Weapon target distribution method based on multilayer fusion network

Abstract

The invention discloses a weapon target distribution method based on a multilayer fusion network, comprising the following steps: constructing a three-layer network model, GT representing each element unit of a node entity layer; gN represents element units and connections of a communication network layer; gC represents element units and connections of a command and control relation layer; establishing a first weapon target distribution model; establishing a second weapon target distribution model based on a multi-layer network, and evaluating the damage efficiency of a node entity layer, a communication network layer and a command and control layer in a weighted form; and carrying out linearization processing on the target function and the constraint, and solving to obtain an optimal solution of the problem. According to the method, the retention condition of the control link and the command and control capability after the system node loss is expressed in a linearized dynamic update constraint, and the networked system breaking technology and the weapon target allocation problem are effectively fused.

Description

Technical Field

[0001] The invention belongs to the technical field of network analysis, and in particular relates to a weapon target allocation method based on a multi-layer fusion network. Background Art

[0002] With the development of network science, network analysis technology represented by complex networks has become an effective method for studying military systems. Compared with common civilian networks such as transportation, virus transmission and social networks, combat networks are more special in nature. First, the nodes in military combat networks have more significant heterogeneity and greater complexity. The physical units (nodes) at all levels play different roles and have different functions in the combat system. A node can also play multiple roles at the same time; secondly, the degree of interaction between nodes is higher, and there are one or more interactive relationships between nodes. Third, the network is more dynamic. During the combat process, as the nodes disappear, the edges will also change dynamically according to the actual situation. These characteristics bring a series of challenges to the research on system attack based on complex networks.

[0003] The existing networked combat system attack scheme design usually consists of two relatively independent decision-making stages, namely, key node identification based on network modeling and weapon target allocation based on the strike list. Such a phased optimization method is relatively easy to implement and can quickly form a feasible combat plan. However, the lack of overall consideration of the target system may lead to two consequences depending on the ammunition inventory: one is that when the ammunition is insufficient, it is not enough to cover all the targets in the list; the other is that when the ammunition is sufficient, the final strike planning plan may not be able to destroy the system capability to the greatest extent. The combat system is an indivisible whole, and the system attack should be an organic combination of target selection and weapon and ammunition matching. Therefore, it is a key issue for the system attack to comprehensively consider the generation of the target strike list under the networked combat system and the allocation of weapon targets under the constraint of limited ammunition to form an overall scheme design method. The present invention performs networked modeling on various physical combat units, and solves the problem of achieving the maximum degree of system capability attack based on limited ammunition.

[0004] Figure 1The interactive relationship between entity units and entities in the networked target system is shown. Each node represents an entity. The letters I, C, and K marked in the node indicate that the node has the corresponding functions of intelligence reconnaissance, command and control, and killing. An entity can have one or more combat capabilities, such as a reconnaissance and strike drone, an early warning aircraft with command capabilities, and a ship with command and strike capabilities. The threat level, reconnaissance capability, and command and control capability of the node are represented by a bar chart. The arrows between the nodes represent the command and control relationship between the entities. The simplest kill chain consists of a reconnaissance node, a command and control node, a strike node, and a reconnaissance node, forming a complete closed loop between the nodes. The dotted lines in the figure indicate two different kill chains. To strike such a networked system target, it is necessary to consider the capabilities of each node, the structure of the command and control network, and the impact of the formation of the kill chain and kill network on the overall combat capability. Summary of the invention

[0005] In view of this, the present invention studies the problem of target strike optimization in a multi-layer networked system, analyzes the target network from three levels: target unit value, network connectivity status, and combat capability, and explores how to maximize the damage effect on the enemy based on limited weapons and equipment in a multi-layer networked target system. First, a mixed integer programming model for the target of a multi-layer networked system under the problem of weapon target allocation is constructed, and an improved linear model that can be directly solved by commercial software is provided using linearization technology. Through simulation experiments, the solving ability of the mathematical model is verified, and the system target strike optimization problem of a maximum of 150 network nodes can be accurately solved within a few seconds. This is the first application of the joint optimization of networked system target selection and weapon target allocation, which provides theoretical and practical support for the subsequent research on optimization problems related to system destruction.

[0006] Specifically, the weapon target allocation method based on the multi-layer fusion network disclosed in the present invention comprises the following steps:

[0007] Construct a three-layer network model G = (V, E) = (G T ,G N ,G C ), where G T =(V T ) represents each element unit of the node entity layer, which is presented in the form of a set of nodes without network relationships; G N =(V N ,E N ) represents the element units and connections of the communication network layer, which are presented in the form of undirected network relationships; G C =(V C ,E C ) represents the element units and connections of the charge relationship layer, which are presented in the form of directed network relationships;

[0008] A first weapon target allocation model is established, wherein the first weapon target allocation model includes m different weapons, the set of weapon types is represented as W = {1, ..., m}, and the number of any weapon i available is w i , the set V = {1,…,n} contains all the target entities to be attacked, each target has a different threat level, corresponding to the function f 1 (V T ), different benefits will be generated after destruction; the threat capability of the jth target is α j , for the communication network layer G N , whose capacity is similar to E N The weights of the edges in are related, and the edges are<i,j> The corresponding value / threat weight is β ij ;

[0009] A second weapon target allocation model based on a multi-layer network is established to evaluate the damage effectiveness of the three-layer network of node entity layer, communication network layer, and command and control layer in a weighted form;

[0010] The objective function and constraints are linearized and solved to obtain the optimal solution to the problem.

[0011] Furthermore, in the three-layer network model, They represent the unit value of the node entity layer, the network capability of the communication network layer, and the comprehensive capability of the command and control relationship layer of the same combat entity i. The total capability F(G) of the fusion network G is the integration of the three-layer capabilities, namely, the unit value of the node entity layer, the network capability of the communication network layer, and the comprehensive capability of the command and control relationship layer. It can be expressed in mathematical form as:

[0012] F(G)=λ1×f 1 (V T )+λ2f 2 (E N )+λ3×f 3 (V C ,E C )

[0013] where λ i represents the weight of the i-th capability, f i represents the capacity calculation function of the i-th layer;

[0014] To maximize the strike effect based on ammunition limitations, it is necessary to form a strike unit list v so that the ability after the strike decreases the most, which can be expressed in mathematical form as:

[0015]

[0016] Where G′=(V―v,E―e), e represents the set of edges associated with node v.

