Military airport runway toughness analysis method under air-guided confrontation

Abstract

The invention discloses a military airport runway toughness analysis method under air-guided confrontation. A military facility toughness analysis framework considering uncertainty of five links including a red and blue attack and defense strategy, a weapon confrontation result, facility physical damage, facility function loss and facility function recovery is created. An airport runway function is used as an evaluation object, and an evaluation-optimization-analysis three-stage toughness analysis method is provided. A function index evaluation model is established in the evaluation stage to describe the expression forms and the transmission processes of uncertainty of the rear four links of the military facility toughness analysis framework, and the first link in the optimization stage is described; establishing an attack and defense strategy optimization model to describe a first link of the military facility toughness analysis framework; and inputting a balance strategy result obtained by the attack and defense strategy optimization model into the function index evaluation model to obtain a toughness analysis result capable of visually, quantitatively and reliably representing the functional loss and recovery condition of the military facility.

Description

Technical Field

[0001] The present invention relates to the technical field of airport damage assessment, and in particular to a method for analyzing the toughness of a military airport runway under missile-to-air confrontation. Background Art

[0002] Military facilities are primarily exposed to the risk of deliberate enemy attacks. Assessing their functional loss and recovery time directly impacts subsequent operational planning and may even influence overall campaign decision-making. Existing Damage Effect Evaluation (DEE) methods typically analyze the extent of physical damage to military facilities after weapon strikes through numerical simulation, reliability-based damage trees, and artificial intelligence. These methods then map physical damage to functional capabilities to assess the facilities' functionality. However, these methods fail to account for the uncertainties inherent in offensive and defensive strategies and the actual weapon engagement process, nor do they consider the uncertainties inherent in the functional recovery process. Since opposing forces often flexibly adapt their strategies based on the enemy's, the uncertainty surrounding the risks faced by military facilities is significant. This, coupled with numerous uncertainties such as weapon performance and the operational environment, poses significant challenges to assessing the functionality of military facilities. These aforementioned damage effect assessments only represent the functional loss of facilities under a specific attack scenario, resulting in a simplistic and incomplete assessment that fails to provide reliable data support for command decision-making.

[0003] Resilience provides a theoretical framework for addressing the uncertainties of various aspects of system functionality under disasters. Natural disaster-related resilience research typically involves conducting resilience assessments and studies based on the following steps: disaster risk analysis, physical damage to facilities, functional loss, and functional recovery. Each step is analyzed based on the prerequisites of the previous step, ensuring that the impact of each step and its uncertainties on functional loss and recovery is fully considered and communicated. Current research on the resilience of military facilities primarily focuses on conceptual definition, indicator evaluation, and practical application methods. However, there is a lack of systematic research on the full-process uncertainty transfer analysis and quantitative assessment. Summary of the Invention

[0004] In order to overcome the deficiencies of the above technologies, the present invention provides a method for analyzing the toughness of a military airport runway under missile-to-air confrontation.

[0005] Explanation of terms:

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

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

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

[0009] 4. CEP: Circular Error Probable, which means that there is a 50% probability that the missile will fall within the 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.

[0015] 10. RE: Restoration.

[0016] 11. SSPK: Single-shot Probability of Kill, the probability of a blue 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, offensive and defensive strategy optimization model.

[0021] The technical solution adopted by the present invention to overcome the technical problems is:

[0022] A method for analyzing the toughness of a military airport runway under missile-to-air confrontation comprises the following steps:

[0023] S1. Based on resilience disaster prevention theory, a framework for analyzing the resilience of military facilities, denoted as AFRMI, is created to consider the uncertainties in five aspects: the offensive and defensive strategies of the red and blue sides, the results of weapon confrontation, the physical damage to the facilities, the loss of facility functions, and the restoration of facility functions.

[0024] S2. Taking the airport runway function in the context of missile-air confrontation as the evaluation object, the resilience analysis of the airport runway is divided into three stages: "assessment-optimization-analysis" based on AFRMI;

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

[0026] S4. In the "Evaluation" 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 evaluation model, denoted as FMEM, is established to obtain the functional index of each airport runway under different attack and defense strategies;

[0027] S5. In the "optimization" stage, the "offensive and defensive strategies of the red and blue teams" are analyzed to establish an offensive and defensive strategy optimization model, 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 indicator evaluation model to obtain the resilience analysis results considering the offensive and defensive strategies of the red and blue teams during combat.

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

[0029] Based on the probability analysis framework of building structure function loss under natural disasters considering the transmission of various types of uncertainty, the probability distribution of military facility function indicators under attack and defense confrontation is obtained, as shown in formula (1). Through the probability distribution of military facility function indicators, a military facility resilience analysis framework is obtained that considers the uncertainties of five links: "attack and defense strategies of the red and blue sides - weapon confrontation results - facility physical damage - facility function loss - facility function recovery":

[0030] (1)

[0031] In formula (1), is the conditional probability density function of the attack and defense actions AC chosen by the red and blue sides under the current combat situation SI; To determine the conditional probability density function of the weapon confrontation result EF under the attack and defense action AC; To determine the conditional probability density function of facility physical damage DA under the countermeasure result 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 function loss LO.

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

[0033] When analyzing the resilience of airport runways in the context of missile-to-air confrontation operations, we first evaluate the functional indicators of each airport runway under all possible attack and defense strategies, and construct an attack and defense benefit matrix; then, based on the attack and defense benefit matrix data, we optimize and solve the attack and defense equilibrium strategies of the red and blue sides for each airport runway; finally, we analyze the attack and defense equilibrium strategies of the red and blue sides for each airport runway, thereby obtaining the resilience analysis results that take into account the attack and defense strategies.

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

[0035] The resilience of an airport runway refers to its combat support capability after being damaged and its ability to recover to a level of combat support higher than that required in the damaged state.

[0036] The resilience index of the airport runway is quantified and reflected by two resilience indicators. Specifically, under the attack and defense strategies of the red and blue sides, the probability distribution of the functional state of the airport runway at the initial moment of damage is According to the repair strategy, the functional state after repair changes from the initial functional state Restore to target functional state The total time for ,Will and As two indicators to quantify the resilience of airport runways;

[0037] based on and , the probability distribution of functional status that changes with recovery time is obtained through the transition probability matrix , and As the airport runway function curve, it is used to describe the change law of the airport runway function from damage to recovery after the attack and defense confrontation;

[0038] The resilience index of the airport runway and and the airport runway function curve As three results of resilience analysis.

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

[0040] S41. Missile-to-air weapon confrontation simulation: The Red team's actions include at least the performance of the missile launcher, the number of missiles, and the penetration method; the Blue team's actions 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 - true and false warhead identification - interceptor missile launch - missile penetration hit" to obtain the weapon confrontation results. The weapon confrontation results are expressed as the probability of the combination of the number of missiles actually hitting the aiming point. The combination of the number of missiles actually hitting the aiming point is referred to as the hit combination.

[0041] S42. Random Damage Scenario Simulation: A damage scenario is defined as the damage to the runway surface caused by the submunitions after the Red Army missile completes the penetration. Due to the discrepancy between the submunitions' impact probability and the bullet dispersion error, random damage scenarios still exist under the same hit combination. Using the random damage scenario modeling method, a Monte Carlo simulation is performed on a random damage scenario under this hit combination to form a random damage scenario sample set. Each sample in the random damage scenario sample set contains the actual impact point coordinate information of all bullets.

