非平衡指派模型及其在航空机务维修中的应用

非平衡指派问题是最优平衡指派问题的推广与深化,在航空机务维修工作中,维修任务的合理配置对及时完成维修任务,保障训练作战计划非常重要.本文从装备完好率和人力资源的优化配置角度出发,按照不考虑维修任务等待时间和考虑维修任务等待时间两种情况分别建立了非平衡指派优化模型,并给出了这两种情况下效益矩阵的构造方法,进而将优化模型转化为最优平衡指派模型进行求解,从而为航空机务维修工作中维修人员的优化配置提供了一种科学、合理的决策方法.     

一种求布尔矩阵传递闭包的基于自反矩阵构造的平方算法

首先,介绍布尔矩阵传递闭包的概念及计算问题;随后,分析布尔矩阵的传递闭包和由该布尔矩阵与单位矩阵取并所得到的自反矩阵的传递闭包之间的关系;最后,利用上述结果给出一种求解布尔矩阵传递闭包的基于自反矩阵构造的平方算法,并通过实例说明了其具体计算过程.     

航空机务维修的优化管理

在航空机务维修工作中,科学的管理、人力资源的合理配置对及时完成维修任务,保障训练作战计划至关重要.从装备完好率和完成任务的及时性出发,分别建立了数学优化配置模型,并给出了这两种情况下效益(成本)矩阵的构造方法,进而将优化模型转化为最优线性指派问题来处理,从而为航空机务维修工作中人力资源的优化配置提供了一种科学、合理的决策方法.     

子矩阵约束下矩阵方程组的双对称最小二乘解

矩阵方程组l∑j=1在控制与系统领域中具有广泛应用.该文构造了一种算法求解这个矩阵方程组,其中X_j∈R~(n_j×n_j)(j=1,2,…,l)为带有特殊中心主子矩阵约束的双对称矩阵.在没有舍入误差的情况下,该算法经过有限步迭代得到[X_1,X_2,…,X_l],使得t∑i=1||l∑j=1A_(ij)X_jB_(ij)-C_i||=min.实例表明这种方法是有效的.     

矩阵序列的有理外推法

基于矩阵的广义逆,本文给出了关于矩阵序列加速收敛的三个有理外推方法.它们包括:(i)基于广义逆的矩阵Pade逼近[4];(ii)矩阵Epsilon算法;(iii)矩阵Aitken △~2-算法。对三种方法之间的内在联系进行了讨论。关于Markov过程的一个实例给出以说明本文的结果。     

码本在压缩感知矩阵中的应用

压缩感知矩阵的构造在压缩感知理论中起着举足轻重的作用.基于线性码和最优码本构造了一类新的压缩感知矩阵,并与DeVore构造的压缩感知矩阵和Gaussian随机矩阵的进比较,从不同的角度分别证明了当参数满足一定的条件时,新构造的压缩感知矩阵具有更好的性能.     

(,z)-矩阵的卡氏积及其性质

二元叠加码(,z)-析取矩阵是Pooling设计理论的一个极其重要的数学模型,定义了两个已知(,z)-析取矩阵的卡氏积并证明了它的性质,最后,对这一定义进行了推广.     

AHP中互反判断矩阵一致性调整的新简易方法及交互式实现

提出Satty′s互反判断矩阵一致性调整的一种新的简易方法,并借助EXCEL电子表格的动态引用功能,实现了判断者在建立判断矩阵时可即时地进行一致性的互动式调整;不但改善了现有不合理的强制性调整方法,而且提供了一种有效辅助判断矩阵建立的途径.其中,借鉴欧式距离原理构造了"行和列偏离程度系数"来衡量一致性优化调整过程有效性,同时实现了权重向量相对于互反判断矩阵调整的灵敏度分析.最后通过一个具体实例演示方法的操作过程,并检验了其有效性.     

(d,r,z]-析取矩阵的卡氏积及其性质

二元叠加码[d,r,z]-析取矩阵是Pooling设计理论的一个极其重要的数学模型,定义了两个已知(d,r,z]-析取矩阵的卡氏积并计算了它的参数,最后介绍了它的检纠错性质.     

生灭矩阵重构三弦乐谱

三弦乐谱可由三对角复矩阵生成,人们听到的声音由其谱决定,自然要求此矩阵有实谱.如同量子力学,模型的描述是复算子,可观测量是实的.换言之,所述的三对角复矩阵应是关于某个测度的复内积空间上的自共轭算子.熟知,生灭Q矩阵可配称,自然就是自共轭的.我们将要介绍最新的一个代表性成果:对于相当广泛的自共轭三对角复矩阵,总可以构造出一个生灭Q矩阵,使得两者等谱(简单地说,两者有完全相同的特征值).这个问题浅显易懂,但我们曾在不同时期,从概率论、统计物理和计算数学三个不同的角度研究过,经历了漫长的求索岁月.本文是根据作者在"International Conference on Probability Theory and Its Applications"(湖南文理学院,2018年7月)的报告整理而成.共分三部分:1)生灭过程的新应用;2)从(非对角线元素非负的)实可配称矩阵到复可配称矩阵;3)此课题的来源(计算)及其判别准则的应用(统计物理及量子力学).     

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.     

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.     

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.     

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.     

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.     

有资格限制的指派问题的求解方法

在实际的指派工作中，常会遇到某个人有没有资格去承担某项工作的问题，因此，本建立了有资格限制的指派问题的数学模型。在此数学模型中，将效益矩阵转化为判定矩阵，由此给出了判定此种指派问题是否有解的方法；在有解的情况下，进一步将效益矩阵转化为求解矩阵，从而将有资格限制的指派问题化为传统的指派问题来求解。最后给出了一个数值例子来说明这样的处理方法是有效的。     

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.     

多目标指派问题在潜艇兵力配置中的应用

运用模糊数学的思想，首先将各目标下的属性值矩阵转化为模糊关系矩阵，再将模糊关系合成矩阵与解决传统指派问题的匈牙利法相结合，提出一种求解多目标指派问题的综合方法：模糊匈牙利法，并结合优化潜艇兵力配置问题进行了应用分析。     

矩阵博弈的胜利度和“僵局”

矩阵博弈又称有限对抗博弈，而对抗博弈的结果必是一方胜利，失败或双方和局，本首先给出胜利、失败和和局的数学模型，接着给出描述胜利程度的概念-胜利度并讨论了胜利度的几个简单性质，依此分析了传统矩阵博弈出现“僵局”的原因并指出排除“僵局”的方法。     

益智游戏“汉诺塔”中的矩阵运算

汉诺塔(Tower of Hanoi)问题源于印度一个古老传说,据此做成了益智游戏,蕴含大量的数学思想与方法.本文采用矩阵描述汉诺塔状态和圆盘移动过程,将圆盘从一个位置移动到另一个位置转化为矩阵的加法,进而构造由若干可能状态矩阵组成的图的邻接矩阵,计算其幂矩阵,由此很方便地求得完成汉诺塔游戏的所有可能的圆盘移动方案,求解过程简单,含义清晰,易于理解和实现.