[0017] Furthermore, in the weapon target allocation model, at the charge relationship layer, for a charge chain R={T1, T2, ..., T k , T1}, define the ath link <T a , T a+1 >The connectivity matrix of all feasible paths in network E is That is, it indicates whether the edge in the network E can form the ath link of the accusation chain; for any pair of nodes <i, j> in the network, It means that node i has the reconnaissance function, node j has the accusation function, and these two nodes are connected in the network;

[0018] The command capability is defined as the condition for a node to be able to perform its function in the command network, which is that there is at least one complete command link containing the node in the current command network. For any node i, the mathematical description of its command capability is expressed as: Let i∈R and The capacity of node i in the network is recorded as γ i .

[0019] Furthermore, in the weapon target allocation model, the damage of a weapon to a target is a random event based on a priori probability; the damage probability p of weapon i∈W to target j∈V is ij It is a fixed parameter. The attacks of multiple weapons on the same target are considered as multiple independent events, and the probabilities are not correlated.

[0020] Due to the uncertainty of random events, a threshold θ is set in the planning stage to determine whether a target j∈V is damaged. j , when the cumulative damage probability exceeds the threshold, the target is judged to have been destroyed; integer decision variable x ij ≥0 indicates the number of weapons i∈W used to attack the target j∈V; the binary decision variable y j ∈{0, 1} is used to determine whether the decision target j∈V has been damaged;

[0021] At the same time, in order to dynamically reflect the capabilities of the nodes in the current command network, a binary auxiliary decision variable z is used j ∈{0, 1} to indicate whether the node is included in the complete charge link; D j It represents the cumulative damage probability of target j∈V being hit, which can be expressed in mathematical form as follows:

[0022]

[0023] For any target j∈N, when D j ≥θ jWhen , the target is considered to have been destroyed, and the decision variable y j =1, otherwise the current attack plan is considered insufficient to destroy the target, and the decision variable satisfies y j =0.

[0024] Furthermore, the objective function of the second weapon target allocation model is expressed as:

[0025] max{λ1×∑ i∈V α i y i -λ2×∑ i∈V ∑ j∈V β ij (1-y i )(1-y j )-λ3×∑ i∈V γ i z i}

[0026] All the λ in the above formula i are all positive numbers. The first part represents the value / threat of maximizing the target of damage; the second part represents the retention status of edge <i, j>. When any node in the edge is damaged, (1-y i )(1-y j ) takes a value of 0, otherwise it takes a value of 1. Therefore, this part represents the weighted sum of the capabilities of all remaining communication relationships in the communication network layer after the attack. The subtraction form is used to represent the need to minimize the remaining capabilities. The third part represents the weighted sum of the units that are still on the command link after the attack. Similarly, we need to minimize the remaining capabilities.

[0027] For a command chain of length 3, it is organized in the form of "reconnaissance-command-attack-reconnaissance". At this time, the mixed integer programming model of this problem is as follows:

[0028]

[0029]

[0030]

[0031]

[0032]

[0033]

[0035] Constraint (1) means that for the destroyed target (y j =1) The cumulative damage probability should be greater than its threshold, while there is no requirement for the cumulative damage probability for targets that are not destroyed;

[0036] Constraint (2) limits the total number of weapons used;

[0037] Constraint (3) stipulates that the destroyed unit must no longer have the command capability, but the undestroyed unit may also lose the command capability due to the destruction of its kill chain;

[0038] Constraints (4) and (5) are conditional assignment constraints, indicating that if a unit still has the command capability, then its next link in the kill chain and the preceding link There should be at least one node that still has the ability to accuse, that is, when z i When the value is 1, and The value of must be no less than 1;

[0039] Constraint (6) is a conditional assignment constraint, which means that for the original kill chain that has not been destroyed, all nodes in it have the capability of the command layer, where , indicating that there is a command link i→j→k→i before the attack. Since the decision variable y represents the target to be destroyed, (1-y i )×(1-y j )×(1-y k )=1, it means that the three nodes i, j, and k are not destroyed; if both conditions are met at the same time, the value on the right side of the inequality is 1; and the decision variable z on the left side of the inequality is a binary decision variable. In order for the inequality to hold, z must be i =z j =z k =1; therefore, this constraint ensures the remaining capacity of the network after destruction and updates the list of nodes that still have strike capability in a dynamic manner.

[0040] Furthermore, finding the optimal solution to the problem includes the following steps:

[0041] First, introduce a binary auxiliary variable Y for the objective function ij , so that its value is equal to (1-y i )(1-y j ) are consistent; therefore, the following two constraints are introduced:

[0042]

[0043]

[0044] Constraint (7) means that when any one of nodes i and j is destroyed, Y ij The value of must be 0; on the other hand, when Y ij=1, nodes i and j must not be destroyed;

[0045] Constraint (8) means that when both nodes i and j are not destroyed, Y ij The value of must be 1;

[0046] By introducing auxiliary variables and linear constraints, the original nonlinear objective function is transformed into the following linear objective function:

[0047] max{λ1×∑ i∈V (α i ×y i )-λ2×∑ i∈V ∑ j∈V (β ij ×Y ij )-λ3×∑ i∈V (γ i × i )}

[0048] Reconstructing this into an improved linear constraint (9):

[0049]

[0050] Because p ij and θ j The values ​​of are all in the open interval (0, 1), so the value on the right side is always not less than 0, that is, when y j = 0, constraint (9) always holds; and when y j = 1, constraint (9) is equivalent to D j ≥θ j , so the original constraint (1) is replaced by constraint (9);

[0051] Applying the “Big-M” method to linearize the constraints, the results of the transformation of constraints (4)-(6) are as follows:

[0052]

[0053]

[0054]

[0055] M is a sufficiently large positive number. For constraint (10), when z i = 0, the inequality holds regardless of the value of the right side, and when z i =1, it is equivalent to limiting That means That is, node i has at least one subsequent node that also has the capability of the accusation layer;

[0056] Constraint (11) stipulates that node i has at least one predecessor node that also has the capability of the charge layer;

[0057] Constraint (12) states that when y i ,y j ,y k The value of is 0, and When the values ​​of are all 1, z must be satisfied i +z j +z k ≥3, and z is a binary variable, so it can only be z i =z i =z k =1, thus ensuring that this constraint is functionally consistent with the original constraint (6);

[0058] Through linearization processing, a mixed integer linear programming model of the problem is finally obtained, and the mixed integer linear programming model is solved.

