IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v298y2022i3p1007-1015.html
   My bibliography  Save this article

Efficiency of the principal eigenvector of some triple perturbed consistent matrices

Author

Listed:
  • Fernandes, Rosário
  • Furtado, Susana

Abstract

In this paper we study the efficiency of the principal eigenvector of a pairwise comparison matrix obtained from a consistent one by perturbing three entries above the main diagonal, and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing three of the perturbed entries and not containing a diagonal entry. Our approach uses the characterization of vector efficiency based on the strongly connection of a certain digraph associated with the matrix. By imposing some conditions on the magnitudes of the perturbations, we identify a class of matrices for which the principal eigenvector is efficient. This class is the union of two subclasses such that the digraph of all the matrices in the same subclass contains a common cycle with all vertices. The matrices we analyse include the simple perturbed consistent matrices, as well as a subfamily of the double perturbed consistent matrices, whose principal eigenvector efficiency has been studied in the literature. For completeness, we also include some examples of triple perturbed matrices for which the principal eigenvector is not efficient. Explicit expressions for the characteristic polynomial and the principal eigenvector of the triple perturbed consistent matrices we consider are also given.

Suggested Citation

  • Fernandes, Rosário & Furtado, Susana, 2022. "Efficiency of the principal eigenvector of some triple perturbed consistent matrices," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1007-1015.
    • Handle: RePEc:eee:ejores:v:298:y:2022:i:3:p:1007-1015
    DOI: 10.1016/j.ejor.2021.08.012
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221721006792
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2021.08.012?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Fichtner, John, 1986. "On deriving priority vectors from matrices of pairwise comparisons," Socio-Economic Planning Sciences, Elsevier, vol. 20(6), pages 341-345.
    2. Saaty, Thomas L., 2003. "Decision-making with the AHP: Why is the principal eigenvector necessary," European Journal of Operational Research, Elsevier, vol. 145(1), pages 85-91, February.
    3. Kułakowski, Konrad & Mazurek, Jiří & Ramík, Jaroslav & Soltys, Michael, 2019. "When is the condition of order preservation met?," European Journal of Operational Research, Elsevier, vol. 277(1), pages 248-254.
    4. Petróczy, Dóra Gréta, 2021. "An alternative quality of life ranking on the basis of remittances," Socio-Economic Planning Sciences, Elsevier, vol. 78(C).
    5. Chao, Xiangrui & Kou, Gang & Li, Tie & Peng, Yi, 2018. "Jie Ke versus AlphaGo: A ranking approach using decision making method for large-scale data with incomplete information," European Journal of Operational Research, Elsevier, vol. 265(1), pages 239-247.
    6. Marcin Anholcer & János Fülöp, 2019. "Deriving priorities from inconsistent PCM using network algorithms," Annals of Operations Research, Springer, vol. 274(1), pages 57-74, March.
    7. László Csató, 2013. "Ranking by pairwise comparisons for Swiss-system tournaments," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(4), pages 783-803, December.
    8. László Csató, 2018. "Characterization of the Row Geometric Mean Ranking with a Group Consensus Axiom," Group Decision and Negotiation, Springer, vol. 27(6), pages 1011-1027, December.
    9. Yoram Wind & Thomas L. Saaty, 1980. "Marketing Applications of the Analytic Hierarchy Process," Management Science, INFORMS, vol. 26(7), pages 641-658, July.
    10. Bozóki, Sándor & Csató, László & Temesi, József, 2016. "An application of incomplete pairwise comparison matrices for ranking top tennis players," European Journal of Operational Research, Elsevier, vol. 248(1), pages 211-218.
    11. Josep Colomer, 2013. "Ramon Llull: from ‘Ars electionis’ to social choice theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(2), pages 317-328, February.
    12. Csató, László & Tóth, Csaba, 2020. "University rankings from the revealed preferences of the applicants," European Journal of Operational Research, Elsevier, vol. 286(1), pages 309-320.
    13. R. Blanquero & E. Carrizosa & E. Conde, 2006. "Inferring Efficient Weights from Pairwise Comparison Matrices," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(2), pages 271-284, October.
    14. Bana e Costa, Carlos A. & Vansnick, Jean-Claude, 2008. "A critical analysis of the eigenvalue method used to derive priorities in AHP," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1422-1428, June.
