Fitting Lanchester equations to the battles of Kursk and Ardennes

Lanchester equations and their extensions are widely used to calculate attrition in models of warfare. This paper examines how Lanchester models fit detailed daily data on the battles of Kursk and Ardennes. The data on Kursk, often called the greatest tank battle in history, was only recently made available. A new approach is used to find the optimal parameter values and gain an understanding of how well various parameter combinations explain the battles. It turns out that a variety of Lanchester models fit the data about as well. This explains why previous studies on Ardennes, using different minimization techniques and data formulations, have found disparate optimal fits. We also find that none of the basic Lanchester laws (i.e., square, linear, and logarithmic) fit the data particularly well or consistently perform better than the others. This means that it does not matter which of these laws you use, for with the right coefficients you will get about the same result. Furthermore, no constant attrition coefficient Lanchester law fits very well. The failure to find a good‐fitting Lanchester model suggests that it may be beneficial to look for new ways to model highly aggregated attrition. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.     

Attrition models of the Ardennes campaign

This paper revisits the modeling by Bracken [3] of the Ardennes campaign of World War II using the Lanchester equations. It revises and extends that analysis in a number of ways: (1) It more accurately fits the model parameters using linear regression; (2) it considers the data from the entire campaign; and (3) it adds in air sortie data. In contrast to previous results, it concludes by showing that neither the Lanchester linear or Lanchester square laws fit the data. A new form of the Lanchester equations emerges with a physical interpretation. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 1–22, 1998     

Applying Lanchester's linear law to model the Ardennes campaign

In modern warfare, many believe the decisive factor in winning a battle is seizing the right moment to shift from defense to attack, or vice versa. This paper attempts to bring that perspective to Lanchester's differential equations of warfare, and continues the application of Lanchester's linear law to the analysis of the World War II battle of Ardennes, as reported in earlier issues of Naval Research Logistics by Bracken and by Fricker. A new variable, shift time, accounting for the timing of the shift between defense and attack is explicitly included in our version of the model, and it helps obtain improved goodness of fit to historical data. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48:653–661, 2001     

“simple-approximate” battle-outcome-prediction conditions for variable-coefficient lanchester-type equations of modern warfare

This article considers combat between two homogeneous forces modeled by variable- coefficient Lanchester-type equations of modern warfare and develops new “simple-approximate” battle-outcome-prediction conditions for military engagements terminated by two different types of prescribed conditions being met (fixed-force-level-breakpoint battles and fixed-force-ratio-breakpoint battles). These battle-outcome-prediction conditions are sufficient (but not necessary) to determine the outcome of battle without having to explicitly compute the force-level trajectories, and they are characterized by their simplicity, requiring no advanced mathematical knowledge or tabulations of “special functions” for their application. Integrability properties of the Lanchester attrition-rate coefficients figure prominently in their results, and involved in their development is a generalization of Lanchester's famous square law to variable-coefficient Lanchester-type combat and several other novel mathematical developments for the analysis of ordinary differential equations. Examples are given, with the attack of a mobile force against a static defensive position (both sides armed with weapons whose firepower is range dependent) being examined in detail.     

The salvo combat model with area fire

This article analyzes versions of the salvo model of missile combat where area fire is used by one or both sides in a battle. Although these models share some properties with the area fire Lanchester model and the aimed fire salvo model, they also display some interesting differences, especially over the course of several salvos. Although the relative size of each force is important with aimed fire, with area fire, it is the absolute size that matters. Similarly, although aimed fire exhibits square law behavior, area fire shows approximately linear behavior. When one side uses area fire and the other uses aimed fire, the model displays a mix of square and linear law behavior. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 652–660, 2013     

