再谈用《超级画板》探索梯子模型

我们在文[1]中讨论了如何用智能化动态几何软件《超级画板》探索“梯子”模型．梯子模型（又称等棍模型）指的是一个定长的梯子靠在墙边，其两端可以沿着墙边滑动．针对这一模型，我们可以引出一系列的数学问题，也有很多人对这一问题进行了深入的研究．我们将在文[1]的基础上继续探讨其他有关的数学问题．     

数学解题追求简单自然

数学问题解答栏中许多数学问题的解答，给人启迪，耐人寻味，引人入胜，颇有创意．但也有个别问题，原作者给出的解答，过程曲折迂回，过度繁琐，不够自然．当然，我们换一个角度看，也许是件好事，它能引起读者的思考，去研究更简单，更自然的解题方法，因为这是我们共同追求的目标．贵刊读刊随笔栏目大量刊出这方面的文章也正好说明这一点．     

周期多尺度分析的特征及它的一个应用

在这篇文章里，我们研究了周期多尺度分析的性质，给出了尺度函数序列的一个特征．这个特征能够使我们从一个尺度函数序列得到另一个尺度函数序列．最后，我们给出了主要结果的一个应用．     

几何图形在代数解题中的应用

代数研究的对象主要是“数”，几何研究的对象主要是“形”．然而两者却有着非常密切的关系．有时，一个代数问题它甚至可能是由一个几何问题演变而来的，如果我们能通过想象，把抽象的代数问题模拟成具体的、直观的几何问题，那么我们便可以根据图形的性质来解决它．本文即是从几个侧面谈谈几何图形在解代数题中的应用．     

一个数论命题和它的概率论背景

现代数学已经发展成为庞大的体系．数学分支的精细，研究方法的艰深，使得关于一个专门问题的研究论文往往只有从事这个专门问题研究的少数数学家可以读懂．有人甚至认为，循此以往，“数学论文的作者要比读者还多”！但这仅仅是问题的一个方面；毕竟“数学科学是一个不可分割的有机整体，它的生命力正是在于各部分之间的深刻联系”（希尔伯特：《数字问题》），数学各分支之间的融合、贯通是值得注意的另一方面．在本文中，通过一个数论命题和它的概率论背景的讨论我们可以看出，即便在比较初等的问题申，数学的不同分支之间的融合、贯通也…     

解赛题 多思考

这是2012年波罗的海数学奥林匹克试题第13题，与笔者读过的一本书中的一个问题非常相近，这引起了我们的思考，下面就分析之，供大家赏析．     

灰色分派问题及其应用

在分派问题中，总假设被分派工作的人（或机器等）做各项工作的效率（或所费时间）都是确定的值．但实际上．对于以往未做过的工作，或做过但情况变化较大的工作都难以定下确切的工作效率，而只能估计出一个大概的范围．这也就是说有些工作效率是有理灰数．我们把这一类分派问题称为灰色分派问题．木文给出了灰色分派问题有关定理的证明以及一个应用实例．     

第六章 一次函数

同学们，函数是研究现实世界变化规律的一个重要模型，对它的学习一直是初中阶段数学学习的一个重要内容．在上学期，我们已经探索了变量之间的关系．在这一章我们将继续通过对变量间关系的探讨，初步体会函数的概念，并研究其中最为简单的一种函数——一次函数．希望同学们通过解剖一次     

关于对偶扩张代数有限维数的注记

没A是一个有限维代数，R为A的对偶扩张代数．本我们讨论R的有限维数findim R of R，证明了，在—般情况下findim R≠2findim A，这就回答了惠昌常教授所提的一个问题．     

网上平衡对策与流向对策值的存在性

在一个网络上可以赋以流量，网上每一个孤可以与一个Ｎ人对策的局中人相对应，有了这种对应之后，我们证明了任何一个可加性对策都是流向对策，且证明了如果Ｖ１，Ｖ２是流向对策，则ｍｉｍ｛Ｖ１，Ｖ２｝也是流向对策．在此基础上，我们证明了θ－全平衡对策是流向对策这一主要定理．最后，我们成功地构造了一个很简单的例子说明了此定理的逆命题不成立．这是其它文献［１，２，３，４］中都没有明确指出的结论．关于θ－全平衡对策与流向对策的关系．目前来看我们的结果是较好的结果［９］．     

Random fuzzy EOQ model with imperfect quality items

This paper investigates an economic order quantity (EOQ) problem with imperfect quality items, where the percentage of imperfect quality items in each lot is characterized as a random fuzzy variable while the setup cost per lot, the holding cost of each unit item per day, and the inspection cost of each unit item are characterized as fuzzy variables, respectively. In order to maximize the expected long-run average profit, a random fuzzy EOQ model is constructed. Since it is almost impossible to find an analytic method to solve the proposed model, a particle swarm optimization (PSO) algorithm based on the random fuzzy simulation is designed. Finally, the effectiveness of the designed algorithm is illustrated by a numerical example.     

