Optimization of warship formation air defense combat model based on dynamic scheduling of combat resources

Abstract

Aiming at the problem of limited resources and operational time combat air defense combat, naval combat formation analysis dynamic resources scheduling needs, on the basis of the establishment of ship dynamic scheduling model of warship formation combat resources dynamic combat formation resource scheduling, the operational resources and operational objectives, relationship between reasonable allocation of resources to play the warship formation combat operations the maximum efficiency. The design of three kinds of air defense command models: centralized platform free attack mode, command mode, request attack mode of air defense strategy, put forward the dynamic scheduling hybrid algorithm for solving the model, according to the naval fleet air defense combat scenario example for simulation. The simulation results show that the air defense strategy with the request attack mode can have effects on multi-target attacks under the shared combat situation, and the hybrid algorithm based on combat resource dynamic scheduling can meet the real-time operational requirements.

Introduction

The scheduling problem of warship formation air defense combat resources ¹ is the optimization plan for the study of the target of intercepting incoming air defense weapon resources. This problem can be described as the combat task of warship formation in air defense operations is divided into four stages: early warning, fire fighting, electronic jamming and maneuver avoidance. There is a certain time constraint and resource constraint between various operational tasks, and each combat mission needs to achieve corresponding combat capability. Fleet air defense combat resources to provide the combat mission complete certain operational capability, but air defense resources are limited, and because the electromagnetic compatibility ² and firepower compatibility ³ constraints of civil air defense can only be allocate processing resources at the same time a combat mission, ⁴ how to quickly and effectively allocate air defense combat resources, the maximum operational effectiveness of the warship formation air defense combat resources is the serious and difficult problem of the current warship formation air defense operational resources. ⁵

From the current research progress, the research approach of warship formation air defense combat resource scheduling problem is gradually shifting from the prewar resource scheduling to the dynamic scheduling of the war. ⁶ Literature ⁷ considering incoming target ability, attack intention and time of arrival as an index, build the threat assessment model, using dynamic firepower access to exit mechanism research resources to plan as a whole, considering the real-time fleet air defense combat resource allocation. Literature^8,9 considering the characteristics of various air defense resources, according to the incoming target chance constrained fire distribution model building area air defense, under the condition of the studied area air defense fleet air defense combat resources to plan as a whole and planning; literature ¹⁰ application multi-intelligence association technology, this article proposes a MAS distribution planning algorithm of center control, breakthrough the traditional top-down centralized decision-making in fleet air defense decision-making way, by the fleet command organization according to the real-time distribution fleet air defense combat; literature ¹¹ designed three air defense combat modes, simplified the regional air defense operation process using process optimization technology, and improved the operational effectiveness of combat resources.

It is of great significance to study the scheduling of air defense combat resources of warship formation, and the ability of the commander to synthesize the air defense combat resources of warship formation ¹² is needed. From the current research progress, the research approach of warship formation air defense combat resource scheduling has been shifted from static scheduling to dynamic scheduling; the ship formation air defense combat command mode is changed from the platform free attack mode to the centralized command mode to the request attack mode to the air defense. Therefore, it is necessary to focus on two points: first, the establishment and solution of the model should meet the high real-time performance so as to make operational decisions in the shortest time. The second is to maximize the planning of all operational resources to maximize operational effectiveness.

Ship formation air defense combat resource scheduling problem

Air defense combat mission

The air defense combat mission is the combat action taken by the warship formation to reach the target. The air defense combat mission set is recorded as $T=\{T_1,T_2,\dots ,T_n\}$ , $n$ is the number of task. For $\forall T_i\in T$ , its attributes include: (1) nature of combat mission $(T_i)$ ; (2) duration of battle $t_i$ ; (3) complete combat task capability vector $(d{t}_{i1},d{t}_{i2},\dots ,d{t}_{\mathrm{il}})$ . If the $d{t}_{\mathrm{il}}>0$ , which means it was completing the combat mission $T_i$ , you need to match the combat task force vector of the $l$ item; if $d{t}_{\mathrm{il}}=0$ , the operational task force vector of the $l$ item contributes is 0 when completing the combat mission $T_i$ . There is a temporal constraint relationship between the various tasks in Air defense combat mission set $T$ . The relationship between combat missions is divided into two categories: the first is the parallel task relationship, which can coordinate the weapons and equipment at the same time. The second is the serial task relationship, which has been completed before the next task is completed. ¹³

