浅谈四类平均数的几何模型

《数学通报》2003年第11期刊登了《四类平均数的几何模型》一文,该文给出了两个正数的调和平均数、几何平均数、算术平均数及平方平均数在圆中的几何模型.下面再给出这四类平均数在四边形中的几何模型,供读者参考.当a>0,b>0时,a2 abb,ab,a 2b,a2 2b2分别叫做a,b的调和平均数、几何平均数、算术平均数及平方平均数,它们的关系是2aba b≤ab≤a2 b≤a22 b2(a>0,b>0).当且仅当a=b时等号成立.下面给出它们在四边形中的几何模型.在四边形ABCD中,设AB∥DC,AB=a,DC=b.1.当a≠b时,不妨设a相似文献   

非恒定概率下基于不完全检查的受限维护优化研究

研究了一个以三个不同状态为特征的单组分体系,针对考虑检查错误的非恒定概率的模型,给出了在给定维护策略下的确切成本和可靠性评估,目标是在可靠性约束下,在无限的时间范围内尽可能降低平均成本.最后,将真实模型与考虑检查错误的常数概率的模型进行比较,通过计算研究发现近似模型的平均成本比真实模型高出26%.此外,当使用近似模型时,发现可靠性约束可能严重受到最大25%的影响.     

四类平均数的几何模型

新教材中关于两个数的算术平均数与几何平均数的几何解释 ,显示了数与形的完美结合 .在新教材数学第二册 (上 )习题 6 2中 ,有这样一个习题 :已知a、b都是正数 ,求证 :21a + 1b≤ab≤ a+b2 ≤ a2 +b22 ,当且仅当a=b时等号成立 .不等式中的四个式子分别称为两个数的调和平均数、几何平均数、算术平均数、平方平均数 .此题描述了这四个平均数之间的关系 ,本文再给出它们的几何模型 .数形结合不仅揭示了数学的内在联系 ,给人以美的享受 ,更能开发学生智力 ,培养学生能力 ,发散学生思维 .1　ab≤ a+b2 的几何模型 .　　如图 1 ,以a+b为直径 (记…     

基于网络优化模型的变电站选址决策研究

利用Cplex优化软件,设计C语言程序,求解有资金约束与无资金约束的变电站最小费用模型。证明在无资金约束条件下修建的变电站数量可能比有资金约束条件下修建的变电站数量要少,建议决策者在进行电网规划时比较使用上述两种模型,使总费用最低,避免主观决策失误。同时指出了Hakimi文章中的计算错误。     

基于教材对比的“数据的集中趋势”的教材改编

当今社会是以信息资源为基本发展资源、信息服务为产业的社会,信息化产业的发展涉及很多与数据相关的知识,数据的采集、处理、应用等就在初中阶段学习.“统计与概率”这一模块在数学中的重要性日益提升,数学课程标准中对“抽样与数据分析”提出了9点要求,其中明确写道:“理解平均数的意义,能计算中位数、众数、加权平均数,了解它们是数据集中趋势的描述.”浙教版数学教学参考书出版于2012年,近年信息发展突飞猛进,对教材的应用也需随之进步,才能达到现在的需求.本文拟以“数据的集中趋势”为例,尝试从更高、更新、更全的角度处理教材,在此与同行分享.     

多阶段并发的协同物流网络任务-资源集成调配模型

协同物流网络作为动态开放的网络需要整合多种资源,以完成多项并行的物流任务。为此,本文从物流服务集成商的角度,研究了带时间窗和资源约束的任务—资源集成调配问题。通过将物流任务分解为不同阶段,构造多阶段并发的协同物流网络任务—资源集成调配模型,并采用遗传算法对模型进行求解,以从候选资源集合中为任务阶段匹配合适的资源。算例结果表明,该模型能够有效实现N-N的任务-资源集成调配,使得协同物流网络的整体效益达到最优。     

基本不等式√ab≤a＋b/2的应用

定理如果a,b是正数，那么a＋b/2≥√ab（当且仅当a=b时取“=”）.这个定理适用的范围：a,b∈R^＋；我们称a＋b/2为a,b的算术平均数，称√ab为a,b的几何平均数。即：两个正数的算术平均数不小于它们的几何平均数．此不等式常称为均值不等式．     

研发资源互补性与合作创新模式

聚焦企业之间研发资源互补性与合作创新模式的关系,通过一个两阶段博弈模型,给出了多个企业之间研发资源互补性大小的临界值.当研发资源的互补性小于该临界值时,企业之间将以共担研发成本为目的,选择成本共享的合作创新模式;当研发资源的互补性大于该临界值时,企业之间将以分享技术成果为目的,选择技术共享的合作创新模式.     