动态贝叶斯网络的无人机炮火校射

动态贝叶斯网络(Dynamic Bayesian Networks-DBNs),是对具有随机过程性质的不确定性问题进行建模和处理的一个有力工具.提出将隐马尔可夫模型(Hidden Markov Models-HMMs)图形模式与贝叶斯网络结合起来构成DBN,将其用于无人机照相侦察情报的推理分析,决定炮火优先打击区域.首先建立动态贝叶斯网络的战场态势变化模型,而后应用HMM的推理算法获得当前隐含序列最优估计,且可预测出未来战场态势.最后应用模糊推理获得优先打击的区域号.仿真结果表明了模型的可行性.该方法有效解决了贝叶斯网络对于瞬间变化战场态势推理的不足的缺陷,为炮兵指挥员更好地运用火力,分出主次奠定了基础.     

Call center arrival modeling: A Bayesian state‐space approach

In this article, we introduce three discrete time Bayesian state‐space models with Poisson measurements, each aiming to address different issues in call center arrival modeling. We present the properties of the models and develop their Bayesian inference. In so doing, we provide sequential updating and smoothing for call arrival rates and discuss how the models can be used for intra‐day, inter‐day, and inter‐week forecasts. We illustrate the implementation of the models by using actual arrival data from a US commercial bank's call center and provide forecasting comparisons. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 28–42, 2011     

Bayesian analysis of queues with impatient customers: Applications to call centers

Queuing models have been extensively used in the literature for obtaining performance measures and developing staffing policies. However, most of this work has been from a pure probabilistic point of view and has not addressed issues of statistical inference. In this article, we consider Bayesian queuing models with impatient customers with particular emphasis on call center operations and discuss further extensions. We develop the details of Bayesian inference for queues with abandonment such as the M/M/s + M model (Erlang‐A). In doing so, we discuss the estimation of operating characteristics and its implications on staffing. We illustrate the implementation of the Bayesian models using actual arrival, service, and abandonment data from call centers. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Approximations and empirics for stochastic war equations

The article develops a theorem which shows that the Lanchester linear war equations are not in general equal to the Kolmogorov linear war equations. The latter are time‐consuming to solve, and speed is important when a large number of simulations must be run to examine a large parameter space. Run times are provided, where time is a scarce factor in warfare. Four time efficient approximations are presented in the form of ordinary differential equations for the expected sizes and variances of each group, and the covariance, accounting for reinforcement and withdrawal of forces. The approximations are compared with “exact” Monte Carlo simulations and empirics from the WWII Ardennes campaign. The band spanned out by plus versus minus the incremented standard deviations captures some of the scatter in the empirics, but not all. With stochastically varying combat effectiveness coefficients, a substantial part of the scatter in the empirics is contained. The model is used to forecast possible futures. The implications of increasing the combat effectiveness coefficient governing the size of the Allied force, and injecting reinforcement to the German force during the Campaign, are evaluated, with variance assessments. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Theoretical analysis of the general linear model for lanchester - type combat between two homogeneous forces

This paper studies Lanchester-type combat between two homogeneous forces modeled by the so-called general linear model with continuous replacements/withdrawals. It demonstrates that this model can be transformed into a simpler canonical form, which is also shown to arise from fixed-force-level-breakpoint battles modeled by Lanchester-type equations for modern warfare. Analytical expressions for the force levels for the general variable coefficient linear model with continuous replacements/withdrawals are constructed out of so-called general Lanchester functions for the model without replacements/withdrawals, for which all solutions are shown to be nonoscillatory in the strict sense. These force-level results are unfortunately so complicated and opaque that the constant-coefficient version of the model must be studied before any insights into the dynamics of combat may be analytically obtained. Thus, fairly complete results are given for the general linear model with constant attrition-rate coefficients and constant rates of replacement/withdrawal. However, the expressions for the force levels are still so complicated that we have not been able to develop battle-outcome prediction conditions directly from them alone but have had to establish general results on the qualitative behavior of solutions. A significant result (and one that greatly complicates the prediction of battle outcome) is that all solutions to the model with replacements/withdrawals are no longer necessarily nonoscillatory in the strict sense, i.e., both sides force levels can take on negative values if the force-on-force attrition equations are not “turned off” at the right time. Thus, this paper shows that the addition of continuous replacements/withdrawals to a Lanchester-type model may significantly change the qualitative behavior of the force-level trajectories. Battle-outcome prediction conditions are nevertheless given, and important insights into the dynamics of combat are briefly indicated.     

