外文翻譯--矩陣聚合方法在群體決策過程的應(yīng)用

1、<p>　　畢 業(yè) 設(shè) 計(jì)（論文）</p><p>　　題 目： Application of Matrix Aggregation Method in Group Decision Making Process</p><p>　　學(xué) 院： 數(shù)理學(xué)院 </p><p>　　專業(yè)名稱： 信息與計(jì)算科學(xué) </p><p>　　學(xué)

2、 號： </p><p>　　學(xué)生姓名： </p><p>　　指導(dǎo)教師： </p><p>　　2013年2月20日</p><p>　　矩陣聚合方法在群體決策過程的應(yīng)用</p><p><b>　　周威廉</b>

3、;</p><p>　　（合肥科技大學(xué)經(jīng)濟(jì)管理學(xué)院，安徽合肥）</p><p>　　摘要：在群體決策過程中，不同的矩陣集合計(jì)劃將產(chǎn)生各種不同的排名關(guān)系和權(quán)值向量。在分析和應(yīng)用兩種凸組合的阿達(dá)瑪基于矩陣聚合方案及圖論之后,本文將從不同矩陣聚合的判斷中探索更合理的方法來測驗(yàn),選擇和優(yōu)化結(jié)果。</p><p>　　關(guān)鍵詞: 群體決策；判斷矩陣；聚合；優(yōu)化</p>

4、;<p><b>　　1引言</b></p><p>　　作為一個(gè)有效的方法用于多目標(biāo)和多因素決策、層次分析法已經(jīng)廣泛應(yīng)用于許多決策方面。它通常涉及多個(gè)決策者,因此,多個(gè)判斷矩陣提供不同的決策者需要匯總,以便達(dá)到更合理的解決方案。領(lǐng)域中的矩陣聚合,李躍進(jìn)和郭欣榮利用連通的無向圖及其理論,通過排除偏見的專家判斷, 從理論的面向電力圖、簡單的m-th無向連通圖想出了一個(gè)互反判斷矩陣

5、聚合方法。然而,劉欣和楊善麗基于判斷矩陣發(fā)展了阿達(dá)瑪凸組合,提供了關(guān)于 “添加法”和“乘法”凸組合一致性明顯改善的證據(jù)。</p><p>　　不同的矩陣集合計(jì)劃將處理專家判斷數(shù)據(jù)、差異和聚合導(dǎo)致判斷矩陣的產(chǎn)生方式不同,因此在計(jì)算重要性和一致性是不同與另一方面。同時(shí),在這個(gè)過程中矩陣的聚合,同一聚合方案也存在不同的判斷矩陣不一致地聚合。在實(shí)踐中解決問題,就必須采用不同的聚合方法和實(shí)施相關(guān)矩陣的驗(yàn)證和選擇。本文將探討

6、矩陣的可行性和存在的問題,從聚合方案啟動圖論和阿達(dá)瑪凸組合,做出相關(guān)的驗(yàn)證、優(yōu)化和選擇。</p><p>　　2 矩陣聚合方法的描述</p><p>　　2.1基于圖論的矩陣聚合方法</p><p>　　基于圖論的矩陣聚合方法：建立一個(gè)水平偏差矩陣E,選擇更一致的因素從不同的專家判斷矩陣A(k),構(gòu)建一個(gè)完整的一致判斷矩陣。詳細(xì)的步驟和解釋見文檔</p>

7、<p>　　步驟1:建立一致性專家判斷矩陣；</p><p>　　步驟2: 在決策過程中設(shè)置品位偏差矩陣</p><p>　　代表專家在年代價(jià)值觀重要性排名比較指標(biāo)i和j</p><p><b>　　(2)</b></p><p>　　步驟3：選擇（n-1）元素，這是有級偏差的最小值，同時(shí)，要求任一項(xiàng)的第（

8、n-1）元素還沒有由其他第（n-2）元素給出；</p><p>　　步驟4：從專家判斷矩陣中，選擇在 相同的位置的所有元素，并記錄為;</p><p>　　步驟5：通過加法或乘法的方法匯總各組，并記錄結(jié)果為;</p><p>　　步驟6：在（N-1）中使用加法合成得到，并建立綜合判斷矩陣A *，應(yīng)用該方法的總結(jié)，計(jì)劃最終排序。</p><p>