Air defense combat resources

The air defense combat resource is the disposal unit of warship formation air defense combat weapon equipment, the ship formation air defense combat resource set is recorded as $P=\{P_1,P_2,\dots ,P_V\}$ , and $V$ is the number of combat resources in the set. For $\forall P_j\in P$ , its attributes include (1) the properties of combat resources $(P_i)$ ; (2) duration of combat resources $(P_i)$ ; (3) combat resource capability vectors $(o{p}_{j1},o{p}_{j2},\dots ,o{p_j}_l)$ , if $o{p_j}_l>0$ . It represents the lth operational capacity of combat resource $P_j$ , if $o{p_j}_l=0$ , the lth operational capability of the combat resource p is 0. When an air defense combat resource $P_j$ is assigned to multiple tasks, $P_j$ will deal with the tasks in order in accordance with the priority order of the combat task, the priority sequence of the combat tasks is ranked according to the target threat ranking. ¹⁴

Air defense combat resource scheduling scheme

The ship formation air defense combat resource scheduling scheme is used to solve the matching relation between operational resources and operational tasks, so as to achieve optimal operational effectiveness. The ship formation air defense combat resource scheduling scheme can be expressed as $M^{T-P}=(m_{\mathrm{ij}}^{T-P}{)}_{N\times V}$ . When combat resources are assigned to task $T_i$ , $m_{\mathrm{ij}}^{T-P}=1$ , or $m_{\mathrm{ij}}^{T-P}=0$ . The quality of the resource scheduling scheme $M^{T-P}$ depends on the quality of the anti-aircraft combat mission and the consumption of combat resources.

As for $\forall T_i\in T$ , the time when the target enters ship-to-air missile weapon system is $s{t}_i$ , the end of the time is $e{t}_i=s{t}_i+t{t}_i$ . When the ship’s fleet of air defense missions starts from 0 moment, the combat mission cycle $\mathrm{TM}$ is

$$\mathrm{TM}=max(e{t}_1,e{t}_2,\dots ,e{t}_N)$$

(1)

As for $\forall T_i\in T$ , the set of combat resources allocated to the target of this batch is $S_{T_i}^P=\{P_j|m_{ij}^{T-P}=1,1\le j\le V.\}$ ; the completed quality of combat mission is defined as

$$Q{T}_i={\left({\displaystyle \prod _{l\in \gamma \left(i\right)}{z}_{il}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\Vert \gamma \left(i\right)\Vert $}\right.}$$

(2)

Among them, $\gamma (i)$ is the collection of the ability to deal with air defense operations, $\gamma (i)=\{k|d{t}_{\mathrm{ik}}>0,1\le k\le L.\}$ , $\Vert \gamma \left(i\right)\Vert$ is the number of combat capabilities of $\gamma (i)$ , and $z_{\mathrm{il}}=min(\sum P_j\in S_{T_i}^P(o{p}_{\mathrm{jl}}/d{t}_{\mathrm{il}}),1)$ is the degree to which the lth operational capability is met.

Ship formation air defense combat resource scheduling model

Demand for dynamic scheduling of air defense combat resources

Fleet of air defense combat when facing complex operational environment, operational events that may occur in the process of operation will make initial warship air defense combat resource scheduling scheme cannot continue, so you need to dynamically adjust air defense combat resources of interception system. These events can be summarized in the following three situations: ¹¹

1. New target appears.

   A sudden threat of attack in the course of a battle, ship formation needs to redistribution air defense combat resources and intercept the target as soon as possible.

2. Invalid air defense combat resources.

   The failure problems caused by the equipment guarantee and operation of combat resources during air defense operations.

