JIA Zhengrong,LU Faxing,and WANG Hangyu,*

1.National Key Laboratory of Science and Technology on Vessel Integrated Power System,Naval University of Engineering,Wuhan 430033,China;2.College of Weaponry Engineering,Naval University of Engineering,Wuhan 430033,China

Abstract: For better reflecting the interactive defense between targets in practical combat scenarios, the basic weapon-target allocation (WTA) framework needs to be improved. A multi-stage attack WTA method is proposed. First, a defense area analysis is presented according to the targets’ positions and the radii of the defense areas to analyze the interactive coverage and protection between targets’defense areas.Second,with the coverage status and coverage layer number, a multi-stage attack planning method is proposed and the multi-stage attack objective function model is established. Simulation is conducted with interactive defense combat scenarios,the traditional WTA method and the multi-stage WTA method are compared, and the objective function model is validated with the Monte-Carlo method. The results suggest that if the combat scenario involves interactive coverage of targets’defense areas, it is imperative to analyze the defense areas and apply the multi-stage attack method to weakening the target defense progressively for better combat effectiveness.

Keywords: weapon-target allocation (WTA), defense area analysis,combat effective analysis.

1.Introduction

Weapon target allocation (WTA) is a key problem for the improvement of combat effectiveness.The main point of WTA is to optimize comprehensive combat effectiveness considering factors including values of different targets, kill probabilities and intercept probabilities between weapons and targets[1–3].

Most of the existing research on WTA concentrates on solving the WTA problem. For example, Rezende et al.[4]and Hu et al. [5]emphasized solving the efficiency in reaching the optimality of the WTA problem, and introduced an ant colony algorithm in optimizing.Liu et al.[6]pointed out that traditional exact methods and heuristic algorithms were all capable of solving the WTA problem, but exact methods performed worse when the problem scale increased in WTA, and introduced the artificial bee colony (ABC) algorithm for solution. In [6], different methods were compared including particle swarm optimization(PSO), genetic algorithm(GA) and ABC. Reference[7]is also based on the ABC method.In[8],a tabu search heuristic method was proposed for WTA, and the attack procedure was described with a tree search model.The expansion of the tree is based on the kill probability and the recognization probability. Nonetheless, some research expands the method reference to the geometric method[9],game theory[10],reinforcement learning[11],the PSO method[12,13],the fuzzy logic method[14]and the rule based method[15]to solve the WTA problem.To conclude, the heuristic and hybrid method [16–22] and the evolutionary method are the most popular methods for solving the WTA problem [23–26]. Other research focuses on other perspectives of the WTA problem,such as distributed algorithms [27,28] and WTA process simulation [29]. Besides, research in [30,31] can also be referenced as comprehensive surveys on the WTA problem.

The existing research provides plenty of solving methods for the WTA problem as references. In most studies,however,the kill probabilities between weapons and targets, the intercept probabilities between targets and weapons are both involved. The kill procedure and intercept procedure are independent of each other and there lacks consideration of the influence on the holistic scenario change by different attack sequences and target distributions.In practical combat scenarios,due to the different importances of different targets, the targets will form closed or reciprocal protection by taking advantage of their own defense capabilities.Thus,a weapon that is allocated to a particular target will possibly be intercepted by multiple targets. Under this condition, the traditional WTA method will not completely describe the interaction of defense areas between different targets.Accordingly,a multistage attack strategy needs to be introduced to weaken the protection between different targets stage by stage,and after each stage of attack, the weakened areas need to be found out as the attack direction for the next stage of attack.

To conclude, for better describing the practical combat scenarios, the WTA problem needs to be improved fundamentally to reflect the interactive protection between different targets. This work proposes a multi-stage attack WTA method (MM method).The remainder of this work is organized as follows.Section 2 presents the basic problem definition and necessity of a multi-stage attack. Section 3 presents the defense area analysis method,which is to analyze the interactive coverage and protection between the targets according to the targets’ positions and radii of the defense areas.Section 4 presents the multi-stage attack planning process,which is to calculate the objective function value and solve the multi-stage attack plan according to the defense area analysis results.