[0042] S43. Constructing a mapping relationship between physical damage and functional status of airport runways: Functional status is divided according to the different types of fighter jets that the runways support. Assuming that the runways at the air force base in the current combat phase need to support the takeoff and landing of three types of fighter jets: large, medium, and small, the functional status of the runways can be divided into four types. The four functional statuses are as follows:

[0043] Fully functional state: can support the take-off and landing of large, medium and small fighters. express;

[0044] Basic functional status: cannot support the take-off and landing of large fighters, can support the take-off and landing of medium and small fighters, express;

[0045] Minimum functional status: cannot support the take-off and landing of large and medium-sized fighters, can only support the take-off and landing of small fighters, through express;

[0046] Total loss state: Unable to support the takeoff and landing of any of the three types of fighters: large, medium, and small. express;

[0047] For the random damage scene 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;

[0048] S44. Airport runway function recovery analysis: Assume that the airport runway repair modes include step-by-step repair and task priority. Construct a function recovery time analysis model based on discrete state Markov process to analyze the law of change of the functional state probability distribution function of the airport runway with the repair time.

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

[0050] With Aim point, the first on the red direction airport runway Aiming point to launch missiles The total number of airport runway hit combinations is ; Assuming that the penetration hit probability of each missile is independent, Aiming points successfully penetrated and hit the missile Probability of hair Follows Bernoulli distribution ; Assuming that the hit probability distribution of each aiming point is independent, the hit combination Probability Expressed as:

[0051] (2)

[0052] In formula (2), Indicates a combination of offensive and defensive actions. and Respectively represent the red side's attack and the blue side's defense. , It represents the probability of a single missile from the Red Army successfully penetrating the defense. ;

[0053] Derived through the penetration probability model:

[0054] (3)

[0055] (4)

[0056] (5)

[0057] In formula (3), It indicates the probability that the detection and tracking accuracy of missiles launched by the Red Army meets the launch requirements of the anti-missile system; It represents the reliability of the interceptor missile platform launch, that is, the probability of successfully launching the interceptor missile; represents the probability that the real warhead is correctly identified; Represents the probability of the three stages of "missile launch detection and tracking - true and false warhead identification - interceptor missile launch";

[0058] Formula (4) uses represents the single-shot interception probability of the blue interceptor missile against the target; assuming that the single-shot interception probability of each interceptor missile against the target is the same, then for each identified warhead, the blue side launches Launch interceptor missiles to intercept, It represents the probability of the missile being successfully intercepted;

[0059] There are only two states for a missile: penetration and being intercepted. Therefore, formula (5) represents the probability of a single missile penetrating.

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

[0061] Random destruction scene sample set established by random destruction scene modeling method Each sample in Contains the actual landing coordinate information of all bullets;

[0062] Missile mother bomb There is a 50% chance of landing on a circle centered on the aiming point. The number of bullets carried by the mother bomb is within the circle with a radius of The coordinates of the landing point are based on the actual landing point of the mother missile as the center and the radius as The circle is uniformly and randomly distributed, and the mother bomb The falling point expression of the bullet is as shown in formula (6). The explosion point coordinate expression is as follows:

[0063] (6)

[0064] (7)

[0065] In formulas (6)-(7), the mother bomb The aiming point coordinates are , The missile circular error probability CEP represents the missile's strike, and Represents a random value of a two-dimensional standard normal distribution, the mother bullet The actual landing point coordinates are ;bullet The coordinates of the explosion point It is a mother bomb Actual landing point coordinates are two independent uniformly distributed random numbers at the origin, and Obeying uniform random distribution, , assume that the damage shape of each bullet crater is based on the coordinates of the explosion point is the center of the circle and the radius is the bullet damage radius The standard circle.

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

[0067] Indicates that the airport runway was damaged at the initial moment 、 、 、 Probability distribution of four functional states; random destruction scenario sample set The samples are input into the functional state classifier, and the highest functional state The corresponding MOS judges and confirms the functional status of the sample in turn;

[0068] Assume that the MOS of small fighter, medium fighter, and large fighter are expressed as 、 、 , < < By optimizing the search algorithm, the random destruction scene sample set Each sample in the , specifically, when there is no complete When Search, if exists , then the functional status of the sample is determined to be If there is no complete , then continue to Search, if exists , then the functional status of the sample is determined to be , if it does not exist , then the functional status of the sample is determined to be ; This will randomly destroy the scene sample set Classified as , corresponding to four functional states, the mapping from physical damage to functional state is completed, and the probability distribution of functional state takes the number of samples of different functional states The total number of samples The ratio is expressed as follows:

[0069] (8)

[0070] right All hit combinations are subjected to Monte Carlo simulation, and we get:

[0071] (9)

[0072] Formula (9) includes the uncertainty of the three links of "weapon confrontation results - facility physical damage - facility function loss" and is used to obtain the resilience index .

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

[0074] S441. Set the runway repair mode to include step-by-step repair and mission priority. Select the runway repair mode based on the urgency of the combat mission. Set the crater repair time to only consider the number of craters. Set all bullets to be non-blocking bullets. Indicated by functional status Restore to functional state Repair required , and the repair time is proportional to the number of craters to be repaired, determined by the functional state Restore to functional state Recovery time Expressed as:

[0075] (10)

[0076] In formula (10), Indicates need for repair Number of craters, It represents the average time to repair one crater;

[0077] When the functional status of the airport runway is restored step by step in order, the step-by-step repair mode is selected; when the functional status of the airport runway is restored in a skipped order, 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 is obtained through Monte Carlo simulation , sample set Represents the sample set under all possible hit combinations, and the sample set Divided into different subsets , for the subset The samples in Search and calculate the minimum number of craters to be repaired for each sample , thus obtaining the total repair time of the sample;

[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 recover from the initial functional state to the target functional state in the step-by-step repair mode is calculated as:

[0081] (11-1)

[0082] The total time taken to recover from the initial functional state to the target functional state in the calculation task priority mode is:

[0083] (11-2)

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

[0085] (12)

[0086] (13)

[0087] Formula (12) and formula (13) are based on the hit combination As a prerequisite, perform weighted summation of formula (12) and formula (13), and the weight is the attack and defense action combination Next hit combination The probability is expressed as follows:

[0088] (14)

[0089] (15)

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

[0091] (16)

[0092] In formula (16), Indicates the initial time of repair Functional state probability distribution when ; Represents the transition probability matrix. Due to the unidirectional nature of the repair, is an upper triangular matrix, represented as follows:

[0093] (17)

[0094] In formula (17), Indicates that at the initial moment Functional status is Under the conditions, The state of the moment function is determined by Convert to The probability of , ;

[0095] Expressed as:

[0096] (18)

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

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

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

[0100] (1) The combat situation space should include at least the types and quantities of the Red side's remaining weapons, the types and quantities of the Blue side's remaining weapons, the dimensions of the airport runway in the current offensive and defensive confrontation, and the combat support mission;

[0101] (2) Represents the action space of the red team, each action Indicates the type and quantity of missiles used;

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

[0103] (4) Indicates status Red player takes action probability;

[0104] (5) Indicates status The blue player takes action probability;

[0105] (6) Represents the red team's reward set, then the blue team's reward is the negative of the red team's reward. The output result of the functional indicator evaluation model is selected for reward setting;

[0106] (7) Represents the state transition probability, which means Moment, Red side takes action and Blue take action Post-state Transition to the next state probability;