消费者环境意识及碳价约束下供应链减排的转移机制及决策

针对在碳价和消费者环境意识约束下由一个制造商和一个零售商组成的供应链系统,建立三阶段Stackelberg博弈模型分析供应链集中决策和分散决策下制造商和零售商的决策过程,并对集中决策和分散决策下的减排效果、消费者剩余、制造商转移减排任务系数进行比较分析.通过改变参数值和成本函数进行数值仿真和敏感性分析,检验模型的合理性和鲁棒性.研究发现,分散决策下,制造商是否转移减排任务取决于碳价和消费者环境意识约束,如果制造商转移减排任务,零售商收益、消费者需求和消费者剩余降低;分散决策和集中决策下减排效果不变,但集中决策下消费者需求和消费者剩余较高;集中决策下,制造商同时考虑减排成本和价格策略,消费者剩余增加.     

基于多目标最优化混合按订单生产和提前生产策略库存模型算法

考虑了包含多阶段和多家零售商在内的混合按订单生产(Produce-to-Order)和提前生产(Produce-in-Advance)策略库存模型.将这一模型推广至多种产品情况下,当需求为正态分布时,将这一问题转化为一类多目标约束最优化问题,并给出两种算法来解决这一问题.     

Optimal allocation policies for testing-resource based on a software reliability growth model

A lot of development resources are consumed during the software testing phase fundamentally consisting of module testing, integration, testing and system testing. Then, it is of great importance for a manager to decide how to effectively spend testing-resources on software testing for developing a quality and reliable software.In this paper, we consider two kinds of software testing-resource allocation problems to make the best use of the specified testing-resources during module testing. Also, we introduce a software reliability growth model for describing the time-dependent behavior of detected software faults and testing-resource expenditures spent during the testing, which is based on a nonhomogeneous Poisson process. It is shown that the optimal allocation of testing-resources among software modules can improve software reliability.     

Flexible software reliability growth model with testing effort dependent learning process

A lot of importance has been attached to the testing phase of the Software Development Life Cycle (SDLC). It is during this phase it is checked whether the software product meets user requirements or not. Any discrepancies that are identified are removed. But testing needs to be monitored to increase its effectiveness. Software Reliability Growth Models (SRGMs) that specify mathematical relationships between the failure phenomenon and time have proved useful. SRGMs that include factors that affect failure process are more realistic and useful. Software fault detection and removal during the testing phase of SDLC depend on how testing resources (test cases, manpower and time) are used and also on previously identified faults. With this motivation a Non-Homogeneous Poisson Process (NHPP) based SRGM is proposed in this paper which is flexible enough to describe various software failure/reliability curves. Both testing efforts and time dependent fault detection rate (FDR) are considered for software reliability modeling. The time lag between fault identification and removal has also been depicted. The applicability of our model is shown by validating it on software failure data sets obtained from different real software development projects. The comparisons with established models in terms of goodness of fit, the Akaike Information Criterion (AIC), Mean of Squared Errors (MSE), etc. have been presented.     

Group Scheduling with Controllable Setup and Processing Times: Minimizing Total Weighted Completion Time

The following single machine scheduling problem is studied. A partition of a set of n jobs into g groups on the basis of group technology is given. The machine processes jobs of the same group contiguously, with a sequence independent setup time preceding the processing of each group. The setup times and the job processing times are controllable through the allocation of a continuously divisible or discrete resource to them. Each job uses the same amount of the resource. Each setup also uses the same amount of resource, which may be different from that for the jobs. Polynomial-time algorithms are constructed for variants of the problem of finding an optimal job sequence and resource values so as to minimize the total weighted job completion time, subject to given restrictions on resource consumption. The algorithms are based on a polynomial enumeration of the candidates for an optimal job sequence and solving the problem with a fixed job sequence by linear programming. This research was supported in part by The Hong Kong Polytechnic University under grant number G-T246 and the Research Grants Council of Hong Kong under grant number PolyU 5191/01E. In addition, the research of M.Y. Kovalyov was supported by INTAS under grant number 00-217.     

带凸资源和恶化效应的单机松弛窗口排序

研究带凸资源和恶化效应的单机窗口指派排序问题,其中窗口指的是松弛窗口,凸资源和恶化效应指的是工件的实际加工时间是其开始加工时间的线性函数,是其资源消耗量的凸函数。目标是确定工件的加工顺序,资源分配量以及窗口的开始加工时间和长度使其在总资源消耗费用(与窗口有关的排序费用)有上界限制的条件下,极小化与窗口有关的排序费用(总资源消耗费用)。获得了求解上述问题的最优算法,证明了该问题是多项式时间可解的。     

Imperfect debugging in software reliability: A Bayesian approach

The objective of studying software reliability is to assist software engineers in understanding more of the probabilistic nature of software failures during the debugging stages and to construct reliability models. In this paper, we consider modeling of a multiplicative failure rate whose components are evolving stochastically over testing stages and discuss its Bayesian estimation. In doing so, we focus on the modeling of parameters such as the fault detection rate per fault and the number of faults. We discuss how the proposed model can account for “imperfect debugging” under certain conditions. We use actual inter-failure data to carry out inference on model parameters via Markov chain Monte Carlo methods and present additional insights from Bayesian analysis.     

