基于末段防空战斗中多型火力部署力量需求模型研究
Research on the demand model of multi-type fire deployment forces in the terminal air defense battle

李梓硕 李俊骞 XXX 韦 哲
Li Zishuo, Li Junxian, XXX, Wei Zhe

摘 要 未来防空战中近程防空导弹通常作为一体化防空体系重要环节，负责关键地区的末段防空任务。准确评估火力需求是规划近程防空导弹作战行动的基础，因此，深入探讨从一体化防空体系视角出发的研究导弹部队火力需求的必要性显得尤为重要。本论文以火力需求的基本理论为指导，深入剖析了近程防空导弹火力需求建模的方法，构建了一套既考虑环形部署也涉及集团部署形式的火力需求定量分析模型。通过具体分析模型的特点与应用可行性，展现了该模型在支持末段防空作战决策方面的实际价值与指导意义。
Abstract: In future air defense battles, short-range air defense missiles are usually an important part of the integrated air defense system, responsible for the terminal air defense tasks in key areas. Accurately assessing the fire power requirements is the basis for planning the combat operations of short-range air defense missiles. Therefore, it is particularly important to deeply explore the necessity of studying the fire power requirements of missile troops from the perspective of the integrated air defense system. This paper is guided by the basic theory of fire power requirements, deeply analyzes the methods of modeling the fire power requirements of short-range air defense missiles, and constructs a quantitative analysis model of fire power requirements that considers both ring deployment and group deployment forms. By analyzing the characteristics and application feasibility of the model, this paper demonstrates the practical value and guiding significance of the model in supporting terminal air defense combat decision-making.

关键词 末段防空；火力部署；力量需求；模型研究
Keywords: Terminal air defense; Fire deployment; Force requirements; Model research

Abstract In future air defense, short-range air defense missiles are usually an important part of the integrated air defense system, responsible for the terminal air defense tasks in key areas. Accurately assessing firepower requirements is the foundation for planning short-range air defense missile combat operations. Therefore, it is particularly important to explore the necessity of studying the firepower requirements of missile forces from the perspective of integrated air defense systems. This paper is guided by the basic theory of firepower demand, and deeply analyzes the modeling method of firepower demand for short-range air defense missiles. It constructs a quantitative analysis model of firepower demand that considers both circular configuration and group configuration forms. By analyzing the characteristics and feasibility of the model, the practical value and guiding significance of the model in supporting decision-making for terminal air defense operations have been demonstrated.
In the future air defense, short-range air defense missiles are usually an important part of the integrated air defense system, responsible for the terminal air defense tasks in key areas. Accurately assessing firepower requirements is the foundation for planning short-range air defense missile combat operations. Therefore, it is particularly important to explore the necessity of studying the firepower requirements of missile forces from the perspective of integrated air defense systems. This paper is guided by the basic theory of firepower demand, and deeply analyzes the modeling method of firepower demand for short-range air defense missiles. It constructs a quantitative analysis model of firepower demand that considers both circular configuration and group configuration forms. By analyzing the characteristics and feasibility of the model, the practical value and guiding significance of the model in supporting decision-making for terminal air defense operations have been demonstrated.

Keywords Terminal air defense; Firepower deployment; Power demand; model research

1 引言
1 Introduction

近程防空导弹火力部署和规模是进行要地防空作战的必要物质基础，同时也是近程防空导弹展开防空作战部署的核心要素。若缺乏合理的火力结构与数量，便难以构建与敌方空袭相匹配的防空作战布局^[1]。末端防空作战中，对近程防空导弹火力的需求涵盖了射击过程控制理论的诸多复杂因素，为解决这一问题，本文尝试综合运用现代防空作战理论、可信性理论以及不确定规划方法，探讨在不同部署模式下，近程防空导弹火力需求的量化问题。所建立的模型不仅能够为指导防空作战提供重要依据，同时也具有显著的现实应用价值。
The deployment and scale of short-range air defense missile fire are essential material foundations for conducting important area air defense operations, and they are also the core elements for the deployment of short-range air defense missile operations. Without a reasonable fire structure and quantity, it is difficult to build an air defense operation layout that matches the enemy's air raids. In the terminal air defense operations, the demand for short-range air defense missile fire encompasses many complex factors of shooting process control theory. To address this issue, this paper attempts to comprehensively apply modern air defense operation theory, credibility theory, and uncertain planning methods to explore the quantification of short-range air defense missile fire requirements under different deployment modes. The established model not only provides an important basis for guiding air defense operations but also has significant practical application value.