[0107] (8) is the discount coefficient, which indicates the importance of future rewards. This shows that the results of each round of confrontation are equally important. This indicates that the results of the current round are more important;

[0108] The analysis process of the attack and defense strategy of the missile-air confrontation of multiple airport runways is as follows: First, the attack and defense Markov decision process simulation is carried out. The initial state It is the initial value of the resource investment of the red and blue parties and any unassigned target airport runway; both parties use strategy and Select initial state The offensive and defensive moves , the offensive and defensive actions and status Input into the functional indicator evaluation model, and select the output of the functional indicator evaluation model as the reward for the first round of confrontation , the status is updated to Repeat the above process until the Red team's resources are exhausted or all airport runway confrontations are completed, reaching the end state. When the whole confrontation process ends, a trajectory is formed, which is expressed as The result of the entire adversarial process is expressed as a cumulative discounted reward, also called a "return", which is expressed as follows:

[0109] (19)

[0110] In formula (19), Indicates the initial state Follow the strategy and The reward for a round of simulated confrontation. Due to the uncertainty of attack and defense strategies, the goal of both parties is the expected reward. The initial state The expected return under is:

[0111] (20)

[0112] In formula (20), Indicates that in the strategy and Lower pair Seek hope, Indicates the initial state Follow the strategy and the expected rewards of engaging in confrontation;

[0113] The Red team's goal is to optimize the offensive strategy Make The blue team's goal is to optimize the defense strategy Make Minimum; there exists a Nash equilibrium joint strategy , so that formula (21) holds true, where and They represent the optimal attack strategy of the red side and the optimal defense strategy of the blue side respectively:

[0114] (twenty one)

[0115] In formula (21), represents the expected return of the red team adopting the optimal attack strategy and the blue team not adopting the optimal defense strategy; represents the expected return of both red and blue players adopting the optimal strategy; represents the expected return of the red team not adopting the optimal attack strategy and the blue team adopting the optimal defense strategy;

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

[0117] (twenty two)

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

[0119] The beneficial effects of the present invention are: 1. The present invention proposes a military facility resilience analysis framework that takes into account the uncertainty of five links, namely "offensive and defensive strategies of the red and blue sides - weapon confrontation results - physical damage to facilities - loss of facility functions - recovery of facility functions", which is denoted as AFRMI. The application method and advantages of AFRMI are introduced by taking the airport runway function evaluation under the background of missile-air confrontation as an example. Compared with the existing military facility damage effect evaluation, the use of AFRMI can not only obtain an offensive and defensive balance strategy that takes into account the resource constraints of the opposing sides under the current combat situation, but also obtain the impact of the weapon confrontation results on the functions of military facilities under the offensive and defensive balance strategy, and use 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 be used as an analysis tool for optimizing higher-level strategic decisions such as combat objectives, weapon resource investment and repair force allocation, providing more accurate, effective and favorable decision-making support for combat command in battlefield environments. 2. This paper uses the airport runway in the context of missile-to-air confrontation as the object of resilience analysis and constructs an airport runway functional index evaluation model (FMEM) for known attack and defense strategies. FMEM consists of four modules: weapon confrontation simulation, random destruction scenario simulation, construction of a mapping relationship between airport runway physical damage and functional status, and airport runway functional recovery analysis. Specifically, the uncertainty in the "weapon confrontation simulation" link is expressed using an analytical formula, and the probability of hit combinations can intuitively and clearly represent the uncertainty of the weapon confrontation results. The uncertainty in the "random destruction scenario simulation" is described using the mother bomb landing point deviation and bullet dispersion error, and random samples are generated through Monte Carlo simulation to ensure the reliability of random samples. The uncertainty in the "mapping relationship between airport runway physical damage and functional status" is classified through an optimized search algorithm, and the probability of functional status is expressed using frequency, which concisely and efficiently reflects the uncertainty of functional status. The uncertainty in the "airport runway functional recovery analysis" is indirectly reflected by the uncertainty of the number of repair craters. Two methods, step-by-step repair and task priority, are proposed to perform statistical analysis on the probability distribution of repair time. 3. This paper studies the uncertainty transfer in the five links of "offensive and defensive strategies of the red and blue teams - weapon confrontation results - physical damage to facilities - loss of facility functionality - and restoration of facility functionality." It uses multiple conditional probability integrals to represent the probability distribution of military facility functional indicators to describe and address the uncertainty transfer process in these four links. 4. This paper analyzes the offensive and defensive strategies of the red and blue teams, reducing them to a zero-sum Markov game between them. It then establishes an offensive and defensive strategy optimization model to describe this, and uses a Q-learning algorithm to solve the offensive and defensive equilibrium strategy. BRIEF DESCRIPTION OF THE DRAWINGS

[0120] Figure 1The figure is a flow chart of the method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to an embodiment of the present invention.

[0121] Figure 2 This is a schematic diagram of the method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to 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 of the simulation process of the random destruction scenario of an airport runway according to an embodiment of the present invention.

[0124] Figure 5 A schematic diagram of the mapping relationship between physical damage to the functional status of an airport runway according to an embodiment of the present invention.

[0125] Figure 6 Schematic diagram of the airport runway function recovery analysis process according to an embodiment of the present invention.

[0126] Figure 7 This is a schematic diagram of a function curve of a step-by-step repair mode for an airport runway according to an embodiment of the present invention.

[0127] Figure 8 This is a schematic diagram of a function curve of an airport runway adopting a task-priority repair mode according to an embodiment of the present invention.

[0128] Figure 9 Schematic diagram of the attack and defense strategy analysis process according to 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 using the functional indicator evaluation model described in an embodiment of the present invention.

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

[0131] Figure 12 This is a schematic diagram of the 9×3 attack and defense benefit matrix of airport runway 3 obtained using the functional indicator evaluation model according to an embodiment of the present invention.

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

[0133] Figure 14 This is a schematic diagram comparing the cumulative benefits of the random attack strategy and the attack-defense balance strategy according to an embodiment of the present invention.

[0134] Figure 15 Schematic diagram of the cumulative distribution function of each functional state transition of the airport runway 2 according to an embodiment of the present invention.

[0135] Figure 16 Schematic diagram of the function curve of airport runway 2 under the attack-defense balance strategy described in an embodiment of the present invention. DETAILED DESCRIPTION

[0136] In order to facilitate those skilled in the art to better understand the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. The following is only exemplary and does not limit the scope of protection of the present invention.

[0137] The present invention discloses a method for analyzing the toughness of a military airport runway under missile-air confrontation. Figure 1 and Figure 2 As shown, the following steps are included:

[0138] S1. Based on the resilience disaster prevention theory, a military facility resilience analysis framework is created that considers the uncertainty of five links: "offensive and defensive strategies of the red and blue sides - weapon confrontation results - physical damage to facilities - loss of facility functions - recovery of facility functions", denoted as AFRMI, as follows: Figure 3 shown.

[0139] In this embodiment, step S1 specifically includes: based on the probability analysis framework of building structure function loss under natural disasters considering the transmission of various types of uncertainty, the probability distribution of military facility function indicators under attack and defense confrontation is obtained, as shown in formula (1). Through the probability distribution of military facility function indicators, a military facility resilience analysis framework is obtained that considers the uncertainties of five links: "attack and defense strategies of the red and blue sides - weapon confrontation results - facility physical damage - facility function loss - facility function recovery":

[0140] (1)

[0141] In formula (1), is the conditional probability density function of the attack and defense actions AC chosen by the red and blue sides under the current combat situation SI; To determine the conditional probability density function of the weapon confrontation result EF under the attack and defense action AC; To determine the conditional probability density function of facility physical damage DA under the countermeasure result 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 function loss LO.