3. Changes in air defense combat resources.

   Because of the continuous invasion of the target, the use of air defense combat resources of the warship formation has affected the completion of the air defense combat mission.

Due to the change of the operational plan caused by the dispatch of naval air defense combat resources, the stability of air defense operational command will be affected. Therefore, the operational decision should be able to deal with three kinds of operational incidents.

In order to solve the above three kinds of combat events, combined with the development of surface warship air defense operations, the three operational command mode resource scheduling models are designed:

1. Resource scheduling of platform free attack mode

   Platform free attack mode is each vessel in the fleet carried out in accordance with the own defense sector autonomous defense, defense effect is determined by their ships air defense capabilities, its operational mode is shown in Figure 1.

2. Centralized command mode resource scheduling.

   The centralized command mode is adopted within connected interoperability, information technology refers to the comprehensive situation of according to hit goal, all fleet air defense combat resources unified scheduling, makes the operational mode from the traditional decentralized command to centralized command, its operational mode as shown in Figure 2.

3. Request resource scheduling for attack mode

   Request resource scheduling for attack mode is through information technology, on the basis of sharing situation maps, each ship trend according to the comprehensive and ships in the fleet air defense combat resource request in air defense combat, refers to the operational mode as shown in Figure 3.

Figure 1.

Platform free attack mode for air defense.

Figure 2.

Air defense under centralized command mode.

Figure 3.

Air defense under attack mode.

Dynamic scheduling model of air defense combat resources

The ship formation sequence is $J=\{J_1,J_2,\dots ,J_G\}$ . At some point, the formation air defense system detected the target sequence within our range of operations $S=\{s_1,s_2,\dots ,s_N\}$ , the ist batch in the sequence attempts to attack the formation ship r. The probability of damage to ship r when we are not fighting is $Q_{\mathrm{ir}}$ , the probability of our fire unit $j$ has a single missile damage to the ist batch target is $p_{\mathrm{ij}}$ , the probability of not hitting the ist batch is $1-Q_{\mathrm{ir}}(1-p_{\mathrm{ij}}{)}^{\mathrm{Nf}}$ , the $\mathrm{Nf}$ is the number of missiles fired from the fire unit. In the case of constraint $C_{\mathrm{ij}}$ , the ship’s survival probability is

$$Q_r\left(D_1\right)=\sum _{i=1}^N(1-Q_{\mathrm{ir}}\sum _{j=1}^M{C}_{\mathrm{ij}}{x}_{\mathrm{ij}}\underset{l=1}{\overset{\mathrm{Nf}}{\Pi }}(1-p_{\mathrm{ij}}))$$

(3)

In the formula, the calculation of damage probability $Q_{\mathrm{ir}}$ satisfies the following formula

$$Q_{\mathrm{ir}}=T{I}_{\mathrm{ir}}{D}_{\mathrm{ir}}$$

(4)

In the formula, $T{I}_{\mathrm{ir}}$ and $D_{\mathrm{ir}}$ respectively represent target i attempt to measure and damage capability measurement of ship r. Fleet to attack and defense both sides of the air defense combat operations are situational change over time, only when the time variable is introduced, the target each hour and weapons are determined, then the quantity of state into the distribution model, updating status updates and distribution solutions, can accurately modeling and simulation process of air defense combat.

Formation from the first batch of incoming target the kill zone far began, to the end of the fleet air defense area without incoming target, according to the operational resources used in air defense combat, air defense combat can be divided into several phases, suppose air defense operations can be divided into $T$ stages, $t=0,1,2,\dots ,T$ . In the $t$ stage, the threat judgment model ¹⁴ can be used to determine the ordering of threats $S^t=\{s_1,s_2,\dots ,s_{N(t)}\}$ , $N(t)$ is the number of batches to hit the target; $s_i^t$ is the state before the $t$ phase of the $i$ batch; ${\delta }_j^t$ is the combat resource $j$ in phase $t$ is occupied. $X(t)=(x_{\mathrm{ij}}^t{)}_{N(t)\times M}$ is the current stage of decision matrix, where

$$x_{\mathrm{ij}}^t=\{\begin{array}{c}\begin{array}{cc}1\hfill & \mathrm{fire\; unit}j\mathrm{assigned\; to\; target}i\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & \mathrm{fire\; unit}j\mathrm{is\; not\; assigned\; to\; target}i\hfill \end{array}}\end{array}$$

