外文翻译--矩阵聚合方法在群体决策过程的应用.docVIP

毕业设计 （外文翻译）毕 业 设 计（论文）题 目： Application of Matrix Aggregation Method in Group Decision Making Process学 院： 数理学院 专业名称： 信息与计算科学 学 号： 200941210104 学生姓名： 石梦弟 指导教师： 明廷桥 2013年2月20日矩阵聚合方法在群体决策过程的应用周威廉（合肥科技大学经济管理学院，安徽合肥）摘要：在群体决策过程中，不同的矩阵集合计划将产生各种不同的排名关系和权值向量。在分析和应用两种凸组合的阿达玛基于矩阵聚合方案及图论之后,本文将从不同矩阵聚合的判断中探索更合理的方法来测验,选择和优化结果。关键词: 群体决策；判断矩阵；聚合；优化1引言作为一个有效的方法用于多目标和多因素决策、层次分析法已经广泛应用于许多决策方面。它通常涉及多个决策者,因此,多个判断矩阵提供不同的决策者需要汇总,以便达到更合理的解决方案。领域中的矩阵聚合,李跃进和郭欣荣利用连通的无向图及其理论,通过排除偏见的专家判断, 从理论的面向电力图、简单的m-th无向连通图想出了一个互反判断矩阵聚合方法。然而,刘欣和杨善丽基于判断矩阵发展了阿达玛凸组合,提供了关于 “添加法”和“乘法”凸组合一致性明显改善的证据。不同的矩阵集合计划将处理专家判断数据、差异和聚合导致判断矩阵的产生方式不同,因此在计算重要性和一致性是不同与另一方面。同时,在这个过程中矩阵的聚合,同一聚合方案也存在不同的判断矩阵不一致地聚合。在实践中解决问题,就必须采用不同的聚合方法和实施相关矩阵的验证和选择。本文将探讨矩阵的可行性和存在的问题,从聚合方案启动图论和阿达玛凸组合,做出相关的验证、优化和选择。2 矩阵聚合方法的描述2.1基于图论的矩阵聚合方法基于图论的矩阵聚合方法：建立一个水平偏差矩阵E,选择更一致的因素从不同的专家判断矩阵A(k),构建一个完整的一致判断矩阵。详细的步骤和解释见文档步骤1:建立一致性专家判断矩阵；步骤2: 在决策过程中设置品位偏差矩阵代表专家在年代价值观重要性排名比较指标i和j (2)步骤3：选择（n-1）元素，这是有级偏差的最小值，同时，要求任一项的第（n-1）元素还没有由其他第（n-2）元素给出；步骤4：从专家判断矩阵中，选择在 相同的位置的所有元素，并记录为;步骤5：通过加法或乘法的方法汇总各组，并记录结果为;步骤6：在（N-1）中使用加法合成得到，并建立综合判断矩阵A *，应用该方法的总结，计划最终排序。2.2基于阿达玛凸组合的矩阵聚合方法由于判断矩阵的群决策的聚集，文献2提供了Hadamard凸组合的概念。如果A1，A2，. Am在数量为m前提下是判断矩阵，相同的问题，假如存在使得 （3） （4）因此，被命名为A1，A2，Am的一个额外的凸组合，是一个阿达玛乘法凸组合。运算符定义如下若C=A B那么；若C=AB则在此基础上,文档3 解释了“加法”和“乘法”凸组合的判断矩阵的基本理论,并认为“加法”和“乘法”凸组合判断矩阵不仅可以消除主观因素的影响,也可以保持和提高判断矩阵的一致性,同时证明了相应过程,因此它证实“加法”和“乘法”凸组合判断矩阵在群体决策支持系统中对判断矩阵是两个有效的聚合方法。3 矩阵聚合方法的应用3.1基于图论的矩阵聚合方法的应用步骤1:根据问题的变化,将对步骤1做一定的调整。原始文档在专家矩阵无法达到一致性时，要求专家判断矩阵重建。相反本文认为, 实际中有各样的困难存在于判断矩阵的重建。该报称,专家数据不一致或不太一致可以忽略和简化问题,专家判断矩阵由评价指标体系的重要性G =(G1、G2、G3、G4,G5,G6,G7)判定,条件是它是符合一致性,符合一致的比率CR由小到大排序,排在前五位的专家判断矩阵如下: 经计算得一致性比率：步骤2:特级偏差矩阵的建立，如上；步骤3:选择根据特级偏差矩阵E,选择更高级别一致性的6个元素并且得到无向连通图。 1图1所示。无向连通图(F1)基于特级偏差矩阵E 因为它是不符合要求的无向连通图,v1 v2 v4 v5,v1 v2 v3都在形成回路。因此它需要遵循破圈法：首先,这些较大的特级偏差的元素都换成了添加元素之后品位偏差仍较小的元素。省略细节流程,得到无向图的连接图2和图3。 2 3基于图2,选择相应的元素从判断矩阵A1-A5是:按照加法原理得: * * 可得：因此,建立初始矩阵如下:基于反射的原理,行和列是成正比的,丢失的元素被填满,因此构造一个一致判断矩阵.根据计算得：W= (0.200, 0.081, 0.419, 0.037, 0.011, 0.032, 0.220)T.同样,对于图3,通过使用相同的计算过程得W=（0.191，0.080，0.408，0.036，0.034，0.031，0.220)T.3.2基于阿达玛凸组合的矩阵聚合方法的应用由阿达玛凸组合的基础理论，根据一致的比率选择专家矩阵A1，A2，A3，A4，A5. 为了便于研究,本文设置不同的专家判断矩阵的权值相同. 令 (0 .2, 0 .2 , 0 .2 , 0 .2 , 0 .2 )对于 有 ，CR=0.0360.1.同理对于 有，CR=0.032G7G1G2G4G6G5； G3G7G1G2G4G5G6 G3G1G7G2G4G5G6； G3G7G1G2G4G5G6基于上面的排序结果,错误是G1和G7中列名的第二、第三的地方,和G5和G的第六和第七.结合上面方案的结果,在排序结果中有G7G1而中为G1G7这可以支持排序结果来自添加凸组合有显著差异的其他方案.结果不合理,应该删除.在中，的G5G6而中G6G5故被排除.因此本文认为,合理的排序方案是:对于聚合的专家矩阵A1，A2，A3，A4，A5, 应该被选为指标体系G= (G1, G2, G3, G4, G5, G6, G7)的权值.4 总结计算权重和索引排序得到来自不同方案的矩阵集合结果的各不相同.在群体决策过程中,如何有效地减少这种差异,达到更合理的结果需要采用多个聚合方法.然后选择和优化必须完成的计算结果产生的那些方法.本文认为, 在群体决策过程,应用多种方法优化和实际选择将有利于提高矩阵聚合的合理性和一致性.5 参考文献1. Lv Yuejin, Guo Xinrong. An Effective Aggregation Method for Group AHP Judgment Matrix. Theory and Practice of Systems Engineering, 2007,20(7):132-136.2. Liu Xin, Yang Shanlin. Hadamard Convex Combinations of Judgment Matrix. Theory and Practice of Systems Engineering, 2000，10(4):83-85.3. Yang Shanlin, Liu Xinbao. Two Aggregation Method of Judgment Matrix in GDSS. Journal of Computers, 2001,24(1):106-111.4. Yang Shanlin, Liu Xinbao. Research on optimizing principle of Convex Combination coefficients of Judgment Matrix. Theory and Practice of Systems Engineering, 2001,21(8):50-52.5. Xu Zeshui. A note in Document 1 and 2 for the Properties of Convex Combinations of Judgment Matrix. Theory and practice of Systems Engineering, 2001,21(1):139-140.6. Wang Jian, Huang Fenggang, Jin Shaoguang. Study on Adjustment Method for Consistency of Judgment Matrix in AHP. Theory and Practice of Systems Engineering, 2005(8):85-91.