represents the red side's action space, and each action... Indicates the type and quantity of missiles used;

[0215] (3) This represents the action space of the blue team, and each action... Indicates the type of interceptor missile used and the method of interception;

[0216] (4) Representing state Red takes action The probability of;

[0217] (5) Representing state The blue team takes action The probability of;

[0218] (6) If the red team's reward set is represented, then the blue team's reward is a negative number of the red team's reward. The reward setting is based on the output of the functional indicator evaluation model.

[0219] (7) Represents the state transition probability, indicating that in At that moment, the red team took action. Take action with the blue team Post-state Transition to the next state The probability of;

[0220] (8) This is a discount factor, indicating the importance of future rewards. This illustrates that the outcome of each round of competition is equally important. This indicates that the results of the current round are given more importance;

[0221] like Figure 9 As shown, the analysis process of air defense and air-guided attack and defense strategies for multiple airport runways is as follows: First, a Markov decision process simulation of attack and defense is performed, with the initial state... It consists of the initial resource inputs of both the red and blue teams and any unassigned airport runway; both sides use strategies... and Select initial state Offensive and defensive actions offensive and defensive actions and state The input is fed into the functional performance evaluation model, and the output of the functional performance evaluation model is selected as the reward for the first round of competition. The status has been updated to Repeat the above process until the Red team's resources are exhausted or all airport runways are used up, at which point the process terminates. When the entire confrontation process ends and a trajectory is formed, it is represented as... The outcome of the entire confrontation process is represented by a cumulative discounted reward, also known as a "reward," as follows:

[0222] (19)

[0223] In formula (19), Indicates the initial state According to the strategy and The reward of a round of simulated combat; due to the uncertainty of offensive and defensive strategies, both sides focus on the expected reward, initial state. The expected return is:

[0224] (20)

[0225] In formula (20), Indicating in strategy and Below Seeking expectations, Represents the initial state According to the strategy and The expected reward of engaging in confrontation.

[0226] The goal of the Red team is to optimize their offensive strategy. Make The goal is to maximize the objective, while the blue team's objective is to optimize their defensive strategy. Make Minimum; exists a Nash equilibrium joint strategy This makes formula (21) true, where and These represent the optimal offensive strategy for the red side and the optimal defensive strategy for the blue side, respectively:

[0227] (twenty one)

[0228] In formula (21), This represents the expected return for the Red team when it adopts the optimal offensive strategy and the Blue team when it does not adopt the optimal defensive strategy; This represents the expected reward when both the red and blue teams adopt the optimal strategy. This represents the expected return for the Red team not adopting the optimal offensive strategy and the Blue team adopting the optimal defensive strategy.

[0229] The expected cumulative reward for multiple airport runways under the Nash equilibrium joint strategy is simplified as formula (22):

[0230] (twenty two)

[0231] The Q-learning algorithm is used to solve the attack and defense strategy optimization model represented by formula (22) to obtain the Nash equilibrium joint strategy. The specific form.

[0232] The following example illustrates the method for analyzing the toughness of military airport runways under air defense countermeasures described in this embodiment.

[0233] I. Case Background

[0234] Referring to information from a certain air force's yearbook for a specific year, two air force bases, A and B, with a total of four runways, were selected as the targets of an offensive and defensive confrontation between two opposing forces. Air Force Base A includes runways 1 and 2, while Air Force Base B includes runways 3 and 4. Specifically, runway 1 measures 2173m × 45m, runway 2 measures 3048m × 45m, runway 3 measures 4100m × 45m, and runway 4 measures 2750m × 45m. In Air Force Base A, one side's combat support aircraft type and MOS (Mean Orbit) are C-141 and 1700m × 15m, respectively, while the other side's combat support aircraft type and MOS are KC-110 and 3000m × 25m, respectively. In Air Force Base B, one side's combat support aircraft type and MOS are C-5 and 2500m × 25m, respectively, while the other side's combat support aircraft type and MOS are KC-135 and 2700m × 25m, respectively.