2 近程防空导弹战斗的部署形式
Deployment forms of short-range air defense missile combat

近程防空导弹作战部署策略需具备应对多方向攻击的全维度防御能力。在人力许可范围内，应优先采取环形布署，确保各区域火力的有效衔接。即便人力有限，未能实现完整环形布署，也应合理分配资源，以增强防空火力的深度或强度^[2-3]。近程防空导弹的战术部署通常包括以下四种形式：
The operational deployment strategy of short-range air defense missiles needs to have a full-dimensional defense capability to cope with multi-directional attacks. Within the scope of human resources, it should prioritize a circular deployment to ensure the effective connection of fire power in each area. Even if human resources are limited and a complete circular deployment cannot be achieved, resources should be allocated reasonably to enhance the depth or intensity of air defense fire. The tactical deployment of short-range air defense missiles usually includes the following four forms:

1）重点环形部署：集中力量形成关键区域的全方位防护；
1) Key circular deployment: concentrate forces to form all-around protection in key areas

2）扇形部署：以一定角度展开，覆盖特定范围的空中威胁；
2) Sector Deployment: Unfolding at a certain angle to cover a specific range of aerial threats;

3）集团部署：多个单位协同，共同构建全面的防御网；
3) Group Deployment: Multiple units coordinating to construct a comprehensive defense network;

4）线形部署：沿特定轴线排列，适用于特定方向的拦截任务；这样的部署旨在最大化利用有限资源，确保高效防御。
4) Linear Deployment: Arranged along a specific axis, suitable for intercepting tasks in a particular direction; such deployment aims to maximize the use of limited resources to ensure efficient defense.

2.1 关键方位的环形布置
2.1 Circular Arrangement at Key Positions

近程防空导弹的关键方位环形布置，着重强调对敌方空中威胁的主要来源及其关键目标进行针对性防御，防空作战分队大致呈环状布局于防空区域的外围。
The key azimuthal arrangement of short-range air defense missiles, emphasizing targeted defense against the main sources and key targets of enemy air threats, with the air defense combat teams roughly arranged in a ring around the periphery of the air defense area.

2.2 扇形布局
2.2 Fan-shaped Layout

扇形布局则是在主要方向上，通过分散设置各个阵位形成扇形结构。在阵位数量有限或受地形限制无法形成环形布局的情况下，采用扇形布局较为适宜。扇面的角度需根据战场指挥官指定的责任射界范围、火力规模、需要保护的目标分布以及地形条件来决定。
The fan-shaped layout is formed by dispersing the positions in the main direction to create a fan structure. It is more suitable when the number of positions is limited or when the terrain restricts the formation of a circular layout. The angle of the sector should be determined based on the battlefield commander's specified responsibility firing range, fire scale, distribution of targets to be protected, and terrain conditions.

2.3 集群部署
2.3 Cluster Deployment

集群部署是指将阵地紧密地安排在一个特定区域内，阵位间的最小间距需确保各阵地之间不会相互干扰电磁信号，并且兵器发射时不会对邻近阵地构成威胁。集群部署的优势在于能显著提高防空效能，但其缺点在于可能在火力运用上不够经济。
Cluster deployment refers to the close arrangement of positions within a specific area, with the minimum spacing between positions to ensure that electromagnetic signals between the positions do not interfere with each other, and that the firing of weapons does not pose a threat to adjacent positions. The advantage of cluster deployment is that it can significantly improve air defense effectiveness, but its disadvantage is that it may not be as economical in terms of fire power application.

2.4 线形部署
2.4 Linear Deployment

线性部署指的是在防御对象周围形成一条直线式的防线布局。这种部署方式相对较少使用，一般仅在保护具有特殊价值的目标时才会采取。
Linear deployment refers to forming a straight-line defense layout around the defense object. This type of deployment is relatively rare and is generally only adopted when protecting targets of special value.

