Marginal optimization method to improve the inconsistent comparison matrix in the analytic hierarchy process

To improve the inconsistency in the analytic hierarchy process(AHP), a new method based on marginal optimization theory is proposed. During the improving process, this paper regards the reduction of consistency ratio(CR) as benefit, and the maximum modification compared to the original pairwise comparison matrix(PCM) as cost, then the improvement of consistency is transformed to a benefit/cost analysis problem. According to the maximal marginal effect principle, the elements of PCM are modified by a fixed increment(or decrement) step by step till the consistency ratio becomes acceptable, which can ensure minimum adjustment to the original PCM so that the decision makers’ judgment is preserved as much as possible. The correctness of the proposed method is proved mathematically by theorem. Firstly, the marginal benefit/cost ratio is calculated for each single element of the PCM when it has been modified by a fixed increment(or decrement).Then, modification to the element with the maximum marginal benefit/cost ratio is accepted. Next, the marginal benefit/cost ratio is calculated again upon the revised matrix, and followed by choosing the modification to the element with the maximum marginal benefit/cost ratio. The process of calculating marginal effect and choosing the best modified element is repeated for each revised matrix till acceptable consistency is reached, i.e., CR<0.1. Finally,illustrative examples show the proposed method is more effective and better in preserving the original comparison information than existing methods.

Fuzzy Consistent Matrix and Its Application

This paper presents a kind of fuzzy consistent matrix and introduces its application in analytic hierarchy process and adaptive fuzzy reasoning.

The construction and combined operation for fuzzy consistent matrixes

Fuzziness is one of the general characteristics of human thinking and objective things. Introducing fuzzy techniques into decision-making yields very good results. Fuzzy consistent matrix has many excellent characteristics, especially cen- ter-division transitivity conforming to the reality of the human thinking process in decision-making. This paper presents a new approach for creating fuzzy consistent matrix from mutual supplementary matrix in fuzzy decision-making. At the same time, based on the distance between individual fuzzy consistent matrix and average fuzzy consistent matrix, a kind of combined opera- tion for several fuzzy consistent matrixes is presented which reflects most opinions of experienced experts. Finally, a practical example shows its flexibility and practicability further.

An Algorithm Concerning Adjustment of Unconsistent Judgment Matrix in Ahp

Consistency test of the judgment matrix is an essential step in the aplication of Analytic Hierarchy Process (AHP). This thesis presents an algerithm concerning the adjustment of a judgment matrix when it failed to pass the consistency test.

TWO APPROACHES TO IMPROVING THE CONSISTENCY OF COMPLEMENTARY JUDGEMENT MATRIX

By the transformation relations between complementary judgement matrix and recip- rocal judgement matrix,this paper proposes two methods for improving the consistency of com- plementary judgement matrix and gives two simple practical iterative algorithms.These two al- gorithms are easy to implementon computer,and the modified complementary judgementmatri- ces remain most information that original matrix contains.Thus the methods supplement and develop the theory and methodology for improving consistency of complementary judgement matrix

HOW TO CONSTRUCT A CONSISTENCY JUDGEMENT MATRIX IN HIERARCHICAL ANALYSIS

This paper presents a two-stage method to construct a judgement matrix inhierarchical analysis,which naturally satisfies the consistency condition.The procedure isillustrated by an example.

Greedy calibration method for binocular camera system via characteristic homography matrix

A plane-based and linear camera calibration technique without considering lens distortion is proposed in a greedy and intuitive framework for the binocular camera system. Characteristic homography matrix and consistency constraints in close range are employed in this calibration. First, in order to calculate the internal geometries of the cameras, total least-square fitting as a robust tool for the geometrical cost function is exploited to recover the accurate principal point of each camera from all the characteristic lines of the homography matrices for all model planes. Secondly, generic prior knowledge of the aspect ratio of pixel cells is incorporated into the system to obtain the exact principal length in each camera. Thirdly, extrinsic geometries are accurately computed for all planar patterns with respect to each monocular camera. Finally, the rigid displacement between binocular cameras can be obtained by imposing the consistency constraints in 3-space geometry. Both simulation and real image experimental results indicate that reasonably reliable results can be obtained by this technique. And the proposed method is sufficient for applications where high precision is not required and can be easily performed by common computer users who are not experts in computer vision.