[0059] Furthermore, the simulation data is generated according to the weapon target allocation method, including the steps of:

[0060] A certain number of target nodes are randomly generated in a two-dimensional plane area, and then the node type and edge relationship are determined based on certain rules, that is, the ability assignment and communication and command and control relationship assignment are assigned; the attack direction starts from the origin and extends along the y=x coordinate axis in the positive direction. The reconnaissance nodes are distributed in the attack area, and their assignment probability is positively correlated with the distance from the origin. The deeper into the enemy's central area, the more reconnaissance nodes there are; the command and control nodes, as the central control center of the kill chain or kill network, are scattered in the area far away from the defender, and their assignment probability is negatively correlated with the distance from the origin; considering that the attack nodes include long-range, medium-range and short-range nodes, it is believed that the attack nodes are relatively evenly distributed in the entire space;

[0061] Along the direction of attack, the combat area is divided into four areas, namely A, B, C, and D areas;

[0062] The threat value assignment intervals of the command, reconnaissance and attack nodes are For dual-capability nodes or triple-capability nodes, the threat value is processed by linear addition. For triple-capability nodes, the threat value calculation formula is:

[0063]

[0064] At the same time, each target has a certain damage threshold. If the weapon cannot reach the damage threshold of the target, then it is considered that the kill chain formed by the target has not been destroyed;

[0065] Set the target damage threshold in the open interval (0, 1), and the attack threshold intervals of the command, reconnaissance and attack nodes are For dual-capability nodes or triple-capability nodes, since the damage threshold value range is (0, 1), a multiplication method is used for comprehensive processing. For triple-capability nodes, the threat value calculation formula is:

[0066]

[0067] The interconnection relationship of the combat unit includes two levels: the communication layer and the command and control layer. The communication network is a fully interconnected network. There is a direct communication relationship between any two nodes i and j. The weight of the communication relationship is a randomly generated number, represented by β ij ∈[0, 1]; at the command capability layer, whether there is a command relationship is related to the distance. In the command link of "reconnaissance-command-attack-reconnaissance", any node randomly generates a command link with all the next link type nodes according to its function; the generation probability is negatively correlated with the distance, that is, the closer the distance, the more likely it is to form a command relationship; based on the command relationship, the parameter The value of

[0068] For any node i, when it has the corresponding capabilities of charge, reconnaissance, and attack, its charge capability is randomly assigned a value in a certain interval, and the value intervals are For nodes with multiple capabilities, their command and control capabilities should have a multiplied benefit. Therefore, for nodes with three capabilities, the command and control capability simulation data is generated according to the following formula:

[0069]

[0070] In the multi-layer network system target strike optimization problem, a total of m different weapons are considered to be distributed simultaneously, and the total number of each weapon i used is w i , the probability p of any weapon i damaging target j ij satisfy

[0071] The beneficial effects of the present invention are as follows:

[0072] The retention of control links and command capabilities after the loss of system nodes is expressed as linearized dynamic update constraints. Through the use of reasonable simplification and linearization technology, the networked system attack technology and weapon target allocation problem are effectively integrated. The application example proves to a certain extent that the multi-layer network target allocation model proposed in this invention can generate networked system target attack plans with high efficiency and quality, providing a basis for the engineering application of system target attack plans and providing support for system attack target selection. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1Schematic diagram of the networked target system;

[0074] Figure 2 The distribution of target entities in a certain area along the attack direction;

[0075] Figure 3 Changes in the objective function as the number of weapons changes. DETAILED DESCRIPTION

[0076] The present invention is further described below in conjunction with the accompanying drawings, but the present invention is not limited in any way. Any changes or substitutions made based on the teachings of the present invention belong to the protection scope of the present invention.

[0077] The research objects of the present invention can be divided into two categories according to the confrontation relationship. One category is the opponent's entity unit, including reconnaissance, accusation and attack entities. Different entities pose different threats to our or friendly military facilities and assets, which can be quantified by numerical values. In the network model, there is a certain communication relationship (undirected graph) between the entities, which represents the interaction of information between the entities and is also the premise for the existence of the accusation relationship. The accusation node has the strongest accusation capability and the lowest threat capability; the reconnaissance node has the strongest reconnaissance capability and the lowest threat capability; the target threat of the attack node is the greatest, and the accusation capability and reconnaissance capability are relatively weak. The other category is our weapon unit. The available number of weapons or ammunition is uncertain, and our weapons need to be distributed to the opponent's target unit to destroy the target network to the greatest extent, thereby achieving the purpose of system destruction.

[0078] Building a mathematical model

[0079] (1) Network Modeling

[0080] In actual combat scenarios, the enemy target system usually includes multiple network layers, such as platform entity layer, reconnaissance detection layer, communication network layer, command and control layer, firepower strike layer, etc. According to the relevant knowledge of graph theory, these network structures can be divided into three categories: isolated nodes, undirected networks, and directed networks. The platform entity units and firepower strike units usually do not have communication or command and control relationships, which are shown as isolated nodes in the graph; the communication network is shown as an undirected network relationship, and the communication relationship between each entity is bidirectional, that is, the two nodes corresponding to any edge of the graph are equivalent in terms of communication relationship. For example, in a communication network, each communication node can transmit information bidirectionally based on the information transmission channel, without distinguishing between superiors and subordinates. The capability evaluation at this level is usually a measure of the connection status of each node or a calculation of the degree of the overall network; from the perspective of reconnaissance relationship and command and control relationship, the relationship between entities can be represented by a directed graph. For example, in the command and control network, there is a superior-subordinate relationship between each command node; and in the reconnaissance and detection network, there is a sequence relationship between each detection node. The capability evaluation at this level is usually associated with the entire information flow process, and the node's capability depends on the integrity of the directed link.

[0081] Through the above analysis, in order to simplify the problem to a certain extent, the present invention constructs a three-layer network model G = (V, E) = (G T , G N , G C ). T =(V T ) represents each element unit of the node entity layer, which is presented in the form of a set of nodes without network relationships; G N =(V N , E N ) represents the element units and connections of the communication network layer, which are presented in the form of undirected network relationships; G C =(V C , E C ) represents the element units and connections of the charge relationship layer, which are presented in the form of directed network relationships.

