“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.     

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.     

On the relationship between the force ratio and the instantaneous casualty-exchange ratio for some lanchester-type models of warfare

A “local” condition of winning (in the sense that the force ratio is changing to the advantage of one of the combatants) is shown to apply to all deterministic Lanchester-type models with two force-level variables. This condition involves the comparison of only the force ratio and the instantaneous force-change ratio. For no replacements and withdrawals, a combatant is winning “instantaneously” when the force ratio exceeds the differential casualty-exchange ratio. General outcome-prediction relations are developed from this “local” condition and applied to a nonlinear model for Helmbold-type combat between two homogeneous forces with superimposed effects of supporting fires not subject to attrition. Conditions under which the effects of the supporting fires “cancel out” are given.     

An examination of the effects of the criterion functional on optimal fire-support policies

This paper examines the dependence of the structure of optimal time-sequential fire-support policies on the quantification of military objectives by considering four specific problems, each corresponding to a different quantification of objectives (i.e. criterion functional). We consider the optimal time-sequential allocation of supporting fires during the “approach to contact” of friendly infantry against enemy defensive positions. The combat dynamics are modelled by deterministic Lanchester-type equations of warfare, and the optimal fire-support policy for each one-sided combat optimization problem is developed via optimal control theory. The problems are all nonconvex, and local optima are a particular difficulty in one of them. For the same combat dynamics, the splitting of supporting fires between two enemy forces in any optimal policy (i.e. the optimality of singular subarcs) is shown to depend only on whether the terminal payoff reflects the objective of attaining an “overall” military advantage or a “local” one. Additionally, switching times for changes in the ranking of target priorities are shown to be different (sometimes significantly) when the decision criterion is the difference and the ratio of the military worths (computed according to linear utilities) of total infantry survivors and also the difference and the ratio of the military worths (computed according to linear utilities) of total infantry survivors and also the difference and the ratio of the military worths of the combatants' total infantry losses. Thus, the optimal fire-support policy for this attack scenario is shown to be significantly influenced by the quantification of military objectives.     

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.     

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     

影响高寒山地作战效能的气象环境模糊推理评估模型

高寒山地作战将是我军在西北方向的主要作战形式,而气象环境严重制约其作战效能的发挥。针对高寒山地典型作战样式进行研究,探讨西北地区气象环境与作战单元和作战样式的影响,并由此提炼评估模型的模糊推理规则,构建模糊推理框架,采用多级评估决策的思想和模糊逻辑推理系统,建立影响高寒山地作战的气象环境效能三级模糊推理评估模型。经试验仿真检验,其评估结果与客观实际基本相符,对高寒山地作战效能和武器作战性能的充分发挥以及最大限度利用战场气象环境优势,具有重要军事价值。     

An indicator of combat success

Assessing the effects of changes in weapons systems or battle tactics is difficult because of the variations in battles and the resulting instability of measures of combat effectiveness. Even under the relatively stable conditions of designed combat exercises, traditional measures may fail to reflect important battle events and dynamics. This variation in results makes the design, conduct, and evaluation of combat experiments, combat simulations, and combat training exercises a challenging endeavor, indeed. We develop and examine a combat measure of effectiveness, based on Lanchester models, which we call the battle trace. The battle trace is a measure of ongoing battle results, measured as a function of time into the battle. We describe how such measures might contribute to assessing effects of changing certain battle parameters. We suggest applications of these ideas to developing combat simulations and to enhancing combat training exercises.     

基于兰彻斯特方程的大区域防空作战效能评估模型

针对大区域防空参战兵力多,体系结构、交战过程复杂的特点,引入作战效能、战斗编组、火力毁伤及任务分配等矩阵,建立了基于兰彻斯特方程的大区域防空作战效能评估模型,包括战斗力指数、战斗编组、毁伤指数、任务分配、指挥控制和作战实力损耗等部分。每个模型的建立均通过设定相应矩阵,分析战斗力指数,得出其计算模型。并以相关抗击率以及安全率表征总体效能指标,较好地解决了对大区域防空作战效能的评估问题。模型简洁,便于理解,易于计算。     

基于复杂网络的作战描述模型研究*

针对传统作战模型重点关注兵力的毁伤,不能充分地反映信息时代作战的特点,提出了一个基于复杂网络的作战描述模型,把作战单元抽象成节点,把各单元之间的相互作用抽象成有向边,将战场描述为一个由传感器、决策器、影响器、目标四类节点组成的有向网络图。定义了网络模型的若干特征参数,把作战环看成是反映作战能力的指标,并区别分析了标准作战环和广义作战环,能更好地反映信息时代作战的特点。最后实例研究了传统作战条件下决策器连通程度的变化给作战能力带来的影响。     

Cutting,Running, or Otherwise? The US Decision to withdraw from Somalia

Although President Bill Clinton made the decision to withdraw U.S. combat troops from Somalia within three days of the disastrous Mogadishu raid that resulted in the deaths of 18 U.S. servicemen, casualties were not the only consideration. Many other factors were at play. Waning Congressional and public enthusiasm for a new “nation-building” mandate, the strategic insignificance of Somalia in the post-Cold War era, and a host of other foreign policy issues had been eroding U.S. support for the mission in Somalia for months. The “Black Hawk Down” incident merely accelerated the final rupture in public and Congressional support, forcing the president to bring the troops home. It was not a simple case of casualty aversion.     