[0142] Formula (1) is a multi-integral formula, which indicates that the functional recovery analysis of military facilities under the current combat situation needs to be carried out according to the steps of "1. Attack and defense strategy analysis - 2. Confrontation result analysis - 3. Physical damage analysis - 4. Function loss analysis - 5. Function recovery analysis". Each analysis step is carried out on the basis of the completion of the previous analysis step to ensure that the uncertainty of the intermediate links of the functional recovery analysis of military facilities under the known current combat situation is fully considered and transmitted.

[0143] S2. Taking the airport runway function in the context of missile-air confrontation as the evaluation object, the resilience analysis of the airport runway is divided into three stages: "assessment-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 results-facility physical damage-facility function loss-facility function recovery" includes objective factors such as at least weapon performance, facility structure, and environmental conditions; while the uncertainty in the analysis of the "red and blue offensive and defensive strategies" step comes from the decision-making of the commanders on both sides, and the decision-making is based on the evaluation results of the four steps of "weapon confrontation results-facility physical damage-facility function loss-facility function recovery"; the "red and blue offensive and defensive strategies" analysis is supported by the analysis and evaluation results of the four steps of "weapon confrontation results-facility physical damage-facility function loss-facility function recovery", optimizes the final form of the strategies of both sides, and then uses the offensive and defensive strategy analysis results as input to obtain the resilience analysis results considering the offensive and defensive strategies. Therefore, when analyzing the resilience of airport runways in the context of missile-to-air confrontation operations, we first evaluate the functional indicators of each airport runway under all possible attack and defense strategies, and construct an attack and defense benefit matrix; then, based on the attack and defense benefit matrix data, we optimize and solve the attack and defense equilibrium strategies of the red and blue sides for each airport runway; finally, we analyze the attack and defense equilibrium strategies of the red and blue sides for each airport runway, and thus obtain the resilience analysis results that take into account the attack and defense strategies.

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

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

[0148] The resilience of an airport runway refers to its combat support capability after being damaged and its ability to recover to a level of combat support higher than that required in the damaged state.

[0149] The resilience index of the airport runway is quantified and reflected by two resilience indicators. Specifically, under the attack and defense strategies of the red and blue sides, the probability distribution of the functional state of the airport runway at the initial moment of damage is According to the repair strategy, the functional state after repair changes from the initial functional state Restore to target functional state The total time for ,Will and As two indicators to quantify the resilience of airport runways.

[0150] based on and , the probability distribution of functional status that changes with recovery time is obtained through the transition probability matrix , and The airport runway function curve is used to describe the changing law of the airport runway function from damage to recovery after attack and defense confrontation.

[0151] The resilience index of the airport runway and and the airport runway function curve As three results of resilience analysis.

[0152] S4. In the "Evaluation" phase, the uncertainties of the four links of "weapon confrontation results - facility physical damage - facility function loss - facility function recovery" are analyzed, and a functional indicator evaluation model, denoted as FMEM, is established to obtain the functional indicators of each airport runway under different attack and defense strategies.

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

[0154] S41. Simulation of air-to-air weapon confrontation: The Red team's actions shall at least include the performance of the missile weapon launched, the number of missiles, and the penetration method; the Blue team's actions shall at least include the performance of the interception weapon and the interception method; the weapon confrontation process shall be simulated through the four stages of "missile launch detection and tracking - true and false warhead identification - interceptor missile launch - missile penetration hit" to obtain the weapon confrontation result. The weapon confrontation result is expressed as the probability of the combination of the number of missiles actually hitting the aiming point, among which the combination of the number of missiles actually hitting the aiming point is referred to as the hit combination.

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

[0156] With Aim point, red direction airport runway Aiming point to launch missiles The total number of airport runway hit combinations is ; Assuming that the penetration hit probability of each missile is independent, Aiming points successfully penetrated and hit the missile Probability of hair Follows Bernoulli distribution ; Assuming that the hit probability distribution of each aiming point is independent, the hit combination Probability Expressed as:

[0157] (2)

[0158] In formula (2), Indicates a combination of offensive and defensive actions. and Respectively represent the red side's attack and the blue side's defense. , It represents the probability of a single missile from the Red Army successfully penetrating the defense. ;

[0159] Derived through the penetration probability model:

[0160] (3)

[0161] (4)

[0162] (5)

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

[0164] Formula (4) uses represents the single-shot interception probability of the blue interceptor missile against the target; assuming that the single-shot interception probability of each interceptor missile against the target is the same, then for each identified warhead, the blue side launches Launch interceptor missiles to intercept, It represents the probability of the missile being successfully intercepted.

[0165] There are only two states for a missile: penetration and being intercepted. Therefore, formula (5) represents the probability of a single missile penetrating.

[0166] S42. Random destruction scenario simulation: The destruction scenario is defined as the damage caused to the runway surface by the cluster bombs after the Red Army missile completes the penetration. Due to the probability deviation of the mother bomb's hit circle and the bullet dispersion error, random destruction scenarios still exist under the same hit combination. The random destruction scenario modeling method is used 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 landing point coordinate information of all bullets.

[0167] In this example, the blue team's runway is 2800m×30m. The red team launches two missiles at the three aiming points (700,15), (1400,15), and (2100,15) and all of them successfully penetrate the defense. This is a random destruction scenario under this hit combination. like Figure 4 As shown, each hit combination obtained from the simulation of missile-to-air weapons Perform Monte Carlo simulation to generate random damage scenario sample sets , where each sample Contains the actual landing point coordinate information of all bullets. The actual landing point coordinates of each missile bullet are as follows Figure 4 As shown in the black dotted box, specifically, the missile's mother bomb There is a 50% chance of landing on a circle centered on the aiming point. The bullets carried by the mother bomb are within the circle with a radius of The coordinates of the landing point are based on the actual landing point of the mother missile as the center and the radius as The circle is evenly and randomly distributed, as shown in the red dotted box. The falling point expression of the bullet is as shown in formula (6). The explosion point coordinate expression is as follows:

[0168] (6)

[0169] (7)

[0170] In formulas (6)-(7), the mother bomb The aiming point coordinates are , The missile circular error probability CEP represents the missile's strike, and Represents a random value of a two-dimensional standard normal distribution, the mother bullet The actual landing point coordinates are ;bullet The coordinates of the explosion point It is a mother bomb Actual landing point coordinates are two independent uniformly distributed random numbers at the origin, and Obeying uniform random distribution, , assume that the damage shape of each bullet crater is based on the coordinates of the explosion point is the center of the circle and the radius is the bullet damage radius The standard circle.

[0171] S43. Constructing a mapping relationship between physical damage and functional status of airport runways: Functional status is divided according to the different types of fighter jets that the runways support. Assuming that the runways at the air force base in the current combat phase need to support the takeoff and landing of three types of fighter jets: large, medium, and small, the functional status of the runways can be divided into four types. The four functional statuses are as follows:

[0172] Fully functional state: can support the take-off and landing of large, medium and small fighters. express.

[0173] Basic functional status: cannot support the take-off and landing of large fighters, can support the take-off and landing of medium and small fighters, express.