[0082] For any combat entity, its label in each level is consistent, that is, They all represent the same combat entity i. The total capability F(G) of the fusion network G should be a combination of three-layer capabilities, namely, the unit value of the node entity layer, the network capability of the communication network layer, and the comprehensive capability of the command and control relationship layer, which can be expressed in mathematical form as:

[0083] F(G)=λ1×f 1 (V T )+λ2f 2 (E N )+λ3×f 3 (V C, E C ) (1)

[0084] where λ i represents the weight of the i-th capability, f i Represents the ability calculation function of the i-th layer. To maximize the strike effect based on the ammunition limit, it is necessary to form a strike unit list v so that the ability after the strike decreases the most, which can be expressed in mathematical form as:

[0085]

[0086] Where G′=(Vv, Ee), e represents the set of edges associated with node v.

[0087] (2) Weapon Target Allocation Modeling

[0088] The multi-layer networked system target strike optimization problem contains m different weapons, and the set of weapon types is represented by W = {1, ..., m}. The total number of weapons that can be used for any weapon i is w. i The set V = {1, ..., n} contains all target entities that can be attacked. Each target has a different threat level, corresponding to the function f 1 (V T ), and different benefits will be generated after destruction. The threat capability of the jth target is α j For the communication network layer G N , whose capacity is similar to E N The weights of the edges in are related, and the value / threat weight corresponding to edge <i, j> is β ij .

[0089] At the accusation relationship level, for an accusation chain R = {T1, T2, ..., T k , T1}, define the ath link <T a , T a+1 >The connectivity matrix of all feasible paths in network E is That is, it indicates whether the edge in the network E can form the ath link of the accusation chain. For example, for a typical accusation chain of "reconnaissance-accusation-strike-reconnaissance", it can be represented as a directed ring of T1→T2→T3→T1. For any pair of nodes <i, j> in the network, It means that node i has the reconnaissance function, node j has the accusation function, and these two nodes can be connected in the network. Similarly, we can get meaning.

[0090] Definition of command capability: The condition for a node to be able to exert its capability in the command network is that there is at least one complete command link (kill chain) containing the node in the current command network. For any node i, the mathematical description of its command capability is expressed as: Let i∈R and The capacity of node i in the network is recorded as γ i .

[0091] The damage of a weapon to a target is considered to be a random event based on a priori probability. The present invention assumes that the damage probability p of weapon i∈W to target j∈V is ij It is a fixed parameter and does not change with changes in the command and control network, cumulative strikes, and other related factors. The strikes of multiple weapons on the same target are considered to be multiple independent events, and there is no correlation in probability.

[0092] Due to the uncertainty of random events, a threshold θ is usually set in the planning stage to determine whether a target j∈V is damaged. j , when the cumulative damage probability exceeds the threshold, we judge that the target has been destroyed. Integer decision variable x ij ≥0 indicates the number of weapons i∈W used to attack the target j∈V. The binary decision variable y j ∈{0, 1} is used to determine whether the decision target j∈V has been damaged. At the same time, in order to dynamically reflect the capabilities of the nodes in the current accusation network, we use a binary auxiliary decision variable z j ∈{0, 1} to indicate whether a node is included in the complete command link. j It represents the cumulative damage probability of target j∈V being hit, which can be expressed in mathematical form as follows:

[0093]

[0094] For any target j∈N, when D j ≥θ j When , we believe that the target has been damaged, then the decision variable y j =1, otherwise the current attack plan is considered insufficient to destroy the target, and the decision variable should satisfy y j = 0. Table 1 summarizes the sets, parameters and decision variables used in this paper.

[0095] Table 1 Symbols and their meanings

[0096]

[0097] (3) Weapon target allocation modeling based on multi-layer network

[0098] The core goal of the multi-layer networked system target strike allocation problem is to maximize the comprehensive effectiveness of the strike based on the limited number of weapons and ammunition. The damage effectiveness of the three-layer network of the node entity layer, the communication network layer, and the command and control layer should be evaluated in a weighted form. According to formulas (1), (2) and the definition of decision variables, we can get the objective function of this problem should be expressed as:

[0099] max{λ1×∑ i∈V α i y i -λ2×∑ i∈V ∑ j∈V β ij (1-y i )(1-y j )-λ3×∑ i∈V γ i z i} (4)

[0101] All λ in formula (4) i are all positive numbers. The first part represents the value / threat of maximizing the target of damage. The second part represents the retention status of edge <i,j>. When any node in the edge is damaged, (1-y i )(1-y j ) is 0, otherwise it is 1. Therefore, this part represents the weighted sum of the capabilities of all remaining communication relationships in the communication network layer after the attack, and the subtraction form is used to minimize the remaining capabilities. The third part represents the weighted sum of the units that are still on the command link after the attack, and we also need to minimize the remaining capabilities.

[0102] To simplify the problem, we consider a command chain of length 3, organized in the form of "reconnaissance-command-attack-reconnaissance". At this time, the mixed integer programming model of this problem is as follows:

[0103]

[0104]

[0105]

[0106]

[0107]

[0108]

[0109] Constraint (5) means that for the destroyed target (y j=1) The cumulative damage probability should be greater than its threshold, while there is no requirement for the cumulative damage probability for targets that are not destroyed. Constraint (6) limits the total number of weapons used. Constraint (7) stipulates that the destroyed unit must no longer have the command capability, but the undestroyed unit may also have no command capability due to the destruction of its kill chain. Constraints (8) and (9) are assignment constraints for conditional judgment, indicating that if a unit still has the command capability, then its next link in the kill chain and the preceding link There should be at least one node that still has the ability to accuse. i When the value is 1, and The value of must be no less than 1. The summation form is adopted because a node may play a role in different command links in the command network (for example, a reconnaissance and strike integrated combat unit can be used as both a reconnaissance node and a strike node), so it is impossible to determine in advance its position in the kill chain R = {T1, T2, ..., T k , T1}.

[0110] Constraint (10) is also a typical conditional assignment constraint, indicating that for the original kill chain that has not been destroyed, all nodes in it should have the capability of the command layer. , it indicates that there is a command link i→j→k→i before the attack. Since the decision variable y represents the target to be destroyed, (1-y i )×(1-y j )×(1-y k )=1, which means that the three nodes i, j, and k are not destroyed. If both conditions are met at the same time, the value on the right side of the inequality is 1. The decision variable z on the left side of the inequality is a binary decision variable. For this inequality to hold, z must be i =z j =z k = 1. Therefore, this constraint ensures the remaining capacity of the network after the destruction, and updates the list of nodes that still have strike capability in a dynamic manner.

