An Extension of the Lanchester Square Law to Inhomogeneous Forces with an Application to Force Allocation Methodology

This paper extends the classic Lanchester representation of a battle between an Attacker and a Defender to the case where the former possesses two weapon types, and the latter one. The Defender's force levels can be reduced by both types of Attacker weapons but at different attrition rates. The converse also applies. Victory is assumed to be by total annihilation of the Defender, towards which the battle progresses in one of two ways: where the Attacker is left with some of both weapon types, or where he retains some of one type only. The analysis is then extended to cover the optimization problem of minimizing the Attacker's initial resources. The implications for casualties are also discussed.     

The effects of splitting exponential stochastic Lanchester battles

The Exponential Stochastic Lanchester (ESL) model of attrition is well known and can be found in many combat models. This paper is concerned with exploring the effects of applying the ESL attrition model to battles which have been split into different arrangements of smaller engagements or fire fights. These fire fights were mainly resolved using a Markov chain formulation of the ESL model. Some interesting and non-intuitive trends were uncovered. It is shown that the ESL model predicts an increasing advantage for the defending side as a battle is split into a larger number of smaller parallel engagements.     

Incorporating fractal concepts into equations of attrition for military conflicts

Using an agent-based model as a ‘battlefield laboratory’, we explore equations of combat attrition which extend beyond the conventional Lanchester equations and which endeavour to encapsulate the more complex aspects of warfare. Our approach compares predictions from candidate attrition equations with casualty data generated artificially from an agent-based model. For situations where the initial regimented structure of the fighting forces breaks down, introducing fractal concepts into the attrition equations proves effective at encapsulating complex aspects of the battle; with details in the time dependence of the casualty data able to be reproduced. Furthermore, measuring the fractal dimension of a fighting force's spatial distribution on the battlefield provides a sensitive probe of the combatants’ behaviour. Precise times at which key events occur during a battle can be pinpointed. This study furthers the body of work which considers warfare as a complex adaptive system and where fractal-like structures are expected to emerge.     

Semi-dynamic modelling of heterogeneous land combat

We propose a semi-dynamic approach to tactical level land combat modelling from the attacker's viewpoint. Our approach decomposes a battle between heterogeneous forces into stages and mini battles. For each mini battle in a stage, we use three models: a mathematical programming model for optimizing force allocations, a Lanchester simulation model for predicting whether or not the stage targets are reached under the allocations, and a model for weapon effectiveness update from one stage to the next. These models interact with each other within the framework of a decision support system to help the user with allocation decisions as well as prediction of force requirements to win the battle.     

Naval Combat Logistics Support System

The subject matter is the sustainability of naval forces in combat and the intimate relationships between tactics and logistics. The observation is made that it has been a long time since the United States Navy has worried about logistics in combat and that to some extent combat logistics have become ‘somebody else's’ responsibility. It is argued that combat logistics is inescapably the force commander's responsibility and its dimensions are sketched. Necessarily, the amount of data and calculation is extensive if an accurate picture of the combat logistics states of the force are to be maintained and predictions of future states are to be made. It is in the predictions of future states that the interactions between tactics and logistics become most apparent. The data storage and retrieval and computational aspects suggest a microprocessor-based support system. BGLCSS (battle group logistics co-ordinator support system) is a program of combat logistics data, event routines, and algorithms designed to allow the logistics co-ordinator to track and predict force combat logistics states and provide timely and meaningful insight to the force commander.     

The Lanchester (<Emphasis Type="Italic">n</Emphasis>, 1) problem

We discuss the background of the Lanchester (n, 1) problem, in which a heterogeneous force of n different troop types faces a homogeneous force. We also present a more general set of equations for modelling this problem, along with its general solution. As an example of the consequences of this model, we take the (2, 1) case and solve for the optimal force allocation and fire distribution in a (2, 1) battle. Next, we present examples that demonstrate our model's advantages over a previous formulation. In particular, we point out how different forces may win a battle, depending on the handling and interpretation of the model's solution. Lastly, we present a variant of the Lanchester square law which applies to the (2, 1) case.     

APPLICATION OF LANCHESTER COMBAT MODEL IN THE ARDENNES CAMPAIGN

Since the appearance of F.W. Lanchester's pioneering development of combat models (Lanchester [1916]) during the first World War, his work has been extensively modified to represent a variety of competitions, ranging from isolated battles to entire wars. A great deal of attention has been devoted to applying such models to actual battle situations (Engel [1954], Hueter [1978], Bracken [1995]). The consideration is to focus on the validity of various Lanchesterian models, i.e., to show that there exist models which do indeed fit the data of actual confrontations (Coleman [1983]). The objective of the present paper is to use the “classical” Lanchester combat model for two reinforced armies having homogeneous, linear nonbattle loss rates, with specific application to the Ardennes campaign of World War II (December 15, 1944–January 16, 1945). We observe that the fit is remarkably good. Moreover, using data from the early phases of the encounter, we also found that there is certain agreement between the predicted and actual force strengths of the latter stages of the battle.     