[0235] Assuming the Red Team's objective is to block all types of aircraft takeoffs and landings on all airport runways, each runway can be targeted with high-, medium-, and low-performance missiles, with 2, 4, and 6 missiles of each type to be launched, but only one type of missile can be used per runway. The Blue Team can simultaneously use high- and low-level interceptor missiles, and intercept using 0 (no interception), 1, or 2 times the number of Red Team missiles, with relevant parameters shown in Table 1. The Red Team has 4 high-, 4 medium-, and 6 low-performance missiles, while the Blue Team has 50 high-level and 100 low-level interceptor missiles. The engagement proceeds in the order of runways 1 to 4. The current operational phase ends when all runways are engaged or the Red Team's resources are exhausted. The offensive and defensive rewards for each runway are equally important (i.e.,...). Under the aforementioned operational situation, a runway resilience analysis was conducted.

[0236] Table 1. Values ​​of relevant parameters

[0237]

[0238] According to the "evaluation-optimization-analysis" three-stage analysis method proposed in this invention, the functional indicators of each airport runway under different attack and defense strategies are first evaluated and the attack and defense benefit matrix is ​​constructed. Based on the attack and defense benefit matrix data, the attack and defense strategies of both the red and blue sides are optimized to obtain the attack and defense equilibrium strategies of both sides for each airport runway. Finally, the attack and defense equilibrium strategies of both sides for each airport runway are analyzed to obtain the resilience analysis results considering the attack and defense strategies.

[0239] II. Functional Indicator Evaluation Results and Offense / Defense Benefit Matrix

[0240] In this embodiment, taking airport runway 2 as an example, its functional state is defined as follows:

[0241] Fully functional state ( It can support the take-off and landing of C-141 and KC-110 fighter jets.

[0242] Basic functional status ( ): It can only support the take-off and landing of C-141 fighter jets.

[0243] Complete loss state ( It cannot support the takeoff and landing of either the C-141 or KC-110 fighter jets.

[0244] Taking the scenario where the Red Team launches two high-performance missiles to strike, and the Blue Team employs a two-layer high-low interception system with a 1x multiplier for defense, the resilience index can be obtained based on the Functional Metric Evaluation Model (FMEM). That is, airport runway 2 is located The probabilities are 58.98%, 1.04%, and 39.98%, respectively.

[0245] In the current combat situation, the Red Force's objective is to block all aircraft takeoffs and landings on the runway. Based on the definition of the aforementioned functional states, the probability distribution of the complete loss functional state of the airport runway can be calculated. As a functional indicator that aligns with operational objectives, the Functional Metric Evaluation Model (FMEM) can be used to obtain a 9×3 offensive and defensive benefit matrix for four runways, as shown below. Figure 10 , Figure 11 , Figure 12 , Figure 13 As shown, each value in the attack and defense payoff matrix represents the payoff for the corresponding combination of attack and defense actions. ,For example Figure 10 In the context of the Red Team's actions, which involve launching two high-performance missiles, and the Blue Team's actions, which involve a combination of high and low-level missiles with a double-strength defense, the corresponding defensive action combinations are as follows: =0.4917.

[0246] III. Results of Attack and Defense Strategy Optimization

[0247] Based on the Offensive-Defensive Strategy Optimization Model (OMODS), the functional indicators of the four airport runways were optimized. The cumulative sum serves as the objective of the offensive and defensive game between the two sides, as shown in the following two formulas:

[0248]

[0249]

[0250] Assuming both sides are rational and have complete information, a dynamic programming algorithm is used to solve the attack-defense equilibrium strategy. As shown in Table 2.

