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.     

Bayesian inference for a Lanchester type combat model

We undertake inference for a stochastic form of the Lanchester combat model. In particular, given battle data, we assess the type of battle that occurred and whether or not it makes any difference to the number of casualties if an army is attacking or defending. Our approach is Bayesian and we use modern computational techniques to fit the model. We illustrate our method using data from the Ardennes campaign. We compare our results with previous analyses of these data by Bracken and Fricker. Our conclusions are somewhat different to those of Bracken. Where he suggests that a linear law is appropriate, we show that the logarithmic or linear‐logarithmic laws fit better. We note however that the basic Lanchester modeling assumptions do not hold for the Ardennes data. Using Fricker's modified data, we show that although his “super‐logarithmic” law fits best, the linear, linear‐logarithmic, and logarithmic laws cannot be ruled out. We suggest that Bayesian methods can be used to make inference for battles in progress. We point out a number of advantages: Prior information from experts or previous battles can be incorporated; predictions of future casualties are easily made; more complex models can be analysed using stochastic simulation techniques. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 541–558, 2000     

战役优势参数及其应用研究

数学分析方法在军事行动计划中扮演着越来越显著的角色。对以兰彻斯特作战模型为基础的描述诸兵种合成作战的矩阵微分方程，以及由方程的控制矩阵和状态变量初值，在不解方程的情况下导出的战役优势参数进行了研究；以空战为例讨论了预测战役结局、辅助军事决策、优化兵力部署和规划火力分配等战役优势参数的主要应用；给出了对战役优势参数和数学模型的评价。     

Force-annihilation conditions for variable-coefficient lanchester-type equations of modern warfare

This paper develops a mathematical theory for predicting force annihilation from initial conditions without explicitly computing force-level trajectories for deterministic Lanchester-type “square-law” áttrition equations for combat between two homogeneous forces with temporal variations in fire effectivenesses (as expressed by the Lanchester attrition-rate coefficients). It introduces a canonical auxiliary parity-condition problem for the determination of a single parity-condition parameter (“the enemy force equivalent of a friendly force of unit strength”) and new exponential-like general Lanchester functions. Prediction of force annihilation within a fixed finite time would involve the use of tabulations of the quotient of two Lanchester functions. These force-annihilation results provide further information on the mathematical properties of hyperbolic-like general Lanchester functions: in particular, the parity-condition parameter is related to the range of the quotient of two such hyperbolic-like general Lanchester functions. Different parity-condition parameter results and different new exponential-like general Lanchester functions arise from different mathematical forms for the attrition-rate coefficients. This theory is applied to general power attrition-rate coefficients: exact force-annihilation results are obtained when the so-called offset parameter is equal to zero; while upper and lower bounds for the parity-condition parameter are obtained when the offset parameter is positive.     

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 lanchester-type combat between two homogeneous forces with supporting fires

This paper studies combat between two homogeneous forces modelled with variable-coefficient Lanchester-type equations of modern warfare with supporting fires not subject to attrition. It shows that this linear differential-equation model for combat with supporting fires may be transformed into one without the supporting fires so that all the previous results for variable-coefficient Lanchester-type equations of modern warfare (without supporting fires) may be invoked. Consequently, new important results for representing the solution (i.e. force levels as functions of time) in terms of canonical Lanchester functions and also for predicting force annihilation are developed for this model with supporting fires. Important insights into the dynamics of combat between two homogeneous forces with such supporting fires are discussed.     

Target selection in lanchester combat: Heterogeneous forces and time-dependent attrition-rate coefficients

We develop solutions to two fire distribution problems for a homogeneous force in Lanchester combat against heterogeneous enemy forces. The combat continues over a period of time with a choice of tactics available to the homogeneous force and subject to change with time. In these idealized combat situations the lethality of each force's fire (as expressed by the Lanchester attrition-rate coefficient) depends upon time. Optimal fire distribution rules are developed through the combination of Lanchester-type equations for combat attrition and deterministic optimal control theory (Pontryagin maximum principle). Additionally, the theory of state variable inequality constraints is used to treat the nonnegativity of force levels. The synthesis of optimal fire distribution policies was facilitated by exploiting special mathematical structures in these problems.     

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.     