作战体系优化方法

体系对抗已成为高技术战争的一个基本形态,它的作战能力不仅取决于体系构成要素的强弱,更取决于要素组合状态的优劣,作战体系的优化是实现最大作战效能的必要手段,也是高技术体系对抗作战迫切需要解决的问题.在给出探索性分析和作战体系的基本概念和体系评价准则后,提出了作战体系的优化方法并给出了优化过程描述.该方法主要采用探索性分析方法进行体系对抗仿真,并对仿真结果、作战任务指标、武器费用进行综合分析,最后得出最优的作战体系.     

Optimizing the US Navy's combat logistics force

We study how changes to the composition and employment of the US Navy combat logistic force (CLF) influence our ability to supply our navy worldwide. The CLF consists of about 30 special transport ships that carry ship and aircraft fuel, ordnance, dry stores, and food, and deliver these to client combatant ships underway, making it possible for our naval forces to operate at sea for extended periods. We have modeled CLF operations to evaluate a number of transforming initiatives that simplify its operation while supporting an even larger number of client ships for a greater variety of missions. Our input is an employment schedule for navy battle groups of ships operating worldwide, extending over a planning horizon of 90–180 days. We show how we use optimization to advise how to sustain these ships. We have used this model to evaluate new CLF ship designs, advise what number of ships in a new ship class would be needed, test concepts for forward at‐sea logistics bases in lieu of conventional ports, demonstrate the effects of changes to operating policy, and generally try to show whether and how the CLF can support planned naval operations. Published 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

Effective in battle: conceptualizing soldiers’ combat effectiveness*

How do we understand combat effectiveness – soldiers’ performance in battle? Despite the broad consensus that understanding combat effectiveness is important both for scholars and policymakers, there is widespread disagreement about what combat effectiveness is. More specifically, studies of effectiveness tend to focus on either the skill of soldiers in battle, or their will to fight. Yet both skill and will are essential components of an effective fighting force. This article argues that understanding combat effectiveness requires understanding both of these key components of effectiveness. In other words, combat effectiveness requires both the skill and will to engage the enemy in an organized manner. It then demonstrates the usefulness of this conceptualization by applying it to the cases of British, Indian, and Australian forces fighting the Japanese during the Second World War. Only when scholars are talking about the same concept will our understanding of the conditions under which militaries are effective in battle progress. By comparing different units fighting the same opponent under the same material conditions, I demonstrate that units vary both in their combat skill and their will to fight, and that understanding their effectiveness in battle requires analyzing both of these key factors.     

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.     

A Lanchester-type aggregated-force model of conventional ground combat

This article develops a Lanchester-type model of large-scale conventional ground combat between two opposing forces in a “sector”. It is shown that nonlinear Helmbold-type equations of warfare with operational losses may be used to represent the loss-rate curves that have been used in many aggregated-force models. These nonlinear differential equations are used to model the attrition of combat capability (as quantified by a so-called firepower index) in conjunction with a rate-of-advance equation that relates motion of the contact zone (or FEBA) between the opposing forces to the force ratio and tactical decisions of the combatants. This simplified auxiliary model is then used to develop some important insights into the dynamics of FEBA movement used in large-scale aggregated-force models. Different types of behavior for FEBA movement over time are shown to correspond to different ranges of values for the initial force ratio, for example, an attack will “stall out” for a range of initial force ratios above a specific threshold value, but it will “break out” for force ratios above a second specific threshold value. Such FEBA-movement predictions are essentially based on being able to forecast changes over time in the force ratio.     

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。     

基于改进粒子群算法的联合火力打击目标分配研究

针对联合火力打击目标分配涉及影响因素多、存在大量参数和变量的特点，建立了联合火力打击目标分配模型，采用改进的粒子群算法探讨了目标最优化分配问题。对粒子群算法中解空间存在的可能解进行编码，将可能解进行交叉、变异和选择操作得到新的粒子个体，不断迭代，从而得到可行的最优解。通过对一个作战想定的多次仿真，进一步表明了算法的可行性和有效性，为指挥员科学决策提供了依据。     

An approach for predicting digital material consumption in electronic warfare

Electronic warfare is a modern combat mode, in which predicting digital material consumption is a key for material requirements planning (MRP). In this paper, we introduce an insensitive loss function (ε) and propose a ε-SVR-based prediction approach. First, we quantify values of influencing factors of digital equipments in electronic warfare and a small-sample data on real consumption to form a real combat data set, and preprocess it to construct the sample space. Subsequently, we establish the ε-SVR-based prediction model based on “wartime influencing factors - material consumption” and perform model training. In case study, we give 8 historical battle events with battle damage data and predict 3 representative kinds of digital materials by using the proposed approach. The results illustrate its higher accuracy and more convenience compared with other current approaches. Taking data acquisition controller prediction as an example, our model has better prediction performance (RMSE = 0.575 7, MAPE (%) = 12.037 6 and R² = 0.996 0) compared with BP neural network model (RMSE = 1.272 9, MAPE (%) = 23.577 5 and R² = 0.980 3) and GM (1, 1) model (RMSE = 2.095 0, MAPE (%) = 24.188 0 and R² = 0.946 6). The fact shows that the approach can be used to support decision-making for MRP in electronic warfare.     

基于武器协同共用的无人机反潜作战效能研究

在一个暴露时间有限的反潜作战想定背景下,提出了评估该作战过程的效能指标,建立了相应的模型,并把武器协同共用条件下的作战结果与传统作战模式的结果作比较,结论表明,武器协同共用作战模式能够有效改善各项作战效能指标,是提高作战效能的有效手段。