(5)

$$s_i^t=\{\begin{array}{cc}1\hfill & \mathrm{target}i\mathrm{is\; not\; destroyed\; in\; the\; phase}t\hfill \\ \multicolumn{1}{c}{0}& \mathrm{target}i\mathrm{has\; been\; destroyed\; in\; the\; phase}t\hfill \end{array}$$

(6)

$${\delta }_{\mathrm{j}}^t=\{\begin{array}{cc}1\hfill & \mathrm{the\; fire\; unit}j\mathrm{is\; free\; when\; the\; beginning\; of\; phase}t\hfill \\ \multicolumn{1}{c}{0}& \mathrm{the\; fire\; unit}j\mathrm{is\; occupied\; when\; the\; beginning\; of\; phase}t\hfill \end{array}$$

(7)

If a new target is added at the $t$ stage, or operational resources $k$ have been effectively intercepted, it goes into $t+1$ phase.

The update of $t+1$ in the new phase will enable the formation combat resources to be rescheduled to the optimal intercept effect.

When the combat resource $h$ enters $t+1$ stage

$${\delta }_j^{t+1}=\{\begin{array}{c}\begin{array}{cc}{\delta }_j^t\hfill & j\ne h\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}1\hfill & j=h_1\hfill \end{array}}\end{array},j=1,2,\dots ,M$$

(8)

After the intercept process is completed in phase $t$ , the results of the observation are updated

$$s_i^{t+1}=\{\begin{array}{c}\begin{array}{cc}1\hfill & \mathrm{no\; visual\; on\; target}i\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & \mathrm{target}i\mathrm{has\; been\; destroyed}\hfill \end{array}}\end{array}$$

(9)

Remember $X(t)=(x_{\mathrm{ij}}^t{)}_{N(t)\times M}$ is $t$ stage combat resource allocation matrix, then

$$x{\prime }_{\mathrm{ij}}^t=\{\begin{array}{cc}1\hfill & \mathrm{phase}t+1\mathrm{assigns\; target}i\mathrm{to\; fire\; unit\; j}\hfill \\ \multicolumn{1}{c}{0}& \mathrm{phase}t+1\mathrm{fire\; unit}j\mathrm{distribution\; is\; unchanged}\hfill \end{array}$$

(10)

$$x_{\mathrm{ij}}^{t+1}=\{\begin{array}{cc}{x}_{\mathrm{ij}}^t\hfill & \mathrm{fire\; unit}j\mathrm{distribution\; is\; unchanged}\hfill \\ \multicolumn{1}{c}{x{\prime }_{\mathrm{ij}}^t}& \mathrm{fire\; unit}j\mathrm{distribution\; is\; changed}\hfill \end{array}$$

(11)

According to the redistributive process of combat resources, it can be

$${\delta }_j^{t+1}=\{\begin{array}{c}\begin{array}{cc}{\delta }_j^{t+1}\hfill & {x\prime }_{\mathrm{ij}}^t=0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & {x\prime }_{\mathrm{ij}}^t=1\hfill \end{array}}\end{array}$$

(12)

The new type of combat resource allocation constraint is

$$C_{\mathrm{FS}}^{t+1}=\sum _{i=1}^{N(t+1)}\sum _{j=1}^M{S}_i^{t+1}{\delta }_j^{t+1}{R}_j^{t+1}$$

(13)

Remark $p_{\mathrm{ij}}(t)$ is the probability of damage to target $i$ by firepower unit $j$ , the state vector for the target $s_i^t$ . The probability distribution function can be calculated according to the following formula

$$P_i(t+1)=\underset{k=1}{\overset{t}{\Pi }}\sum _{j=1}^M{C}_{\mathrm{FS}}^{t+1}{x}_{\mathrm{ij}}^k\underset{l=1}{\overset{\mathrm{Nf}}{\Pi }}(1-p_{\mathrm{ij}}^l\left(k\right))$$