[0111] Model Linearization Methods

[0112] There are nonlinear parts in the mixed integer programming model, which are mainly reflected in the objective function, damage probability calculation, and conditional constraints for judging the command layer capability after damage. The optimal solution to the problem can be obtained by linearizing the objective function and constraints and then using commercial software such as CPLEX to solve the problem.

[0113] First, for the objective function, a binary auxiliary variable Y is introduced ij, so that its value is equal to (1-y i )(1-y j ) are consistent. Therefore, the following two constraints are introduced:

[0114]

[0115]

[0116] Constraint (11) means that when any one of nodes i and j is destroyed, Y ij The value of must be 0. On the other hand, when Y ij = 1, nodes i and j must not be destroyed. Constraint (12) means that when nodes i and j are not destroyed, Y ij The value of must be 1. By introducing auxiliary variables and linear constraints, the original nonlinear objective function can be transformed into the following linear objective function:

[0117] max{λ1×∑ i∈V (α i ×y i )-λ2×∑ i∈V ∑ j∈V (β ij ×Y ij )-λ3×∑ i∈V (γ i × i )} (13)

[0119] For constraint (5) representing the damage probability calculation, we can substitute equation (3) first, shift the terms and take the logarithm to obtain:

[0120]

[0121] For the left side of the above inequality, it is easy to find that when y j = 0, the value on the left is always zero, and y j =1, the value on the left is ln(1-θ j ). Then we might as well reconstruct the left side of the inequality as ln(1-θ j )×y j At this point, it is not difficult to find that the reconstructed formula is similar to the original formula in y j Different values ​​of have the same result, so this reconstruction is equivalent. Based on this reconstruction, we get an improved linear constraint (15):

[0122]

[0123] Because p ij and θ jThe values ​​of are all in the open interval (0, 1), so the value on the right side is always not less than 0, that is, when y j = 0, constraint (15) always holds. j = 1, constraint (15) is equivalent to D j ≥θ j Therefore, we can replace the original constraint (5) with constraint (15).

[0124] The conditional assignment constraints in the model have two characteristics: first, the nonlinear part only involves the multiplication of multiple variables, and second, the nonlinear part is composed entirely of binary variables. Under this premise, we can consider applying the "Big-M" method to linearize the constraints. The results of the transformation of constraints (8)-(10) are as follows:

[0125]

[0126]

[0127]

[0128] In this type of method, M is represented by a sufficiently large positive number. For constraint (16), when z i = 0, the inequality holds true no matter what the right side takes. i =1, it is equivalent to limiting That means That is, node i has at least one subsequent node that also has the ability to charge the layer. Similarly, constraint (17) limits node i to have at least one previous node that also has the ability to charge the layer. Constraint (18) states that when y i ,y j ,y k The value of is 0, and When the values ​​of are all 1, z must be satisfied i +z j +z k ≥3, and z is a binary variable, so it can only be z i =z j =z k = 1. This ensures that the constraint is functionally consistent with the original constraint (10).

[0129] Through linearization, we finally get the mixed integer linear programming model of the problem. Specifically, it includes the objective function (13), the constraints (11) and (12) that assist in the linearization of the objective function, the constraint for calculating the weapon damage effectiveness (15), the constraint for limiting the number of weapons (6), and the constraints for determining the command and control layer capabilities of the network after damage (7), (16)-(18).

[0130] In order to prove the effectiveness of the method provided by the present invention, a simulation data generation method is proposed, which randomly generates a certain number of target nodes in a two-dimensional plane area, and then determines the node type and edge relationship of the nodes based on certain rules, that is, the ability assignment and communication and charge relationship assignment. Figure 2 As shown in the figure, the attack direction starts from the origin and extends along the positive direction of the coordinate axis y=x. The reconnaissance nodes are mainly distributed in the attack area, and their probability of being assigned is positively correlated with the distance from the origin. The deeper into the enemy's central area, the more reconnaissance nodes there are; the command and control nodes, as the central control center of the kill chain or kill network, are mainly scattered in the area far away from the defender (near the origin), and their probability of being assigned is negatively correlated with the distance from the origin; considering that the strike nodes include long-, medium- and short-range nodes, it can be considered that the strike nodes are relatively evenly distributed in the entire space.

[0131] Along the attack direction, the combat area is divided into four areas, namely A, B, C, and D. The generation probabilities of the command, reconnaissance, and strike nodes are shown in Table 2.

[0132] Table 2 Probability of ability assignment

[0133]

[0134] Node feature data generation

[0135] As mentioned above, each target node will pose a certain threat to our side or the friendly side, so the present invention randomly assigns threat capabilities to the three types of targets according to actual conditions. The threat value assignment intervals for the accusation, reconnaissance and attack nodes are For dual-capability nodes or triple-capability nodes, the threat value is processed by linear addition. Taking the triple-capability node as an example, the threat value calculation formula is:

[0136]

[0137] At the same time, each target has a certain damage threshold. If the weapon cannot reach the damage threshold of the target, then it is considered that the kill chain (network) formed by the target has not been destroyed. In the target damage threshold interval provided by the present invention, the threshold is specified in the (0, 1) open interval, and the attack threshold intervals of the command, reconnaissance and attack nodes are respectively For dual-capability nodes or triple-capability nodes, since the damage threshold value range is (0, 1), a multiplication method is used for comprehensive processing. Taking the triple-capability node as an example, the threat value calculation formula is:

[0138]

[0139] Target network connection relationship generation

[0140] The interconnection relationship of the combat unit includes two levels: the communication layer and the command and control layer. Assume that the communication network is a fully connected network, and there is a direct communication relationship between any two nodes i and j. The weight of the communication relationship is a randomly generated number, and the symbol is β ij ∈[0, 1]. At the command capability layer, it is assumed that whether there is a command relationship is related to the distance. In the command link of "reconnaissance-command-attack-reconnaissance", any node randomly generates a command link with all the next link type nodes based on its function. The generation probability is negatively correlated with the distance, that is, the closer the distance, the more likely it is to form a command relationship. Based on the command relationship, the parameter The value of .