3 分析与建模策略
3 Analysis and Modeling Strategies

在防空作战中近程防空导弹火力需求的考量涉及火力结构与规模两维度：
The consideration of short-range air defense missile fire power requirements in air defense operations involves two dimensions: fire power structure and scale:

火力结构：侧重于定性分析，探讨在末端防御行动中，近程防空导弹火力结构的构建与优化策略；
Fire power structure: focuses on qualitative analysis, discussing the construction and optimization strategies of short-range air defense missile fire power structure in terminal defense operations;

火力规模：聚焦定量研究，主要通过计算火力数量来确定规模，往往借助数学模型进行深入剖析^[4]。
Fire power scale: focuses on quantitative research, mainly determining the scale through the calculation of fire power quantity, often using mathematical models for in-depth analysis ^[4] .

上述研究均基于近程防空导弹火力需求的固有原则、基本准则以及理论基础展开。在火力结构的研究层面，强调了分析方法和方向，旨在指导如何合理部署火力以应对特定作战环境。
The above research is based on the inherent principles, basic criteria, and theoretical basis of the short-range air defense missile fire power requirements. At the level of fire power structure research, it emphasizes the analysis methods and directions, aiming to guide how to reasonably deploy fire power to cope with specific combat environments.

火力规模的探讨则更侧重于数量计算，通常采用量化方法，构建模型以辅助决策。通过模型求解^[5]，能够获得更为精确合理的结果。模型的构建可分为两大类：
The exploration of fire power scale focuses more on quantitative calculations, usually employing quantitative methods to construct models to assist in decision-making. By solving ^[5] through the model, more precise and reasonable results can be obtained. The construction of the model can be divided into two major categories:

1）环形部署模型：适用于围绕或辐射目标区域的火力布局需求计算；
1) Ring deployment model: Suitable for calculating the fire layout requirements around or radiating from the target area;

2）集团部署模型：针对集中火力进行重点防御或攻击的情况。
2) Group deployment model: Targeted at situations where concentrated fire power is used for key defense or attack.

上述两种构建模型的过程遵循以下步骤：
The process of constructing the aforementioned two types of models follows the following steps:

目标路径与特性分析：研究敌方目标的来袭路线及其特征；
Target Path and Feature Analysis: Researching the incoming route and characteristics of enemy targets;

参数识别与约束条件设定：基于实际对抗中的关键参数（如空袭速度、高度、我方兵器与目标距离、兵器部署间隔等），确定模型需满足的条件；
Parameter Identification and Constraint Condition Setting: Based on key parameters in actual countermeasures (such as air raid speed, altitude, distance between our weapons and targets, and weapon deployment intervals), determine the conditions that the model needs to meet;

3）模型推导与实战应用：结合前两步的结果，构建末端防空作战近程防空导弹火力需求模型，并探讨其在实际战场上的应用方式。这种系统化、分层化的分析与建模策略，为近程防空导弹火力需求的精准预测与高效部署提供了有力支持。
3) Model Derivation and Practical Application: Combining the results of the first two steps, construct a short-range air defense missile fire requirement model for terminal air defense operations, and discuss its application methods in actual combat. This systematic and layered analysis and modeling strategy provides strong support for the accurate prediction and efficient deployment of short-range air defense missile fire requirements.

4.1 基于环形与扇形部署的火力需求模型探讨
4.1 Discussion on Fire Requirement Models Based on Circular and Sectorial Deployment

在战术层面，环形部署与扇形部署之间存在差异，其中环形部署在火力规模需求计算中展现出最优的防空效能^[6-7]。然而，在实战场景中，受限于火力的数量、地形条件等多方面因素，实施环形部署存在一定的局限性。因此，实践中通常采用更为灵活的扇形部署。近程防空导弹在关键地区的空中防御行动中，承担着末端防御层的保护职责，并根据上级指示，在指定方向上采取扇形部署策略。具体而言，实际构建模型时关注的是敌方空袭区域的张角α的大小。当α达到特定角度时，环形部署可以被视为一种特殊的扇形部署，用于研究和分析。以下以扇形部署为例，构建相应的火力需求模型，以便更精确地评估并满足实际作战需求。
On the tactical level, there are differences between circular deployment and fan-shaped deployment, among which circular deployment shows the optimal air defense effectiveness in the calculation of fire scale requirements. However, in actual combat scenarios, due to various factors such as the number of fire power, terrain conditions, and so on, the implementation of circular deployment has certain limitations. Therefore, in practice, a more flexible fan-shaped deployment is usually adopted. Short-range air defense missiles bear the responsibility of protecting the terminal defense layer in the air defense operations in key areas, and according to superior instructions, adopt a fan-shaped deployment strategy in the specified direction. Specifically, when constructing the model, attention is paid to the size of the angle α of the enemy air raid area. When α reaches a certain angle, circular deployment can be regarded as a special type of fan-shaped deployment for research and analysis. The following takes the fan-shaped deployment as an example to construct the corresponding fire scale demand model for more precise evaluation and satisfaction of actual combat requirements.