[0174] Minimum functional status: cannot support the take-off and landing of large and medium-sized fighters, can only support the take-off and landing of small fighters, through express.

[0175] Total loss state: Unable to support the takeoff and landing of any of the three types of fighters: large, medium, and small. express.

[0176] For the random damage scene sample set, this embodiment proposes to use a functional state classifier based on minimum operating plane MOS search to establish a mapping relationship from physical damage to functional state, such as Figure 5 As shown, specifically including:

[0177] Indicates that the airport runway was damaged at the initial moment 、 、 、 Probability distribution of four functional states; random destruction scenario sample set The samples are input into the functional state classifier, and the highest functional state The corresponding MOS is judged and confirmed in turn. Let the MOS of small fighter, medium fighter and large fighter be expressed as 、 、 , < < By optimizing the search algorithm, the random destruction scene sample set Each sample in the , specifically, when there is no complete When Search, if exists , then the functional status of the sample is determined to be If there is no complete , then continue to Search, if exists , then the functional status of the sample is determined to be , if it does not exist , then the functional status of the sample is determined to be ; This will randomly destroy the scene sample set Classified as , corresponding to four functional states, the mapping from physical damage to functional state is completed, and the probability distribution of functional state takes the number of samples of different functional states The total number of samples The ratio is expressed as follows:

[0178] (8)

[0179] right All hit combinations are subjected to Monte Carlo simulation, and we get:

[0180] (9)

[0181] Formula (9) includes the uncertainty of the three links of "weapon confrontation results - facility physical damage - facility function loss" and is used to obtain the resilience index .

[0182] S44. Airport runway function recovery analysis: Assume that the airport runway repair modes include step-by-step repair and task priority. Construct a function recovery time analysis model based on discrete state Markov process to analyze the law of change of the functional state probability distribution function of the airport runway with the repair time.

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

[0184] S441. Set the runway repair mode to include step-by-step repair and mission priority. Select the runway repair mode based on the urgency of the combat mission. Set the crater repair time to only consider the number of craters. Set all bullets to be non-blocking bullets. Indicated by functional status Restore to functional state Repair required , and the repair time is proportional to the number of craters to be repaired, determined by the functional state Restore to functional state Recovery time Expressed as:

[0185] (10)

[0186] In formula (10), Indicates need for repair Number of craters, Indicates the average time required to repair one crater.

[0187] When the functional status of the airport runway is restored step by step in order, the step-by-step repair mode is selected; when the functional status of the airport runway is restored skipping levels, 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 is obtained through Monte Carlo simulation , sample set Represents the sample set under all possible hit combinations, and the sample set Divided into different subsets , for the subset The samples in Search and calculate the minimum number of craters to be repaired for each sample , thus obtaining the total repair time of the sample.

[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 recover from the initial functional state to the target functional state in the step-by-step repair mode is calculated as:

[0191] (11-1)

[0192] The total time taken to recover from the initial functional state to the target functional state in the calculation task priority mode is:

[0193] (11-2)

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

[0195] (12)

[0196] (13)

[0197] Formula (12) and formula (13) are based on the hit combination As a prerequisite, perform weighted summation of formula (12) and formula (13), and the weight is the attack and defense action combination Next hit combination The probability is expressed as follows:

[0198] (14)

[0199] (15)

[0200] Due to the uncertainty, the steps of runway function recovery analysis are as follows Figure 6 As shown, if analysis is required Recovery time, ① from the sample set after functional status classification Select a sample , using optimization algorithms to search Location (such as Figure 6 red rectangle), represented by triangles The origin ( The lower left corner, coordinates are (1400,10)), and the sample is obtained by searching 5; ② Each sample Optimization algorithm search is performed. The spatial distribution of the origin is as follows Figure 6 The triangle distribution of the runway surface is shown and can be statistically analyzed. middle The probability distribution of The sample Substituting into formula (10) we can calculate the sample Recovery time , and All samples in Perform statistics to obtain the probability density function .

[0201] However, for the same sample, even if the final repair target is the same, the recovery time under different repair strategies will be different. For example, if the functional state of the runway after damage is The ultimate repair goal is According to the urgency of the current combat mission, the take-off and landing priorities are small, medium and large fighters, and the repair targets are 、 、 If the current mission requires emergency takeoff and landing of medium-sized fighters, the repair target is first , then In this embodiment, the functional status will be restored step by step in order (eg: ) is defined as a step-by-step repair mode. The function curve of the step-by-step repair mode is as follows: Figure 7 As shown; restore the functional status by leaps and bounds (such as: ) is defined as the task priority mode, and the function curve of the task priority mode is as follows: Figure 8 shown.

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

[0203] (16)

[0204] In formula (16), Indicates the initial time of repair Functional state probability distribution when ; Represents the transition probability matrix. Due to the unidirectional nature of the repair, is an upper triangular matrix, represented as follows:

[0205] (17)

[0206] In formula (17), Indicates that at the initial moment Functional status is Under the conditions, The state of the moment function is determined by Convert to The probability of , ;

[0207] Expressed as:

[0208] (18)

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

[0210] S5. In the "optimization" stage, the "offensive and defensive strategies of the red and blue teams" are analyzed to establish an offensive and defensive strategy optimization model, 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 indicator evaluation model to obtain the resilience analysis results considering the offensive and defensive strategies of the red and blue teams during combat.

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

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

[0213] (1) The combat situation space should include at least the types and quantities of the Red side's remaining weapons, the types and quantities of the Blue side's remaining weapons, the dimensions of the airport runway in the current offensive and defensive confrontation, and the combat support mission;

[0214] (2) Represents the action space of the red team, each action Indicates the type and quantity of missiles used;

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

[0216] (4) Indicates status Red player takes action probability;

[0217] (5) Indicates status The blue player takes action probability;

[0218] (6) Represents the red team's reward set, then the blue team's reward is the negative of the red team's reward. The output result of the functional indicator evaluation model is selected for reward setting;

[0219] (7) Represents the state transition probability, which means Moment, Red side takes action and Blue take action Post-state Transition to the next state probability;

[0220] (8) is the discount coefficient, which indicates the importance of future rewards. This shows that the results of each round of confrontation are equally important. This indicates that the results of the current round are more important;

[0221] like Figure 9 As shown in the figure, the attack and defense strategy analysis process of multiple airport runways is as follows: First, the attack and defense Markov decision process simulation is performed. The initial state It is the initial value of the resource investment of the red and blue parties and any unassigned target airport runway; both parties use strategy and Select initial state The offensive and defensive moves , the offensive and defensive actions and status Input into the functional indicator evaluation model, and select the output of the functional indicator evaluation model as the reward for the first round of confrontation , the status is updated to Repeat the above process until the Red team's resources are exhausted or all airport runway confrontations are completed, reaching the end state. When the whole confrontation process ends, a trajectory is formed, which is expressed as The result of the entire adversarial process is expressed as a cumulative discounted reward, also called a "return", which is expressed as follows:

[0222] (19)

[0223] In formula (19), Indicates the initial state Follow the strategy and The reward for a round of simulated confrontation. Due to the uncertainty of attack and defense strategies, the goal of both parties is the expected reward. The initial state The expected return under is:

[0224] (20)

[0225] In formula (20), Indicates that in the strategy and Lower pair Seek hope, Indicates the initial state Follow the strategy and The expected reward for engaging in confrontation.