    15. Csató, László, 2019. "A characterization of the Logarithmic Least Squares Method," European Journal of Operational Research, Elsevier, vol. 276(1), pages 212-216.
    16. László Csató, 2017. "On the ranking of a Swiss system chess team tournament," Annals of Operations Research, Springer, vol. 254(1), pages 17-36, July.
    17. Kristóf Ábele-Nagy & Sándor Bozóki & Örs Rebák, 2018. "Efficiency analysis of double perturbed pairwise comparison matrices," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 69(5), pages 707-713, May.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Csató, László & Petróczy, Dóra Gréta, 2021. "On the monotonicity of the eigenvector method," European Journal of Operational Research, Elsevier, vol. 292(1), pages 230-237.
    2. D'ora Gr'eta Petr'oczy & L'aszl'o Csat'o, 2019. "Revenue allocation in Formula One: a pairwise comparison approach," Papers 1909.12931, arXiv.org, revised Dec 2020.
    3. Szádoczki, Zsombor & Bozóki, Sándor & Tekile, Hailemariam Abebe, 2022. "Filling in pattern designs for incomplete pairwise comparison matrices: (Quasi-)regular graphs with minimal diameter," Omega, Elsevier, vol. 107(C).
    4. Csató, László, 2019. "A characterization of the Logarithmic Least Squares Method," European Journal of Operational Research, Elsevier, vol. 276(1), pages 212-216.
    5. Ágoston, Kolos Csaba & Csató, László, 2022. "Inconsistency thresholds for incomplete pairwise comparison matrices," Omega, Elsevier, vol. 108(C).
    6. Bice Cavallo, 2019. "Coherent weights for pairwise comparison matrices and a mixed-integer linear programming problem," Journal of Global Optimization, Springer, vol. 75(1), pages 143-161, September.
    7. Petróczy, Dóra Gréta, 2021. "An alternative quality of life ranking on the basis of remittances," Socio-Economic Planning Sciences, Elsevier, vol. 78(C).
    8. László Csató, 2019. "Axiomatizations of inconsistency indices for triads," Annals of Operations Research, Springer, vol. 280(1), pages 99-110, September.
    9. Csató, László & Tóth, Csaba, 2020. "University rankings from the revealed preferences of the applicants," European Journal of Operational Research, Elsevier, vol. 286(1), pages 309-320.
    10. László Csató, 2019. "An impossibility theorem for paired comparisons," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 497-514, June.
    11. József Temesi, 2011. "Pairwise comparison matrices and the error-free property of the decision maker," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 19(2), pages 239-249, June.
    12. Jana Siebert, 2019. "Fuzzy eigenvector method for deriving normalized fuzzy priorities from fuzzy multiplicative pairwise comparison matrices," Fuzzy Optimization and Decision Making, Springer, vol. 18(2), pages 175-197, June.
    13. Zein Kallas & Teresa Serra & José Maria Gil, 2010. "Farmers’ objectives as determinants of organic farming adoption: the case of Catalonian vineyard production," Agricultural Economics, International Association of Agricultural Economists, vol. 41(5), pages 409-423, September.
    14. Tekile, Hailemariam Abebe & Brunelli, Matteo & Fedrizzi, Michele, 2023. "A numerical comparative study of completion methods for pairwise comparison matrices," Operations Research Perspectives, Elsevier, vol. 10(C).
    15. József Temesi, 2019. "An interactive approach to determine the elements of a pairwise comparison matrix," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 533-549, June.
    16. L'aszl'o Csat'o & Csaba T'oth, 2018. "University rankings from the revealed preferences of the applicants," Papers 1810.04087, arXiv.org, revised Feb 2020.
    17. Jiří Mazurek & Konrad Kulakowski, 2020. "Information gap in value propositions of business models of language schools," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 30(2), pages 77-89.
    18. Rico, Margarita & González, Andrés, 2015. "Social participation into regional forest planning attending to multifunctional objectives," Forest Policy and Economics, Elsevier, vol. 59(C), pages 27-34.
    19. Fatima Lambarraa-Lehnhardt & Rico Ihle & Khadija Mhaouch, 2021. "Geographical indications for supporting rural development in the context of the Green Morocco Plan: Oasis dates," Agricultural Economics, Czech Academy of Agricultural Sciences, vol. 67(2), pages 70-79.
    20. Alessio Ishizaka & Sajid Siraj, 2020. "Interactive consistency correction in the analytic hierarchy process to preserve ranks," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 443-464, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:298:y:2022:i:3:p:1007-1015. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.