图1 单层环形部署
Figure 1: Single-layer Circular Deployment

4.1.1 单层部署火力规模需求的确定性模型
4.1.1 Deterministic Model of Fire Scale Requirements for Single-layer Deployment

图2 单层扇形部署
Figure 2: Single-layer Fan-shaped Deployment

如图2，空袭目标位于两火力单位连线的中点，旨在对防御目标形成威胁。假设X个火力单位，包括K₁、K₂、直至K_x，呈扇形均匀分布于一层。为了确保这四个火力单位能够共同瞄准同一目标进行射击，必须满足特定条件：
As shown in Figure 2, the target of the air strike is located at the midpoint of the line connecting two fire units, aiming to threaten the defensive target. Assuming X fire units, including K ₁ , K ₂ , up to K _x , are evenly distributed in a fan shape on a single level. To ensure that these four fire units can aim at the same target for shooting, certain conditions must be met:

式（1）中X表示射击火力分队数；R呈现担负执勤捍卫目标的间距值；α表示全部进攻领域的角度值；V是进攻目标的飞行速度值；t_c表示防空武器战斗的最短周期值。
In equation (1), X represents the number of firing teams; R represents the spacing value for the duty of defense; α represents the angle value of the entire offensive area; V is the flight speed value of the offensive target; t _c represents the minimum combat cycle value of the air defense weapons.

图3 火力需求计算立体图
Figure 3: Stereogram of fire requirement calculation

综合图2和图3情况可知，末段防空导弹的水平毁伤间距与四个防空火力分队同步对指示目标实施精确火力毁伤需要的函数法则关系：
By integrating the situations in Figures 2 and 3, it can be known that the horizontal destruction spacing of the terminal air defense missile is in a functional relationship with the simultaneous precise fire destruction of the indicated target by four air defense fire teams:

在上述情境下，公式中的参数定义如下：D_max表示导弹的最大毁伤距离；H表示空袭目标的飞行高度。这样的定义有助于更精确地描述导弹作战效能与目标位置的关系。
In the above context, the parameter definitions in the formula are as follows: D _max represents the maximum destructive distance of the missile; H represents the flight altitude of the air raid target. Such definitions help to more accurately describe the relationship between the combat effectiveness of the missile and the target position.

通过分析图2和图3，我们可以得出在不同火力单位数量X下，目标航线优化路径的关系表达式：
By analyzing Figures 2 and 3, we can derive the relationship expression between the optimized path of the target route and the number of fire units X.

该表达式中的系数和变量等元素与之前所述保持一致。在图 2中，基于战术策略^[8]，相邻火力单位间的间距需遵循特定规则：
The coefficients and variables in this expression are consistent with those previously described. In Figure 2, based on the tactical strategy ^[8] , the spacing between adjacent fire units must follow specific rules.

在战斗攻击导向上，火力覆盖射击目标点位K_i+1和K_i+2之间行程距离为：
In terms of combat attack orientation, the travel distance between the shooting target points K _i+1 and K _i+2 is:

K_i及其加三火力范围的路径长度为：
The path length of the fire range of K and its addition of three is:

K_i+1 和 K_i+2 可以用于表示对空袭目标的射击次数：
K _i+1 and K _i+2 can be used to represent the number of times of shooting at air raid targets:

在公式中，t_s为对空射击靶标的战斗周期；K_i和 K_i+3为对空射击靶标的次数。
In the formula, t _s is the combat cycle for shooting at air targets; K and K _i+3 are the number of times for shooting at air targets.