New Method for Constructing Comparison Matrix Based on the Proportion Scales in the AHP

The main faults existing in current scale methods are that the scales do not represent the real importance of alternatives and their relations. This paper presents a proportion judgment scale and introduces a new method based on the proportion scale for construction comparison matrix in the analytic hierarchy process (AHP). The proportion judgment scales do not have the faults existing in current scale methods and the comparison matrix constructed by the new scale

Behavior Consistency Computation for Workflow Nets With Unknown Correspondence

Consistency degree calculation is established on the basis of known correspondence, but in real life, the correspondence is generally unknown, so how to calculate consistency of two models under unknown correspondence has become a problem. For this condition, we should analyze unknown correspondence due to the influence of different correspondences.In this paper we obtain the relations of transitions based on event relations using branching processes, and build a behavioral matrix of relations. Based on the permutation of behavioral matrix, we express different correspondences, and define a new formula to compute the maximal consistency degree of two workflow nets. Additionally, this paper utilizes an example to show these definitions, computation as well as the advantages.

Mixed Least Square Method for Priority of Complementary Judgement Matrix and Its Algorithm

Based on the concept of multiplicative fuzzy consistent complementary judgement matrix, the mixed least square method （MLSM） for priority of complementary judgement matrix is proposed and proved. Then, the corresponding convergent iterative algorithm is given and its convergence is proved. Finally, some main properties of the developed priority method, such as rank preservation under strong condition, etc., ate introduced. The theoretical analyses show that the MLSM can sufficiently reflect the preference information of the decision maker, and is easy to realize on a computer.

EFFECTS OF MICROSTRUCTURE ON THE MACROSCOPIC ELASTO-PLASTIC PROPERTIES OF METAL MATRIX COMPOSITES

The generalized self-consistent finite-element iterative averaging method was adopted to analyze the elasto-plastic tensile properties of SiC whiskers reinforced aluminum matrix composites. The effects of varying fiber's aspect ratio and volume fraction on the macroscopic elasto-plastic deformation of the composites were studied. By the analysis of microscopic stress fields, the relation between the propagation of the elasto-plastic region in the matrix and the macroscopic elasto-plastic deformation of composites was discussed. It was found that the propagation of the plastic region in the matrix between the fiber's ends would affect prominently the elasto-plastic tensile behaviour of the composites. It was shown that the characterization of the stress-strain response in terms of the 0.2% offset yield strength is incomplete.

乘型模糊判断矩阵排序向量的递推方法

文章首先在模糊判断矩阵乘型一致性以及矩阵元素和权重之间关系的基础上,结合模糊判断矩阵的上三角矩阵元素,构建了一个关于权重和矩阵上三角元素的方程组,并证明了该方程组存在唯一的正解。随后指出方程组的证明过程就是模糊判断矩阵排序向量的求解过程,从而给出了乘型一致性模糊判断矩阵排序向量的一种递推方法。然后,在偏差函数基础上,通过构造并求解一个优化模型,求出了非乘型一致性模糊判断矩阵的排序向量,结果显示,其解的形式与采用乘型一致性模糊判断矩阵递推方法得到的排序向量完全一样。最后,通过实例以及相关方法对比说明排序向量递推方法是可行有效的。

考虑剪切变形的欧拉梁单元一致质量矩阵

为明确考虑剪切变形的欧拉梁单元一致质量矩阵推导方法,以形函数为基础,基于虚功原理,区分伸缩、扭转以及是否考虑剪切变形的弯曲受力状态,给出了欧拉梁单元一致质量矩阵表达式。研究表明:不考虑剪切变形时,欧拉梁受弯状态的一致质量分析中一般不计单元上微元体水平方向的惯性力,此时仅需弯曲变形引发的竖向位移形函数即可;考虑剪切变形时,则需计入水平方向的惯性力,且同时需要完整的竖向位移形函数和纯弯曲转角位移形函数;对于变截面欧拉梁单元,其一致质量矩阵表达式过于复杂,可在等截面梁的基础上,根据元素位置对矩阵元素匹配左右端或平均截面面积及截面极惯性矩近似给出实用的质量矩阵表达式;考虑剪切变形时,欧拉梁的刚度矩阵亦可经竖向位移和纯弯曲转角位移形函数计算得到,这一过程与使用纯弯曲竖向位移和纯剪切竖向位移形函数的过程在本质上是一致的。

基于一致性的赋权方法在课程过程性评价中的应用

针对课程平时成绩的权重比例,构建采用客观赋权法确定权重的动态代数模型.基于平时成绩的各项得分,分析项目之间的一致度,应用矩阵理论对权重进行一致性检验,给出一致性最强的权重比例.结合实例计算最优权重.