An analytical model for close combat dynamics

A coherent dynamic combat model is developed from basic principles. The governing set of equations has a striking resemblance to the continuity equation in fluid dynamics with an additional term for the losses of combat units. The salient features of the model are a moving battle front, the replenishment of losses, and the withdrawal of combat units while others are still engaged. A basic example shows that the often used force ratio of three can produce a frontline movement up to 90% of the speed of the attacker. Another example simulates a well documented battle from the American Civil War. It is shown that terrain influences and the absence of reconnaissance had a large adverse effect on the outcome of the battle for the Confederate forces.     

Firepower concentration in cellular automaton combat models—an alternative to Lanchester

One of a new generation of combat models is examined to determine how its behaviour differs from older approaches based on first-order linear differential equations. This new methodology, which uses ‘cellular automaton’ or ‘agent-based’ models, has been around for a decade, prompting closer scrutiny. The method gives entities within a combat simulation the autonomy to react to circumstances in their local area. The reaction is determined by each entity's ‘personality’. It is found that the automata tend to either fight as a massed force, or form dispersed patterns of clusters within clusters. Such a pattern is known as a ‘fractal’. By adopting this pattern, a non-intuitive relationship between the kill probability of the automata and the force attrition rate develops. This provides a compelling example of how the result presented by earlier workers—that automaton models may evolve into fractal distributions—can have significance for operational researchers.     

A Proposed Entropy Measure for Assessing Combat Degradation

This paper describes a method for assessing the combat degradation of an army, based on the Shannon entropy measure. It is shown that this measure is useful for analysing combat degradation due to casualties, since it is in broad agreement with the values established from past battle experience. It is also shown that the measure can be used to evaluate the degree of confidence in the outcome of training manoeuvres when weapons simulators are employed.     

综合作战实力建设优化模型研究

本文根据战争(战斗)的风险与收益共存准则、作战理论以及现代资产投资组合理论(Portfolio理论),提出了综合作战实力建设优化的概念,并建立了综合作战实力建设优化模型,解决了在有限的资源情况下进行战争(战斗)各种力量建设的最优投资比例,使有限的资源发挥最大的效益。     

Graphical displays of synchronization of tactical units

To maximize combat readiness, the U.S. Army employs highly instrumented combat ranges for training troops under the most realistic possible conditions. Many of these ranges accommodate force-on-force battles, including simulated firings of weapons and of kills against the opponent; the physical variables used in these simulated kills (times of events, locations of players) are then available for computer replays, as well as investigations of the effects of changes to some battle details. This paper describes the use of player and event time-location data to visually portray some aspects of synchronization of forces, one of the classic (and current) battlefield tenets.     

Measuring the effects of knowledge in military campaigns

A particularly difficult problem in command and control is that of identifying the relationship between intelligence, decision and combat outcome. The problem centres on three things: (1) an adequate representation of the situation confronting the commander on the battlefield; (2) an adequate measure of combat outcome; and (3) an appropriate metric linking knowledge of the first to the second. In this paper, we focus on the third of these by developing a measure of the knowledge possessed by the commander at the time he takes his decision and by relating this to combat outcomes. Combat outcomes are represented using traditional attrition-based metrics and the combat situation is simply the size, location and identity of enemy units. Therefore, the possible number of identified enemy units arrayed against the friendly commander constitutes the set of hypotheses on alternative situations. Surveillance assets provide the commander with evidence that is used to update the probability distribution. Knowledge if then represented as the product of two components: residual knowledge, the knowledge gained from the updated probability distribution, and detection knowledge, the knowledge from the detection itself. Information entropy was used to develop a metric that reflects the degree to which the commander understands the situation confronting him. The metric was applied in a UK Ministry of Defence study of a proposed Airborne STand-Off Radar (ASTOR) to measure the effects of improved surveillance on combat outcomes.     