文章来源：The third session of the teaching management and course construction of academic conference proceedings，2012Application of Matrix Aggregation Method in Group Decision Making ProcessesWeilian Zhou（School of Management Hefei University of Technology Anhui Economic Management Institute Hefei, China）Abstract The different matrix aggregation schemes will lead to various ranking relations and weighting vector results in group decision making processes. After analyzing and applying two kinds of the Hadamard convex combination based on matrix aggregation schemes and utilizing the graph theory, this paper will explore the more rational approaches to exam, select and optimize the results developed from different judgment matrix aggregations.Keywords Group Decision Making; Judgment Matrix; Aggregation; Optimization1 IntroductionAs an effective method utilized in multi-objective and multifactor decision making, Analytic Hierarchy Process has been widely applied in many decision making aspects. It normally involves several decision makers, therefore, multiple judgment matrixes provided by different decision maker need to be aggregated so that to reach a more reasonable solution. In the field of matrix aggregation, Lv Yuejin and Guo Xinrong utilized the Connected Undirected Graph and its theories, by excluding the biased expert judgments, have come up with a reciprocal judgment matrix aggregation method which was oriented from the theory of m-th power graph of simple undirected connected graph 1. Nevertheless,Liu Xin and Yang Shanlin developed Hadamard convex combination based on judgment matrix 2-3, provided the evidence on “additive”and“multiplicative” convex combinations consistency improvement as well. The document 4 has studied on the optimization principle related with the convex coefficients of judgment matrix, and provided solution to the convex combination coefficients of judgment matrix.Different matrix aggregation schemes will process the expert judgment data with discrepancy and aggregation results of judgment matrix are produced differently, therefore the weight and consistency after calculation are differential from another. While, in the process of matrix aggregation, the same aggregation schemes also present discrepantly in different judgment matrix aggregation. In the practice of solving problems, it is necessary to adopt different matrix aggregation methods and implement relevant verification and choice. This paper will investigate the feasibility and problems of matrix aggregation schemes that are initiated from graph theory and Hadamard convex combinations, andrelevant verification, optimizing and choice have been made.2 The Description of Matrix Aggregation Method2.1 The Solution of Matrix Aggregation Method Based on Graph TheoryMatrix aggregation solution based on graph theory, is to set up a level deviation matrix E, and select more consistent factors from the different expert judgment matrix A(k), so as to construct a complete consistent judgment matrix A*. The detailed construction steps and explanation see document 1.Step1: Setting up the expert judgment matrix with consistency A1-Am；Step2: Setting up the grade deviation matrix in decision-making process， Supported by colleges and universities natural science research project of AnhuiProvince, china (KJ2012Z054) ;Enterprise development special fund project of Anhui province china in 2011. represents that the