Sensitivity analysis of release time of software reliability models incorporating testing effort with multiple change-points

To accurately model software failure process with software reliability growth models, incorporating testing effort has shown to be important. In fact, testing effort allocation is also a difficult issue, and it directly affects the software release time when a reliability criteria has to be met. However, with an increasing number of parameters involved in these models, the uncertainty of parameters estimated from the failure data could greatly affect the decision. Hence, it is of importance to study the impact of these model parameters. In this paper, sensitivity of the software release time is investigated through various methods, including one-factor-at-a-time approach, design of experiments and global sensitivity analysis. It is shown that the results from the first two methods may not be accurate enough for the case of complex nonlinear model. Global sensitivity analysis performs better due to the consideration of the global parameter space. The limitations of different approaches are also discussed. Finally, to avoid further excessive adjustment of software release time, interval estimation is recommended for use and it can be obtained based on the results from global sensitivity analysis.     

Resource-constrained project scheduling with concave processing rate functions

In this paper, we address a resource-constrained project scheduling problem involving a single resource. The resource can be applied at varying consumption rates to the activities of the project. The duration of each activity is defined by a convex, non-increasing time-resource trade-off function. In addition, activities are not preemptable (ie, the resource consumption rate of an activity cannot be altered while the activity is being processed). We explicitly consider variation of the rate at which an activity is performed with variation in resource consumption rate. We designate the number of units (amount of an activity) performed per unit time with variation in resource consumption rate as the processing rate function, and assume this function to be concave. We present a tree-search-based method in concert with the solution of a nonlinear program and the use of dominance properties to determine: (i) the sequence in which to perform the activities of the project, and (ii) the resource consumption rate to allocate to each activity so as to minimize the project duration (makespan). We also present results of an experimental investigation that reveal the efficacy of the proposed methodology. Finally, we present an application of this methodology to a practical setting.     

Tabu search for discrete–continuous scheduling problems with heuristic continuous resource allocation

Problems of scheduling non-preemptable, independent jobs on parallel identical machines under an additional continuous renewable resource to minimize the makespan are considered. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. The processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation that minimize the makespan. A heuristic procedure for allocating the continuous resource is used. The tabu search metaheuristic to solve the considered problem is presented. The results produced by tabu search are compared with optimal solutions for small instances, as well as with the results generated by simple search methods – multi-start iterative improvement and random sampling for larger instances.     

Resource level minimization in the discrete–continuous scheduling

A discrete–continuous problem of non-preemptive task scheduling on identical parallel processors is considered. Tasks are described by means of a dynamic model, in which the speed of the task performance depends on the amount of a single continuously divisible renewable resource allotted to this task over time. An upper bound on the completion time of all the tasks is given. The criterion is to minimize the maximum resource consumption at each time instant, i.e., the resource level. This problem has been observed in many industrial applications, where a continuously divisible resource such as gas, fuel, electric, hydraulic or pneumatic power, etc., has to be distributed among the processing units over time, and it affects their productivity. The problem consists of two interrelated subproblems: task sequencing on processors (discrete subproblem) and resource allocation among the tasks (continuous subproblem). An optimal resource allocation algorithm for a given sequence of tasks is presented and computationally tested. Furthermore, approximation algorithms are proposed, and their theoretical and experimental worst-case performances are analyzed. Computer experiments confirmed the efficiency of all the algorithms.     

The effect of testing errors on a repetitive testing process

Repetitive testing is a fairly common practice in the final testing stage of a chip manufacturing. Decisions on setting initial lot size and the number of testing repetitions are crucial to the effectiveness of the testing process. The task of setting optimal parameters for a testing process is often difficult in practical situations due to uncertainties in both incoming product yield and testing equipment condition during the testing process. In this paper, we investigate a repetitive testing process where the testing equipment may shift randomly from an in-control state to an inferior state during the testing process which, correspondingly, results in different testing errors. We develop a quantitative model that helps us to find optimal test parameters that maximizes system performance. Based on the model, we performed extensive numerical experiments to test the effects of incoming product defective rate, testing equipment shift rate, especially, type II testing errors on decision and system performance. We find that test equipment condition may significantly affect the optimal decisions on the number of test repetitive and initial testing batch size. Further, we find that, while a small type II testing error may have negligible negative effect of system performance, the effect increases as the error or the incoming product yield increases. The results of this research may potentially provide practitioners with insights and a quantitative tool for designing an efficient repetitive testing process.