Survivability Optimization of the Silo Group Deployment by Agent Modeling and Simulation

Survivability is an important indicator for the nuclear weapon deployed by silo group. From the view of survivability optimization, firstly, a brief analysis for Aside factors, B-side factors and environmental factors that affect survivability of the silo group is completed, extract the main factors and make corresponding limitations. Secondly, agent modeling method is introduced to establish the silo agent model, detail the model rules, analyze the changes in the deployment status of the silo group using the Markov process, and construct an objective function model for the optimization of the silo group survivability. Finally, simulation based on the objective function is performed, and the simulation result shows that the problem of survivability optimization deployment of the silo group get some sort of solution, which may provide reference for similar military target deployment problems.


Introduction
Looking at the development trajectory of the nuclear countries in the world and increasing the overall deterrent capability and the second counterattack capability of nuclear weapons have always been a major thread for the research and development of nuclear weapons in various countries. The overall deterrence capability, in turn, depends to a large extent on reliable secondary counterattacks. In order to improve the ability to counterattack twice, one side must ensure that after the first wave of attacks by the enemy, it can still retain enough nuclear weapons. This makes the survivability of nuclear weapons an important indicator of the effectiveness of nuclear weapons. Silo-based nuclear missile also place great emphasis on survivability. During the Cold War between the United States and the Soviet Union, both countries invested a lot of manpower and material resources to upgrade the silos and improve the protection capabilities of the silos in technical point of view. But with the arrival of the bottleneck of the silo reinforcement technology and the constant renewal of the enemy's attack means that this idea is considered to be extremely low cost-effectiveness ratio and extremely passive [1~2]. As a result, the cluster deployment pattern of silos has attracted more attention.
Taken together, the silo group deployment pattern has the following salient features: it facilitates the deployment of false targets within the silo group; it can technically reduce the engineering protection requirements for the silo itself; it can achieve cluster deployment of multiple nuclear missiles. The silo group deployment can be thought as a tactical remedy technique to achieve a deployment style with survivability. Therefore, it is of great significance to study the placement of silos in the silo group to optimize survivability.

Analysis of Influencing Factors on Survivability of the Silo Group
The silo group survivability refers to the ability of nuclear missile silos to maintain technical and tactical performance in a specific battlefield environment and under attack conditions. The core of the silo group survival is the survival of nuclear missile the silo group. The factors that affect the survivability of the silo group are complex and diverse. Both static and invariable, real-time updates, man-made and objective. It can be summed up in three categories: A-side factors, B-side factors and environmental factors (A-side is the side to deploy the silo group and B-side is the side to attack the silo group).
A-side factors. Including the method of the silo group deployment, the proportion, structure and scale of the real and false targets within the silo group, the camouflage level, the engineering protection capability, the force's rapid response ability and the readiness level, etc. B-side factors. It is mainly Bside's reconnaissance ability and strike ability. The stronger the reconnaissance ability and the strike ability of B-side is, the more detrimental to the survival of the silo group. Environmental factors. It mainly includes weather, topography, and electromagnetic environment and so on. In general, weather conditions with low visibility and complex topographic features are helpful to improve the survivability of silo group.

Limiting factors
In order to study the deployment of silos in the silo group and improve the survivability of the silo group, some factors must be limited.
Limiting B-side's reconnaissance measures. The reconnaissance methods that B-side may take mainly include: close to reconnaissance, aerial reconnaissance and satellite reconnaissance. Although close to reconnaissance can obtain the most accurate information, while the fact that the silo group is located in the territory of A-side and is heavily guarded, it is dangerous and difficult for the enemy to infiltrate and carry out reconnaissance activities; the method of aerial reconnaissance is better in flexibility and pertinence, but the B-side detectors rely on the platform (mainly reconnaissance aircraft). Under the condition that the A-side air defense system is complete and the wartime is highly guarded, the B-side platform is difficult to penetrate and the reconnaissance cost is relatively high. The accuracy of satellite reconnaissance is not so high, but it has the advantage of accumulation of reconnaissance information for fixed targets, and the security of the platform is better. The analysis believes that the satellite reconnaissance threat is the most important and most realistic threat. The reasons are as follows: First, the B-side satellite reconnaissance technology is increasingly advanced, and some satellites have an accuracy of 0.1 meters. Second, the number of B-side satellites is huge and military and civilian satellites can be mobilized to carry out intensive reconnaissance activities when necessary. Therefore, the B-side's reconnaissance method is limited to satellite reconnaissance.
Limiting B-side's attack methods. Possible attack methods that B-side may take against A-side's silo group mainly include: air-drop drilling of ground bombs, precision guided attack outside the defense zone, and nuclear missile attack. Among them, the method of air-dropping ground-bombing is highly targeted and has a good destruction effect. However, the B-side platform is also threatened by the A-side's airdefense system, which poses a greater risk of action. The method of accurate guided attack, although highly safe, However, when A-side deploys anti-missile weapons in key areas, its attack effect will be reduced and the cost will increase [3]. Although the accuracy of nuclear missile attack is poor, it has a long range and a good damage effect. It is not easily intercepted by B-side. After analysis, it is considered that the B-side nuclear missile attack is the most important attack method. The reasons are as follows: First, the B-side rocket carrier technology and guidance control technology are increasingly advanced, and the accuracy of hitting can be further improved. Secondly, nuclear missile attacks have a good effect on the damage, and underground targets can also be subjected to a nuclear-burst attack, which poses a greater threat. Finally, the silo group deployment areas are mostly away from densely populated areas, which can reduce collateral damage and humanitarian pressure from the nuclear missile attack. Therefore, the B-side attack is limited to nuclear missile attacks. According to the above assumption, the survival confrontation problem of the silo group has been simplified, laying the foundation for the modeling of the later.