2.Problem definition

The WTA problem can be described as follows. Allocatenwweapons tonttargets; the WTA plan is annw × ntmatrixA= [aω,τ],whereaω,τ ∈{0,1}(aω,τ= 1 means a weaponωis allocated to a targetτ), each weapon can only be allocated to one target,namelyand different weapons can be allocated to one target;each target is qualified with a value, and the value vector of all targets is a 1× ntvectorV= [vτ]; different weapons have different kill probabilities for different targets,which is annw×ntmatrixPk=[pk,ω,τ],wherepk,ω,τ ∈[0,1];different targets have different intercept probabilities for different weapons as annw × ntmatrixPc= [pc,ω,τ],wherepc,ω,τ ∈[0,1].The traditional WTA formulates the objective functionHtas

where 1?PA(A)is the conditional cumulative kill probability to the targetτand the weapons allocated to the targetτare not intercepted byτ.Htis the conditional sum of values of all allocated targets.

However, in practical combat scenarios, the defense areas of the targets will overlap with each other as illustrated in Fig.1.Without loss of generality,all units in this paper are normalized.

Fig.1 Typical combat scenario with overlapping defense areas

In Fig.1,each target has a defense area with a radiusρτcentered at its positionXτ.T-7 and T-8 are major targets,and T-1 to T-6 are minor targets. In this scenario, T-1 to T-6 form a closed circle end to end,and every weapon that is allocated to T-7 or T-8 must pass through one or more defense areas of T-1 to T-6.Besides,T-7 is also covered by the defense area of T-8.

Assume a salvo attack with four weapons W-1 to W-4,and the value vectorV,the kill probabilities matrixPkand the intercept probabilities matrixPcare shown in Table 1–Table 3. The WTA plan that is solved with the traditional method is in Table 4.

Because the weapons allocated to T-7 and T-8 will pass through defense areas that belong to other targets, which will increase the probability being intercepted, and the practical objective function value is 47.26.

Table 1 Value vector

Table 2 Kill probabilities matrix

Table 3 Intercept probabilities matrix

Table 4 Traditional WTA plan

Instead, if the salvo attack is conducted stage by stage with the same WTA plan, then, (i) the first stage is W-2 to T-1 and W-3 to T-4; (ii)the second stage is W-1 to T-8 passing through T-1;and(iii)the third stage is W-4 to T-7 passing through T-4 and T-8. With a multi-stage strategy,the objective function value can reach 72.17.The objective function value is calculated with the Monte-Carlo method in Section 5.1.

Obviously,in this combat scenario with overlapping defense areas,the traditional WTA method will not be capable of describing the interaction between targets and their defense areas, and it is harder still to acquire a solution with optimality.Thus,a multi-stage attack strategy and its corresponding solving method are imperative.

To conclude, multi-stage attack WTA has two main problems.

(i)Defense area analysis.Analyze the interrelationships between defense areas, including overlapping and coverage according to each defense area’s position and radius,for further objective function value estimation and multistage attack planning.

(ii)Objective function formulation of multi-stage attack and multi-stage planning. Generate the multi-stage plan with defense area analysis results,and simultaneously calculate the objective function value of a multi-stage attack.

Besides, the multi-stage WTA problem will become more complicated considering path planning. For simplicity,this study focuses on the WTA problem without path planning,and assumes that the initial positions of weapons can be arbitrarily configured outside the defense areas of targets.

3.Defense area analysis

The distribution of targets and their corresponding defense areas forms a reciprocal protecting combat scenario. It is necessary to analyze the defense areas for finding out the weak points for improving attack effectiveness.

3.1 Fundamentals

Definition 1Closed area.

A closed area is formed with one or multiple defense areas.A closed area separates the space into two parts(internal part and external part). Any curve connecting the internal part and the external part will pass through the defense area(s)that form(s)this closed area.

A closed area can protect other targets.By Definition 1,there are two basic forms of closed areas: (i)independent closed area, which is formed with single defense areas,as shown in Fig. 2(a); (ii) circular closed area, which is formed with multiple defense areas connected with each other end to end,as shown in Fig.2(b).

Fig.2 Basic forms of closed area

Besides these two basic forms,any union of basic closed areas is still a close area.

Theorem 1The union of any closed areas is a closed area.

Definition 2Coverage.

A target covered by a closed area is that the target is in the internal part of the closed area,and that any curve connecting the target with any point in the external part must pass through the defense area of this closed area.

It is obvious that by finding out all closed areas,whether a target is covered or not can be further judged.If a target is in the internal part of a closed area, any weapon allocated to this target will pass through the defense area of this closed area.