基于权重的AHP判断矩阵一致性调整方法

传统的判断矩阵调整方法计算得到的权重不能最大限度地保留原始判断矩阵的信息。为此,提出一种基于权重的判断矩阵调整方法。该方法根据权重的性质,构造完全一致矩阵,利用AdaGrad最速下降法对矩阵进行拟合,使该矩阵相对于原始判断矩阵的总改变量最小,以其对应的权重作为矩阵的计算结果。数值计算结果表明,该方法相比于其他方法,权重对原始判断信息的保留度更高,且不需要反复进行矩阵一致性检验。

层次分析法中判断矩阵应用的场景化教学设计——以供应商综合评价为例

判断矩阵是层次分析法的核心,在多因素综合评价决策中有着重要应用。然而,现有关于判断矩阵的教学,忽略了一致性检验未通过应当如何重新构建的问题,致使学生难以形成系统认知,进而影响学生的实际应用。文章根据学生对判断矩阵的学习与掌握逻辑,从判断矩阵的概念、性质、构建、检验与调整等知识点进行了系统化梳理,并通过编制的“供应商综合评价”案例,特意设置一些有违一致矩阵性质的数据,让学生体会判断矩阵的构造原则、检验步骤和调整思路,加深对判断矩阵的构建与检验要点的理解,能熟练应用判断矩阵解决多因素综合决策问题。

基于特征点动态选择的三维人脸点云模型重建

针对典型的点云配准方法中伪特征点过多导致配准效率低和配准结果不精确的问题,提出一种基于特征点动态选择的三维人脸点云模型重建方法。该方法在粗配准阶段,采用动态特征矩阵求解法获取粗匹配特征变换矩阵以避免伪特征点的干扰。在精配准过程中,采用二次加权法向量垂直距离法在人脸流形表面选择更有效的特征点以减少伪特征点的数量,并采用基于特征融合与局部特征一致性的迭代最近点方法进行精配准。经过对比实验验证了算法的可行性,实验结果表明,该算法能够实现高精度且快速的三维人脸点云模型重建,且均方根误差达到1.8165 mm,相较其他算法,其在模型重建精度和效率方面都有所提升,具有良好的应用前景。

基于改进模糊层次分析法的智能电能表运行质量评估策略

文章将传统模糊层次分析法进行改进并应用于配电网电能表运行质量评价。模型将电能表运行质量采用准确率指标、故障率指标、可靠率指标,事件记录可靠性四类宏观指标进行评估并分别制定方案层指标。在数据的预处理工作之后通过专家打分法形成两两之间的成对比较矩阵,并基于反馈过程对主观性较强的评分权重进行调节;针对成对比较矩阵计算其最大特征值以及相应的最大特征向量,将该向量进行归一化处理得到各指标的评价权重,并基于最大特征值构建相关并进行一致性检验。最后采用所建立的模型对某地区配电网内电能表运行质量进行改进模糊层次评估,针对实际电能表的评估结果表明所建立的模型能够有效评价电能表的运行状态。

概率犹豫模糊判断矩阵的一致性调整与多重最优解分析

针对备选方案优先权重及多重最优解的计算,建立了一个概率犹豫模糊互补判断矩阵的离散化一致性调整模型,通过该模型调整后的矩阵能够较好地维持偏好值的具体含义,利于后续决策者的反馈修改。而后提出了多重最优解筛选机制,采用距离测度量化各个调整后的矩阵的决策信息扭曲程度及缺失状况,解决了一致性调整过程中的潜在解缺失导致的信息流失及排序互逆问题。最后,给出一个算例验证了模型及筛选机制的有效性。

基于一致性和多样性的多尺度自表示学习的深度子空间聚类

深度子空间聚类(DSC)基于原始数据位于低维非线性子空间的集合中的假设。其中深度子空间聚类多尺度表示学习方法在深度自编码器的基础上,将每一层的编码器与对应的解码器之间都添加全连接层,并以此捕获多尺度的特征,但它没有深度分析多尺度特征的性质,也没有考虑输入数据和输出数据之间多尺度的重构损失。为了解决上述问题,首先建立每个网络层的重构损失函数,监督不同级别编码器参数的学习;然后利用多尺度特征共有的自表示矩阵和特有的自表示矩阵的和具有块对角性,提出更有效的多尺度自表示模块;最后分析不同尺度特征特有的自表示矩阵之间的多样性,有效地利用了多尺度的特征矩阵。在此基础上,提出一种基于一致性和多样性的多尺度自表示学习的深度子空间聚类(MSCD-DSC)方法。在数据集Extended Yale B、ORL、COIL20和Umist上的实验结果表明,相较于次优的MLRDSC(Multi-Level Representation learning for Deep Subspace Clustering),MSCD-DSC的聚类错误率分别降低了15.44%、2.22%、3.37%和13.17%,表明MSCD-DSC的聚类效果优于已有的方法。