Weibull tolerance intervals associated with moderate to small survival proportions for use in a new formulation of lanchester combat theory

Given herein is an easily implemented method for obtaining, from complete or censored data, approximate tolerance intervals associated with the upper tail of a Weibull distribution. These approximate intervals are based on point estimators that make essentially most efficient use of sample data. They agree extremely well with exact intervals (obtained by Monte Carlo simulation procedures) for sample sizes of about 10 or larger when specified survival proportions are sufficiently small. Ranges over which the error in the approximation is within 2 percent are determined. The motivation for investigation of the methodology for obtaining the approximate tolerance intervals was provided by the new formulation of Lanchester Combat Theory by Grubbs and Shuford [3], which suggests a Weibull assumption for time-to-incapacitation of key targets. With the procedures investigated herein, one can use (censored) data from battle simulations to obtain confidence intervals on battle times associated with given low survivor proportions of key targets belonging to either specified side in a future battle. It is also possible to calculate confidence intervals on a survival proportion of key targets corresponding to a given battle duration time.     

战斗建模中的作战Agent模型研究

围绕战斗建模中的作战Agent模型的实现问题,通过对坦克战斗过程的分析,提出了以属性建模、能力建模和行为机制建模为要点的作战Agent建模方法,并在此基础上建立了面向实际应用的作战Agent模型;通过推演实验和与兰切斯特方程的比较实验,验证了作战Agent模型在模拟和解释战斗活动时的可行性、有效性和可操作性。     

军事欺骗战法分析的数学方法-应用

针对军事欺骗战法定量分析困难的问题,采用基于贝叶斯统计推断和博弈均衡分析的数学方法,在理论上为解决此难题提供了一种可行的方法。该方法首先定义基本概率矩阵,然后根据贝叶斯原理为交战双方构造用于统计推断的主观概率矩阵,并在由两个主观概率矩阵元素构成的偏序链与实施军事欺骗机会之间建立对应关系,再将此关系推广到基本概率矩阵,即是否存在实施军事欺骗的机会将由基本概率矩阵元素构成的偏序链决定。对理论成果的初步应用表明：通过分析由战场情报信息构成的基本概率矩阵的偏序链,可以科学地运用军事欺骗和反军事欺骗战法。     

Waypoint analysis for command and control

Command and Control (C2) in a military setting can be epitomized in battles‐of‐old when commanders would seek high ground to gain superior spatial‐temporal information; from this vantage point, decisions were made and relayed to units in the field. Although the fundamentals remain, technology has changed the practice of C2; for example, enemy units may be observed remotely, with instruments of varying positional accuracy. A basic problem in C2 is the ability to track an enemy object in the battlespace and to forecast its future position; the (extended) Kalman filter provides a straightforward solution. The problem changes fundamentally if one assumes that the moving object is headed for an (unknown) location, or waypoint. This article is concerned with the new problem of estimation of such a waypoint, for which we use Bayesian statistical prediction. The computational burden is greater than an ad hoc regression‐based estimate, which we also develop, but the Bayesian approach has a big advantage in that it yields both a predictor and a measure of its variability. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Knowledge-oriented modeling for influencing factors of battle damage in military industrial logistics: An integrated method