expert ranked at s values the importance by comparing indicator i with j. (2)Step3: Selecting (n-1) elements , which are got minimum value of grade deviation, meanwhile, require any one of (n-1) elements is not given by the other (n-2) elements;Step4: From expert judgment matrix A1-Am, choosing all the elements that locate in the same position of in,and recording as ；Step5: Aggregating selected every group of by using additive or multiplicative method, and recording the results as；Step6: Using additive synthesis to get at (n-1) and establishing comprehensive judgment matrix A*, applying the method of summation to sort the schemes eventually.2.2 Matrix Aggregating Method Based on Hadamard Convex CombinationFor the aggregation of judgment matrix in group decision making, document 2 provides the concept of Hadamard convex combinations, i.e. if A1，A2，.Am are judgment matrixes at number of m for the same problem, if it is,let (3) (4) Therefore, is named as an added convex combination for A1，A2，Am, and is a Hadamard multiplicative convex combination. For the operators, ,are defined as follows:If C=A B，so，if C=AB，we will get the results that On this basis, document 3 explained the basic theory for “addition ”and “multiplication” convex combinations of judgment matrix, it is believed that addition ”and “multiplication” convex combinations of judgment matrix not only can eliminate the effects caused on subjective factors , but also can keep and improve consistency of judgment matrix, in the meanwhile the correspondent proving process was provided, therefore it is confirmed that “addition ”and “multiplication” convex combinations of judgment matrix are two effective aggregating methods for judgment matrix in groups decision support system.3 The Application of Matrix Aggregation Method3.1 The Application of Matrix Aggregation Based on Graph TheoryStep1: According to the problem change in reality and environment, this paper will make certain adjustment on step1. The original document required the expert reconstructed judgment matrix when expert matrix cannot reach the consistency, on the contrary this paper argue that there are various difficulties exist in real practice for reconstructing judgment matrix. The paper claim that the expert data with inconsistency or less consistent could be ignored directly and to simplify the problem, as to the expert judgment matrix formed by evaluating the importance of index system G=(G1,G2,G3,G4,G5,G6,G7),on the condition that it is complied with the consistency, and in accordance with the consistent ratio CR sorted from small to large, the top 5 expert judgment matrix is listed as follows :After calculating, the consistency ratios are: (0.041, 0.045, 0.047, 0.048, 0.050)TStep2: Grade deviation matrix E is established as above.Step3: Choosing According to the grade deviation matrix E, 6 elements with higher level of consistency are selected: and the undirected connected graph is produced. Figure1. Undirected Connected Graph (F1)Based on Grade deviation Matrix EBecause it does not meet the requirements of undirected connected graph, V1-V2-V4-V5, and V1-V2-V3 are all in the formation of loop, thus it is needed to follow the broken circle method. Firstly, the elements with larger grade deviation are replaced by adding the remained elements with smaller grade deviation. Omitting the detail processes, the undirected connected graphs Figure 2 and Figure 3 are obtained. 