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 矩陣上的容忍度分析是十分的複雜，然而近幾年來卻是廣泛的討論，特別是在決策評估的成對倒數矩陣（positive reciprocal matrix）上。成對倒數矩陣的容忍度分析是在固有系統中，隨著一致性和倒數性質的要求下所執行的。在矩陣和可被視為在評估問題中屬性重要性權重的固有向量之間的關係，透露了在容忍度分析中高度的複雜性卻有趣的含意。因此，我們所提出容忍度分析的第一個主題是給定一個成對倒數矩陣和所導致的權重向量，在可接受的一致性要求下，對於這一個權重向量而言，成對倒數矩陣最大的容忍範圍為何？既然在給定成對倒數矩陣上的容忍度分析中，同一個權重向量未必是有意義的，因此，我們的第二個主題是給定一個一致性的成對倒數矩陣，在要求達到有效度的前提下，成對比較矩陣的最大容忍範圍為何？在得到容忍範圍之後，我們進一步提出了可能的應用。從給定有效的容忍範圍，以建構在固有系統中同一或不同權重的一致性成對倒數矩陣。我們將以例子來驗證所提出的步驟。結果顯示了容忍度分析提供充分的資訊給決策者做一個較佳的決策。
 The tolerance analysis on a matrix is highly complicated and has been widely discussed in recent years, especially on a positive reciprocal matrix (PRM) for an evaluation decision. With the requirement of the consistency and reciprocal, tolerance analysis of a PRM can be performed in an eigensystem. The relation between the matrix and the resultant eigenvector that can be regarded as weights of importance of the attributes in an evaluation problem reveals the high complexity yet interesting interpretation in tolerance analysis. Therefore, the 1st issue of our tolerance analysis is given a PRM and the resultant weighting vector, what is the largest tolerance levels of PRM with respective to the vector under the acceptable consistent requirement? Since tolerance analysis on a given PRM is not necessarily significant in relation to the weighting vector, therefore our 2nd issue is given a consistent PRM, what would be the largest tolerance levels such that the required effectiveness will be obtained. After the tolerance levels are acquired, we further propose the possible applications for constructing a consistent PRM with/without the same weight in an eigensystem from the given effective tolerance levels. The procedure has been demonstrated by an illustrative example. The result shows that the tolerance analysis provides the sufficient information to make a better decision.
 CONTENTSABSTRACT ICHINESE ABSTRACT IIACKNOWLEDGEMENT IIICONTENTS IVFIGURE AND TABLE CAPTIONS VICHAPTER 1. INTRODUCTION 1CHAPTER 2. LITERATURE REVIEW 52.1 Basic Concept and Definitions 62.2 Calibration Methods of An Inconsistent Matrix 82.3 Calibration Effectiveness 92.4 Interval Computation 102.5 Summary 11CHAPTER 3. TOLERANCE ANALYSIS OF A PRM BY CONSISTENCY CONDITION 123.1 Tolerance Analysis in an Eigensystem 123.1.1 Interval Derivation 123.1.2 Construction of a New PRM with the Same Eigenvector 153.1.3 Construction of a New PRM with Different Eigenvector 153.2 Procedure of Tolerance Analysis and Reconstruction of An Inconsistent PRM 193.3 Numerical Example 213.4 Summary 22CHAPTER 4. TOLERANCE ANALYSIS OF A CONSISTENT PRM BY EFFECTIVENESS CONDITIONS 244.1 Tolerance Analysis with Single Parameter 254.1.1 Theoretical Development 254.1.2 Pattern of the Tolerance Matrix 264.1.3 The Procedure 284.1.4 Numerical Example 294.2 Tolerance Analysis with Different Parameters 324.2.1 The Procedure 324.2.2 Numerical Example 334.3 Summary 35CHAPTER 5. AN ILLUSTRATIVE CASE 395.1 A Numerical Example For Tolerance Analysis 395.2 A Numerical Example of Effectiveness For Tolerance Analysis 415.2.1 Tolerance Analysis For The Same Variation 415.2.2 Tolerance Analysis For The Different Variations 445.3 Summary 46CHAPTER 6. SUMMARY AND CONCLUSION 49REFERENCE 50FIGURE AND TABLE CAPTIONSFigure 4.1 Range of “a” 26Figure 4.2 Flow Chart of the Algorithm (1) 37Figure 4.3 Flow Chart of the Algorithm (2) 38Table3.1 THE MEAN CONSISTENCY INDEX OF RANDOMLY GENERATEDMATRICES 20
 REFERENCE1.Franklin J.N., Matrix Theory, Prentice-hall, N.J., 19682.Hillier F.S., Lieberman G.J., Introduction to OperationsResearch 6th edition, McGraw-Hill Inc, Singapore, 19953.Kwiesielewicz M., Uden, E.V., Inconsistent andcontradictory judgements in pairwise comparison method inthe AHP, Computers & Operations Research 31 (2004) 713-7194.Ma. Weiyi, X. Jiangyue, W. Yixiang, A practical approachto modifying pairwise comparison matrices and twocriteria of modificatory effectiveness, Systems Scienceand Systems Engineering 2 (1993) 334-3385.Moore R.E., Interval Analysis, Prentice-hall, N.J., 19666.Moore R.E., Methods And Application of Interval Analysis,SIAM, Philadelphia,19797.Saaty T.L., The Analytic Hierarchy Process, McGraw-Hill,New York, 19808.Wilkinson J.H., The Algebraic Eigenvalue Problem, OxfordUniversity Press, London, 19659.Xu Zeshui, Wei Cuiping, A consistency improving method inthe analytic hierarchy process, European Journal ofOperational Research 116 (1999) 443-44910.Xu Z., On consistency of the weighted geometric meancomplex judgment matrix in AHP, European Journal ofOperational Research 126 (2000) 683-687
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