上述解算得出，K_i、K_i+1、K_i+2、K_i+3对防空导弹攻击目标点位的总数量为：
The calculation above gives the total number of attack target positions for anti-aircraft missiles of K, K _i+1 , K _i+2 , and K _i+3 :

防空火力分队呈扇形区域合理部署，1-(1-p)C（p为一个射击分队对进攻目标点位一次毁伤率）为进攻目标点位完成战斗任务前被毁伤率，如数量值大则表示毁伤效果好，设：
Anti-aircraft fire teams are deployed in a fan-shaped area in a reasonable manner, and the rate of destruction before the combat task of the attacking target position is completed is 1-(1-p)C, where p is the one-time destruction rate of a shooting team against the attacking target position. If the number value is large, it indicates good destruction effect, and it is set as follows:

根据以上各式可以得出基于每个层级部署的防空射击单元火力需求模型：
Based on the above formulas, a fire requirement model for anti-aircraft shooting units at each level of deployment can be obtained:

4.1.2 双层部署火力规模需求模型
4.1.2 Double-layer Deployment Fire Scale Requirement Model

图4 双层扇形部署
Figure 4: Double-layer Fan-shaped Deployment

在采用双层部署场景下，外部部署数量设定为x₁，这些火力单位位于距离保卫目标R₁的扇形区域内；内部部署数量则为x₂，它们位于距离保卫目标R₂的另一扇形区域内。我们的目标是确保每一层都具备两个可以同时进行射击的火力单位^[9]。如图4所示，外部防线由x₀个火力单位构成，而内部防线则由 x₁个火力单位组成，单层防线的部署分析相似，外层火力单位对空袭目标的射击次数将基于特定的条件和计算方法得出。
Under the double-layer deployment scenario, the number of external deployments is set to x ₁ , and these fire units are located within a sector area at a distance R ₁ from the protected target; the internal deployment number is x ₂ , and they are located within another sector area at a distance R ₂ from the protected target. Our goal is to ensure that each layer has two fire units that can fire simultaneously ^[9] . As shown in Figure 4, the external defense line consists of x ₀ fire units, while the internal defense line consists of x ₁ fire units. The deployment analysis of a single-layer defense line is similar, and the number of times the outer fire units will shoot at air raid targets will be based on specific conditions and calculation methods.

内层E_j和E_j+1单位对空中袭击目标进行射击的次数涉及以下参数：x₁表示外层防御单位的数量；R₁表示外层与保卫对象之间的距离；x₂表示内层防御单位的数量；R₂表示内层与保卫对象之间的距离，其中R₁大于R₂。
The number of times the inner E and E _j+1 units fire at air raid targets involves the following parameters: x ₁ represents the number of outer defense units; R ₁ represents the distance between the outer layer and the protected object; x ₂ represents the number of inner defense units; R ₂ represents the distance between the inner layer and the protected object, where R ₁ is greater than R ₂ .

根据式子中的参数保持一致，可计算得出，对于空袭目标的射击总次数C等于C₁与C₂之和。
Based on the consistency of the parameters in the formula, the total number of times C for shooting at air raid targets can be calculated as the sum of C ₁ and C ₂ .

由此，可以构建基于双层部署的火力规模需求的确定性模型。
Therefore, a deterministic model for the fire scale requirements based on double-layer deployment can be constructed.

在实际操作中，模型具备一定的指导价值。假设指挥官已知以下参数：我方对目标的毁伤概率p，各层次防御系统与保护对象间的距离R₁和R₂，火力单位的射击频率t_s，以及其最小射击周期 t_c；高射炮的最大有效打击范围D_max；敌方空袭武器的飞行角度a；以及预设的置信水平。即便在面对敌方空袭兵器的具体飞行高度H、速度V和我方兵器之间的部署间距r等信息缺失的情况下，通过运用这些已知参数，可以进行计算以确定所需的两层火力部署规模，从而指导作战部署。
In practical operations, the model has certain guiding value. Assuming the commander is aware of the following parameters: the probability of damage to the target by our side p, the distances R ₁ and R ₂ between each level of defense system and the protected object, the shooting frequency t _s of the fire units, and their minimum shooting cycle t _c ; the maximum effective striking range D _max of anti-aircraft guns; the flight angle a of the enemy's air strike weapons; as well as the preset confidence level. Even in the absence of specific information such as the flight altitude H, speed V of the enemy's air strike weapons, and the deployment spacing r between our weapons, these known parameters can be used to calculate the required scale of the two-layer fire deployment, thereby guiding the operational deployment.