Considering that a closed area is formed with one or multiple defense areas,we can introduce the 1×ntvectorRs= [rτ]to describe which defense areas are selected in the closed area.rτ ∈{0,1}, and thatrτequals 1 means theτth target is selected,and 0 means not.

WithRs,we can further define the following operation

is the logical AND operation,andRsis the intersection of defense areas ofRs,1andRs,2.

is the logical OR operation,andRsis the union of defense areas ofRs,3andRs,4.

is the sum of 1-elements inRs, or the number of defense areas inRs.meansRs,1andRs,2share common defense areas.

3.2 Circular closed area solving

Obviously,the independent closed area is the corresponding defense area itself, and inRs, there is only one element equal to 1.It is relatively difficult to solve the circular closed area.

Considering that the circular closed area involves multiple overlapping defense areas, we introduce annt ×ntmatrixC= [cij] as a connection matrix to describe it.cij ∈{0,1},and thatcijequals 1 means the defense areaioverlaps with the defense areaj,and 0 means not,that is

For all defense areas,only some parts of them can form a circular closed area,and thus we introduce the transformation operatorS:

whereM(Rs) is formulated withRs,M(Rs) is anns ×ntmatrix,nsis the number of 1-elements inRsorns=and we have

Simply speaking, the operatorStransformsCinto a new square matrix by selecting columns and rows corresponding to 1-elements inRs.

Theorem 2For a circular closed area,the transformed connection matrixS(C|Rs) = [sis,js] has the property that for?js,there is

ProofSince in a circular closed area,the defense areas connect with each other end by end, and for any defense area, there are two other defense areas connected with it,that is

Theorem 3For a givenS(C|Rs) = [sis,js], if for?js,there isand the corresponding defense areas form a circular closed area.

ProofWe prove this theorem with reduction to absurdity.

Assume that for?js, there isand the defense area selected byRscannot form a circular closed area.

Select a defense area inRsmarked asis, withand there are two other defense areas inRsconnected withis,marked asis,1andis,2.

Foris,1, two defense areas connect with it, namelyisandis,3.

Further,we can haveis,nthat connects withis,n?1.Notice thatand there are in total four possibilities.

(i)is,nconnects withis,2,but with assumption,the defense areas selected byRscannot form a circular closed area, sois,ncannot connect withis,2, which contradicts the assumption.

(ii)is,nconnects withis,p, wherep ∈{3,...,n ?2},sois,pconnects withis,p?1,is,p+1andis,n,which contradicts

(iii)is,nonly connects withis,n?1, which contradicts

(iv)A new defense areais,n+1connects withis,n.Notice that the number of defense areas is finite,ns < ∞,which will terminally reach cases(i)–(iii).

To conclude,all possibilities contradict the assumption.

From Theorem 3, forthat satisfeisthe corresponding defense areas can form a circular closed area,but the number of the formed circular closed areas is not regulated.In fact,whenis satisfied,ifis a partitioned matrix as

Rscorresponds to two disjoint circular closed areas.Therefore,Rswhich satisfiesmay contain multiple circular closed areas.

With the analysis above,we can find out all circular defense areas with Algorithm 1 and Algorithm 2.First,Algorithm 1 gives all circular closed areas and their union sets.Second, with Algorithm 2, all union sets are deleted, and only circular closed areas remain.

Algorithm 1Circular closed areas and their union sets.

Input:a connection matrixC.

Step 1Formulate a mapping formRsto a binary integernR,and every bit ofnRcorresponds to an element ofRssequentially,wheren

Step 2TraversenRin its value range,and for eachnR,there isRs(nR).Judge whetherS(C|Rs)satisfies that for?js,there is

Step 3For all satisfyingRs,mark it asand construct a set

Output: All circular closed areas and their union sets

With Algorithm 2,the union sets will be deleted and we can reach a set with only single circular closed areas.

Algorithm 2Circular closed areas.

Step 1Mark

Step 2In ΩB,kB,for theand

it means there exists other closed areas marked ascontaining defense areas inis a union set of multiple circular closed areas.

Step 3Repeat Step 2 until forthere is

Output: ΩB={Rs,B}.

3.3 Coverage judge

After finding out all independent closed areas and circular closed areas, we can judge whether a target is covered by any closed areas.

(i)Coverage judge by independent closed areas.