(14)

In the formula $l=1,2,\dots ,\mathrm{Nf}$ is the number of missiles fired from the fire unit, $p_{\mathrm{ij}}^l(k)$ is the probability of damage to target $i$ by the $l$ weapon of firepower unit $j$ . Based on the survival probability model of ship formation in static case, considering the working state of combat resources, the probability distribution of the target in different stages and the suitability factor for the target of combat resources, the dynamic allocation model ¹⁵ can be established at the stage of $t+1$

$$\begin{array}{l}F(t+1)=\underset{X\in \Theta }{\max }\left\{{\displaystyle \sum _{r=1}^G{\eta }_r{\displaystyle \prod _{i=1}^{N(t+1)}\left(1-Q_{ir}{P}_i(t+1)\right)}}\right\}\\ =\max \left\{{\displaystyle \sum _{r=1}^G{\eta }_r}{\displaystyle \prod _{i=1}^{N(t+1)}\left(1-Q_{ir}{\displaystyle \prod _{k=1}^{t+1}{\displaystyle \sum _{j=1}^M{C}_{FS}^k{x}_{ij}^k{\displaystyle \prod _{l=1}^{Nf}}(1-p_{ij}^l(k))}}\right)}\right\}\end{array}$$

$$s.t.\{\begin{array}{c}{x}_{\mathrm{ij}}^t\in \{0,1\},\sum _{i=1}^{N(t)}{x}_{\mathrm{ij}}^t\in \{0,1\}\hfill \\ \multicolumn{1}{c}{\forall i\in \{1,2,\dots ,N(t)\},\sum _{j=1}^M{x}_{\mathrm{ij}}^t\le \mathrm{E}}\\ \multicolumn{1}{c}{\sum _{i=1}^{N(t)}\sum _{j=1}^M{x}_{\mathrm{ij}}^t\le M}\end{array}$$

(15)

In this formula, $i=1,2,\dots ,N,j=1,2,\dots ,M$ , ${\eta }_r$ is the value coefficient of ship r in fleet formation, and E is the largest number of combat resources allocated for fleet formation.

Algorithm flow based on different command modes

If the current combat resources of the ship formation is $P(t\prime )$ , the current combat task set is $T(t\prime )$ , the set of new interception tasks at this time is $T^{\mathrm{new}}$ , because the combat resources have not been able to achieve the corresponding combat capabilities, the mass of the overall combat mission is affected by a set of $T^{\mathrm{impact}}$ , $T^{\mathrm{new}}\cup T^{\mathrm{impact}}=\{T_{w1},T_{w2},\dots ,T_{\mathrm{wB}}\}$ , B is the number of elements in $T^{\mathrm{new}}\cup T^{\mathrm{impact}}$ . As for $\forall T_{\mathrm{wi}}\in T^{\mathrm{new}}\cup T^{\mathrm{impact}}$ , the composition of combat resources for $T_{\mathrm{wi}}$ is $S{T}_{wi}^P$ :

1. Algorithm flow of platform free attack mode

2. (b) $\forall T_{\mathrm{wi}}\in T(t\prime )-T^{\mathrm{new}}\cup T^{\mathrm{impact}}$ , $M_{\mathrm{current}}=M_1$ .

Step 2: From ${\mathrm{ST}}_{w1}^P$ , ${\mathrm{ST}}_{w2}^P$ ,…, ${\mathrm{ST}}_{\mathrm{wB}}^P$ choose a set of ${\mathrm{ST}}_{\mathrm{wi}}^P$ . And then we can remove a redundant combat resource from ${\mathrm{ST}}_{\mathrm{wi}}^P$ , we can get a new solution for a model, according to selected battle resource collection and delete combat redundant resources, the new solution corresponds to the optimal model optimization target, call this new solution $M_2$ .