4.2 基于集群部署的作战资源需求模型
4.2 A Combat Resource Requirement Model Based on Cluster Deployment

4.2.1 飞行路径优化计算
4.2.1 Calculation of Flight Path Optimization

如图5所示，高射炮部队对目标的水平打击范围以点D为远界半径，最大攻击角度设定为q_max，对应的最远打击距离即最大航路捷径为d_j。
As shown in Figure 5, the horizontal striking range of the anti-aircraft artillery units against the target is set with point D as the far boundary radius, and the maximum attack angle is set to q _max , corresponding to the longest striking distance, i.e., the maximum shortcut of the flight path, as d.

为了有效对抗来袭目标，每个火力单位的打击区域需具备一定的纵深覆盖能力。通过分析可知，水平打击纵深^10]的最小值可通过以下步骤计算得出：
In order to effectively counter the incoming targets, each fire unit's striking area must have a certain depth of coverage. Analysis shows that the minimum value of the horizontal striking depth ^10] can be calculated through the following steps:

图5 有效航路捷径计算示意图
Figure 5: Schematic diagram for calculating the effective route shortcut

假如对空射击战斗中，防空火力分队有n次调转方向调转方向射击目标点位，调转方向射击时间D_t，则n次调转方向射击的时间间隔为(n-1)Dt，设射击目标点位速率为V，得出：
In an air defense shooting battle, if the air defense fire team turns its direction to shoot at the target position n times, the time interval between the n turns is (n-1)Dt, where V is the speed of the target position, we can derive:

假设a=|OA|，根据余弦定理法则得到：
Assuming a=|OA|, according to the cosine law, we get:

解算这个方程，取消负根得到：
Solve this equation, and eliminate the negative root to get:

从而
Thus

得到：
We obtain:

可以得到最大航路捷径：
We can obtain the maximum shortcut route:

4.2.2 集团部署火力规模需求模型
4.2.2 Group Deployment Fire Scale Requirement Model

为了有效抵御密集来袭的目标群，或集中火力对关键目标实施攻击，需要实现火力覆盖的重叠^[11]。
In order to effectively resist a dense group of incoming targets or concentrate fire on key targets, it is necessary to achieve overlapping fire coverage ^[11] .

根据图6和图7显示的内容，为了建立包含X个单位和宽度为L的有效射击区域，需要确保火力单位之间的部署间距 D_I 满足以下条件：
According to the content shown in Figures 6 and 7, in order to establish an effective firing area containing X units and a width of L, it is necessary to ensure that the deployment spacing D _I between fire units meets the following conditions:

图6 火力重叠数为2时的集团部署示意图
Figure 6: Schematic diagram of group deployment when the fire overlap is 2

当X = 1时，D_I≤ 2d_j；
When X = 1, D _I ≤ 2d;

当X = 2 时，D_I≤ 2d_j - L，显然L≤2d_j；
When X = 2, D _I ≤ 2d - L, it is obvious that L ≤ 2d;

当X = 3时，2D_I ≤2dj - L；
When X = 3, 2D _I ≤ 2dj - L;

当X = n时，(n - 1)D_I≤2d_j - L。
When X = n, (n - 1)D _I ≤ 2d - L.

即构成X次重叠的最大部署间隔为
The maximum deployment interval for X times of overlap is formed

图7 火力重叠数为3时的集团部署示意图
Figure 7: Schematic diagram of group deployment when the fire overlap number is 3

若将单个火力单位的火力密度设定为μ，那么X次火力重叠区的总体火力密度则变为X_μ。在防空作战场景下，高射炮的数量以及技术性能都是有限的，这导致防空系统仅能对部分来袭的目标进行拦截与打击 ^[12]。假设来袭目标的密度为 λ，而上级的要求是使目标遭受破坏的概率达到P。为了有效地对抗空袭，需要
If the fire density of a single fire unit is set to μ, then the overall fire density of the X times fire overlap area becomes X _μ . In the air defense combat scenario, the number and technical performance of anti-aircraft guns are limited, which leads to the air defense system being able to intercept and strike only a part of the incoming targets ^[12] . Assuming the density of incoming targets is λ, and the superior requirement is to achieve a probability of P for target destruction. In order to effectively counter the air raids,

找到火力重叠数的最小数量为：
Find the minimum number of fire overlap:

为了确保有效抗击能力，火力单位数量X需要满足的重叠数最小值为^[13]。
The minimum overlap value required for the number of fire units X to ensure effective combat capability is ^[13] .