[0251] Table 2 Balanced Offense and Defense Strategies for Both Red and Blue Teams

[0252]

[0253] As can be seen from Table 2, this solution is a pure strategy Nash equilibrium solution. =2.5627. Simultaneously, a random attack strategy is constructed for comparison: the red team consistently selects an attack action that satisfies the constraints for each airport runway with the same probability, while the blue team takes an attack action based on the red team's current choice. Minimal action. A diagram comparing the cumulative payoffs of a random attack strategy and a balanced attack-defense strategy is shown below. Figure 14 As shown, the expected value of the simulation results for the random attack strategy is =1.8384, as shown by the blue horizontal line. This illustrates that without offensive and defensive strategy analysis, it is impossible to determine the specific risks faced by each airport runway, and thus impossible to quickly identify the worst-case scenario for military facilities. A random attack strategy, similar to current damage assessments, requires evaluating all possible offensive actions to obtain the minimum value of facility functional loss, but this is difficult to achieve quickly when the offensive and defensive action space is large.

[0254] IV. Results of Airport Runway Resilience Analysis

[0255] By inputting the offensive and defensive balance strategy obtained above back into the Functional Metric Evaluation Model (FMEM), we can obtain the runway resilience analysis results considering the offensive and defensive strategies (taking airport runway 2 as an example again):

[0256] (1) Resilience index Resilience index of airport runway 2 That is, airport runway 2 is located The probabilities should be as follows: %, 1 .

[0257] (2) Resilience index Cumulative distribution function of the functional state transitions of airport runway 2 like Figure 15 As shown. Figure 15 In the middle, green represents the tiered repair mode. of , 30 units of time; blue indicates task priority mode. of , The recovery period is 28 units of time. Although the mission-priority mode can reach the target state faster, it has a longer state dwell time, during which no type of aircraft can take off or land; the yellow area in the image represents the step-by-step recovery mode. of , The time interval is 12 units, indicating that although the step-by-step repair mode takes a relatively long time, it is within 12 units of time. It can accommodate C-141 fighter jets. The results show that selecting an appropriate repair mode can improve the resilience of the airport runway to some extent.

[0258] The comparison results of attack and defense strategies in this embodiment ( Figure 15 The results show that the assessment of military facility functions under different offensive and defensive strategies are significantly different. The resilience assessment index corresponding to the offensive and defensive equilibrium strategy is the theoretical lower limit of the resilience index (the worst case) considering all possible offensive and defensive combinations, which can provide reliable data support for the defender's decision-making and command.

[0259] (3) Runway function curve The function curve of airport runway 2 under the balanced offensive and defensive strategy is as follows: Figure 16 As shown, from Figure 16 It can be seen that as the recovery time progresses, Gradually decreasing, As it increases, due to Relatively small, Recovery is relatively fast, so the function curve No major changes.

[0260] As can be seen from this embodiment, using AFRMI can not only obtain an offensive and defensive equilibrium strategy that conforms to the current combat situation and considers the resource constraints of both sides, but also obtain the impact of the weapon confrontation results on the function of military facilities under the offensive and defensive equilibrium strategy. It also uses a function curve based on probability distribution to characterize the loss and recovery of military facility functions, providing quantitative and comprehensive resilience analysis results. At the same time, it can also serve as an analytical tool for higher-level strategic decisions such as optimizing combat objectives, weapon resource investment, and repair force configuration, providing more efficient, accurate, and advantageous decision support for combat command in the battlefield environment.

[0261] The above description only outlines the basic principles and preferred embodiments of the present invention. Those skilled in the art can make many changes and modifications based on the above description, and these changes and modifications should fall within the protection scope of the present invention.

