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.     

Quantifying the value of reconnaissance using Lanchesterian type equations

This paper presents a method to quantify the value of reconnaissance for both direct and indirect fire weapons for the defense in sector battle scenario. The Lanchester area fire model and the Helmbold equations were modified to allow the lethality of the defending blue force to be increased as they gained more combat intelligence about the attacking red force, thus modeling intelligence as a true combat multiplier. By adjustments made to parameters in the model, the lethality of the blue's direct and indirect fire weapons could be adjusted based on the quantity and quality of their intelligence assets. With information from a computer database, and the COMAN model, maximum likelihood attrition rate estimates were calculated for both red and blue forces for ten heavy defensive battles conducted at the Army's National Training Center. In each battle, the red force attrition rate was fit to a curve which represented a percentage of blue's full potential, represented here by the square law. Using this model in a combat simulation, and with some preliminary work with comparable systems, one could implement a change in blue's intelligence assets and then provide a quantitative measure of the effect that this had on the outcome of a battle     

一线救治伤员后送资源配置策略研究

参考国内外历史案例和数据,构建战场伤员产生的仿真模型。基于该模型,利用综合评价、资源配置优化算法,研究探讨连抢救组及营救护站的救治、后送运力配置及路径规划策略。利用假设战例验证模型,基于减员预计模型计算减员情况,分析了相应的营、连救治策略。结果验证一线伤员收拢后送运力配置及路径优化模型可靠,可为一线救治信息化、自动化、智能化发展提供技术支撑。本研究为一线伤员救治、收拢、后送力量配置及路径规划提供理论方法。     

Modelling medieval military logistics: an agent-based simulation of a Byzantine army on the march

Although historical studies are frequently perceived as clear narratives defined by a series of fixed events; in reality, even where critical historical events may be identified, historic documentation frequently lacks corroborative detail to support verifiable interpretation. Consequently, interpretation rarely rises above the level of unproven assertion and is rarely tested against a range of evidence. Agent-based simulation can provide an opportunity to break these cycles of academic claim and counter-claim. This paper discusses the development of an agent-based simulation designed to investigate medieval military logistics so that new evidence may be generated to supplement existing historical analysis. It uses as a use-case the Byzantine army’s march to the battle of Manzikert (AD 1071), a key event in medieval history. The paper focuses primarily on the design of the agents and the environment they interact with, as well as how the agent-based simulation as a whole can be used to generate new parameters with which historical evidence can be situated.     

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.     

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.     

A new look at the 3:1 rule of combat through Markov Stochastic Lanchester models

The 3:1 rule of combat states that in order that for the attacker to win the battle, his forces should be at least three times the force of the defender. This somewhat vague statement has resulted in numerous interpretations and discussions from historical and military science points of view. In this paper we attempt to examine this rule by utilising a number of Markov Stochastic Lanchester models that correspond to various basic combat situations and to draw some conclusions from their implementations. We identify general combat situations where the 3:1 rule is reasonable as well as situations where the force ratio should be either smaller or larger. Since the analysis is performed in the formal and somewhat ‘sterile’ setting of (pure) mathematical modeling, the results should be appropriately interpreted as reasoning of a certain abstraction of the battlefield.     

A fractal graph model of capillary type systems

Abstract

We consider blood flow in a vessel with an attached capillary system. The latter is modelled with the help of a corresponding fractal graph whose edges are supplied with ordinary differential equations obtained by the dimension-reduction procedure from a three-dimensional model of blood flow in thin vessels. The Kirchhoff transmission conditions must be satisfied at each interior vertex. The geometry and physical parameters of this system are described by a finite number of scaling factors which allow the system to have self-reproducing solutions. Namely, these solutions are determined by the factors’ values on a certain fragment of the fractal graph and are extended to its rest part by virtue of these scaling factors. The main result is the existence and uniqueness of self-reproducing solutions, whose dependence on the scaling factors of the fractal graph is also studied. As a corollary we obtain a relation between the pressure and flux at the junction, where the capillary system is attached to the blood vessel. This relation leads to the Robin boundary condition at the junction and this condition allows us to solve the problem for the flow in the blood vessel without solving it for the attached capillary system.     