对式（21）、（23）进行反向解算得出：
Reverse calculate equations (21) and (23).

假设 X = 1，λ = μ ，可导出 P = 1；
Assuming X = 1, λ = μ, it can be derived that P = 1.

假设 X > 1，λ > μ ，解算得出
Assuming X > 1, λ > μ, the calculation results are as follows.

如上述情况进行综合解算考量，则建立起根据作战单位群组设定的火力力量需求模型：
Considering the comprehensive calculation in the above situation, a fire power demand model based on the group settings of combat units is established:

在实际操作中，模型提供了一定的参考依据，以辅助指挥员实施对应决策。具体而言，若已知防空导弹水平毁伤区远界半径为D、目标最大航路角为q_max、转火射击次数为n、转火时间D_t、正面宽度为L，并设定相应的置信水平值，即便在敌方空袭武器的飞行速度V与我方武器部署间隔r尚不明朗的情况下，可以解算出群团战斗情况下防空作战火力规模。
In actual operations, the model provides a certain reference basis to assist commanders in making corresponding decisions. Specifically, if the far boundary radius of the anti-aircraft missile's horizontal damage area is D, the maximum flight path angle of the target is q _max , the number of relay shooting is n, the relay shooting time is D _t , the front width is L, and the corresponding confidence level values are set, it is possible to calculate the fire power scale of air defense operations in group combat situations even when the flight speed V of the enemy's air strike weapons and the deployment spacing r of our weapons are not yet clear.

五 总结
Summary

基于本文提出的三层梯次搭配防空背景，为强化有效执行文中构想的末端防空任务，合理的火力结构显得至关重要。上述情况分析可得，被射击空中目标点位均处在低空时突然被末段防空导弹击中的概率低，低空来袭击目标被有效拦截系统长期在各国进行实验研究。所以，在防空战斗中要全部射击火力有效涵盖多层级纵深和方向，但是当前防空导弹针对低空目标来袭时很难有效拦截。在低空战斗领域中，防空分队射击时多层级搭配叠加情况少，还有不少射击区域死角，这时部署多型防空火力进行及时有效拦截凸显重要作用。本文搭建多型火力部署力量需求模型，具有很好的理论研究方向价值，在防空射击分队实操中也具备重要指导意义。
Based on the three-level hierarchical anti-aircraft background proposed in this paper, a reasonable fire power structure is crucial to strengthen the effective execution of the terminal air defense tasks envisioned in the text. The analysis of the above situation shows that the probability of being hit by the terminal air defense missile is low when the target points being shot are at low altitude, and the low-altitude attack targets have been long-term experimentally studied by the effective interception system in various countries. Therefore, in air defense combat, it is necessary to cover multiple layers of depth and direction with all shooting fire power. However, it is difficult for the current anti-aircraft missiles to effectively intercept low-altitude targets. In the low-altitude combat field, the multi-layered combination of air defense detachments is rare, and there are still many shooting blind spots. At this time, deploying multiple types of air defense fire power for timely and effective interception plays a significant role. This paper builds a multi-type fire deployment demand model, which has great theoretical research value and also has important guiding significance in the practical operation of air defense shooting detachments.