9、;　　2.2基于阿達(dá)瑪凸組合的矩陣聚合方法</p><p>　　由于判斷矩陣的群決策的聚集，文獻(xiàn)[2]提供了Hadamard凸組合的概念。如果A1，A2，..... Am在數(shù)量為m前提下是判斷矩陣，相同的問題，假如存在使得</p><p><b>　?。?）</b></p><p><b>　　（4）</b></p&

10、gt;<p>　　因此，被命名為A1，A2，…Am的一個(gè)額外的凸組合，是一個(gè)阿達(dá)瑪乘法凸組合。運(yùn)算符⊕定義如下</p><p>　　若C=A⊕ B那么；若C=A·B則</p><p>　　在此基礎(chǔ)上,文檔[3] 解釋了“加法”和“乘法”凸組合的判斷矩陣的基本理論,并認(rèn)為“加法”和“乘法”凸組合判斷矩陣不僅可以消除主觀因素的影響,也可以保持和提高判斷矩陣的一致性,同時(shí)

11、證明了相應(yīng)過程,因此它證實(shí)“加法”和“乘法”凸組合判斷矩陣在群體決策支持系統(tǒng)中對判斷矩陣是兩個(gè)有效的聚合方法。</p><p>　　3 矩陣聚合方法的應(yīng)用</p><p>　　3.1基于圖論的矩陣聚合方法的應(yīng)用</p><p>　　步驟1:根據(jù)問題的變化,將對步驟1做一定的調(diào)整。原始文檔在專家矩陣無法達(dá)到一致性時(shí)，要求專家判斷矩陣重建。相反本文認(rèn)為, 實(shí)際中有各樣的

12、困難存在于判斷矩陣的重建。該報(bào)稱,專家數(shù)據(jù)不一致或不太一致可以忽略和簡化問題,專家判斷矩陣由評價(jià)指標(biāo)體系的重要性G =(G1、G2、G3、G4,G5,G6,G7)判定,條件是它是符合一致性,符合一致的比率CR由小到大排序,排在前五位的專家判斷矩陣如下:</p><p>　　經(jīng)計(jì)算得一致性比率：</p><p>　　步驟2:特級偏差矩陣的建立，如上；</p><p>

13、<b>　　步驟3:選擇</b></p><p>　　根據(jù)特級偏差矩陣E,選擇更高級別一致性的6個(gè)元素并且得到無向連通圖。</p><p><b>　　1</b></p><p>　　圖1所示。無向連通圖(F1)基于特級偏差矩陣E</p><p>　　因?yàn)樗遣环弦蟮臒o向連通圖,v1 v2 v4

14、 v5,v1 v2 v3都在形成回路。因此它需要遵循破圈法：首先,這些較大的特級偏差的元素都換成了添加元素之后品位偏差仍較小的元素。省略細(xì)節(jié)流程,得到無向圖的連接圖2和圖3。</p><p><b>　　2</b></p><p><b>　　3</b></p><p>　　基于圖2,選擇相應(yīng)的元素從判斷矩陣A1-A5是:

15、</p><p><b>　　按照加法原理得:</b></p><p><b>　　*</b></p><p><b>　　*</b></p><p><b>　　可得：</b></p><p>　　因此,建立初始矩陣如下:<

16、;/p><p>　　基于反射的原理,行和列是成正比的,丟失的元素被填滿,因此構(gòu)造一個(gè)一致判斷矩陣.根據(jù)計(jì)算得：W= (0.200, 0.081, 0.419, 0.037, 0.011, 0.032, 0.220)T.同樣,對于圖3,通過使用相同的計(jì)算過程得W=（0.191，0.080，0.408，0.036，0.034，0.031，0.220)T.</p><p>　　3.2基于阿達(dá)瑪凸組合