Spatial Lanchester models

Lanchester equations have been widely used to model combat for many years, nevertheless, one of their most important limitations has been their failure to model the spatial dimension of the problems. Despite the fact that some efforts have been made in order to overcome this drawback, mainly through the use of Reaction–Diffusion equations, there is not yet a consistently clear theoretical framework linking Lanchester equations with these physical systems, apart from similarity. In this paper, a spatial modeling of Lanchester equations is conceptualized on the basis of explicit movement dynamics and balance of forces, ensuring stability and theoretical consistency with the original model. This formulation allows a better understanding and interpretation of the problem, thus improving the current treatment, modeling and comprehension of warfare applications. Finally, as a numerical illustration, a new spatial model of responsive movement is developed, confirming that location influences the results of modeling attrition conflict between two opposite forces.     

"兰氏"平方律作战毁伤过程兵力损耗率系数战例求解理论

如何求得著名的“兰氏”平方律战斗动力学方程中双方兵力损耗率系数，这是作战模拟应用研究领域中一个久攻未克的难题。本文提出了以作战结果来逆向研究作战过程中双方兵力损耗率系数的思想。阐述了对于不变的作战双方在相同（相近）的作战环境与作战条件下相继进行的作战序列里，双方各自的兵力损耗率系数不变（波动不大）的公理，论证了揭示作战序列内部规律的两条定理。据此，建立了兵力损耗率系数的战例求解理论与方法。运用这一理论与方法，据以往发生的作战过程其数值特征可以求得未来相似或相同作战过程中双方兵力损耗率系数的具体取值，首次解决了作战模拟研究领域中兵力损耗率系数的具体取值这一难题。     

Lanchester model for three-way combat

Lanchester (1960) modeled combat situations between two opponents, where mutual attrition occurs continuously in time, by a pair of simple ordinary (linear) differential equations. The aim of the present paper is to extend the model to a conflict consisting of three parties. In particular, Lanchester’s main result, i.e. his Square Law, is adapted to a triple fight. However, here a central factor – besides the initial strengths of the forces – determining the long run outcome is the allocation of each opponent’s efforts between the other two parties. Depending on initial strengths, (the) solution paths are calculated and visualized in appropriate phase portraits. We are able identify regions in the state space where, independent of the force allocation of the opponents, always the same combatant wins, regions, where a combatant can win if its force allocation is wisely chosen, and regions where a combatant cannot win itself but determine the winner by its forces allocation. As such, the present model can be seen as a forerunner of a dynamic game between three opponents.     

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.     

基于灰色可拓优度的联合火力打击陆军弹药消耗预测方案评估研究

弹药是构成武器系统的重要组成部分,是军队战斗力的重要构成因素,弹药消耗预测是一个复杂的问题,它是弹药保障的基本依据,是我军做好军事斗争装备准备弹药物资精确化保障的重要条件.在联合火力打击中的陆军弹药消耗预测影响因素多,不确定性强,许多因素难以在预测过程中定量地计算,如何确定和优选联合火力打击陆军弹药消耗预测方案是否达到规定要求或从中优选出最佳的方案是一个复杂的评估问题.在分析联合火力打击陆军弹药消耗影响因素的基础上,构建了包含打击目标、作战持续时间、作战任务、参战部队综合战斗力指数、指挥信息系统信息化程度、武器装备及弹药性能等6个2级指标共26个底层指标的评估指标体系,基于灰色模糊的思想和层析分析法(AHP)综合确定指标权重,在此基础上进一步构建了联合火力打击陆军弹药消耗预测方案的可拓优度评估模型,运用此模型对联合火力打击陆军弹药消耗预测方案的进行总体评估,评估结果表明模型是一种有效的联合火力打击陆军弹药消耗预测方案评估方法,可为联合火力打击陆军弹药消耗预测方案的确定和评估优选提供新的有效途径.     