参 考 文 献
Reference

［1］代进进,李相民,薄宁,吴小鹤.要地防空作战中防空武器掩护能力评估方法[J].现代防御技术,2018(04)
[1] Dai Jinjin, Li Xiangmin, Bo Ning, Wu Xiaohuo. Evaluation Method of Air Defense Weapon Covering Ability in Important Air Defense Operations [J]. Modern Defense Technology, 2018(04)

［2］马新星,滕克难,侯学隆.岛礁防空火力单元部署距离计算模型[J].兵工自动化,2017(10)
[2] Ma Xinxing, Teng Kena, Hou Xuelong. Calculation Model for Deployment Distance of Air Defense Fire Units on Island Reefs [J]. Military Automation, 2017(10)

［3］许金宇,赵德辉. 防空阵地网的理论及应用［M］. 西安：
[3] Xu Jinyu, Zhao Dehui. Theory and Application of Air Defense Position Network [M]. Xi'an:

西北工业大学出版社，2005：98-102.
Northwestern Polytechnical University Press, 2005: 98-102.

［4］刘立佳,李相民,李亮.多防线要地防空体系目标拦截能力评估[J].海军航空工程学院学报,2014(05)
[4] Liu Lijia, Li Xiangmin, Li Liang. Evaluation of Target Interception Capability of Multi-defense Important Airfield Air Defense System[J]. Journal of Naval Aeronautical Engineering College, 2014(05)

［5］缪旭东,王永春.舰艇编队协同防空任务规划理论及应用[M]. 国防工业出版社.2013
[5] Miao Xudong, Wang Yongchun. Theory and Application of Air Defense Task Planning for Fleet Coordination[M]. National Defense Industry Press. 2013

［6］ 孙兴龙;马克茂;姜宇;侯振乾.临近空间高速目标拦截策略设计.空天防御,2022(04)
[6] Sun Xinglong; Ma Kemaoh; Jiang Yu; Hou Zhenqian. Design of Near-Space High-Speed Target Interception Strategy. Space Defense, 2022(04)

［7］胡小利;王炳.基于高射频舰炮武器系统的射击效力分析[J]. 指挥控制与仿真,2015(02)
[7] Hu Xiaoli; Wang Bing. Analysis of Shooting Effectiveness of High-Radio Frequency Ship Gun Weapon Systems [J]. Command Control and Simulation, 2015(02)

［8］王炳华.信息化条件下防空兵作战编组模式［J］.防空
[8] Wang Binghua. Combat Formation Patterns of Air Defense Troops under Informationization Conditions [J]. Air Defense Academy Journal

兵学院学报,2015，32（3）：33-35.
Journal of the National Defense Academy, 2015, 32(3): 33-35.

［9］ 李相民;刘立佳;颜骥;刘蕊.要地防空阵地部署前空袭兵器进袭方向预先判断[J]. 系统工程理论与实践,2014(05)
[9] Li Xiangmin; Liu Lijia; Yan Ji; Liu Rui. Preliminary Judgment of the Direction of Air Strikes Before Deployment of Defense Positions in Key Areas [J]. Systems Engineering - Theory & Practice, 2014(05)

［10］ 赵文飞;陈健;王䶮;滕克难.基于强化学习的海上要地群协同防空动态火力分配[J].兵工学报,2023(11)
[10] Zhao Wenfei, Chen Jian, Wang Yanyan, Teng Kena. Dynamic 火力分配 of Coastal Important Sites based on Reinforcement Learning[J]. Journal of Ordnance Industry, 2023(11)

［11］王䶮;赵文飞;陈健;高松;周璐.海上要地防空反导阵地优度排序研究[J].系统工程与电子技术,2022(06)
[11] Wang Yanyan, Zhao Wenfei, Chen Jian, Gao Song, Zhou Lu. Study on the Optimal Sorting of Anti-Aircraft and Anti-Missile Positions of Coastal Important Sites[J]. Systems Engineering and Electronics Technology, 2022(06)

［12］ 李相民;刘立佳;朱绍强;颜骥.要地防空阵地网低空补盲部署模型及优化[J].火力与指挥控制,2014(08)
[12] Li Xiangmin, Liu Lijia, Zhu Shaokang, Yan Ji. Model and Optimization of Low Altitude Blind Spot Deployment of Air Defense Positions of Important Sites[J]. Firepower and Command Control, 2014(08)

［13］ 傅从义;何俊;王斌.要地防空作战雷达干扰任务分配模型研究.现代防御技术,2013(02)
[13] Fu Congyi, He Jun, Wang Bin. Research on the Task Allocation Model of Radar Interference in Air Defense Operations of Important Sites. Modern Defense Technology, 2013(02)