Modeling influencing factors of battle damage is one of essential works in implementing military industrial logistics simulation to explore battle damage laws knowledge. However, one of key challenges in designing the simulation system could be how to reasonably determine simulation model input and build a bridge to link battle damage model and battle damage laws knowledge. In this paper, we propose a novel knowledge-oriented modeling method for influencing factors of battle damage in military industrial logistics, integrating conceptual analysis, conceptual modeling, quantitative modeling and simulation implementation. We conceptualize influencing factors of battle damage by using the principle of hierarchical decomposition, thus classifying the related battle damage knowledge logically. Then, we construct the conceptual model of influencing factors of battle damage by using Entity-Relationship approach, thus describing their interactions reasonably. Subsequently, we extract the important influencing factors by using social network analysis, thus evaluating their importance quantitatively and further clarifying the elements of simulation. Finally, we develop an agent-based military industry logistics simulation system by taking the modeling results on influencing factors of battle damage as simulation model input, and obtain simulation model output, i.e., new battle damage laws knowledge, thus verifying feasibility and effectiveness of the proposed method. The results show that this method can be used to support human decision-making and action.     

Lanchester方程与火力指数的内在联系

本文从Lanchester 方程出发,利用正矩阵的特性,将矩阵特征向量与火力指数有机地联系起来。通过例子进一步说明火力指数是Lanchester 方程中威力系数概念的推广;它不仅与武器本身有关,而且依赖于整个战斗的武器配系。当某些种类的武器被摧毁,战斗格局发生一定变化时,火力指数也会发生变化。火力指数还决定于火力分配原则。     

Communications intelligence and the battle for convoy OG 71, 15–23 August 1941

In 1941, the British instituted a comprehensive system of convoys in the Atlantic. One of the first convoys, OG 71, while sailing between Britain and Gibraltar, was shadowed by German aircraft and attacked by U‐boats. In the ensuing August battle, while no U‐boats were sunk, the British lost, out of a convoy consisting of 22 vessels, two escorts and eight merchant ships. The German victory was not complete owing to the inability of the German aircraft and U‐boats to cooperate successfully as well as the failure of the U‐boats to fight the battle effectively and aggressively. The British also made many mistakes during the defence of Convoy OG 71. The surface escorts made many errors of tactics; as for the British aircraft, owing to an inability to cooperate with the surface forces, they became almost totally ineffective. One bright spot for the British during the battle, however, was communications intelligence. The battle saw the first use of high frequency direction finders and on several occasions skillful use was made of information obtained from enemy radio transmissions. In fact important lessons were learned by the British from such use of communications intelligence which would pave the way for a more effective implementation of such information in future convoy battles.     

一类带时滞的非线性兰彻斯特战斗模型的分析

基于非线性兰彻斯特方程的一般形式和现代战争的特点,考虑到时间因素在现代战争中的巨大作用,建立并讨论了一类带时滞的非线性兰彻斯特战斗模型。通过定性分析,得到了模型的平衡点及其稳定性,证明了原模型解的存在唯一性,并给出了解的存在区域。战例分析结果表明该模型能用来描述现代战争。因此,该模型对研究现代战争的战斗进程、武器发展规划、现代军事练兵等都具有一定的参考价值。     

从三次战役看《孙子兵法》在《三国演义》中的体现

《三国演义》是中国古代军事战争小说的经典之作,它深受《孙子兵法》的影响。文章以赤壁之战、智取荆州、彝陵之战为窗口,以期窥探《三国演义》中所体现的《孙子兵法》的军事思想。     

基于ANFIS的目标毁伤等级预测模型

分析了神经网络和模糊推理系统的优缺点,研究了自适应神经模糊推理系统（ANFIS）结构模型及后向传播和递归最小二乘算法相结合的混合算法．在分析了目标毁伤等级主要影响因素的基础上,构建了目标毁伤等级预测ANFIS模型,利用毁伤试验样本数据训练该模型,得到了与实际一致的目标毁伤等级,并将预测结果与基于BP神经网络的预测结果进行了仿真对比分析．仿真结果表明,该目标毁伤等级预测模型能够准确地预测出目标的毁伤等级,并且其预测精度较BP神经网络方法高,为目标毁伤等级预测提供了一种有效的方法．