Assume the center of thektth independent closed area isthe radius of the defense area isand the targetXτis in its internal part if

(ii)Coverage judge by circular closed areas.

Since circular closed areas are formed with connected defense areas, and whether a target is in each single defense area is judged in (i), here we only need to judge whether the target is in the polygon area formed by connecting the centers of intersecting defense areas.There are well researched methods to judge whether a point is in the internal part of a polygon.

3.4 Pivotal node vector,closed node vector and coverage layer number

With coverage judging, for each targetτ, we can present a set that contains all closed areas covering it, marked asΨτ={Rs,τ,q}. That is, when a weapon is allocated toτ,it will definitely pass through some defense areas designated byΨτ.In theseRs,τ,q,some defense areas are avoidable while others are not,we can introduce a pivotal node vector to describe the inevitable defense areas.

Definition 3Pivotal node vectorRse,τ.

ForΨτ={Rs,τ,q}, there existsRse,τthat separatesΨτinto two disjoint sets namelyΨse,τ(Rse,τ) andand

Forandthere is

InforΨse,τ(Rse,τ),whereqaandqbare indices,there is

Thus,we have the pivotal node vector as

whereqeis an index.Obviously,any weapon assigned toτwill definitely pass through at least one defense area in

Correspondingly, there is a union set inthat describes all defense areas connected with

Definition 4Closed node vectorRsa,τ.

The closed node vectorRsa,τis represented by

Here we solveRse,τandRsa,τinΨτ={Rs,τ,q}with Algorithm 3.

Algorithm 3SolveRse,τandRsa,τ.

Input:Ψτ={Rs,τ,q}.

Step 1Note the number of elements inΨτasnq, and assumeand

Step 2From the second element inΨτ,traverseqif

We have

Step 3After the traverse, we have the finalandmarked asRse,τ,landRsa,τ,l. Delete allRs,τ,qinΨτthat satisfies

If

the remaining set contains closed areas that do not intersect withRse,τ,landRsa,τ,l.

By inputting

into Step 1 and repeating all procedures,we haveRse,τ,l+1and

Output: allRse,τ,landRsa,τ,l.

With Algorithm 3, allRse,τ,landRsa,τ,linΨτare found out, and we can formΨse,τ={Rse,τ,l}andΨsa,τ={Rsa,τ,l}. Forthere is= ?,so different pivotal node vectors do not intersect with each other.Thus, for any weapon assigned to the targetτ, it is inevitable to pass through at leastnse,τdefense areas, wherense,τis the number ofRse,τ,l, andnse,τcan be noted as the coverage layer number.

4.Multi-stage attack planning process

With the defense area analysis, the traditional WTA plan matrixA=[aω,τ]is insufficient to describe the holistic attack plan.We need to find an attack path for those weapons allocated to targets that are protected by other closed areas.

Besides, with the following Theorem 4, when the allocation matrixAis given, an optimal path can be determined analytically,and thus,it is still viable to useAas a parameter for optimization in the multi-stage attack WTA problem.

4.1 Optimal attack path

The optimal attack path is comprised of 1-elements in each pivotal node vector covering the targetτ,which means the weapon allocated toτneeds to pass through those defense areas above sequentially.

Theorem 4For a weaponωthat is allocated to the targetτ, the optimal attack path is comprised ofnse,τnodes,and each node corresponds to the pivotal node vectorRse,τ,lcovering the targetτwhose defense area has the lowest intercept probability to the weaponω.

ProofFor each layer of the pivotal node vectorRse,τ,lcovering the targetτ,the corresponding closed node vector isRsa,τ,l,and we have

Rsu,τ,ldesignates defense areas that are not pivotal but still cover the targetτ. If a weapon passes through some defense areas inRsu,τ,l, it still needs to pass through at least one defense area inRse,τ,lto penetrate this layer of close areas.

Assume in thelth layer of closed areas,is the lowest intercept probability inRse,τ,l,andis the lowest intercept probability inRsu,τ,l.

We have

and

Thus,for all closed areas covering the targetτ,we have the optimal(minimal)cumulative intercept probability for the weaponωas

whereτl,mindesignates the defense area with the lowest intercept probability in thelth layer.