17、的矩陣聚合方法的應(yīng)用</p><p>　　由阿達(dá)瑪凸組合的基礎(chǔ)理論，根據(jù)一致的比率選擇專家矩陣A1，A2，A3，A4，A5. 為了便于研究,本文設(shè)置不同的專家判斷矩陣的權(quán)值相同. </p><p>　　令 (0 .2, 0 .2 , 0 .2 , 0 .2 , 0 .2 )</p><p>　　對于 有 ，CR=0.036<0.1.</p>&

18、lt;p>　　同理對于 有，CR=0.032<0.1</p><p>　　4 矩陣聚合的選擇和優(yōu)化</p><p>　　總之,通過使用4個(gè)不同的方法, 獲得4個(gè)權(quán)向量：</p><p>　　分析上面列出的權(quán)向量得索引系統(tǒng)G=(G1,G2,G3,G4,G5,G6,G7)的重要性排序列表</p><p>　　① G3>G7>

19、;G1>G2>G4>G6>G5；② G3>G7>G1>G2>G4>G5>G6</p><p>　?、?G3>G1>G7>G2>G4>G5>G6； ④ G3>G7>G1>G2>G4>G5>G6</p><p>　　基于上面的排序結(jié)果,錯(cuò)誤是G1和G7中列名的第二

20、、第三的地方,和G5和G的第六和第七.</p><p>　　結(jié)合上面方案的結(jié)果,在排序結(jié)果①②④中有G7>G1而③中為G1>G7這可以支持排序結(jié)果來自添加凸組合有顯著差異的其他方案.結(jié)果不合理,應(yīng)該刪除.在①②④中，②④的G5>G6而①中G6>G5故①被排除.因此本文認(rèn)為,合理的排序方案是:</p><p>　　對于聚合的專家矩陣A1，A2，A3，A4，A5, &l

21、t;/p><p>　　應(yīng)該被選為指標(biāo)體系G= (G1, G2, G3, G4, G5, G6, G7)的權(quán)值.</p><p><b>　　4 總結(jié)</b></p><p>　　計(jì)算權(quán)重和索引排序得到來自不同方案的矩陣集合結(jié)果的各不相同.在群體決策過程中,如何有效地減少這種差異,達(dá)到更合理的結(jié)果需要采用多個(gè)聚合方法.然后選擇和優(yōu)化必須完成的計(jì)算結(jié)果

22、產(chǎn)生的那些方法.本文認(rèn)為, 在群體決策過程,應(yīng)用多種方法優(yōu)化和實(shí)際選擇將有利于提高矩陣聚合的合理性和一致性.</p><p><b>　　5 參考文獻(xiàn)</b></p><p>　　1. Lv Yuejin, Guo Xinrong. An Effective Aggregation Method for Group AHP Judgment Matrix. Theor

23、y and Practice of Systems Engineering, 2007,20(7):132-136.</p><p>　　2. Liu Xin, Yang Shanlin. Hadamard Convex Combinations of Judgment Matrix. Theory and Practice of Systems Engineering, 2000，10(4):83-85.<

24、;/p><p>　　3. Yang Shanlin, Liu Xinbao. Two Aggregation Method of Judgment Matrix in GDSS. Journal of Computers, 2001,24(1):106-111.</p><p>　　4. Yang Shanlin, Liu Xinbao. Research on optimizing prin

25、ciple of Convex Combination coefficients of Judgment Matrix. Theory and Practice of Systems Engineering, 2001,21(8):50-52.</p><p>　　5. Xu Zeshui. A note in Document [1] and [2] for the Properties of Convex C

26、ombinations of Judgment Matrix. Theory and practice of Systems Engineering, 2001,21(1):139-140.</p><p>　　6. Wang Jian, Huang Fenggang, Jin Shaoguang. Study on Adjustment Method for Consistency of Judgment Ma

27、trix in AHP. Theory and Practice of Systems Engineering, 2005(8):85-91.</p><p>　　文章來源：The third session of the teaching management and course construction of academic conference proceedings，2012</p>&

28、lt;p>　　Application of Matrix Aggregation Method in Group Decision Making Processes</p><p>　　Weilian Zhou</p><p>　　（School of Management Hefei University of Technology Anhui Economic Managemen