Silo Agent Modeling
Agent modeling is one of the commonly used methods for bottom-up modeling of complex systems. It can analyze the operating mechanism and emerging characteristics of complex systems and is widely used in the fields of economy, military, and management [3~5].
For the silo group survivability agent modelling, the silos in the silo group are first divided into three categories, namely real silos, empty silos and false silos. Among them, real silos are silos that have launching capabilities and are equipped with nuclear missiles; empty silos have launching capabilities but there are no nuclear missiles in silos, which can be considered as spare silos for real silos; and false silos are silos that do not have launching capabilities, they are purely false targets. When a nuclear missile is moved from a real silo to an empty silo, the original real silo becomes an empty silo, and the empty silo becomes a real silo. Overall, this movement will not change the number of real silos and false silos, but it will change the distribution of real silos and empty silos in the silo group. At the same time, in order to achieve better shielding effect, the position of the false silos need to be adjusted. It can be imagined that when there are a large number of silos in the silo group, the conversion of such real silos and empty silos is very complex, and the distribution of points brought about by this is also very complicated.
In order to facilitate the study, it is considered that each silo (including real silos, empty silos, and false silos) in the silo group is an agent that can move in point, and the missile's moving process can be characterized by the movement of the agent. The movement of the silo is used to describe the movement of the missile and the changes in the position of the real silo and the empty silo. The establishment of silo agent model is shown in Fig.1.

Figure 1. Agent model
In a grid coordinate system, the position of any silo i in a silo group in the moment t is expressed as ( , ) t i i W x y , and define the movement rules of the agent as follows: Each agent can only move one grid to the neighboring grid within one beat (it can also remain still). If it moves up one grid, the new position is , the upper right is neighboring cells of each agent. That is, there must be at least one space between any two silos (to prevent B-side's one missile from destroying two silos at the same time); all agents are only required to move within the deployment areas.
Through the above analysis, the movement of each agent has the following characteristics: The status of the next beat of the silo agent is related only to its current status, and has nothing to do with the status of the previous beat, that is, it has no aftereffect, so the law of agent's movement can be approximated as the Markov process [6] on a two-dimensional plane.
Among them, t W is the status of distribution at the moment t , t K is the status of the agent at the moment t . Since each agent's next position status space has 9 elements (constraints are not considered here), denoted as   For all agents in the entire silo group, on the basis of the above mobility rules, the constraints of not exceeding the specified range and at least one grid interval between two adjacent agents are added. The one-step transition probability of all agents in the silo group is described as a set:   Among them i k is the number of elements for silo i in the next status space, S is the total number of silos in the silo group, according to the characteristics of the Markov process, the next status of a certain silo i is: Assume that the current status of the silo group is: ) ..
The next status of the silo group is:

Silo group survivability optimization modeling
The survivability optimization problem of the silo group is to maximize the preservation of nuclear missiles in the silo group. Assume the nuclear missile number in a silo group is N (real silos number are also N ), the number of empty silos is M ,number of false silos is F , the total number of silos is S N M F    , and use H indicates the number of nuclear missiles survived by B-side's attack. The survivability of the silo group is usually measured in the form of probability. The survivability of the silo group is: In the equation, i n is the judge value of the real silo; i P is the survival probability of real silo i ; 0 P is the probability threshold to judge the real silo survive or not; T P is the real silo's detected probability; . Analysis believe that the probability of real silo being detected and the probability of being destroyed and attacked both have nothing to do with the existence of empty and false silos. The former depends on whether the B-side's satellite scans the real silo, and the latter depends on the B-side weapon's attack ability and the A-side real silo's protection capabilities. Therefore, the A-side real silo is basically in the case of unchanged parameters, the former depends on B-side's reconnaissance strength and environmental protection, while the latter depends on B-side's attack strength and environmental protection. For the strength of B-side's reconnaissance and attack, according to expert experience, the two may be blurred. The three descriptions and three quantized values corresponding to them shown in Table 1 and Table 2. The influence of the environment on detection and damage is mainly due to the influence of topography and landform. According to the experience of experts, for common three deployment landforms, the values of the factors are shown in Table 3.  Table   Ⅲ, so / For equation (2), Assume the following analysis: It is assumed that all empty silos and false silos in the silo group have a cover effect on the real silo and show as cooperative force; between the real silo show as competitiveness force, and the strength of cooperation or competition affects the B-side identification, and this impact can be characterized as a competitive or cooperative factor  ,  get positive value when cooperation and negative value when competitiveness [1~6]. In addition, it is assumed that the similarities between empty silos and real silos in the silo group are greater than the similarities between false silos and real silos, showing a greater cooperation force, and the camouflaging level of real silos and empty silos is a pair of interrelationships with the false silo's camouflaging levels, all to increase similarity. So factor  is: Among them,  is the recognition factor from terrain and landscape recognition; the value of . Based on the above analysis, the survivability equation (9) is improved to: The objective function of silo group survivability optimization is to obtain max P (the maximum value of P ) .When the number of real silos in a silo group is fixed, the objective function is equivalent to obtain the maximum value max H .

