A two-on-one stochastic duel with maneuvering and fire allocation tactics

The large body of work on stochastic duels represents an attempt to model combat situations, or parts of it, by means of formal probability models. Most, but not all, of the existing stochastic duel models, however, relate to static posture and fail to capture dynamic aspects as well as tactical considerations that may be present. In this article we propose a simple model of a two-on-one duel in which dynamic and tactical aspects are considered. The model represents a combat situation that is typical of a battle in which a maneuvering force attacks a smaller defending unit that is static.     

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.     

Modeling the operation of multireservoir systems using decomposition and stochastic dynamic programming

Stochastic dynamic programming models are attractive for multireservoir control problems because they allow non‐linear features to be incorporated and changes in hydrological conditions to be modeled as Markov processes. However, with the exception of the simplest cases, these models are computationally intractable because of the high dimension of the state and action spaces involved. This paper proposes a new method of determining an operating policy for a multireservoir control problem that uses stochastic dynamic programming, but is practical for systems with many reservoirs. Decomposition is first used to reduce the problem to a number of independent subproblems. Each subproblem is formulated as a low‐dimensional stochastic dynamic program and solved to determine the operating policy for one of the reservoirs in the system. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

A stochastic air combat logistics decision model for Blue versus Red opposition

Technologically advanced aircraft rely on robust and responsive logistics systems to ensure a high state of operational readiness. This paper fills a critical gap in the literature for combat models by closely relating effectiveness of the logistics system to determinants of success in combat. We present a stochastic diffusion model of an aerial battle between Blue and Red forces. The number of aircraft of Blue forces aloft and ready to be aloft on combat missions is limited by the maximum number of assigned aircraft, the reliability of aircraft subsystems, and the logistic system's ability to repair and replenish those subsystems. Our parsimonious model can illustrate important trade‐offs between logistics decision variables and operational success.     

A simple Markov model for the analysis of multiple air combat

It is proposed to describe multiple air-to-air combat having a moderate number of participants with the aid of a stochastic process based on end-game duels. A simple model describing the dominant features of air combat leads to a continuous time discrete-state Markov process. Solution of the forward Kolmogorov equations enables one to investigate the influence of initial force levels and performance parameters on the outcome probabilities of the multiple engagement. As is illustrated, such results may be useful in the decision-making process for aircraft and weapon system development planning. Some comparisons are made with Lanchester models as well as with a semi-Markov model.     

Markovian models of three‐on‐one combat involving a hidden defender

In this paper, Markovian models of three‐on‐one stochastic firefights between ground‐based weapon systems are developed. These models address a common scenario of interest to the military, but one which has been much neglected in analytic combat models, that of combat between a hidden defender and an exposed attacking force. Each combatant must detect an opponent before commencing their firing cycle, a task which is considerably more difficult for the attacker. In the models developed here, the defender detects the exposed attacking group after an exponentially distributed time interval, while each attacker has a fixed probability of detecting the defender via the flash signature produced after each shot fired by him. The utility of the approach is demonstrated by investigating what impact the introduction of a coordinated gun‐laying system for the attacking force might have, a system made possible by battlefield digitization. The method used here allows models to be developed incrementally. This and other advantages of the Markovian approach are discussed. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 627–646, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10041     

The validity of assumptions underlying current uses of Lanchester attrition rates

This study examines critically the various assumptions, results, and concepts that exist to date in the literature and scientific community concerning the relationships among the Lanchester, stochastic Lanchester, and the general renewal models of combat. Many of the prevailing understandings are shown to be erroneous.     

基于广义随机Petri网GSPN的系统安全性研究

针对基于广义随机Petri网GSPN的动态系统安全性建模与分析方法进行了系统研究。基于Petri网、GSPN模型的基本原理,提出了关键状态搜索算法,并进行了安全性仿真分析.将广义随机Petri网的扩展模型应用于安全性分析领域,丰富了动态系统安全性问题的研究手段。     

Some two-on-two homogeneous stochastic combats

In this article we consider two versions of two-on-two homogeneous stochastic combat and develop expressions, in each case, for the state probabilities. The models are natural generalizations of the exponential Lanchester square law model. In the first version, a marksman whose target is killed resumes afresh the killing process on a surviving target; in the second version, the marksman whose target is killed merely uses up his remaining time to a kill on a surviving target. Using the state probabilities we then compute such important combat measures as (1) the mean and variance of the number of survivors as they vary with time for each of the sides, (2) the win probabilities for each of the sides, and (3) the mean and variance of the battle duration time. As an application, computations were made for the specific case of a gamma (2) interfiring time random variable for each side and the above combat measures were compared with the appropriate exponential and deterministic Lanchester square law approximations. The latter two are shown to be very poor approximations in this case.     

The multivariate Polya distribution in combat modeling

This paper investigates the multivariate Polya distribution (MPD) for application in combat models for which enemy targets must be randomly assigned to fighting units. Six sets of actual combat data motivate the allocation model. Scenarios include air‐to‐air combat, tank, and submarine warfare. Goodness of fit tests are derived which verify the validity of the MPD, and reinforce the notion that multinomial allocation is too simplistic for such combat models. Simulations to determine optimal allocations for minimal combat cost are applied in order to illustrate the role of the MPD in combat modeling. © 2001 John Wiley & Sons,Inc. Naval Research Logistics 48: 1–17, 2001     

超视距机群空战效能评估模型研究(英文)

针对超视距机群空战仿真的应用场合,建立了一种超视距机群空战的效能评估模型.该模型首先通过分析超视距空战的特点,在导弹攻击典型时间序列的基础上,对抢先发射导弹这一随机事件分三种情况进行了具体研究,得出计算飞机损失数的解析表达式.然后讨论了超视距空战中抢先情况的条件,提出了抢先概率的计算方法.最后着重对抢先发射、抢先发射区域的影响因素以及飞机机动性对空战结果的影响进行了分析.     