An agent-based model of organizational ambidexterity decisions and strategies in new product development

Organizational ambidexterity, defined as the pursuit of both exploitation and exploration, has become an important topic in the study of organizations, especially in innovation management theory. Previous literature has not focused on the strategic (game-theoretic) aspects of organizational ambidexterity or on its decision-making aspects. Little is known about how or even whether the decision to adopt ambidexterity is competitively advantageous in the presence of diverse strategies that competitors may adopt. This facet of the subject is inherently game-theoretic; the value of a decision by one firm depends in part on decisions made by other firms. This paper initiates systematic investigation of these strategic aspects, including the overall performance of available strategies. Specifically, this study examines questions of ambidexterity-related strategy performance in the context of new product development. The main contributions are (1) to introduce and make available to the research community an agent-based model and decision support system that captures many of the key aspects and tradeoffs, which have been identified in the literature, of the exploration–exploitation dilemma faced by firms in the new product development process, with a focus on organizations’ product investment decisions and (2) to report on results obtained from the model, calibrated with available data from the literature, augmented by new data collected from interviews with practitioners.     

基于聚类分析的非监督多属性目标分群方法

随着现代战争节奏的加快,以及传感器所产生战场监视数据量的剧增,指挥人员面临着越来越大的认知压力.目标分群作为一种重要的高级数据融合技术,能够减轻指挥员的认知负担,但目前的目标分群算法都需要人为地给出一些参数,且分群的依据较为单一,不能满足联合作战指挥的需要.为解决多属性目标分群问题,首先确定了目标分群问题的描述方式,其次以多目标属性为基础,通过计算相似度和网络最佳分类判定函数,提出了基于层次聚类的非监督目标分群算法,最后给出了算法的实现描述和例子.     

Fractal dimension of a random invariant set

In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.     

Verifying agent-based models with steady-state analysis

Agent-based modeling has been well received in the simulation community. Complex systems are simulated by many autonomous agents whose behavior is defined by a conceptual model. However, the model can be improperly implemented or misinterpreted resulting in an implementation that does not reflect the conceptual rules. It is imperative that the implementation’s function be tested against the model’s expected outcome. In this paper, we present certain steady-state techniques that can be used to verify the operation of agent-based simulations. These methods are introduced and then applied to an ecological model which simulates reproductive dynamics of mosquitoes.     

Modeling complex systems macroscopically: Case/agent-based modeling,synergetics, and the continuity equation

Recently, the continuity equation (also known as the advection equation) has been used to study stability properties of dynamical systems, where a linear transfer operator approach was used to examine the stability of a nonlinear equation both in continuous and discrete time (Vaidya and Mehta, IEEE Trans Autom Control 2008, 53, 307–323; Rajaram et al., J Math Anal Appl 2010, 368, 144–156). Our study, which conducts a series of simulations on residential patterns, demonstrates that this usage of the continuity equation can advance Haken's synergetic approach to modeling certain types of complex, self-organizing social systems macroscopically. The key to this advancement comes from employing a case-based approach that (1) treats complex systems as a set of cases and (2) treats cases as dynamical vsystems which, at the microscopic level, can be conceptualized as k dimensional row vectors; and, at the macroscopic level, as vectors with magnitude and direction, which can be modeled as population densities. Our case-based employment of the continuity equation has four benefits for agent-based and case-based modeling and, more broadly, the social scientific study of complex systems where transport or spatial mobility issues are of interest: it (1) links microscopic (agent-based) and macroscopic (structural) modeling; (2) transforms the dynamics of highly nonlinear vector fields into the linear motion of densities; (3) allows predictions to be made about future states of a complex system; and (4) mathematically formalizes the structural dynamics of these types of complex social systems.     

Strategic bidding of offer curves: An agent-based approach to exploring supply curve equilibria

We model a market in which suppliers bid step-function offer curves using agent-based modeling. Our model is an abstraction of electricity markets where step-function offer curves are given to an independent system operator that manages the auctions in electricity markets. Positing an elementary and computationally accessible learning model, Probe and Adjust, we present analytic results that characterize both the behavior of the learning model and the properties of step-function equilibria. Thus, we have developed a framework for validating agent-based models prior to using them in situations that are too complicated to be analyzed using traditional economic theory. In addition, we demonstrate computationally that, by using alternative policies, even simple agents can achieve monopoly rewards for themselves by pursuing more industry-oriented strategies. This raises the issue of how participants in oligopolistic markets actually behave.