A model for the determination of optimal interfiring times

Various models of stochastic duels with round dependent hit probabilities have appeared in the literature [1]. However, none of them analyzed the effect interfiring times will have on the hit probability. In this paper we formulate a model for a marksman versus a passive target where the probability of a hit at a given round is a function of the interfiring times. We show how to solve for the optimal interfiring times and prove that under certain assumptions the optimal rate of fire is a non-decreasing function of the round fired.     

Stochastic duels involving reliability

The reliability of weapons in combat has been treated by Bhashyam in the context of a stochastic duel characterized by fixed ammunition supplies. negative exponentially distributed firing times and weapon lifetimes, and a fixed number of spare weapons for each duelist. The present paper takes a different approach by starting with the fundamental duel of Ancker and Williams, characterized by unlimited ammunition and by ordinary renewal firing times, and adding to it weapon lifetimes which can be functions of time or of round position in the firing sequence. Probabilities of winning and tieing are derived and it is shown that under certain conditions the weapon lifetimes are equivalent to random time and ammunition limits.     

Stochastic duels with round-dependent hit probabilities

The effect of round dependent hit probabilities in the fundamental stochastic duel are examined. The general solution and several specific examples are derived where one side's hit probabilities are improved from round to round. For these specific cases the advantages of round to round improvement are explicitly displayed.     

The two-on-one stochastic duel

The one-on-one stochastic duel is extended to the general two-on-one duel for the first time. The state equations, win probabilities, mean value, and variance functions are derived. The case where one side has Erlang (2) firing times and the other is negative exponential is compared with the corresponding “Stochastic Lanchester” and Lanchester models to demonstrate their nonequivalence.     

New results on a stochastic duel game with each force consisting of heterogeneous units

Two forces engage in a duel, with each force initially consisting of several heterogeneous units. Each unit can be assigned to fire at any opposing unit, but the kill rate depends on the assignment. As the duel proceeds, each force—knowing which units are still alive in real time—decides dynamically how to assign its fire, in order to maximize the probability of wiping out the opposing force before getting wiped out. It has been shown in the literature that an optimal pure strategy exists for this two‐person zero‐sum game, but computing the optimal strategy remained cumbersome because of the game's huge payoff matrix. This article gives an iterative algorithm to compute the optimal strategy without having to enumerate the entire payoff matrix, and offers some insights into the special case, where one force has only one unit. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 56–65, 2014     

Damage accumulation and probability of kill for gun and target engagements

Determination of the gunfire probability of kill against a target requires two parameters to be taken into consideration: the likelihood of hitting the target (susceptibility) and the conditional probability of kill given a hit (vulnerability). Two commonly used methods for calculating the latter probability are (1) treating each hit upon the target independently, and (2) setting an exact number of hits to obtain a target kill. Each of these methods contains an implicit assumption about the probability distribution of the number of hits‐to‐kill. Method (1) assumes that the most likely kill scenario occurs with exactly one hit, whereas (2) implies that achieving a precise number of hits always results in a kill. These methods can produce significant differences in the predicted gun effectiveness, even if the mean number of hits‐to‐kill for each distribution is the same. We therefore introduce a new modeling approach with a more general distribution for the number of hits‐to‐kill. The approach is configurable to various classes of damage mechanism and is able to match both methods (1) and (2) with a suitable choice of parameter. We use this new approach to explore the influence of various damage accumulation models on the predicted effectiveness of weapon‐target engagements.     

直升机载制导火箭弹命中精度评定研究

针对直升机载制导火箭弹特点,根据命中精度概念,分析出直升机载制导火箭命中精度的合理表示方法;考虑研制成本、试验条件等因素,提出研制阶段基于点估计和区间估计的制导火箭命中精度评定方案,并通过试验数据验证其正确性。鉴于产品批量生产交付阶段抽检质量时实弹难以统计弹着点和用弹数量有限的困难,提出了CEP评定转换为命中率考核,采用截尾序贯法,并进行优化的评定方案。     

Hit probability for a chain-like series of shots—a simple formula

To find the hit probability for a series of correlated shots is, with conventional methods, a very tedious job even for the simplest statistical distributions. Experimental data, however, commonly show the shots lying chain-like on smooth curves about the target. For this case this paper introduces a new method of finding the hit probability without the use of correlation coefficients. The method is based on a transformation, where the target is transformed into a point and each bullet into an area the size of the target.     

Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

A simultaneous non‐zero‐sum game is modeled to extend the classical network interdiction problem. In this model, an interdictor (e.g., an enforcement agent) decides how much of an inspection resource to spend along each arc in the network to capture a smuggler. The smuggler (randomly) selects a commodity to smuggle—a source and destination pair of nodes, and also a corresponding path for traveling between the given pair of nodes. This model is motivated by a terrorist organization that can mobilize its human, financial, or weapon resources to carry out an attack at one of several potential target destinations. The probability of evading each of the network arcs nonlinearly decreases in the amount of resource that the interdictor spends on its inspection. We show that under reasonable assumptions with respect to the evasion probability functions, (approximate) Nash equilibria of this game can be determined in polynomial time; depending on whether the evasion functions are exponential or general logarithmically‐convex functions, exact Nash equilibria or approximate Nash equilibria, respectively, are computed. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 139–153, 2017     

飞行器易损性指标计算的基本理论与方法(Ⅰ)

给出了与易损性指标计算有关的一些基本概念,推导了在单击中情况下无余度非重叠模型、无余度重叠模型、余度非重叠模型、余度重叠模型４个模型及其相应的易损性指标计算公式。若干算例说明了方法的有效性。     

Stochastic duels with displacements (suppression)

The stochastic duel is extended to include the possibility of a near-miss on each round fired, which causes the opponent to displace. During displacement, the displacing contestant cannot return the fire but is still a target for his opponent. An alternative interpretation of this model is to consider the displacement time as the time a contestant's fire is suppressed by his opponent's fire and that he does not move, but merely ceases fire temporarily. All times are exponentially distributed.     

对目标打击概率的数学模型及分析

通过分析识别跟踪攻击目标的过程,建立了攻击目标打击概率的数学模型和基本算法,通过计算机采用蒙特卡罗法模拟计算了不同条件下对目标的打击概率,以此研究影响打击概率的因素.计算结果表明,武器系统的作战半径和威力半径两个因素对打击概率影响较大,误差因素在一定范围内影响较小,各参数之间存在优化匹配的空间.     

A duel complicated by false targets and uncertainty as to opponent type

An attacker, being one of two types, initiates an attack at some time in the interval [-T, 0]. The a priori probabilities of each type are known. As time elapses the defender encounters false targets which occur according to a known Poisson process and which can be properly classified with known probability. The detection and classification probabilities for each type attacker are given. If the defender responds with a weapon at the time of attack, he survives with a probability which depends on the number of weapons in his possession and on attacker type. If he does not respond, his survival probability is smaller. These probabilities are known, as well as the current number of weapons in the defender's possession. They decrease as the number of weapons decreases. The payoff is the defender's survival probability. An iterative system of first-order differential equations is derived whose unique solution V₁(t),V₂(t),…,V_k(t) is shown to be the value of the game at time t, when the defender has 1, 2,…, k,… weapons, respectively. The optimal strategies are determined. Limiting results are obtained as t→-∞, while the ratio of the number of weapons to the expected number of false targets remaining is held constant.     

减少航程损失的尾流自导新方案

提出了一种基于三元换能器组的尾流自导导引方案,它对尾流的左右舷判别能力使其可以采用较小的尾流出入角度,从而可以较大程度降低鱼雷航程损失.仿真结果表明,该方案可以在不降低鱼雷末弹道命中概率的前提下,将尾流导引段的航程损失由通常的15%降低到5%左右.     

Periodic inspection policy with preventive maintenance

This article considers a modified inspection policy with periodic check intervals, where the unit after check has the same age as before with probability p and is as good as new with probability q. The mean time to failure and the expected number of checks before failure are derived, forming renewal-type equations. The total expected cost and the expected cost per unit of time until detection of failure are obtained. Optimum inspection policies which minimize the expected costs are given as a numerical example.     

射击门与命中概率分析

详细讨论了火控系统设置射击门前后的命中概率,针对不同的距离区间,给出了在射击门下命中概率的计算公式。     

射击门与命中概率分析

详细讨论了火控系统设置射击门前后的命中概率,针对不同的距离区间,给出了在射击门下命中概率的计算公式。     

直升机悬停投雷命中概率仿真

为研究投雷阵位对直升机悬停投雷命中该率的影响,根据直升机悬停投雷过程,建立投雷参数求解模型、鱼雷水下弹道模型和潜艇机动模型,仿真不同阵位下直升机悬停投雷的命中概率。仿真结果表明投雷阵位对鱼雷命中概率有较大影响,直升机投雷存在一最优投雷区域。仿真结论对空投鱼雷作战使用有一定指导意义。     

坦克火控首发命中率试验脱靶弹数据处理方法

通过对出现脱靶弹情况下的散布特征参数估计方法的研究 ,建立了正态双边截尾样本均值和方差估计数学模型 ,并给出了数值计算方法 ,为坦克火控系统首发命中概率试验脱靶弹数据处理提供了理论依据。     

直升机装备子母弹对坦克群作战的效能分析

分析了几个影响坦克威胁程度的主要因素,采用新的归一化效用函数法,对各目标的威胁程度进行评估,确定智能子母弹的攻击瞄准点.然后建立智能子母弹对目标攻击过程的数学模型及毁伤概率的计算模型,通过蒙特卡洛方法计算各目标命中的子弹数目.综合目标的特性进行多次仿真,得出了子母弹对整个坦克群目标的作战效能.最后模拟实现了整个攻击过程...