29、t Institute Hefei, China）</p><p>　　Abstract— The different matrix aggregation schemes will lead to various ranking relations and weighting vector results in group decision making processes. After analyzing a

30、nd 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 d

31、eveloped from different judgment matrix aggregations.</p><p>　　Keywords— Group Decision Making; Judgment Matrix; Aggregation; Optimization</p><p>　　1 Introduction</p><p>　　As an eff

32、ective 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, mult

33、iple 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 Undirect

34、ed Graph and its theories, by excluding the biased </p><p>　　Different matrix aggregation schemes will process the expert judgment data with discrepancy and aggregation results of judgment matrix are produce

35、d 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 judgme

36、nt matrix aggregation. In the practice of solving problems, it is necessary to adopt different matrix aggregation methods and implemen</p><p>　　2 The Description of Matrix Aggregation Method</p><p

37、>　　2.1 The Solution of Matrix Aggregation Method Based on Graph Theory</p><p>　　Matrix aggregation solution based on graph theory, is to set up a level deviation matrix E, and select more consistent facto

38、rs 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].</p><p>　　Step1: Setting up th

39、e expert judgment matrix with consistency A1-Am；</p><p>　　Step2: Setting up the grade deviation matrix in decision-making process， </p><p>　　Supported by colleges and universities natural scienc

40、e research project of Anhui</p><p>　　Province, china (KJ2012Z054) ;Enterprise development special fund project of Anhui province china in 2011.</p><p>　　represents that the expert ranked at s va

41、lues the importance by comparing indicator i with j.</p><p><b>　　(2)</b></p><p>　　Step3: Selecting (n-1) elements , which are got minimum value of grade deviation, meanwhile, require

42、 any one of (n-1) elements is not given by the other (n-2) elements;</p><p>　　Step4: From expert judgment matrix A1-Am, choosing all the elements that locate in the same position of in,and recording as ；<

43、/p><p>　　Step5: Aggregating selected every group of by using additive or multiplicative method, and recording the results as；</p><p>　　Step6: Using additive synthesis to get at (n-1) and establish

44、ing comprehensive judgment matrix A*, applying the method of summation to sort the schemes eventually.</p><p>　　2.2 Matrix Aggregating Method Based on Hadamard Convex Combination</p><p>　　For

45、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&

46、lt;/p><p><b>　　(3)</b></p><p><b>　　(4) </b></p><p>　　Therefore, is named as an added convex combination for A1，A2，…Am, and is a Hadamard multiplicative convex

47、 combination. For the operators, ⊕ ,·are defined as follows:</p><p>　　If C=A⊕ B，so，if C=A·B，we will get the results that </p><p>　　On this basis, document [3] explained the basic theor

48、y 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 subject

49、ive 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 combinat

50、ions of</p><p>　　3 The Application of Matrix Aggregation Method</p><p>　　3.1 The Application of Matrix Aggregation Based on Graph Theory</p><p>　　Step1: According to the problem cha

51、nge 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 t

52、his 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 s

53、implify the problem, as t</p><p>　　After calculating, the consistency ratios are: (0.041, 0.045, 0.047, 0.048, 0.050)T</p><p>　　Step2: Grade deviation matrix E is established as above.</p>

54、<p>　　Step3: Choosing </p><p>　　According to the grade deviation matrix E, 6 elements with higher level of consistency are selected: and the undirected connected graph is produced.</p><p&

55、gt;　　Figure1. Undirected Connected Graph (F1)Based on Grade deviation Matrix E</p><p>　　Because it does not meet the requirements of undirected connected graph, V1-V2-V4-V5, and V1-V2-V3 are all in the forma

56、tion 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 process

57、es, the undirected connected graphs Figure 2 and Figure 3 are obtained.</p><p><b>　　2</b></p><p><b>　　3</b></p><p>　　Based on Figure2, the elements selected

58、correspondently from the judgment matrix A1-A5 are:</p><p>　　In accordance with the additive principle, it is:</p><p><b>　　*</b></p><p><b>　　*</b></p>

