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On a Maximum Eigenvalue of Third-Order Pairwise Comparison Matrix in Analytic Hierarchy Process and Convergence of Newton’s Method

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  • Shunsuke Shiraishi

    (University of Toyama)

  • Tsuneshi Obata

    (Oita University)

Abstract

Nowadays, the analytic hierarchy process is an established method of multiple criteria decision making in the field of Operations Research. Pairwise comparison matrix plays a crucial role in the analytic hierarchy process. The principal (maximum magnitude) eigenvalue of the pairwise comparison matrix can be utilized for measuring the consistency of the decision maker’s judgment. The simple transformation of the maximum magnitude eigenvalue is known to be Saaty’s consistency index. In this short note, we shed light on the characteristic polynomial of a pairwise comparison matrix of third order. We will show that the only real-number root of the characteristic equation is the maximum magnitude eigenvalue of the third-order pairwise comparison matrix. The unique real-number root appears in the area where it is greater than 3, which is equal to the order of the matrix. By applying usual Newton’s method to the characteristic polynomial of the third-order pairwise comparison matrix, we see that the sequence generated from the initial value of 3 always converges to the maximum magnitude eigenvalue.

Suggested Citation

  • Shunsuke Shiraishi & Tsuneshi Obata, 2021. "On a Maximum Eigenvalue of Third-Order Pairwise Comparison Matrix in Analytic Hierarchy Process and Convergence of Newton’s Method," SN Operations Research Forum, Springer, vol. 2(3), pages 1-11, September.
    • Handle: RePEc:spr:snopef:v:2:y:2021:i:3:d:10.1007_s43069-021-00077-w
    DOI: 10.1007/s43069-021-00077-w
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    References listed on IDEAS

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    1. V.M. Rao Tummala & Hong Ling, 1998. "A Note on the Computation of the Mean Random Consistency Index of the Analytic Hierarchy Process (Ahp)," Theory and Decision, Springer, vol. 44(3), pages 221-230, June.
    2. Yoram Wind & Thomas L. Saaty, 1980. "Marketing Applications of the Analytic Hierarchy Process," Management Science, INFORMS, vol. 26(7), pages 641-658, July.
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