Step 3: $M_{\mathrm{current}}=M_2$ , Update the collection ${\mathrm{ST}}_{w1}^P$ , ${\mathrm{ST}}_{w2}^P$ ,…, ${\mathrm{ST}}_{\mathrm{wB}}^P$ according $M_{\mathrm{current}}$ . Determining whether $M_{\mathrm{current}}$ is the most important solution to combat resources, if true, make the $M_{\mathrm{current}}$ is the optimal output, otherwise return to execute Step 2.

The algorithm flow of centralized command mode

Step 1: Build the initial solution of the model $M_1$ , $M_1$ content:

(a) As for $\forall T_{\mathrm{wi}}\in T^{\mathrm{new}}\cup T^{\mathrm{impact}}$ , ${\mathrm{ST}}_{\mathrm{wi}}^P=P(t\prime )$ ;

(b) For $\forall T_{\mathrm{wi}}\in T(t\prime )-T^{\mathrm{new}}\cup T^{\mathrm{impact}}$ , $M_{\mathrm{current}}=M_1$ .

Step 2: element in ${\mathrm{ST}}_{\mathrm{wi}}^P$ , $M_{\mathrm{current}}^{w_i}=M_{\mathrm{current}}$ , according to $M_{\mathrm{current}}^{w_i}$ . Update ${\mathrm{ST}}_{w1}^P$ , ${\mathrm{ST}}_{w2}^P$ ,…, ${\mathrm{ST}}_{\mathrm{wB}}^P$ .

A new solution can be obtained by removing a redundant combat resource from ${\mathrm{ST}}_{\mathrm{wi}}^P$ , according to selected battle resource collection and delete combat redundant resources, the new solution corresponds to the optimal model optimization target, call this new solution $M_2$ ; $M_{\mathrm{current}}^{w_i}=M_2^{w_i}$ , according to $M_{\mathrm{current}}^{w_i}$ update ${\mathrm{ST}}_{\mathrm{wi}}^P$ , determine whether ${\mathrm{ST}}_{\mathrm{wi}}^P$ is the optimal processing mode for $T_i$ , if true, keep the matrix $M_{\mathrm{current}}^{w_i}$ and end, otherwise return to execute Step 2.

Step 3: in the $\{M_{\mathrm{current}}^{w_i}|{\mathrm{ST}}_{\mathrm{wi}}^P\in \mathrm{ST}\}$ , if $M_{\mathrm{current}}^{w_i}$ is the optimal solution, then $M_{\mathrm{current}}=M_{\mathrm{current}}^{w_i}$ , update $\mathrm{ST}=\mathrm{ST}-\{{\mathrm{ST}}_{w_j}^P\}$ , determine whether $\mathrm{ST}$ is empty, if true, the $M_{\mathrm{current}}$ is the optimal output, otherwise return to execute Step 2.

Request an algorithm flow under attack mode

Step 1: Build the initial solution of the model $M_1$ , $M_1$ content:

(a) As for $\forall T_{\mathrm{wi}}\in T^{\mathrm{new}}\cup T^{\mathrm{impact}}$ , ${\mathrm{ST}}_{\mathrm{wi}}^P=P(t\prime )$ ;

(b) $\forall T_{\mathrm{wi}}\in T(t\prime )-T^{\mathrm{new}}\cup T^{\mathrm{impact}}$ , $M_{\mathrm{current}}=M_1$ .

Rank the tasks of $T^{\mathrm{new}}\cup T^{\mathrm{impact}}=\{T_{w1},T_{w2},\dots ,T_{\mathrm{wB}}\}$ in descending order of magnitude, the order of the tasks after permutation is $T_{z1},T_{z2},\dots ,T_{\mathrm{zB}}$ , $M_{\mathrm{current}}=M_1$ to the current solution $M_{\mathrm{current}}=M_1$ , counter $j=1$ .

Step 2: If $T_{\mathrm{zj}}=T_{\mathrm{wi}}(1\le i\le B)$ , when a combat resource is removed from ${\mathrm{ST}}_{\mathrm{wi}}^P$ , a new solution can be obtained. Judging the combat resource platform removed from ${\mathrm{ST}}_{\mathrm{wi}}^P$ . The new solution corresponding model optimization target $\mathrm{TM}$ is the largest, remark $M_2$ .