59、<p>　　The results is listed as follows：</p><p>　　Therefore the initial matrix is established as below:</p><p>　　Based on the reflexive principle, and principle that rows and columns are pr

60、oportional to the others, the missing elements are filled in, hence a consistent judgment matrix A* is constructed.</p><p>　　According to the calculation, it comes to W= (0.200, 0.081, 0.419, 0.037, 0.011, 0

61、.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)T</p><p>　　3.2 The Application of Matrix Aggregation Based on Hadamard

62、Convex Combination</p><p>　　On 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 aggr

63、egation. 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 )</p><p>　　According to the calculation from,i

64、t comes to,By using the square law, it comes to CR=0.036<0.1. </p><p>　　Similarly, According to the calculation from，By using the square law, it comes to CR=0.032<0.1</p><p>　　4 Selection

65、and Optimization on Weighting the Results after Matrix Aggregation</p><p>　　In summary, by using 4 different kinds of methods, 4 weight vectors are acquired as follows:</p><p>　　It is could be s

66、een by comparing and analyzing the weight vectors listed above, the importance sorting list on the index system G=(G1,G2,G3,G4,G5,G6,G7) is shown as:</p><p>　?、?G3>G7>G1>G2>G4>G6>G5；② G3>

67、;G7>G1>G2>G4>G5>G6</p><p>　?、?G3>G1>G7>G2>G4>G5>G6； ④ G3>G7>G1>G2>G4>G5>G6</p><p>　　Based on the sorting results above, the errors are focus on

68、 G1 and G7 which are listed in the second, third places, and the G5 and G6, in the sixth and seventh respectively.</p><p>　　Accordingly and by combining with the 5 sorting results from above schemes, among t

69、he sorting results, ①②④ all have the outcomes G7>G1 (G7 is more important than G1), while ③ has the contrary result: G1>G7, which can also support that the sorting results coming from adding convex combination have

70、 significant difference from that of the other schemes, this result is attached with irrationality and should be removed. Meanwhile, in the results ①②④, ② and ④ both show G5>G6, only ① shows G6>G5, so th</p>

71、<p>　　The abandon of sorting scheme ① and ③ is the discard of undirected connected graph and convex combination in this matrix aggregation essentially; the rationality of this abandon has been supported in correspo

72、nding 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 consisten

73、cy exists, but it will show clear randomness that arithmetic average met</p><p>　　are more reasonable to be selected as the weights of index system G= (G1, G2, G3, G4, G5, G6, G7).</p><p>　　5 Co

74、nclusion</p><p>　　The calculation results of weights and indexes sorting getting from different schemes of matrix aggregation differ from each other, how to reduce this difference effectively and reach more

75、reasonable results requires to adopt multiple aggregation methods in group decision making process, then selection and optimization must be done on the calculation results produced by those methods. This paper believes t

76、hat the application of multiple methods and optimization and selection in practice will ben</p><p><b>　　Reference</b></p><p>　　1. Lv Yuejin, Guo Xinrong. An Effective Aggregation Met

77、hod for Group AHP Judgment Matrix. Theory and Practice of Systems Engineering, 2007(7):132-136.</p><p>　　2. Liu Xin, Yang Shanlin. Hadamard Convex Combinations of Judgment Matrix. Theory and Practice of Syst

78、ems Engineering, 2000,20(4):83-85.</p><p>　　3. Yang Shanlin, Liu Xinbao. Two Aggregation Method of Judgment Matrix in GDSS. Journal of Computers, 2001, 24(1):106-111.</p><p>　　4. Yang Shanlin, L

79、iu Xinbao. Research on optimizing principle of Convex Combination coefficients of Judgment Matrix. Theory and Practice of Systems Engineering, 2001,21(8):50-52.</p><p>　　5. Xu Zeshui. A note in Document [1]

80、and [2] for the Properties of Convex Combinations of Judgment Matrix. Theory and practice of Systems Engineering, 2001,21(1):139-140.</p><p>　　6. Wang Jian, Huang Fenggang, Jin Shaoguang. Study on Adjustment