作战时耗指派矩阵取胜指派矩阵兵力耗损指派矩阵的一体构造

指派矩阵构造是指派问题应用研究的难点,在作战应用领域展开指派矩阵构造专题研究.文中回望了1914年Lanchester关于"兰氏"平方律作战过程取胜条件与剩余兵力的分析结果,以及1996年本文第一作者提出的关于"兰氏"平方律作战过程存在胜负的情况下其作战持续时间计算的数学模型,提出了关于"兰氏"平方律作战过程在作战双方势均力敌的情况下作战持续时间的数学模型.综合运用上述的已有理论与新建理论,建立了取胜矩阵、时耗矩阵、兵力耗损矩阵的一体构造模型.该一体构造模型从作战系统的4类可知数据出发,对于具体的多部队参战的作战过程均能构造出具体的取胜、时耗、兵力耗损数值矩阵.最后给出了取胜、时耗、兵力耗损矩阵的一个一体构造实例,并运用(n×m)-k缺省指派问题理论对该实例求得了其最多K胜条件下的最短时限最少耗费缺省指派最优解.     

Bits or shots in combat? The generalized Deitchman model of guerrilla warfare

We generalize Deitchman’s guerrilla warfare model to account for trade-off between intelligence (‘bits’) and firepower (‘shots’). Intelligent targeting leads to aimed fire; absence of intelligence leads to unaimed fire, dependent on targets’ density. We propose a new Lanchester-type model that mixes aimed and unaimed fire, the balance between these being determined by quality of information. We derive the model’s conserved quantity, and use it to analyze the trade-off between investments in intelligence and in firepower—for example, in counterinsurgency operations.     

Supporting acquisition decisions through effective experimental design

Weapon acquisition is a very expensive process, especially in today's cost reduction environment. Techniques must be used to conduct operational testing using a minimum of resources while not sacrificing the adequacy and credibility of the test. A coordinated use of a combat simulation and a design of experiment procedure, the Taguchi method, shows promise as an acquisition strategyThis research focused on the Javelin medium antitank system which just completed operational testing in the Fall of 1993 and was intended to give the Javelin Project Manager's office information regarding the probable outcome of critical design characteristics prior to the test. Using the Taguchi method, many different design parameters were analyzed at several different levels of performance The method reduced the number of trials required to obtain a desired level of statistical significance while still obtaining the necessary data for each parameter. Once the required number of trials were identified, the Janus combat model simulated the operational test trials.Results suggest what weapon parameters are critical to the specific measures of effectiveness of survivability, lethality, and engagement range. In a broader view, this tandem use of an experimental design technique and a combat simulation can provide acquisition managers with insights on critical system parameters prior to actual testing.     

Identifying indicators of chemical,biological, radiological,and nuclear (CBRN) weapons development activity in sub-national terrorist groups

Data collected on known terrorist organizations allow intelligence agencies to build a statistical database of features for each group and an observed level of development of chemical, biological, radiological, or nuclear (CBRN) weapons. For the intelligence analyst, a statistical exploration of the structure of the multivariate data is helpful for determining which subset of features—and the relative contribution of each feature in the subset—best discriminate between levels of CBRN weapons development. The resulting function that is used to discriminate between CBRN development levels is called the ‘classifier’. Once the appropriate subset of indicators has been identified and a classifier developed, intelligence agencies will be better able to focus their information gathering and to assess the effect that changes in a terrorist group's features will have on their CBRN weapons development. Additionally, the classifier will enable the intelligence agency to predict the CBRN weapons development level of terrorist group where the feature set of the group is known but the level is unknown. In this analysis, we compare three approaches for building a classifier that best predicts CBRN weapons development levels using a training set with 45 observations; (1) heuristic pattern recognition approach that couples a weighted Minkowski distance metric with a nonparametric kernel-based classification method, (2) classification trees, and (3) discriminant analysis. Where possible, cross-validation is conducted on the data to ensure that the resulting classifier is not overly dependent on the training set. This initial analysis provides some interesting results and suggests a reasonable starting point for finding structure in the data as more observations are added.     

Force Deployment in a Conventional Theatre-level Military Engagement

This paper is concerned with the study of force deployment in a conventional theatre-level military engagement. First, it discusses some of the governing parameters that affect the modelling. A non-linear programming model is then developed to describe the combat dynamics. Sensitivity analysis on the optimal solution is also discussed. The merit of the model is demonstrated by applying the formulation to study the effect of various sustainment policies on the outcome of a battle.     

多维约束下的军队战斗力增长模型

战斗力是军事科学研究中的一个基本概念，其基本要素是人与武器及其相互关系。已有的研究大多采用定性分析的方法探讨了战斗力的重要性、战斗力的组成、影响战斗力的因素等，缺少关于战斗力的量化分析模型。本采用量化分析方法，建立了一个基本的反映军队战斗力大小的模型，在对单因素约束下战斗力增长模型进行较详细的优化分析基础上，建立了多维约束下的战斗力增长模型。