基于随机规划的建筑集群冷热电联供系统优化研究

面向建筑集群的冷热电联供系统的设计和优化是实现建筑楼宇能源成本节约的重要途径。随机因素对该联供系统的优化决策,具有显著的影响。考虑建筑楼宇的能源需求为随机变量,构建随机混合整数规划模型,解决以最小化建筑楼宇总费用为目标时建筑集群冷热电联供系统的优化问题;其次,提出采用Benders多割平面方法求解多目标规划问题,从而寻找冷热电联供系统的设备配置和系统运行的Pareto最优决策;最后,通过实验验证了模型和算法的有效性。实验结果表明建筑集群在协作模式下,相比于非协作模式,具有更低的总费用。     

随机环境下基于成本经济的项目调度权衡优化研究

针对随机环境下项目前摄性调度与反应性调度在应对不确定因素过程中起到的不同作用,从成本经济角度出发,研究了如何通过两种调度方法的权衡实现项目计划与执行的最优配合。在此基础上构建了基于成本的前摄反应调度权衡模型,通过对鲁棒性成本与调整成本进行量化分析,实现两种调度方案的最佳权衡。考虑到问题的NP难属性,设计了基于混合变邻域禁忌搜索的随机两点启发式算法,并通过大规模算例测试验证了算法的有效性。结果表明,根据承包商对成本的敏感度,前摄性调度与反应性调度在应对不确定性因素干扰中承担的工作量会随着成本权衡比的变化而发生改变,逐渐从前摄性方法为主过渡到以反应性方法为主。最后,从项目管理角度给出了有价值的管理启示。     

A production-repairing inventory model with fuzzy rough coefficients under inflation and time value of money

In this paper, a production-repairing inventory model in fuzzy rough environment is proposed incorporating inflationary effects where a part of the produced defective units are repaired and sold as fresh units. Here, production and repairing rates are assumed as dynamic control variables. Due to complexity of environment, different costs and coefficients are considered as fuzzy rough type and these are reduced to crisp ones using fuzzy rough expectation. Here production cost is production rate dependent, repairing cost is repairing rate dependent and demand of the item is stock-dependent. Goal of the research work is to find decisions for the decision maker (DM) who likes to maximize the total profit from the above system for a finite time horizon. The model is formulated as an optimal control problem and solved using a gradient based non-linear optimization method. Some particular cases of the general model are derived. The results of the models are illustrated with some numerical examples.     

需求率依赖于库存水平的离散性库存费的库存模型

传统的库存控制模型都视需求率为固定不变的,放松了这个假定,通过考虑库存费为存储时间的阶梯函数的情形:(1)全单位库存费用,(2)增量库存费用,并且在需求率依赖于库存水平,当库存水平下降到一定程度时,需求率变为常数的形式下,把变化的订购费引入,发展了两个离散性库存费的变质物品的库存控制模型。在模型中允许周期末库存水平不为零,并且提出了最优解的算法。     

顺序相依拆卸线平衡多目标优化模型及算法研究

拆卸是产品回收过程最关键环节之一,拆卸效率直接影响再制造成本。本文在分析现有模型不足基础上,考虑最小化总拆卸时间,建立多目标顺序相依拆卸线平衡问题优化模型,并提出了一种自适应进化变邻域搜索算法。所提算法引入种群进化机制,并采用一种组合策略构建初始种群,通过锦标赛法选择个体进化;在局部搜索时,设计了邻域结构自适应选择策略,并采用基于交叉的全局学习机制加速跳出局部最优,以提高算法寻优能力。对比实验结果,证实了所提模型的合理性以及算法的高效性。     

Barzilai–Borwein method with variable sample size for stochastic linear complementarity problems

A smoothing method for solving stochastic linear complementarity problems is proposed. The expected residual minimization reformulation of the problem is considered, and it is approximated by the sample average approximation (SAA). The proposed method is based on sequential solving of a sequence of smoothing problems where each of the smoothing problems is defined with its own sample average approximation. A nonmonotone line search with a variant of the Barzilai–Borwein (BB) gradient direction is used for solving each of the smoothing problems. The BB search direction is efficient and low cost, particularly suitable for nonmonotone line search procedure. The variable sample size scheme allows the sample size to vary across the iterations and the method tends to use smaller sample size far away from the solution. The key point of this strategy is a good balance between the variable sample size strategy, the smoothing sequence and nonmonotonicity. Eventually, the maximal sample size is used and the SAA problem is solved. Presented numerical results indicate that the proposed strategy reduces the overall computational cost.     

Cooperation in one machine scheduling

A combinatorial optimization problem can often be understood as the problem to minimize cost in a complex situation. If more than one party is involved, the solution of the optimization problem is not the end of the story. In addition it has to be decided how the minimal total cost has to be distributed among the parties involved. In this paper cost allocation problems will be considered arising from one-machine scheduling under additive and weakly increasing cost functions. The approach of the problem will be game theoretical and we shall in fact show that in many cases the games related to the cost allocation problems are balanced.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

可变产量的理性产出范围研究

本从机会成本损失最小化的角度出发，对可变立量的三个阶段的两种划分方法所确定的两种不同的理性产出范围进行了分析，分析结果表明。可变产量的理性产出范围应在最小边际成本所对应的产出量到边际成本与产品的预期价格相等所对应的产出量之间。