[0226] The goal of the Red team is to optimize the offensive strategy Make The blue team's goal is to optimize the defense strategy Make Minimum; there exists a Nash equilibrium joint strategy , so that formula (21) holds true, where and They represent the optimal attack strategy of the red side and the optimal defense strategy of the blue side respectively:

[0227] (twenty one)

[0228] In formula (21), represents the expected return of the red team adopting the optimal attack strategy and the blue team not adopting the optimal defense strategy; represents the expected return of both red and blue players adopting the optimal strategy; represents the expected return of the red side not adopting the optimal attack strategy and the blue side adopting the optimal defense strategy.

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

[0230] (twenty two)

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

[0232] The following is an explanation of the military airport runway toughness analysis method under missile-to-air confrontation described in this embodiment through a specific case.

[0233] 1. Case Background

[0234] Referencing information from a certain year's yearbook of an air force, two air bases, A and B, with a total of four runways, are selected as the targets of a Red and Blue offensive and defensive confrontation. Air Base A includes Runways 1 and 2, while Air Base B includes Runways 3 and 4. Specifically, Runway 1 measures 2173m x 45m, Runway 2 measures 3048m x 45m, Runway 3 measures 4100m x 45m, and Runway 4 measures 2750m x 45m. At Air Base A, the combat support aircraft types and MOSs for one side are C-141s and 1700m x 15m, respectively, while those for the other side are KC-110s and 3000m x 25m. At Air Base B, the combat support aircraft types and MOSs for one side are C-5s and 2500m x 25m, respectively, while those for the other side are KC-135s and 2700m x 25m.

[0235] Assume that the current combat goal of the Red side is to block the takeoff and landing of all types of fighter jets on all airport runways. Each airport runway can choose to attack with three types of missiles: high, medium, and low, and each type can launch 2, 4, and 6 rounds, but each airport runway can only use one type of missile; the Blue side can use high and low-level interceptor missiles at the same time, and use 0 times (i.e. no interception), 1 times, or 2 times the number of Red side missiles for interception. The relevant parameter values are shown in Table 1. The Red side has 4 high, 4 medium, and 6 low-performance missiles, respectively, and the Blue side has 50 high-level and 100 low-level interceptor missiles, respectively. Confrontation will be carried out in the order of Airport Runway 1 to Airport Runway 4. The current combat phase ends when all airport runway confrontations are completed or the Red side's resources are exhausted. The offensive and defensive confrontation reward results of each airport runway are equally important (i.e. ). Under the above operational situation, complete the runway resilience analysis.

[0236] Table 1 Related parameter values

[0237]

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

[0239] 2. Functional Index Evaluation Results and Attack-Defense Profit Matrix

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

[0241] Fully functional state ( ): Can support the takeoff and landing of C-141 and KC-110 fighter jets.

[0242] Basic functional status ( ): Can only support the takeoff and landing of C-141 fighter jets.

[0243] Total loss status ( ): It cannot support the takeoff and landing of any type of fighter jets, including C-141 and KC-110.

[0244] Taking the example of the Red side launching two high-performance missiles to attack and the Blue side taking two-layer 1x interception to defend, the resilience index can be obtained based on the functional index evaluation model FMEM. , that is, the airport runway 2 is at The probabilities are: 58.98%, 1.04%, and 39.98% respectively.

[0245] In the current combat situation, the Red Army's combat goal is to block all fighter takeoffs and landings on the runway. According to the definition of the functional state mentioned above, the probability distribution of the complete loss of the airport runway function state can be expressed as As a functional indicator that meets the combat purpose. At the same time, using the functional indicator evaluation model FMEM, we can get the 9×3 attack and defense benefit matrix of the four runways, as follows: Figure 10 、 Figure 11 、 Figure 12 、 Figure 13 As shown, each value in the attack and defense benefit matrix is ,For example Figure 10 In the middle, the red side uses 2 high-performance missiles and the blue side uses the high and low double layers. =0.4917.

[0246] 3. Offense and Defense Strategy Optimization Results

[0247] According to the attack and defense strategy optimization model OMODS, the functional indicators of the four airport runways are The cumulative sum of is the offensive and defensive game goal of both parties, as shown in the following two formulas:

[0248]

[0249]

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

[0251] Table 2 Red and Blue attack and defense balance strategies

[0252]

[0253] As can be seen from Table 2, the solution is a pure strategy Nash equilibrium solution =2.5627. At the same time, a random attack strategy is constructed for comparison. That is, the red team always maintains the same probability of selecting an attack action that meets the constraints for each airport runway, while the blue team takes the action based on the current red team's choice. Minimum action. The cumulative benefits comparison diagram of random attack strategy and attack and defense equilibrium strategy is as follows Figure 14 As shown, the expectation of the simulation results of the random attack strategy is =1.8384, as shown by the blue horizontal line. This demonstrates that without attack and defense strategy analysis, it is impossible to determine the specific risks faced by each airport runway, nor is it possible to quickly identify the most unfavorable conditions for military facilities. A random attack strategy, similar to current damage assessments, requires evaluating all possible attack actions to minimize facility functional loss. However, this is difficult to achieve quickly when the attack and defense action space is large.

[0254] 4. Results of Airport Runway Resilience Analysis

[0255] By inputting the attack and defense balance strategy obtained above into the functional index evaluation model FMEM again, we can obtain the runway resilience analysis results considering the attack and defense strategy (still taking airport runway 2 as an example):

[0256] (1) Resilience index :Resilience index of airport runway 2 , that is, the airport runway 2 is at The probabilities should be: %, 1 .

[0257] (2) Resilience index : Cumulative distribution function of each functional state transition of airport runway 2 like Figure 15 shown. Figure 15 In the middle, green is the step-by-step repair mode of , 30 units of time; blue is task priority mode of , The time is 28 units. Although the mission priority mode can reach the target state faster, it has a longer state retention period, and no type of aircraft can be taken off or landed during the entire recovery period; the yellow in the figure is the step-by-step repair mode of , It is 12 time units, which means that although the step-by-step repair mode takes longer, it is still effective at 12 time units ( ), capable of taking off and landing C-141 fighter jets. The results show that selecting an appropriate repair mode can improve the resilience of an airport runway to a certain extent.

[0258] The comparison results of the attack and defense strategies in this embodiment ( Figure 15 ) shows that the functional evaluation results of military facilities under different attack and defense strategies are significantly different. The resilience evaluation index corresponding to the attack and defense balance strategy is the theoretical lower limit value of the resilience index (the most unfavorable case) considering all possible attack and defense combinations, which can provide reliable data support for the decision-making and command of the defender.

[0259] (3) Runway function curve The function curve of airport runway 2 under the attack and defense balance strategy is as follows: Figure 16 As shown, from Figure 16 It can be seen that as the recovery time progresses, gradually decreases, and It increases accordingly, due to Relatively small, Recovery is faster, so the functional curve No major changes.

[0260] It can be seen from this embodiment that the use of AFRMI can not only obtain an attack-defense balance strategy that is consistent with the resource constraints of the opposing sides under the current combat situation, but also obtain the impact of the weapon confrontation results on the functions of military facilities under this attack-defense balance strategy, and use a function curve based on probability distribution to characterize the loss and recovery of the functions of military facilities, providing quantitative and comprehensive resilience analysis results. At the same time, it can also be used as an analysis tool for optimizing higher-level strategic decisions such as combat objectives, weapon resource investment and repair force allocation, providing more efficient, more accurate and more favorable decision-making support for combat command in battlefield environments.