Claims

1. A method for analyzing the resilience of a military airfield runway under air defense, characterized in that, Comprise the following steps: S1, based on the resilience disaster prevention theory, create a military facility resilience analysis framework considering the uncertainty of the five links of "red and blue attack and defense strategy-weapon confrontation result-facility physical damage-facility function loss-facility function recovery", denoted as AFRMI; S2, taking the runway function of the airport in the air defense background as the evaluation object, dividing the runway resilience analysis of the airport into "evaluation-optimization-analysis" three stages based on AFRMI; S3, define and quantify the resilience index of the airport runway, determine the runway function curve of the airport based on the resilience index of the airport runway, and take the resilience index of the airport runway and the runway function curve of the airport as the resilience analysis result; S4, in the "evaluation" stage, the uncertainty of the four links of "weapon confrontation result-facility physical damage-facility function loss-facility function recovery" is analyzed, a function index evaluation model is established, denoted as FMEM, and the function index under different attack and defense strategies of each airport runway is obtained; S5, in the "optimization" stage, the "red and blue attack and defense strategy" link is analyzed to establish an attack and defense strategy optimization model, denoted as OMODS, and the attack and defense equilibrium strategy result obtained by solving the attack and defense strategy optimization model is input into the function index evaluation model, so as to obtain the resilience analysis result considering the attack and defense strategy of red and blue in combat.

2. The method of claim 1, wherein the method is a method of runway vulnerability analysis of a military airfield under air defense. Step S1 specifically comprises the following: Based on the building structure function loss probability analysis framework considering various uncertainties under natural disasters, the function index probability distribution of military facilities under attack and defense confrontation is obtained, as shown in formula (1), and the military facility resilience analysis framework considering the uncertainty of the five links of "red and blue attack and defense strategy-weapon confrontation result-facility physical damage-facility function loss-facility function recovery" is obtained through the military facility function index probability distribution: （1） In formula (1), is the conditional probability density function of the attack-defense action AC selected by the red and blue sides under the current situation SI; is the conditional probability density function of the weapon countermeasure result EF under the attack-defense action AC; is the conditional probability density function of the physical damage DA of the facility under the countermeasure result EF; is the conditional probability density function of the functional loss LO of the facility under the physical damage DA; is the cumulative distribution function of the functional recovery RE of the facility under the functional loss LO.

3. The method of claim 1, wherein the method is a method of runway vulnerability analysis of a military airfield under air defense. Step S2 specifically comprises the following: In the resilience analysis of the airport runway under the background of air defense combat, the function index under all possible attack and defense strategies of each airport runway is first evaluated, and an attack and defense benefit matrix is constructed; then the attack and defense equilibrium strategy of red and blue for each airport runway is obtained by data optimization and solving based on the attack and defense benefit matrix; finally, the attack and defense equilibrium strategy of red and blue for each airport runway is analyzed, so as to obtain the resilience analysis result considering the attack and defense strategy.

4. The method of claim 1, wherein the method is a method of runway vulnerability analysis for a military airfield under air control. Step S3 specifically comprises the following: The resilience of the airport runway refers to the combat support capability of the airport runway after being damaged and the capability of recovering to the level of combat support requirements above the damaged state; The resilience indicators of the airport runway are quantified, which are embodied by two resilience indicators, specifically, under the attack and defense strategies of the red and blue sides, the probability distribution of the initial function state of the damaged airport runway is , the total time for the function state to recover from the initial function state to the target function state after repair according to the repair strategy is , and and are taken as the two resilience indicators for quantifying the airport runway; Based on and , the functional state probability distribution varying with recovery time is obtained through the transition probability matrix , and is taken as the airport runway function curve to describe the change law of the airport runway function from damage to recovery after attack and defense confrontation. resilience index of an airport runway and and a functional curve of an airport runway as a result of the resilience analysis.