兰切斯特方程在水雷战模型中的应用与分析

根据对抗模拟的特点,以兰切斯特方程为基础建立了水雷战的解析数学模型,并进行了仿真,得到了被封锁方的兵力组合优化曲线,以定量分析的方法阐明了该模型在战役全局和战术局部上对水雷战对抗双方的指导意义。     

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。     

基于SD的两栖坦克连水上火力支援行动分析

围绕两栖坦克连水上火力支援行动的作战特点,着眼于充分发挥两栖坦克火力和机动的优势,在"非接触作战、非线性作战"理论的指导下,运用系统动力学方法和兰切斯特战斗理论,建立两栖坦克连水上火力支援行动的系统动力学模型,探索装甲兵在应急作战中运用的新模式和作战的新方法,为指挥员科学使用这一战术手段提供有效地辅助决策支持。     

基于信息优势的决策能力分析

随着信息优势评估的不断深入,信息优势支持下的决策优势能力,已成为C4ISR系统作战效能评估的又一重要研究内容.从决策优势的基本概念入手,通过分析决策优势与信息优势的关系,提出了决策周期时间、决策质量和决策循环速度三个衡量决策优势的基本度量,并结合典型作战样式和信息优势的准确性、完备性指标,分析了这些度量的基本构成和量化方法.通过兰彻斯特方程,导出了一种可用于仿真评估的决策优势能力量化公式,进而指明了夺取决策优势应采取的主要措施.     

突袭敌方指挥所兵力需求仿真

甲方以特种部队突袭乙方师(旅)指挥所的行动是未来渡海登岛作战中的重要内容,能大大加快战役进程.战前通过侦察已知对方实力,通过作战模拟的方法确定规定时间内完成任务所需派遣特种部队的规模有决定性意义.采用指数-Lanchester理论模拟作战过程,在考虑气象、士气因素对作战过程影响的基础上,以甲方在消灭乙方50%有生力量后作战结束,通过MATLAB7 0仿真工具得出在一定的作战想定条件下甲方需要投入兵力的综合战斗力指数.     

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     

空防对抗武器装备对比优势评估模型

针对战役对抗准备和兵力部署的需要,根据兰切斯特动态方程,从作战能力入手,建立了空防对抗武器装备对比优势评估模型,提出了一种异类武器装备对比优势的定量计算方法。给出了应用该方法的具体步骤,通过算例分析,验证了模型的合理性,该模型对其他武器装备的对比优势评估具有一定的参考价值。     

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.     

联合火力打击弹药需求计算动态模型研究

弹药是火力打击和火力毁伤的基础.传统的火力毁伤弹药需求计算主要有两种思路,一种是不考虑对抗,单纯基于目标的幅员和弹药的毁伤概率静态计算弹药的需求,一种是考虑对抗,运用兰切斯特方程计算弹药的消耗,这样求得的预测结果与实际需求均有较大的差距.研究发现,将基于目标打击的弹药需求和在对抗条件下武器损耗因素结合起来考虑,可以有机地将两种弹药消耗的计算思路融合在一起,建立新的数学模型,所得结果反映了弹药实际需求与外部因素的内在关系,与实际作战更加相符,对战时的弹药供应决策具有重要意义.     

海战中信息支援的影响分析

进入信息时代,信息成为作战取胜的第一要素,信息的作用与运行流程的掌握也就成为现代作战分析与决策的重要任务。以信息在作战应用中的特点进行分类为基础,建立了新的作战信息指标体系;想定了具体的典型海战对抗模式,通过构建基于信息支援的兰彻斯特方程模型,经作战仿真模拟,定量描述了在现代海战中信息支援能力的增强对作战能力的提升的显著程度;信息支援能力的提高使敌方损耗呈指数曲线上升。     

“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.     

高精度差分格式WNND的构造及数值实验

基于二阶NND格式,通过引入Jiang和Shu的加权思想以及具有TVD性质的三阶Runge Kutta方法,构造了一种时间、空间均达到三阶精度的WNND格式。分别以波动方程、一维Euler方程和三维全Navier Stokes方程为例,通过对WNND格式的数值结果分析表明,WNND格式引起的耗散和波动较小,并且能够高精度地分辨场间断。