THE SUPERMATRIX EIGENPROBLEM: AN INTERPRETATION OF THE PRIORITY VECTORS IN THE ANALYTIC NETWORK PROCESS
##plugins.themes.bootstrap3.article.main##
##plugins.themes.bootstrap3.article.sidebar##
Abstract
Continuing the previous articles on interpretation of the solutions obtained in the Analytic Hierarchy Process (AHP) and in the Analytic Network Process (ANP) with outer dependence, the current work considers a general ANP problem and shows that the solution for its supermatrix, obtained by raising the supermatrix to powers is equivalent to solving the eigenproblem for this matrix. Thus, the global priority solution for an ANP model is an eigenproblem of its supermatrix, and the principal eigenvector of the supermatrix itself gives the mean priorities for the whole network of the compared items. This approach provides an easy way to describe complex ANP interconnections, and gives an explicit interpretation of the priority results convenient for practical managerial decisions.
http://dx.doi.org/10.13033/ijahp.v5i1.132
How to Cite
Downloads
##plugins.themes.bootstrap3.article.details##
Analytic Network Process, Supermatrix Eigenvectors
International Journal of Operations and Quantitative Management, 1, 183-195.
Lipovetsky, S. and Conklin, M. (2003). Priority estimations by pair comparisons: AHP,
Thurstone scaling, Bradley-Terry-Luce, and Markov stochastic modeling. Proceedings of
the Joint Statistical Meeting, The American Statistical Association, August 2003, San
Francisco, CA, 2473-2478.
Lipovetsky, S. (2005). Analytic Hierarchy Processing in Chapman-Kolmogorov
equations. International Journal of Operations and Quantitative Management, 11, 219-228.
Lipovetsky, S. (2010). An interpretation of the AHP eigenvector solution for the lay
person. International Journal of the Analytic Hierarchy Process, 2, 158-162.
Lipovetsky. S. (2011a). An interpretation of the AHP global priority as the eigenvector
solution of an ANP supermatrix. International Journal of the Analytic Hierarchy
Process, 3, 70-78.
Lipovetsky. S. (2011b). Priority eigenvectors in Analytic Hierarchy/Network Processes
with outer dependence between alternatives and criteria. International Journal of the
Analytic Hierarchy Process, 3, 172-179.
Saaty, T.L. (1994). Fundamentals of Decision Making and Priority Theory with the
Analytic Hierarchy Process. Pittsburgh, PA: RWS Publications.
Saaty, T.L. (1996). Decision Making with Dependence and Feedback: The Analytic
Network Process. Pittsburgh, PA: RWS Publications.
Appendix I. Additional bibliography
Gogh, Vincent van, The complete letters, published by the Van Gogh Museum, Letter to
his brother Theo, 1882.
Lipovetsky, S. (1996). The synthetic hierarchy method: an optimizing approach to obtaining
priorities in the AHP. European Journal of Operational Research, 93, 550 - 564.
Lipovetsky, S., Tishler, A., Dvir, D., & Shenhar, A. (1997). The relative importance of
project success dimensions. R&D Management, 27, 97-106.
Lipovetsky, S., & Tishler, A. (1999). Interval estimation of priorities in the AHP. European
Journal of Operational Research, 114, 153-164.
Lipovetsky, S., & Lootsma F. A. (2000). Generalized golden section, repeated bisections,
and aesthetic preference. European Journal of Operational Research, 121, 213-216.
Lipovetsky, S., & Conklin, M. (2001). Dual priority-antipriority Thurstone scales as AHP
eigenvectors. Engineering Simulation, 18, 631-648.
Lipovetsky, S., & Conklin, M. (2002). Robust estimation of priorities in the AHP.
European Journal of Operational Research, 137, 110-122.
Lipovetsky, S. (2008a). Bradley–Terry Choice Probability in Maximum Likelihood and
Eigenproblem Solutions. International J. of Information Technology & Decision Making,
7, 395-405.
Lipovetsky, S. (2008b). Comparison among different patterns of priority vectors
estimation methods. International J. of Mathematical Education in Science and
Technology, 39, 301-311.
Lipovetsky, S. (2009a). Global priority estimation in multiperson decision making.
Journal of Optimization Theory and Applications, 140, 77-91.
Lipovetsky, S. (2009b). Optimal hierarchy structures for multi-attribute-criteria decisions.
Journal of Systems Science and Complexity, 22, 228-242.
Magnan, S., Lipovetsky, S. (2010). Key driver analysis with very small samples using the
Analytic Hierarchy Process method. Joint Statistical Meeting, The American Statistical
Association, August 2010, Vancouver, BC, Canada.
Saaty, T.L. (1977). A scaling method for priorities in hierarchical structures.
Mathematical Psychology, 15, 234-281.
Saaty, T.L. (1980). The Analytic Hierarchy Process. New York: McGraw-Hill.
Saaty, T.L. (2000). Decision Making for Leaders. Pittsburgh, PA: RWS Publications.
Saaty, T.L. (2010). Principia Mathematica Decernendi: Mathematical Principles of
Decision Making: Generalization of the Analytic Network Process to Neural Firing and
Synthesis. Pittsburgh, PA: RWS Publications.
Saaty, T.L., & Kearns, K.P. (1985). Analytical Planning. New York: Pergamon.
Saaty, T.L., & Vargas, L.G. (1994). Decision Making in Economic, Political, Social and
Technological Environment with the Analytic Hierarchy Process. Pittsburgh, PA:RWS
Publications.
Wasil, E.A., & Golden, B.L. (2003). Celebrating 25 years of AHP-based decision
making. Computers and Operations Research, 30, 1419-1420.
Whitaker, R. (2007). Validation examples of the Analytic Hierarchy Process and Analytic
Network Process. Mathematical and Computer Modelling, 46(7-8), 840-859.
Whitaker, R., (2007). Criticisms of the Analytic Hierarchy Process: Why they often make
no sense. Mathematical and Computer Modelling, 46(7-8), 948-961.
Copyright of all articles published in IJAHP is transferred to Creative Decisions Foundation (CDF). However, the author(s) reserve the following:
- All proprietary rights other than copyright, such as patent rights.
- The right to grant or refuse permission to third parties to republish all or part of the article or translations thereof. In case of whole articles, such third parties must obtain permission from CDF as well. However, CDF may grant rights with respect to journal issues as a whole.
- The right to use all or parts of this article in future works of their own, such as lectures, press releases, reviews, textbooks, or reprint books.
- The authors affirm that the article has been neither copyrighted nor published, that it is not being submitted for publication elsewhere, and that if the work is officially sponsored, it has been released for open publication.
The only exception to the statements in the paragraph above is the following: If an article published in IJAHP contains copyrighted material, such as a teaching case, as an appendix, then the copyright (and all commercial rights) of such material remains with the original copyright holder.
CDF will receive permission for publication of copyrighted material in IJAHP. This permission is not transferable to third parties. Permission to make electronic and paper copies of part or all of the articles, including all computer files that are linked to the articles, for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage.
This permission does not apply to previously copyrighted material, such as teaching cases. In paper copies of the article, the copyright notice and the title of the publication and its date should be visible. To copy otherwise is permitted provided that a per-copy fee is paid.
To republish, to post on servers, or redistribute to lists requires that you post a link to the IJAHP article, which is available in open access delivery mode. Do not upload the article itself.
Authors are permitted to present a talk, based on a paper submitted to or accepted by IJAHP, at a conference where the paper would not be published in a copyrighted publication either before or after the conference and where the author did not assign copyright to the conference or related publisher.