4.2 Multi-stage attack objective function calculation principle

Theoretically, an optimal multi-stage attack plan is companioned with a path parameter,compared with thenw×ntallocation matrixA, and the dimension of the parameter with a path is max{nse,τ}×nw ×nt,which complicates the multi-stage attack WTA problem.However,with Theorem 4,whenAis given,the attack path can also be determined analytically.Thus we can still useAfor optimization.Moreover,in the optimization procedure,we need to calculate the objective function value in the order from the target with a small coverage layer number to those with a large coverage layer number.The reason for this is as follows.

For two targetsτ1andτ2, the weaponω1is allocated toτ1andω2toτ2. Ifthe coverage layer number ofτ1is smaller thanτ2’s.

(i)Ifτ1belongs to the pivotal node vector ofτ2,the attack toτ1should be conducted in advance,so the intercept probability fromτ1toω2will be

(ii) On the contrary, if the attack toτ2is conducted ahead ofτ1,

which will reduce the penetrate probability for the following weapons.

(iii) Ifτ1does not belong to the pivotal node vector ofτ2, the order to attack these two targets will not affect the penetrate probability.

4.3 Multi-stage attack objective function model and multi-stage planning

With the analysis above, note the multi-stage attack objective function asHm, and the calculation ofHmis as follows.

Step 1Set the initial value as

andQis annw ×max{nse,τ}zero matrix to mark the optimal penetration path.

Step 2For all targetsτsatisfyingcalculateas

where

Find the optimal path.Iffor a targetτallocated with the weaponω, sequentially(lfrom 1 tonse,τ)selectfromwhereτmis theτmth element ofRse,τ,l,and we have

SetQ=[qω,l]as

Updatefor each column ofwe have

Repeat Step 2 untiland outputHmandQ.

4.4 Multi-stage attack planning flow

To conclude,we can present the complete flow for multistage attack planning as follows.

(i) According to the target positionXτand its defense area radiiρτ,solve all independent closed areas,and solve all circular closed areas with Algorithm 1 and Algorithm 2.

(ii) For each targetτ, traverse all closed areas, judge whether the closed area covers this target, and form the closed area set covering the targetτasΨτ={Rs,τ,q}.

(iii) For each targetτ, solve all pivotal node vectorsΨse,τ={Rse,τ,l},and cover the layer numbernse,τ.

(iv)For a given WTA plan matrixA,calculate the multistage attack objective function valueHmand penetrate the pathQ.

(v)Optimize the matrixAwith a particular optimization algorithm to maximizeHm,and output the correspondingA?andQ?.

5.Numerical results

5.1 Methods and Monte-Carlo objective function value calculation

To validate the MM method, we present the traditional WTA method and calculate the objective function of both methods with the Monte-Carlo method.

(i)MM method.

The multi-stage attack plan is solved according to Section 4.4.

The path of any weapon is expressed as T-X1→T-X2→T-XN, and the path length of the node number is equal to the attack stage or the attack order.For an attack stage larger than 1, the weapon will pass through T-X1, T-X2and T-(XN?1),where there areN ?1 defense areas of targets in total.Therefore,the Monte-Carlo objective function value calculation should be in order from the small attack stage to the large attack stage.

In each attack stage, for the weaponω, when it passes through the defense area of thektth targetτkt,it generates a random numberpc,r,wherepc,r ∈[0,1],and if

the weaponωis intercepted by the targetτkt,and calculate another weapon.If not,continue the intercept judge of the next target.

If this weapon is not intercepted by the defense areas passing through,and reaches the allocated targetτ,it generates a random numberpc,r,and if

the weaponωis intercepted by targetτ, and calculate another weapon. If not, it generates a random numberpk,r.If

the weaponωdestroys the targetτsuccessfully. Add the value of the targetτto the objective function value, and set the intercept probability of the targetτto 0.

(ii) Traditional WTA method with single attack stage(TS method).

Solve the WTA plan with the traditional WTA method.In the Monte-Carlo procedure, if the allocated target is covered by other defense areas, select a defense area randomly from the closed node vector for this weapon to pass through,until this weapon reaches the target allocated.

In the TS method,Ψse,τ={Rse,τ,l}andΨsa,τ={Rsa,τ,l}are both used in the Monte-Carlo objective function value calculation for that the TS method cannot provide an attack path.

For each weaponω, acquireΨse,τ={Rse,τ,l}andΨsa,τ={Rsa,τ,l}of the targetτallocated. Sequentially select a defense area fromRsa,τ,l, and conduct the intercept judge. If the defense area selected is also inRse,τ,land the weapon is not intercepted, this weapon will pass through this layer of coverage, until this weapon reaches the targetτ.