5. The method of claim 4, wherein the method further comprises: Step S4 specifically comprises the following: S41, air defense weapon confrontation simulation: the red side action at least includes missile weapon performance, missile quantity, penetration method; the blue side action at least includes interception weapon performance, interception method; the weapon confrontation process is simulated through "missile launch detection and tracking-true and false warhead identification-interception missile launch-missile penetration hit" four stages to obtain the weapon confrontation result, and the weapon confrontation result is expressed by the probability of the combination of the actual hit missile number of the aiming point, wherein the combination of the actual hit missile number of the aiming point is referred to as hit combination; S42, random damage scene simulation: the damage scene definition is the damage of the runway by the sub-munitions after the penetration of the missile, due to the existence of the probability deviation of the mother bomb hitting and the dispersion error of the sub-bomb, there are still random damage scenes under the same hit combination, the random damage scene modeling method is adopted to simulate a random damage scene under the same hit combination to form a random damage scene sample set, each sample in the random damage scene sample set contains the actual landing point coordinate information of all sub-bombs; S43, construction of the mapping relationship between the physical damage of the airport runway and the functional state: according to the functional state division of the airport runway supporting the take-off and landing of different types of aircraft, it is assumed that the airport runway of the air base in the current combat stage needs to support the take-off and landing of large, medium and small aircraft, then the functional state of the airport runway is divided into four kinds, and the four functional states are as follows: Full function state: can support large, medium, small three types of warplanes take off and land, through indicates; Basic function state: unable to support large warplane take-off and landing, able to support medium and small warplane take-off and landing, pass indicates; Minimum capability: unable to support large and medium-sized fighter aircraft take-off and landing, only able to support small fighter aircraft take-off and landing, through indicates; Complete loss state: unable to support the take-off and landing of any of the large, medium, and small types of warplanes, through indication; For the random damage scene sample set, a functional state classifier based on the minimum operation plane MOS search is adopted to establish the mapping relationship between the physical damage and the functional state; S44, airport runway function recovery analysis: assuming that the repair mode of the airport runway includes two kinds of step-by-step repair and task priority, a functional recovery time analysis model based on a discrete state Markov process is constructed to analyze the law of the functional state probability distribution function of the airport runway changing with the repair time.

6. The method of claim 5, wherein the method further comprises: In step S41, the air defense weapon confrontation simulation includes: It has The first aiming point, the red direction on the airport runway. Missiles launched from a single aiming point At this point, the total number of combinations hit by the airport runway is... Assuming the penetration hit probability of each missile is independent, the th... The missile successfully penetrated the defenses and hit the target at the aiming point. The probability of sending Follows Bernoulli distribution Assuming the hit probability distribution for each aiming point is independent, then the hit combination... probability Represented as: （2） In formula (2), denotes the attack-defense action combination, and denote the attack action of the red side and the defense action of the blue side, respectively, , denotes the probability of successful penetration of the single-shot missile of the red side, ; Through the penetration probability model: （3） （4） （5） In formula (3), P (T) represents the probability that the detection and tracking accuracy of the red missile meets the requirements of the anti-missile system; P (L) represents the reliability of the platform launch of the interceptor, i.e., the probability of successful launch of the interceptor; P (I) represents the probability that the real warhead is correctly identified; P (T, I, L) represents the probability of the three stages of "missile launch detection and tracking-real and false warhead identification-interceptor launch". The formula (4) uses to represent the single-missile interception probability of the blue side interceptor missile to the target. Assuming that the single-missile interception probability of each interceptor missile to the target is the same, then for each identified warhead, the blue side launches interceptor missiles to intercept, to represent the probability that the missile is successfully intercepted. The missile has only two states of penetration and interception, so formula (5) represents the probability of single missile penetration.