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

Claims

1. A method for analyzing the toughness of a military airport runway under missile-air confrontation, characterized in that: The steps include: S1. Based on resilience disaster prevention theory, a framework for analyzing military facility resilience, denoted as AFRMI, is created that considers the uncertainties in five aspects: offensive and defensive strategies of the Red and Blue sides, weapon confrontation outcomes, physical damage to facilities, loss of facility functionality, and restoration of facility functionality. S2. Taking the airport runway function in the context of missile-air confrontation as the evaluation object, the resilience analysis of the airport runway is divided into three stages: "assessment-optimization-analysis" based on the AFRMI. S3. Define and quantify the runway resilience index, determine the runway function curve based on the runway resilience index, and use the runway resilience index and the runway function curve as the resilience analysis results; S4. In the "Evaluation" 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 evaluation model, denoted as FMEM, is established to obtain the functional index of each airport runway under different attack and defense strategies; S5. In the "Optimization" phase, the "offensive and defensive strategies of the red and blue teams" are analyzed to establish an attack and defense strategy optimization model, denoted as OMODS. The attack and defense equilibrium strategy results obtained by solving the attack and defense strategy optimization model are input into the functional indicator evaluation model to obtain the resilience analysis results considering the attack and defense strategies of the red and blue teams during combat.

2. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 1 is characterized in that: Step S1 specifically includes the following: Based on the probability analysis framework of building structure function loss under natural disasters, which considers the transmission of various types of uncertainty, the probability distribution of military facility function indicators under attack and defense confrontation is obtained, as shown in formula (1). Through the probability distribution of military facility function indicators, a military facility resilience analysis framework is obtained that considers the uncertainty of five links: "attack and defense strategies of the red and blue sides - weapon confrontation results - facility physical damage - facility function loss - facility function recovery": （1） In formula (1), is the conditional probability density function of the attack and defense actions AC chosen by the red and blue sides under the current combat situation SI; To determine the conditional probability density function of the weapon confrontation result EF under the attack and defense action AC; To determine the conditional probability density function of facility physical damage DA under the countermeasure result 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 function loss LO.

3. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 1 is characterized in that: Step S2 specifically includes the following: When analyzing the resilience of airport runways in the context of missile-to-air confrontation operations, we first evaluate the functional indicators of each airport runway under all possible attack and defense strategies, and construct an attack and defense benefit matrix; then, based on the attack and defense benefit matrix data, we optimize and solve the attack and defense equilibrium strategies of the red and blue sides for each airport runway; finally, we analyze the attack and defense equilibrium strategies of the red and blue sides for each airport runway, thereby obtaining the resilience analysis results that take into account the attack and defense strategies.

4. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 1 is characterized in that: Step S3 specifically includes the following: The resilience of an airport runway refers to its combat support capability after being damaged and its ability to recover to a level of combat support higher than that required in the damaged state. The resilience index of the airport runway is quantified and reflected by two resilience indicators. Specifically, under the attack and defense strategies of the red and blue sides, the probability distribution of the functional state of the airport runway at the initial moment of damage is According to the repair strategy, the functional state after repair changes from the initial functional state Restore to target functional state The total time for ,Will and As two indicators to quantify the resilience of airport runways; based on and , the probability distribution of functional status that changes with recovery time is obtained through the transition probability matrix , and As the airport runway function curve, it is used to describe the change law of the airport runway function from damage to recovery after the attack and defense confrontation; The resilience index of the airport runway and and the airport runway function curve As three results of resilience analysis.

5. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 4 is characterized in that: Step S4 specifically includes the following: S41. Missile-to-air weapon confrontation simulation: Red team actions include at least the performance of the missile launcher, the number of missiles, and the penetration method; Blue team actions 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 - true and false warhead identification - interceptor missile launch - missile penetration hit" to obtain the weapon confrontation results. The weapon confrontation results are expressed as the probability of the combination of the number of missiles actually hitting the aiming point. The combination of the number of missiles actually hitting the aiming point is referred to as the hit combination. S42. Random Damage Scenario Simulation: A damage scenario is defined as the damage to the runway surface caused by the submunitions after the Red Army missile completes the penetration. Due to the discrepancy between the submunitions' impact probability and the bullet dispersion error, random damage scenarios still exist under the same hit combination. Using the random damage scenario modeling method, a Monte Carlo simulation is performed on a random damage scenario under this hit combination to form a random damage scenario sample set. Each sample in the random damage scenario sample set contains the actual impact point coordinate information of all bullets. S43. Constructing a mapping relationship between physical damage and functional status of airport runways: Functional status is divided according to the different types of fighter jets that the runways support. Assuming that the runways at the air force base in the current combat phase need to support the takeoff and landing of three types of fighter jets: large, medium, and small, the functional status of the runways can be divided into four types. The four functional statuses are as follows: Fully functional state: can support the take-off and landing of large, medium and small fighters. express; Basic functional status: cannot support the take-off and landing of large fighters, can support the take-off and landing of medium and small fighters, express; Minimum functional status: cannot support the take-off and landing of large and medium-sized fighters, can only support the take-off and landing of small fighters, through express; Total loss state: Unable to support the takeoff and landing of any of the three types of fighters: large, medium, and small. express; For the random damage scene 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; S44. Airport runway function recovery analysis: Assume that the airport runway repair modes include step-by-step repair and task priority. Construct a function recovery time analysis model based on discrete state Markov process to analyze the law of change of the functional state probability distribution function of the airport runway with the repair time.

6. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 5 is characterized in that: In step S41, the air-to-air weapon confrontation simulation specifically includes: With Aim point, the first on the red direction airport runway Aiming point to launch missiles The total number of airport runway hit combinations is ; Assuming that the penetration hit probability of each missile is independent, Aiming points successfully penetrated and hit the missile Probability of hair Follows Bernoulli distribution ; Assuming that the hit probability distribution of each aiming point is independent, the hit combination Probability Expressed as: （2） In formula (2), Indicates a combination of offensive and defensive actions. and Respectively represent the red side's attack and the blue side's defense. , It represents the probability of a single missile from the Red Army successfully penetrating the defense. ; Derived through the penetration probability model: （3） （4） （5） In formula (3), It indicates the probability that the detection and tracking accuracy of missiles launched by the Red Army meets the launch requirements of the anti-missile system; It represents the reliability of the interceptor missile platform launch, that is, the probability of successfully launching the interceptor missile; represents the probability that the real warhead is correctly identified; Represents the probability of the three stages of "missile launch detection and tracking - true and false warhead identification - interceptor missile launch"; Formula (4) uses represents the single-shot interception probability of the blue interceptor missile against the target; assuming that the single-shot interception probability of each interceptor missile against the target is the same, then for each identified warhead, the blue side launches Launch interceptor missiles to intercept, It represents the probability of the missile being successfully intercepted; There are only two states for a missile: penetration and being intercepted. Therefore, formula (5) represents the probability of a single missile penetrating.

7. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 6 is characterized in that: In step S42, random destruction scenario simulation specifically includes: Random destruction scene sample set established by random destruction scene modeling method Each sample in Contains the actual landing coordinate information of all bullets; Missile mother bomb There is a 50% chance of landing on a circle centered on the aiming point. The number of bullets carried by the mother bomb is within the circle with a radius of The coordinates of the landing point are based on the actual landing point of the mother missile as the center and the radius as The circle is uniformly and randomly distributed, and the mother bomb The falling point expression of the bullet is as shown in formula (6). The explosion point coordinate expression is as follows: （6） （7） In formulas (6)-(7), the mother bomb The aiming point coordinates are , The missile circular error probability CEP represents the missile's strike, and Represents a random value of a two-dimensional standard normal distribution, the mother bullet The actual landing point coordinates are ;bullet The coordinates of the explosion point It is a mother bomb Actual landing point coordinates are two independent uniformly distributed random numbers at the origin, and Obeying uniform random distribution, , assume that the damage shape of each bullet crater is based on the coordinates of the explosion point is the center of the circle and the radius is the bullet damage radius The standard circle.

8. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 5 is characterized in that: In step S43, for the random damage scene sample set, a functional state classifier based on minimum operating plane MOS search is used to establish a mapping relationship between physical damage and functional state, specifically including: Indicates that the airport runway was damaged at the initial moment 、 、 、 Probability distribution of four functional states; random destruction scenario sample set The samples are input into the functional state classifier, and the highest functional state The corresponding MOS judges and confirms the functional status of the sample in turn; Assume that the MOS of small fighter, medium fighter, and large fighter are expressed as 、 、 , < < By optimizing the search algorithm, the random destruction scene sample set Each sample in the , specifically, when there is no complete When Search, if exists , then the functional status of the sample is determined to be If there is no complete , then continue to Search, if exists , then the functional status of the sample is determined to be , if it does not exist , then the functional status of the sample is determined to be ; This will randomly destroy the scene sample set Classified as , corresponding to four functional states, the mapping from physical damage to functional state is completed, and the probability distribution of functional state takes the number of samples of different functional states The total number of samples The ratio is expressed as follows: （8） right All hit combinations are subjected to Monte Carlo simulation, and we get: （9） Formula (9) includes the uncertainty of the three links from "weapon confrontation results - facility physical damage - facility function loss" and is used to obtain the resilience index .

9. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 5 is characterized in that: In step S44, the airport runway function recovery analysis specifically includes: S441. Set the runway repair mode to include step-by-step repair and mission priority. Select the runway repair mode based on the urgency of the combat mission. Set the crater repair time to only consider the number of craters. Set all bullets to be non-blocking bullets. Indicated by functional status Restore to functional state Repair required , and the repair time is proportional to the number of craters to be repaired, determined by the functional state Restore to functional state Recovery time Expressed as: （10） In formula (10), Indicates need for repair Number of craters, It represents the average time to repair one crater; When the functional status of the airport runway is restored step by step in order, the step-by-step repair mode is selected; when the functional status of the airport runway is restored in a skipped order, the task priority mode is selected; 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 is obtained through Monte Carlo simulation , sample set Represents the sample set under all possible hit combinations, and the sample set Divided into different subsets , for the subset The samples in Search and calculate the minimum number of craters to be repaired for each sample , thus obtaining the total repair time of the sample; 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 ; The total time taken to recover 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 taken to recover from the initial functional state to the target functional state in the calculation task priority mode is: （11-2） Statistical sample data, get The probability density function and cumulative distribution function of the step-by-step repair mode and the task priority mode are calculated in the same way. The probability density function and cumulative distribution function of are shown in formula (12) and formula (13), respectively, as follows: （12） （13） Formula (12) and formula (13) are based on the hit combination As a prerequisite, perform weighted summation of formula (12) and formula (13), and the weight is the attack and defense action combination Next hit combination The probability is expressed as follows: （14） （15） S443. Construct a functional recovery time analysis model based on a discrete-state Markov process to analyze how the functional state probability distribution function of the runway changes with the repair time. Specifically, the functional state probability distribution is expressed as a function of the repair time as follows: （16） In formula (16), Indicates the initial time of repair Functional state probability distribution when ; Represents the transition probability matrix. Due to the unidirectional nature of the repair, is an upper triangular matrix, represented as follows: （17） In formula (17), Indicates that at the initial moment Functional status is Under the conditions, The state of the moment function is determined by Convert to The probability of , ; Expressed as: （18） In formula (18), Indicates the function status A higher level of functional status, Indicates functional status Restore to functional state Cumulative distribution function of the total time.

10. The method for analyzing the toughness of a military airport runway under missile-to-air confrontation according to claim 1 is characterized in that: Step S5 specifically includes the following: The attack and defense strategy optimization model is configured to solve the zero-sum Markov game problem between the red and blue sides, denoted as TZMG, and represented by the following 8-tuple: (1) The combat situation space should include at least the types and quantities of the Red side's remaining weapons, the types and quantities of the Blue side's remaining weapons, the dimensions of the airport runway in the current offensive and defensive confrontation, and the combat support mission; (2) Represents the action space of the red team, each action Indicates the type and quantity of missiles used; (3) Represents the action space of the blue team, each action Indicates the type of interceptor missile used and the interception method; (4) Indicates status Red player takes action probability; (5) Indicates status The blue player takes action probability; (6) Represents the red team's reward set, then the blue team's reward is the negative of the red team's reward. The output result of the functional indicator evaluation model is selected for reward setting; (7) Represents the state transition probability, which means Moment, Red side takes action and Blue take action Post-state Transition to the next state probability; (8) is the discount coefficient, which indicates the importance of future rewards. This shows that the results of each round of confrontation are equally important. This indicates that the results of the current round are more important; The analysis process of the attack and defense strategy of the missile-air confrontation of multiple airport runways is as follows: First, the attack and defense Markov decision process simulation is carried out. The initial state It is the initial value of the resource investment of the red and blue parties and any unassigned target airport runway; both parties use strategy and Select initial state The offensive and defensive moves , the offensive and defensive actions and status Input into the functional indicator evaluation model, and select the output of the functional indicator evaluation model as the reward for the first round of confrontation , the status is updated to ; Repeat the above process until the Red team's resources are exhausted or all airport runway confrontations are completed, reaching the end state. When the whole confrontation process ends, a trajectory is formed, which is expressed as The result of the entire adversarial process is expressed as a cumulative discounted reward, also called "return", which is expressed as follows: （19） In formula (19), Indicates the initial state Follow the strategy and The reward for a round of simulated confrontation. Due to the uncertainty of attack and defense strategies, the goal of both parties is the expected reward. The initial state The expected return under is: （20） In formula (20), Indicates that in the strategy and Lower pair Seek hope, Indicates the initial state Follow the strategy and the expected rewards of engaging in confrontation; The Red team's goal is to optimize the offensive strategy Make The blue team's goal is to optimize the defense strategy Make Minimum; there exists a Nash equilibrium joint strategy , so that formula (21) holds true, where and They represent the optimal attack strategy of the red side and the optimal defense strategy of the blue side respectively: （21） In formula (21), represents the expected return of the red team adopting the optimal attack strategy and the blue team not adopting the optimal defense strategy; represents the expected return of both red and blue players adopting the optimal strategy; represents the expected return of the red team not adopting the optimal attack strategy and the blue team adopting the optimal defense strategy; The expected cumulative reward of multiple airport runways under the Nash equilibrium joint strategy is abbreviated as formula (22): （22） The Q-learning algorithm is used to solve the attack and defense strategy optimization model expressed in formula (22) to obtain the Nash equilibrium joint strategy specific form.

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