Book contents
- Frontmatter
- Contents
- Preface
- PART I Theoretical and Methodological Aspects of the AHP/ANP Methods
- 1 Priorities Originate from Dominance and Order Topology in AHP/ANP: The Fundamental Scale, Relative Scales and When to Preserve Rank
- 2 Validation Examples for the Analytic Hierarchy Process and the Analytic Network Process
- 3 Rank from Comparisons and from Ratings in the Analytic Hierarchy/Network Processes
- PART II Application of the AHP in Solving Economic, Organizational and Management Problems 77
- PART III Application of the AHP in Connection with Other Methods
- PART IV Application of the ANP in Solving Economic, Organizational, Social and Political Problems
- PART V Other Multicriteria Methods
1 - Priorities Originate from Dominance and Order Topology in AHP/ANP: The Fundamental Scale, Relative Scales and When to Preserve Rank
from PART I - Theoretical and Methodological Aspects of the AHP/ANP Methods
Published online by Cambridge University Press: 01 February 2018
- Frontmatter
- Contents
- Preface
- PART I Theoretical and Methodological Aspects of the AHP/ANP Methods
- 1 Priorities Originate from Dominance and Order Topology in AHP/ANP: The Fundamental Scale, Relative Scales and When to Preserve Rank
- 2 Validation Examples for the Analytic Hierarchy Process and the Analytic Network Process
- 3 Rank from Comparisons and from Ratings in the Analytic Hierarchy/Network Processes
- PART II Application of the AHP in Solving Economic, Organizational and Management Problems 77
- PART III Application of the AHP in Connection with Other Methods
- PART IV Application of the ANP in Solving Economic, Organizational, Social and Political Problems
- PART V Other Multicriteria Methods
Summary
“A scientific truth does not triumph by convincing its opponents and making them see the light. But rather because its opponents eventually die, and a new generation grows up that is familiar with it.”
Max PlanckKey words: Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), supermatrix, fundamental scale, pairwise comparison, priority, inconsistency
Abstract
The Analytic Hierarchy and Network Processes perform paired comparisons to derive priorities. It is shown in three ways that it is necessary that these priorities take the form of the principal eigenvector of the matrix of paired comparisons when it is inconsistent. One way is to show that if the priorities generated in any way are used to weight the corresponding rows of the matrixes that are then summed, they must reproduce themselves and this leads to the principal eigenvector. The second way is to obtain the priorities by considering all order dominance and use Cesaro summability to show that the limit is the principal eigenvector. The third and final way uses perturbation of the judgments of a consistent matrix, whose priorities are trivially known to be the principal right eigenvector, to obtain the same kind of solution for an inconsistent matrix with small inconsistency. It is also shown how to derive the fundamental scale of the AHP from the solution (in the real domain) of the generalization of the eigenvalue problem to Fredholm's equation of the second kind and its corresponding stimulus response first order approximation. A discussion of scales of measurement, of relative scales and of the absence of a unique unit and an absolute zero is included. Finally, the paper contains a brief discussion of rank preservation and reversal with and without conditional independence.
INTRODUCTION
People have thought written books about and taught us that the only way to measure things is by having a scale that has a zero and a unit and applying such a scale to objects concrete or abstract. But that is not true. Our genetics have endowed us with the ability to make comparisons, and from such comparisons we can also derive scales that have no predetermined zero and unit.
- Type
- Chapter
- Information
- The Analytic Hierarchy and Network ProcessesApplication in Solving Multicriteria Decision Problems, pp. 13 - 38Publisher: Jagiellonian University PressPrint publication year: 2009