2 3Based on Figure2, the elements selected correspondently from the judgment matrix A1-A5 are:In accordance with the additive principle, it is: * * The results is listed as follows：Therefore the initial matrix is established as below:Based on the reflexive principle, and principle that rows and columns are proportional to the others, the missing elements are filled in, hence a consistent judgment matrix A* is constructed.According to the calculation, it comes to W= (0.200, 0.081, 0.419, 0.037, 0.011, 0.032, 0.220)T Similarly, for Figure3, by using the same calculation processes, it comes to W=（0.191，0.080，0.408，0.036，0.034，0.031，0.220)T3.2 The Application of Matrix Aggregation Based on Hadamard Convex CombinationOn the basis of foundational theory on Hadamard Convex Combination, expert matrix A1，A2，A3，A4，A5 above is selected according to consistent ratio, to take correspondent matrix aggregation. For the purposes of facilitating study, this paper set various expert judgment matrixes on equal weights, that is, let (0 .2, 0 .2 , 0 .2 , 0 .2 , 0 .2 )According to the calculation from,it comes to,By using the square law, it comes to CR=0.0360.1. Similarly, According to the calculation from，By using the square law, it comes to CR=0.032G7G1G2G4G6G5； G3G7G1G2G4G5G6 G3G1G7G2G4G5G6； G3G7G1G2G4G5G6Based on the sorting results above, the errors are focus on G1 and G7 which are listed in the second, third places, and the G5 and G6, in the sixth and seventh respectively.Accordingly and by combining with the 5 sorting results from above schemes, among the sorting results, all have the outcomes G7G1 (G7 is more important than G1), while has the contrary result: G1G7, which can also support that the sorting results coming from adding convex combination have significant difference from that of the other schemes, this result is attached with irrationality and should be removed. Meanwhile, in the results , and both show G5G6, only shows G6G5, so that should be removed likewise. Therefore this paper supposes that the reasonable sorting scheme is:The abandon of sorting scheme and is the discard of undirected connected graph and convex combination in this matrix aggregation essentially; the rationality of this abandon has been supported in corresponding documents. The document 5 believed that the integrated judgment matrix contributed by weighted arithmetic average method cannot extend the reciprocity of original judgment matrix, therefore, there is no consistency exists, but it will show clear randomness that arithmetic average method is used partially to maintain reciprocity, and then the new judgment matrix is constructed on a reciprocal basis. The document 6 by utilizing specific example also proved that the additive convex combination is not valid as well，however, it also believes that by using the simple geometric average, the generated matrices convert all the consistent and inconsistent information from expert judgments to a complete consistent positive reciprocal matrix, and also claim that it would be more reasonable to use weighted geometric average to get adjusted matrix by adopting weighted coefficients, which are generated on principle of consistency and rule of minority yield. Thus, this paper believed that the aggregation of judgment matrixes should be mutual proved and selected in variety of schemes to choose more consistent results. For aggregation of expert matrix A1，A2，A3，A4，A5, are more reasonable to be selected as the weights of index system G= (G1, G2, G3, G4, G5, G6, G7).