[0141] For any node i, when it has the corresponding capabilities of charge, reconnaissance, and attack, its charge capability is randomly assigned a value in a certain interval, and the value intervals are For nodes with multiple capabilities, their command and control capabilities should have a multiplied benefit. Therefore, taking a three-capability node as an example, the command and control capability simulation data is generated according to the following formula:

[0142]

[0143] (1) Weapon Target Division

[0144] Configuration parameter generation

[0145] In the multi-layer network system target strike optimization problem, a total of m different weapons are considered to be distributed simultaneously, and the total number of each weapon i that can be used is w i The probability p of any weapon i damaging target j is ij satisfy For a node with multiple capabilities, the damage probability calculation formula is:

[0146]

[0147] Formulas (19)-(22) reflect the calculation of the capabilities of composite units during the parameter setting process, and are generally related to real conditions. Formula (19) reflects the cumulative effect of the threat of composite units. Formula (20) indicates that for a target to be destroyed, all of its capability units must be destroyed, and also reflects the realistic scenario that composite functional units are more difficult to attack than single functional units. Formula (21) shows the multiplier effect of the command and control capabilities of composite functional units. The more capabilities a node has, the stronger its command and control capabilities. Formula (22) indicates that the more complex the unit, the more difficult it is to attack. At the same time, it should be noted that if it only has two capabilities, then only the product of the two should be taken when calculating (for example, if it only has command and reconnaissance capabilities, the calculation should be Application Examples

[0148] Since the multi-layer fusion network system target strike optimization problem is a new problem, there is currently no public data set that can be used to test the model performance, so the present invention verifies the model's solving ability based on the above-mentioned data generation scheme. In addition, since CPLEX may have a phenomenon of long convergence time, the maximum solving time for a single case is set to 3600s.

[0149] (1) First, in order to verify the solving performance of the linearized model proposed in this paper on large-scale problems, ten cases containing 150 target entities are simulated. Table 3 shows some experimental results of solving the linear model using the CPLEX solver. The experimental results show that the model can solve the large-scale multi-layer network weapon target allocation problem in about 60 seconds, and the solving ability is stable.

[0150] Table 3. Experimental results of large-scale cases

[0151] Case Number Objective function value Solution time (unit: S) 1 -646.4 28.38 2 -661.1 62.30 3 -626 59.55 4 -652.6 44.00 5 -636.9 56.45 6 -595.4 56.88 7 -690.7 50.83 8 -700.5 35.16 9 -621.9 32.20 10 -672.5 48.28

[0152] (2) In order to further verify the model's solving ability, a set of medium-sized examples were designed to solve the weapon target allocation model based on complex networks and the weapon target allocation model based on target threats. The number of targets in this example is 50. The number of targets, parameters, and variables such as communication between targets and command and control relationships remain unchanged, and the number of weapons is increased from 1 to 90. The experimental results show that the model has stable solution quality when applied in the solver and can solve medium-sized networked weapon target allocation problems within an acceptable time. Figure 3 It shows the changes in the comprehensive strike effectiveness of the network system.

[0153] From the experimental comparison results, the modeling method proposed by the present invention is obviously superior to the traditional threat-based weapon target allocation model. And when the number of weapons is in different situations, it shows differentiated strike results. When the number of weapons is less than 15, the objective function has a more significant trend of increasing with the increase of the number of weapons; when the number of weapons is between 15 and 45, the growth trend is relatively slow, and there is a situation where the increase of weapons but the objective function does not change significantly or even does not increase; when the number of weapons exceeds 45, there will be an obvious leap-like growth, and then it will grow slowly in a longer growth process. For the solution results of the comparison model, its growth process is similar to the overall trend of the direct solution results, but there is a lack of a leap-like growth stage, and in some local areas, the number of weapons is increased but the corresponding objective function does not change or even decreases. The reason for this trend is mainly because the objective function in the solution process of the comparison model only considers the threat capability. Therefore, it may choose to attack nodes with greater threat capability but not high contribution to the system command capability after adding weapon units. In order to destroy these more threatening nodes, it is possible to divert the weapons previously allocated to attack other nodes, so that the judgment of the system comprehensive damage shows a downward trend.

[0154] From the results of the solution, in the early stage of the increase in the number of weapons, the model can accurately find the weak links of the system (easy to attack and with a large system contribution rate), so that the system's capabilities can be affected relatively quickly in the initial stage. However, as the weak links are damaged, the remaining attackable nodes have relatively stronger anti-destruction capabilities. Therefore, it is possible that the number of weapons is increased but the damage threshold of important nodes is not reached, resulting in a situation where there is no effect on the objective function. As the number of weapons reaches a saturation threshold, the attack plan can consider destroying all nodes in a certain command and control link in the entire system, thereby completely paralyzing its system capabilities and causing a leap-forward change scenario of system collapse; and in the final stage, not all targets in the system are actually destroyed (only when the two curves overlap, it means that the targets of the attack are completely the same, that is, the situation of complete destruction). According to the analysis of the solution, some nodes are still retained due to their own strong anti-destruction capabilities. These nodes are usually multifunctional and complex nodes that integrate command and control, killing, and reconnaissance. In real environments, they are indeed the parts of the system targets that are difficult to completely attack and destroy. However, in order to make it lose its command and control capability, it is not necessary to completely destroy it. It is only necessary to use relatively small amounts of ammunition to destroy other nodes in the target system that can constitute the system capability of the unit to achieve the combat objective.

[0155] The beneficial effects of the present invention are as follows:

[0156] The retention of control links and command capabilities after the loss of system nodes is expressed as linearized dynamic update constraints. Through the use of reasonable simplification and linearization technology, the networked system attack technology and weapon target allocation problem are effectively integrated. The application example proves to a certain extent that the multi-layer network target allocation model proposed in this invention can generate networked system target attack plans with high efficiency and quality, providing a basis for the engineering application of system target attack plans and providing support for system attack target selection.

[0157] As used herein, the word "preferred" is intended to be used as an example, instance, or illustration. Any aspect or design described herein as "preferred" is not necessarily to be construed as being more advantageous than other aspects or designs. On the contrary, the use of the word "preferred" is intended to present concepts in a specific way. The term "or" as used in this application is intended to mean an inclusive "or" rather than an exclusive "or". That is, unless otherwise specified or clear from the context, "X uses A or B" means any one of the naturally included permutations. That is, if X uses A; X uses B; or X uses both A and B, then "X uses A or B" is satisfied in any of the foregoing examples.