7. The method of claim 6, wherein the method further comprises: In step S42, random damage scene simulation includes: The random damage scene sample set established by the random damage scene modeling method Each sample in the random damage scene sample set Contains all the actual bullet landing coordinate information; Missile mother warhead There is a 50% probability that the target will land in a circle centered on the aiming point. Within a circle of radius , the mother missile carries the submunitions. The impact point coordinates are based on the actual impact point of the mother missile as the center, within a radius of... The projectiles are uniformly and randomly distributed within the circle. The expression for the bullet's impact point is shown in formula (6). The coordinates of the explosion point are expressed as in formula (7), and formulas (6) and (7) are as follows: （6） （7） In formulas (6)-(7), the mother missile The coordinates of the aiming point are , The circular probability deviation (CEP) of a missile strike is represented by the number of missiles that strike it. and The random value representing the standard normal distribution in two dimensions, the mother missile The actual landing point coordinates are ;bullet Explosion point coordinates Therefore, the mother bomb Actual landing point coordinates Let be two independent, uniformly distributed random numbers originating from the origin. and Follows a uniform random distribution. Assume that the shape of each bullet crater is relative to the coordinates of the explosion point. Centered on a circle with radius equal to the bullet's damage radius. A standard circle.

8. The method of claim 5, wherein the method further comprises: In step S43, for the random damage scene sample set, a functional state classifier based on the minimum operation plane MOS search is adopted to establish the mapping relationship between the physical damage and the functional state, which includes: representing the probability distribution of the four functional states at the initial time of runway damage , , , the sample set of random damage scenarios is input to the functional state classifier, and the sample functional state is determined and confirmed in turn according to the corresponding MOS of the highest functional state . Let the MOS of small fighter aircraft, medium fighter aircraft and large fighter aircraft be represented as , , , By optimizing the search algorithm, each sample in the random damage scene sample set is judged to determine its functional state Specifically, when there is no complete , the search for continues, if there is , the functional state of the sample is determined as , if there is no complete , the search for continues, if there is , the functional state of the sample is determined as , if there is no , the functional state of the sample is determined as ; in this way, the random damage scene sample set is classified as , corresponding to four functional states, at this time, the mapping from physical damage to functional state is completed, and the probability distribution of the functional state is the proportion of the number of samples in different functional states to the total number of samples , which is represented as follows:​​​​​​​ （8） To A Monte Carlo simulation is performed for each hit combination, i.e. （9） Equation (9) includes the uncertainty from the three links of "weapon engagement result-facility physical damage-facility function loss" to obtain the resilience index .

9. The method of claim 5, wherein the method further comprises: In step S44, the airport runway function recovery analysis includes: S441、The airport runway repair mode includes two kinds of step-by-step repair and task priority, the repair mode of the airport runway is selected according to the emergency degree of the combat task, the repair time of the bomb pit is only considered the number of the bomb pit, all bullets are not set as blocking type bullets, and the repair time of the airport runway is calculated by represents the required repair of the functional state restored to the functional state at the time , and the repair time is proportional to the number of bomb pits required for repair, the recovery time from the functional state to the functional state restored to the functional state is represented as: （10） In equation (10), represents the number of craters that need to be repaired represents the number of craters that need to be repaired represents the average time to repair one crater; When the functional state of the airport runway is recovered in order, the step-by-step repair mode is selected; when the functional state of the airport runway is recovered by level, the task priority mode is selected; S442, the goal of blue square is to find the shortest , i.e. the one with the least . , denoted as , get a sample set by Monte Carlo simulation , the sample set represents all possible hit combinations, divide the sample set into different subsets , search the samples in the subset , calculate the least number of repair pits for each sample , and thus get the total repair time of the sample; Let the initial functional state of the runway after the strike be , let the target functional state be , i.e. the final repair target, let the functional state between the initial functional state and the target functional state be ; The total time for recovering from the initial functional state to the target functional state in the step-by-step repair mode is calculated as: （11-1） The total time for recovering from the initial functional state to the target functional state in the task priority mode is calculated as: （11-2） The statistical sample data is obtained The probability density function and the cumulative distribution function of the step-by-step repair mode are the same as the probability density function and the cumulative distribution function of the task priority mode, The probability density function and the cumulative distribution function are respectively shown in formula (12) and formula (13), and are specifically as follows: （12） （13） Equation (12) and Equation (13) are weighted sums of hit combinations Equation (12) and Equation (13) are weighted sums of hit combinations Equation (12) and Equation (13) are weighted sums of hit combinations Equation (12) and Equation (13) are weighted sums of hit combinations （14） （15） S443, a functional recovery time analysis model based on a discrete state Markov process is constructed to analyze the law of the functional state probability distribution function of the airport runway changing with the repair time, which is specifically represented by the function of the functional state probability distribution to the repair time, as follows: （16） In Equation (16), denotes the functional state probability distribution at the initial time of repair; denotes the transition probability matrix, which is upper triangular due to the unidirectionality of repair, and is given by​ （17） In equation (17), denotes the initial time The functional state is under the condition that The probability that the functional state at time is changed to , , ; is represented by: （18） In equation (18), denotes a higher functional state than the functional state denotes a functional state restoring to the functional state cumulative distribution function of the total time.