When the weaponωreaches the targetτ, conduct the intercept judge and the kill judge.If the kill judge passes,add the value of the targetτto the objective function value and set the intercept probability of the targetτto 0.

5.2 Methods comparison

In the following combat scenario, 24 weapons launch an attack to 18 targets.The value vectorVin this scenario is shown in Table 5 and the target distribution is in Fig.3.

Table 5 Value vector in the combat scenario

Fig.3 Target distribution

With the method given above,the WTA plan is solved as shown in Table 6 and the multi-stage attack plan is given in Table 7.

In Table 6,the expression of the WTA plan is the same as that of the traditional method,while in Table 7,the path of each weapon is given.

Table 6 WTA plan

Table 7 Multi-stage attack plan

After each stage of attack,the intercept probabilities to weapons will be changed. Take W-6 as an example (W-6 is in the third attack stage), the changed intercept probabilities and the target distribution are in Figs. 4–6. The number behind the target label is the intercept probability to W-6,and the changed intercept probabilities are marked with the red font.

The color of the circles around the target indicates the intercept probability.A higher intercept probability tends to be red,while a lower one blue.

Fig.4 Intercept probabilities to W-6(non attack)

In Fig. 4 to Fig. 6, the intercept probabilities of the outer targets are weakened after each stage of attack.The circles of defense areas of the outer targets are changed into blue.The objective function value calculated with the Monte-Carlo method is shown in Fig.7,where the objective function value is calculated with the Monte-Carlo experiment.In each experiment,a certain objective function value is obtained as an example. With a large amount of experiments,all examples are counted into a value range,and the probability of the value falling in a value range is computed.It is obvious that the statistical results of the MM method fall into higher value ranges.That is,the MM method is likely to reach high combat performance.

Fig.5 Intercept probabilities to W-6(after the first attack stage)

Fig.6 Intercept probabilities to W-6(after the second attack stage)

Fig. 7 Ojbective function value distribution with Monte-Carlo method

5.3 Result analysis

To further compare the multi-stage attack method and the traditional WTA method, the objective function value is calculated with the Monte-Carlo method.The scenarios include the complex scenario(in Section 5.2)and the simple scenario(in Section 2).

The mean value and confidence interval of the Monte-Carlo objective function value and the corresponding analytical value are shown in Table 8 and Table 9.The confidence coefficient is 0.95.

Table 8 Objective function value comparison(simple scenario)

From the results we have the following conclusions.

Table 9 Objective function value comparison(complex scenario)

(i)The analytical value is consistent with the mean value by the Monte-Carlo method,and falls into the confidence interval of 0.95 confidence coefficient.

(ii)The MM method is better than the TS method.The more complex the combat scenario is, the larger the gap between MM and TS is. According to the objective value distribution of the Monte-Carlo method, the value of the MM method distributes mainly in the high value range,while the value of the TS method distributes mainly in the low value range. It suggests that in complicated combat scenarios, it is imperative to analyze the defense areas of targets and use the multi-stage attack strategy to weaken the interactive defense between targets progressively.

(iii)If there is no interrelationship between the targets’defense areas,then the MM method is equivalent to the TS method, for in this case, the MM method only calculates the first stage of attack that involves all weapons,and the solving procedure is equal to the TS method.That means the MM method includes the TS method.

(iv) With Figs. 4–6, after each stage of attack of the MM method, the intercept probabilities will be updated,and therefore the MM method can also be applied to online or real-time combat scenario estimation for weapons with communication capabilities. After each stage of attack, a re-allocation can also be conducted for better further combat effectiveness. Besides, the MM method may also be used as a framework of scenario analysis for commanders to find out the weak area in further attack.

6.Conclusions

First, if the combat scenario involves interactive coverage of the targets’ defense areas, it is imperative to analyze the defense areas and use the multi-stage attack method to weaken the target defense progressively for better combat effectiveness. Second, if there is no interrelationship between the targets’ defense areas, the multistage WTA method is equivalent to the traditional WTA method, and the traditional method can be regarded as a special case in the multi-stage WTA problem.Finally,the multi-stage WTA framework including defense area analysis and defense scenario update can be used as a reference for decision-making.