Optimal force mix in heterogeneous combat

This article presents models for determining the optimum number of Red weapons required to win a heterogeneous combat in which m(m> 1) types of Red weapons face a single type of Blue weapon under a newly defined termination policy. Red aims at either minimizing the total cost or maximizing the aggregated remaining force strength. Kuhn-Tucker and simulated annealing techniques are used for obtaining the optimal solution. The methodology is illustrated by analysing heterogeneous combat to determine (i) the feasibility of introducing new types of weapons and (ii) the number of weapons required to win if a specific type of weapon, say infantry, dominates. © 1995 John Wiley & Sons, Inc.     

基于仿真推演的导弹作战实验框架分析与设计

针对导弹武器论证设计高成本、高复杂度的问题,本文梳理了导弹作战实验设计服务的多种功能需求,阐述了导弹作战实验设计的数理方法;为构建具有较好柔性的导弹作战实验仿真框架,针对导弹系统多个性能指标,辨析了各类导弹作战实验仿真变量;针对作战实验中的环境和装备特性,梳理了导弹作战实验模型体系,提出了仿真新模型开发框架;结合导弹作战实验案例,构建了实验设计优化方法,提出了作战实验检验设计方案;从运行实验出发,阐述了设定作战想定、建立控制机制、开展结果分析等仿真环节的具体设计。通过这些工作,形成了一种支撑导弹作战实验的柔性仿真框架,为进一步开展基于仿真推演的导弹作战实验提供参考。     

不确定信息下反导拦截效果评估

针对不确定信息下反导拦截效果评估过程中指标信息和权重信息都不确定,传统评估方法很难加以解决的问题,使用区间数和随机多属性可接受性分析(SMAA)相结合的方法对反导拦截效果进行了评估。首先,根据来袭弹道导弹目标的毁伤程度划分了拦截效果等级;然后,针对观测信息的不确定性,采用区间数和隶属函数对特征指标进行了表示;在此基础上,通过引入SMAA方法,计算和比较拦截效果为各拦截等级的可接受性指标和信任因子,对不确定信息下反导拦截效果进行了评估。实例计算结果表明所介绍的不确定信息下反导拦截效果评估方法是可行的。     

基于复杂网络理论的作战系统结构拓扑模型研究

基于复杂网络理论,分析了信息化作战系统网络结构,构建了传统作战系统和信息化作战系统拓扑模型的生成算法,并通过对传统作战系统和信息化作战系统结构拓扑模型度量性质的比较,说明了信息化战争下的作战系统结构特点,为进一步研究网络化战争提供了良好的模型基础．     

战斗机远距空战建模与仿真分析

针对中远距一对一的空战问题,从效能分析的角度建立了一对一中远距空战及作战效能分析的模型,包括飞机运动模型、飞机机动动作控制模型、推力模型、火控系统模型、导弹运动模型、导弹制导与控制模型、信息系统模型和效能评估模型等。依据该模型进行了空战仿真并给出效能分析结果,仿真结果与实际吻合,证明模型的有效性,为进一步研究多对多空战仿真打下基础。     

Pareto-optimality in classical inventory problems

In this paper the inventory problem with backorders both deterministic and stochastic is studied using trade-off analysis in the context of vector optimization theory. The set of Pareto-optimal solutions is geometrically characterized in both the constrained and unconstrained cases. Moreover, a new way of utilizing Pareto-optimality concepts to handle classical inventory problems with backorders is derived. A new analysis of these models is done by means of a trade-off analysis. New solutions are shown, and an error bound for total inventory cost is provided. Other models such as multi-item or stochastic lead-time demand inventory problems are addressed and their Pareto-optimal solution sets are obtained. An example is included showing the additional applicability of this kind of analysis to handle parametric problems. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 83–98, 1998     

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     

A Lanchester-type model of combat with stochastic rates

The effects of environmental stochasticity in a Lanchester-type model of combat are investigated. The methodology is based on a study of stochastic differential equations with random parameters characterized by dichotomous Markov processes. Exact expressions for the Laplace transforms of the time evolution of the first- and second-order moments of the system are obtained. A special case when the fluctuations in the parameters occur with great rapidity in comparison with the natural time scale of the system is also analyzed. The stochastic stability in the mean-square sense is discussed by using the Routh–Hurwitz criterion and it is found that the stochastic perturbations tend to destabilize the system.     

The (s,Q) inventory model with Erlang lead time and deterministic demand

We develop a simple, approximately optimal solution to a model with Erlang lead time and deterministic demand. The method is robust to misspecification of the lead time and has good accuracy. We compare our approximate solution to the optimal for the case where we have prior information on the lead‐time distribution, and another where we have no information, except for computer‐generated sample data. It turns out that our solution is as easy as the EOQ's, with an accuracy rate of 99.41% when prior information on the lead‐time distribution is available and 97.54–99.09% when only computer‐generated sample information is available. Apart from supplying the inventory practitioner with an easy heuristic, we gain insights into the efficacy of stochastic lead time models and how these could be used to find the cost and a near‐optimal policy for the general model, where both demand rate and lead time are stochastic. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004