[0158] Moreover, although the present disclosure has been shown and described with respect to one or implementations, those skilled in the art will think of equivalent variations and modifications based on the reading and understanding of this specification and the accompanying drawings. The present disclosure includes all such modifications and variations, and is limited only by the scope of the appended claims. In particular, with respect to the various functions performed by the above-mentioned components (such as elements, etc.), the terms used to describe such components are intended to correspond to any component (unless otherwise indicated) that performs the specified function of the component (such as it is functionally equivalent), even if the structure is not equivalent to the disclosed structure of the function in the exemplary implementation of the present disclosure shown herein. In addition, although the specific features of the present disclosure have been disclosed with respect to only one of several implementations, such features can be combined with one or other features of other implementations that may be desired and advantageous for a given or specific application. Moreover, insofar as the terms "including", "having", "containing" or their variations are used in specific embodiments or claims, such terms are intended to be included in a manner similar to the term "comprising".

[0159] The functional units in the embodiments of the present invention may be integrated into a processing module, or each unit may exist physically separately, or multiple or more units may be integrated into one module. The above-mentioned integrated module may be implemented in the form of hardware or in the form of a software functional module. If the integrated module is implemented in the form of a software functional module and sold or used as an independent product, it may also be stored in a computer-readable storage medium. The above-mentioned storage medium may be a read-only memory, a disk or an optical disk, etc. The above-mentioned devices or systems may execute the storage method in the corresponding method embodiment.

[0160] To sum up, the above embodiment is an implementation mode of the present invention, but the implementation mode of the present invention is not limited by the embodiment. Any other changes, modifications, substitutions, combinations, and simplifications that deviate from the spirit and principles of the present invention should be equivalent replacement methods and are included in the protection scope of the present invention.

Claims

1. A weapon target allocation method based on a multi-layer fusion network, characterized in that: The following steps are involved: Construct a three-layer network model G = (V, E) = (G T , G N , G C ), where G T =(V T ) represents each element unit of the node entity layer, which is presented in the form of a set of nodes without network relationships; G N =(V N , E N ) represents the element units and connections of the communication network layer, which are presented in the form of undirected network relationships; G C =(V C , E C ) represents the element units and connections of the charge relationship layer, which are presented in the form of directed network relationships; A first weapon target allocation model is established, wherein the first weapon target allocation model includes m different weapons, the set of weapon types is represented as W = {1, ..., m}, and the number of weapons i available is w i , the set V = {1, ..., n} contains all the target entities to be attacked, each target has a different threat level, and the corresponding function is f 1 (V T ), different benefits are generated after the target entity is destroyed; the threat capability of the jth target is α j , for the communication network layer G N , whose capacity is similar to E N The weights of the edges in are related, and the edges are<i,j> The corresponding value / threat weight is β ij ; A second weapon target allocation model based on a multi-layer network is established to evaluate the damage effectiveness of the three-layer network of node entity layer, communication network layer, and command and control layer in a weighted form; The objective function and constraints are linearized and solved to obtain the optimal solution to the problem.

2. The weapon target allocation method based on a multi-layer fusion network according to claim 1 is characterized in that: In the three-layer network model, for any combat entity, They represent the unit value of the node entity layer, the network capability of the communication network layer, and the comprehensive capability of the command and control relationship layer of the same combat entity i. The total capability F(G) of the fusion network G is the integration of the three-layer capabilities, namely, the unit value of the node entity layer, the network capability of the communication network layer, and the comprehensive capability of the command and control relationship layer. It can be expressed in mathematical form as: F(G)=λ1×f 1 (V T )+λ2f 2 (E N )+λ3×f 3 (V C ,E C ) where λ i represents the weight of the i-th capability, f i represents the capacity calculation function of the i-th layer; Based on the ammunition limitation, the strike effect is maximized, that is, the strike unit list v is formed so that the ability after the strike is reduced the most, which can be expressed in mathematical form as: Where G′=(Vv, Ee), e represents the set of edges associated with node v.

3. The weapon target allocation method based on the multi-layer fusion network according to claim 2 is characterized in that: In the weapon target allocation model, at the command relationship layer, for a command chain R={T1, T2, ..., T k , T1}, define the ath link <T a , T a+1 >The connectivity matrix of all feasible paths in network E is That is, whether the edge in the network E can form the ath link of the accusation chain; for any pair of nodes in the network<i,j> , It means that node i has the reconnaissance function, node j has the accusation function, and these two nodes are connected in the network; The command capability is defined as the condition for a node to be able to perform its function in the command network, which is that there is at least one complete command link containing the node in the current command network. For any node i, the mathematical description of its command capability is expressed as: Let i∈R and The capacity of node i in the network is recorded as γ i .

4. The weapon target allocation method based on a multi-layer fusion network according to claim 3 is characterized in that: In the weapon target allocation model, the damage of a weapon to a target is a random event based on a priori probability; the probability p of a weapon i∈W damaging a target j∈V is ij It is a fixed parameter. The attacks of multiple weapons on the same target are considered as multiple independent events, and the probabilities are not correlated. Due to the uncertainty of random events, a threshold θ is set in the planning stage to determine whether a target j∈V is damaged. j , when the cumulative damage probability exceeds the threshold, the target is judged to have been destroyed; Integer decision variable x ij ≥0 indicates the number of weapons i∈W used to attack the target j∈V; the binary decision variable y j ∈{0, 1} is used to determine whether the decision target j∈V has been damaged; At the same time, in order to dynamically reflect the capabilities of the nodes in the current command network, a binary auxiliary decision variable z is used j ∈{0, 1} to indicate whether the node is included in the complete command link; D j It represents the cumulative damage probability of target j∈V being hit, which can be expressed in mathematical form as follows: For any target j∈N, when D j ≥θ j When , the target is considered to have been destroyed, and the decision variable y j =1, otherwise the current attack plan is considered insufficient to destroy the target, and the decision variable satisfies y j =0.