Step 3: Make $M_{\mathrm{current}}=M_2$ , according to $M_{\mathrm{current}}^{w_i}$ update ${\mathrm{ST}}_{w1}^P$ , determine whether ${\mathrm{ST}}_{\mathrm{wi}}^P$ is the maximum processing mode for $T_i$ , if true, carry out Step 4, otherwise return to execute Step 2.

Step 4: Update $j=j+1$ , to determine whether $j>B$ is true or not, if true, the $M_{\mathrm{current}}$ is the optimal output, otherwise return to execute Step 2.

Simulation experiment

Simulation parameter setting

In the ship’s formation air defense operations, our fleet is composed of three anti-aircraft destroyers and two anti-aircraft frigates, the air defense formation is ring-shaped, with destroyer 3 as the formation command ship. In early warning airplane, early warning and helicopters as well as a variety of intelligence information under the guarantee of our fleet can realize each other information and formation of skirmishes and weapon system performance parameters such as shown in Table 1.

Table 1.

Configuration and equipment of warship formation.

Ship number	Combat resource type	Number of fire channels	Kill probability	Channel occupancy time (s)
Destroyer 1	Ship-to-air missile	4	0.90	25
Frigates 2	Ship-to-air missile	4	0.85	20
Destroyer 3	Ship-to-air missile	4	0.90	25
Frigates 4	Ship-to-air missile	4	0.85	20
Destroyer 5	Ship-to-air missile	4	0.90	25

According to the threat judgment algorithm ¹⁴ determines the threat measurement, incoming time and attack direction of the target relative to the command ship as shown in Table 2.

Table 2.

Target threat and attack.

Target batch number	Threat to measure	Direction	Time window
1	0.76	1	[0 s, 50 s]
2	0.84	1	[5 s, 35 s]
3	0.88	2	[10 s, 40 s]
4	0.78	2	[15 s, 50 s]
5	0.94	1	[16 s, 40 s]
6	0.79	3	[17 s, 65 s]
7	0.93	3	[18 s, 45 s]
8	0.89	2	[19 s, 45 s]
9	0.85	1	[20 s, 60 s]
10	0.77	2	[21 s, 80 s]
11	0.80	1	[22 s, 70 s]
12	0.81	3	[24 s, 65 s]
13	0.82	3	[26 s, 70 s]
14	0.83	1	[29 s, 70 s]
15	0.88	3	[30 s, 60 s]
16	0.88	2	[75 s, 115 s]
17	0.82	2	[88 s, 125 s]
18	0.89	1	[91 , 130 s]

According to the anti-air combat resources of my ship formation and the target tactical performance, the time of the intercept phase is 130 s. After the distribution of fire, the global decision matrix $M_{\mathrm{current}}=|M_{1,i},M_{2,i},M_{3,i},M_{4,i},M_{5,i}|$ is determined in this stage. According to the global decision matrix $M_{\mathrm{current}}$ and the incoming target threat measurement $w$ is expected to obtain the mathematical expectation of warship formation air defense operations

$$E=\sum _{i=1}^n\sum _{j=1}^m{M}_{\mathrm{ij}}\times w_j$$

(16)

Analysis of simulation results

Simulation is based on the dynamic distribution model of three different command modes; the dynamic allocation of combat resources was carried out on 18 batches of targets:

1. Allocation of air defense resources for free attack mode of platform.

According to the algorithm flow of the free attack mode of the platform, the $M_{\mathrm{current}}$ should be shown in Table 3.

• 2. Centralized command mode of air defense resource allocation.

Table 3.

Ship formation air defense resource allocation scheme.