Model Simulation and Result Analysis
According to the agent model and optimization model established above, the agent simulation flow based on silo group survivability optimization is designed as shown in Fig.2 Outside initial Simulation carry out times: sim=0； Optimal dis tribution in m ultiple sim ulation : Wkmax=W0； Survival probability when achieve optim al di stribution: Pkmax=P0； Sum of beats when achieve optimal distribut ion: Ss um=0； Average of beats when achieve optimal distribution: Saverage=0； Inside initi al Number of beats that agent moved : step=0； Number of beats when achieve optim al s urvival probability :Mstep=0； Optimal dis tribution in once si mulation: Wmax=W0； Survival probabi lity when achieve Wmax : Pm ax=P0； The main input parameters are the proportion, number, and initial distribution of silos in the silo group, agent parameters, B-side reconnaissance and attack conditions, deployment terrain, simulation times, and the number of simulated agent movements per simulation.

Simulation
Using the Anylogic8.2.3 software platform, a simulation model is set up, where the input parameters are set as follows: There are 12 real silos, 13 empty silos, and 7 false silos in the silo group. The initial distribution is shown in Fig.3 (Z in the figure represents the real silo and K represents the empty silo. J represents the false silo). The level of reconnaissance strength and attack strength of B-side is medium, and the simulation times is 100. The number of movements of each simulated agent is divided into two levels, 1000 and 2000. The deployment terrain is divided into three kinds (open areas, jungle lands and mountain forest lands). The simulation results are shown in Table 4.
The distribution of silo group survivability optimized for each group of experiments is shown in Fig.  4 and Fig. 5. Comparing Fig.3, Fig.4 and Fig.5, it can be seen that the initial distribution of silos in the silo group is approximately a cluster distribution, but after the simulation, the distribution of silos in Fig.4 and Fig.5 tend to be evenly distributed, and this is a change of entropy increasing. It shows that the overall distribution of different types of silos in the form of cross-mixing is beneficial to the improvement of the survivability of the silo group.

Analysis of Simulation Results
Comparing the two sets of experiments shown in Fig. 4 and Fig.5, and comparing the data in Table 4, it can be seen that comparing 2000 times to 1000 times, the average number of beats needed to reach the optimal distribution is larger, and finally reaches the optimal distribution. The higher survival probability indicates that the agent's moving number of beats objectively limits, the optimal distribution that the simulation experiment can achieve. It can be imagined that if the simulation times is increased, the optimal distribution of the survivability of the silo group may also be updated.  Assuming that the jungle land is a more complex landform than the open land, and the mountain forest land is a more complex landform than the jungle land, then according to the results of the optimized distribution of the landforms in Table 4, it can be seen that with the complexity of topography and landform, The average number of beats required to achieve an optimal distribution increases, and the probability of survival is higher when the optimization distribution is finally achieved. Looking at Fig. 4 and Fig. 5, as topography and geomorphology become more complex, the distribution of silos in the silo group tends to spread evenly across the surface evenly distributed, which explain that deploying silo group in open areas should be deployed in a planar manner, while In complex terrain and landscapes, deployment in strips is conducive to improving survivability.

Conclusion
The simulation results show that the method of agent modeling and simulation can be used to obtain a plan to deploy the silo group in different terrains to optimize the survivability. It has guiding significance for the deployment of the silo group, and this method of using agent modeling to find the optimal deployment problem Can be extended to similar military deployment research.
It must be pointed out that, first of all, the research in this paper has some artificial restrictions on the terrain environment and agent rules. It only considers a certain single terrain, and does not involve multiple terrain crossover situations. Secondly, the proposed silo model in this paper is not really an agent. It is only a mobile semi-autonomous force model. Its movement rules are also a blind exploration, an exhaustive selection method, which results in experimental redundancy [7]. Finally, this paper only studied the distribution of silos within a single silo group and did not consider the situation of multiple silo groups.