5. The weapon target allocation method based on the multi-layer fusion network according to claim 4 is characterized in that: The objective function of the second weapon target allocation model is expressed as: max{λ1×∑ i∈V a i y i -λ2×∑ i∈V ∑ j∈V b ij (1-y i )(1-y j )-λ3×∑ i∈V c i z i } All the λ in the above formula i are all positive numbers. The first part represents the value / threat of maximizing the target damage; the second part represents the edge<i,j> The retention status of the edge is that when any node in the edge is damaged, (1-y i )(1-y j ) takes a value of 0, otherwise it takes a value of 1. Therefore, this part represents the weighted sum of the capabilities of all remaining communication relationships in the communication network layer after the attack. The subtraction form is used to represent the need to minimize the remaining capabilities. The third part represents the weighted sum of the units that are still on the command link after the attack. Similarly, we need to minimize the remaining capabilities. For a command chain of length 3, it is organized in the form of "reconnaissance-command-attack-reconnaissance". At this time, the mixed integer programming model of this problem is as follows: Constraint (1) means that the cumulative damage probability of the destroyed target is greater than the preset threshold, while there is no requirement for the cumulative damage probability of the undestroyed target; Constraint (2) limits the total number of weapons used; Constraint (3) stipulates that the destroyed unit must no longer have the command capability, but the undestroyed unit may also lose the command capability due to the destruction of its kill chain; Constraints (4) and (5) are conditional assignment constraints, indicating that if a unit still has the command capability, then its next link in the kill chain and the preceding link There should be at least one node that still has the ability to accuse, that is, when z i When the value is 1, and The value of must be no less than 1; Constraint (6) is a conditional assignment constraint, which means that for the original kill chain that has not been destroyed, all nodes in it have the capability of the command layer, where , indicating that there is a command link i→j→k→i before the attack. Since the decision variable y represents the target to be destroyed, (1-y i )×(1-y j )×(1-y k )=1, it means that the three nodes i, j, and k are not destroyed; if both conditions are met at the same time, the value on the right side of the inequality is 1; and the decision variable z on the left side of the inequality is a binary decision variable. In order for the inequality to hold, z must be i =z j =z k =1; therefore, this constraint ensures the remaining capacity of the network after destruction and updates the list of nodes that still have strike capability in a dynamic manner.

6. The weapon target allocation method based on a multi-layer fusion network according to claim 5 is characterized in that: Finding the optimal solution to the problem includes the following steps: First, introduce a binary auxiliary variable Y for the objective function ij , so that its value is equal to (1-y i )(1-y j ) are consistent; therefore, the following two constraints are introduced: Constraint (7) means that when any one of nodes i and j is destroyed, Y ij The value of must be 0; on the other hand, when Y ij =1, nodes i and j must not be destroyed; Constraint (8) means that when both nodes i and j are not destroyed, Y ij The value of must be 1; By introducing auxiliary variables and linear constraints, the original nonlinear objective function is transformed into the following linear objective function: max{λ1×∑ i∈V (a i ×y i )-λ2×∑ i∈V ∑ j∈V (b ij ×Y ij )-λ3×∑ i∈V (c i ×z i )} Reconstructing this into an improved linear constraint (9): Because p ij and θ j The values ​​of are all in the open interval (0, 1), so the value on the right side is always not less than 0, that is, when y j = 0, constraint (9) always holds; and when y j = 1, constraint (9) is equivalent to D j ≥θ j , so the original constraint (1) is replaced by constraint (9); The "Big-M" method is used to linearize the constraints. The results of the transformation of constraints (4)-(6) are as follows: M is a sufficiently large positive number. For constraint (10), when z i = 0, the inequality holds regardless of the value of the right side, and when z i =1, it is equivalent to limiting ≥1, which means z j =1, That is, node i has at least one subsequent node that also has the capability of the accusation layer; Constraint (11) stipulates that node i has at least one predecessor node that also has the capability of the charge layer; Constraint (12) states that when y i ,y j ,y k The value of is 0, and When the values ​​of are all 1, z must be satisfied i +z j +z k ≥3, and z is a binary variable, so it can only be z i =z j =z k =1, thus ensuring that this constraint is functionally consistent with the original constraint (6); Through linearization processing, a mixed integer linear programming model of the problem is finally obtained, and the mixed integer linear programming model is solved.

7. The weapon target allocation method based on a multi-layer fusion network according to claim 6 is characterized in that: The simulation data is generated according to the weapon target allocation method, including the steps of: A certain number of target nodes are randomly generated in a two-dimensional plane area, and then the node type and edge relationship are determined based on certain rules, that is, the ability assignment and communication and command and control relationship assignment are assigned; the attack direction starts from the origin and extends along the y=x coordinate axis in the positive direction. The reconnaissance nodes are distributed in the attack area, and their assignment probability is positively correlated with the distance from the origin. The deeper into the enemy's central area, the more reconnaissance nodes there are; the command and control nodes, as the central control center of the kill chain or kill network, are scattered in the area far away from the defender, and their assignment probability is negatively correlated with the distance from the origin; considering that the attack nodes include long-range, medium-range and short-range nodes, it is believed that the attack nodes are relatively evenly distributed in the entire space; Along the direction of attack, the combat area is divided into four areas, namely A, B, C, and D areas; The threat value assignment intervals of the command, reconnaissance and attack nodes are For dual-capability nodes or triple-capability nodes, the threat value is processed by linear addition. For triple-capability nodes, the threat value calculation formula is: At the same time, each target has a certain damage threshold. If the weapon cannot reach the damage threshold of the target, then it is considered that the kill chain formed by the target has not been destroyed; Set the target damage threshold in the open interval (0, 1), and the attack threshold intervals of the command, reconnaissance and attack nodes are For dual-capability nodes or triple-capability nodes, since the damage threshold value range is (0, 1), a multiplication method is used for comprehensive processing. For triple-capability nodes, the threat value calculation formula is: The interconnection relationship of the combat unit includes two levels: the communication layer and the command and control layer. The communication network is a fully interconnected network. There is a direct communication relationship between any two nodes i and j. The weight of the communication relationship is a randomly generated number, represented by β ij ∈[0, 1]; at the command capability layer, whether there is a command relationship is related to the distance. In the command link of "reconnaissance-command-attack-reconnaissance", any node randomly generates a command link with all the next link type nodes according to its function; the generation probability is negatively correlated with the distance, that is, the closer the distance, the more likely it is to form a command relationship; based on the command relationship, the parameter The value of For any node i, when it has the corresponding capabilities of charge, reconnaissance, and attack, its charge capability is randomly assigned a value in a certain interval, and the value intervals are For nodes with multiple capabilities, their command and control capabilities should have a multiplied benefit. Therefore, for nodes with three capabilities, the command and control capability simulation data is generated according to the following formula: In the multi-layer network system target attack optimization problem, a total of m different weapons are considered to be distributed simultaneously, and the total number of each weapon i used is w i , the probability p of any weapon i damaging target j ij satisfy For a node with multiple capabilities, the damage probability calculation formula is:

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