Target batch number	Allocation of combat resources	Target batch number	Allocation of combat resources	Target batch number	Allocation of combat resources
1	$M_{1,1}=0.87$	2	$M_{2,2}=0.83$	3	$M_{3,3}=0.87$
4	$M_{4,4}=0.82$	5	$M_{5,5}=0.81$	6	$M_{1,6}=0.82$
7	$M_{2,7}=0.79$	8	$M_{3,8}=0.84$	9	$M_{4,9}=0.76$
10	$M_{5,10}=0.85$	11	$M_{1,11}=0.84$	12	$M_{2.12}=0.76$
13	$M_{3.13}=0.86$	14	$M_{4.14}=0.76$	15	$M_{5.15}=0.82$
16	$M_{1.16}=0.81$	17	$M_{2.17}=0.71$	18	$M_{3.18}=0.82$

According to the algorithm flow of centralized command mode, the $M_{\mathrm{current}}$ is shown in Table 4.

• 3. Allocation of air defense resources for request resource scheduling for attack mode.

Table 4.

Ship formation air defense resource allocation scheme.

Target batch number	Allocation of combat resources	Target batch number	Allocation of combat resources	Target batch number	Allocation of combat resources
1	$M_{1,1}=0.87$	2	$M_{3,2}=0.85$	3	$M_{4,3}=0.85$
4	$M_{2,4}=0.81$	5	$M_{5,5}=0.85$	6	$M_{2,6}=0.84$
7	$M_{1,7}=0.80$	8	$M_{5,8}=0.86$	9	$M_{4,9}=0.81$
10	$M_{3,10}=0.87$	11	$M_{4,11}=0.85$	12	$M_{5.12}=0.78$
13	$M_{2.13}=0.87$	14	$M_{1.14}=0.78$	15	$M_{3.15}=0.84$
16	$M_{4.16}=0.81$	17	$M_{3.17}=0.73$	18	$M_{5.18}=0.83$

According to the algorithm flow of centralized command mode, the $M_{\mathrm{current}}$ is shown in Table 5.

Table 5.

Ship formation air defense resource allocation scheme.

Target batch number	Allocation of combat resources	Target batch number	Allocation of combat resources	Target batch number	Allocation of combat resources
1	$M_{3,1}=0.89$	2	$M_{4,2}=0.85$	3	$M_{5,3}=0.89$
4	$M_{2,4}=0.83$	5	$M_{1,5}=0.86$	6	$M_{4,6}=0.85$
7	$M_{2,7}=0.82$	8	$M_{5,8}=0.87$	9	$M_{1,9}=0.82$
10	$M_{3,10}=0.88$	11	$M_{1,11}=0.87$	12	$M_{4.12}=0.79$
13	$M_{5.13}=0.87$	14	$M_{2.14}=0.81$	15	$M_{3.15}=0.86$
16	$M_{2.16}=0.83$	17	$M_{4.17}=0.75$	18	$M_{3.18}=0.85$

Three battle command modes corresponding to three different types of air defense fleet air defense combat strategy, in actual combat training, should make refers to according to combat situation in different operational resources itself. According to formula (16), the mathematical expectation of the warship formation air defense operation is shown in Table 6.

Table 6.

Operational mathematical expectations for air defense mode.

Air defense mode	Simulation times	Average battle time	Mathematical expectations of air defense operations
Platform free attack mode	100	144	12.3256
	500	144
	1000	144
Centralized command mode	100	137	12.5473
	500	137
	1000	137
Request attack mode	100	135	12.7927
	500	135
	1000	135

For three kinds of statistical analysis of the air defense mode request shortest average operation time attack mode, mathematical expectation maximum air defense combat, can play a maximum fleet air defense combat resources.

Conclusion

In this article, a dynamic scheduling algorithm for air defense combat resources is proposed for three kinds of air defense combat command modes. The algorithm can allocate the combat resources reasonably in multiple batches of target attack situations and maximize the effectiveness of the overall air defense operation of the fleet. The improvement of warship air defense combat effectiveness depends on the scientific nature of combat command mode. The research shows that the request attack mode in the information condition can greatly shorten the battle time of the warship formation on the basis of the attack target. This model is beneficial to the saturation attack of attacking targets. However, the proposed model algorithm still needs to be improved. How to ensure that all combatants are under active defense posture, the design of the corresponding incentive algorithm is the next problem to be solved.