10. The method of claim 1, wherein the method is a method of runway vulnerability analysis for a military airfield under air control. Step S5 specifically includes the following: The attack and defense strategy optimization model is configured to solve the problem of red and blue zero-sum Markov game, denoted as TZMG, which is represented by the following 8-tuple: (1) represents the battle situation space, including at least the remaining weapon type and quantity of the red side, the remaining weapon type and quantity of the blue side, the runway size of the current attack and defense confrontation, and the battle support task; (2) represents the action space of the red party, each action represents the missile model and number used; (3) represents the action space of blue, each action represents the use of interceptor missile model and interception method; (4) representing a state the probability that the red agent will take an action the probability that the red agent will take an action (5) representing a state the probability that the blue agent will take an action the probability that the blue agent will take an action (6) Let R represent the red side reward set, then the blue side reward is the negative of the red side reward, and the output result of the selection function index evaluation model is used to set the reward. (7) denote state transition probabilities, denote the probability that after the red player takes action at time and the blue player takes action the state transitions to the next state ; (8) is a discount factor indicating the importance of future rewards, indicates that the results of each round of confrontation are equally important, indicates that more importance is attached to the results of the current round; The analysis process of air defense and air-guided attack and defense strategies for multiple airport runways is as follows: First, a Markov decision process simulation of attack and defense is performed, with the initial state... It consists of the initial resource inputs of both the red and blue teams and any unassigned airport runway; both sides use strategies... and Select initial state Offensive and defensive actions offensive and defensive actions and state The input is fed into the functional performance evaluation model, and the output of the functional performance evaluation model is selected as the reward for the first round of competition. Status updated to ; The above procedure is repeated until the red resource is depleted or all airport runways are engaged, reaching a terminal state The entire engagement process is over, and a trajectory is formed, denoted as The result of the entire engagement process is expressed in terms of cumulative discounted rewards, also known as "returns", and is expressed as follows: （19） In formula (19), represents the initial state According to the strategy and The return of a round of simulated confrontation, due to the uncertainty of attack and defense strategy, both sides focus on the target for expected return, initial state The expected return under the initial state is: （20） In equation (20), denotes the strategy and the expectation under the strategy denotes the initial state and the expected return under the strategy and The goal of the Red team is to optimize their offensive strategy. Make The goal is to maximize the objective, while the blue team's objective is to optimize their defensive strategy. Make Minimum; exists a Nash equilibrium joint strategy This makes formula (21) true, where and These represent the optimal offensive strategy for the red side and the optimal defensive strategy for the blue side, respectively: （21） In Equation (21), represents the expected return when the red team takes the optimal offensive strategy and the blue team does not take the optimal defensive strategy; represents the expected return when both the red and blue teams take the optimal strategies; represents the expected return when the red team does not take the optimal offensive strategy and the blue team takes the optimal defensive strategy; The expected cumulative reward of the multi-airport runway under the Nash equilibrium joint strategy is simply written as formula (22): （22） The attack-defense strategy optimization model expressed by formula (22) is solved by using a Q-learning algorithm to obtain a Nash equilibrium joint strategy in a specific form.

Images