447. Theoretical study of characteristic polynomial of 4th order PCM in AHP
Invited abstract in session TC-10: Pairwise comparisons and preference relations 1, stream Multiple Criteria Decision Aiding.
Tuesday, 12:30-14:00Room: Clarendon SR 1.06
Authors (first author is the speaker)
| 1. | Shunsuke Shiraishi
|
| Faculty of Applied Information Science, Hiroshima Institute of Technology | |
| 2. | Tsuneshi Obata
|
| Faculty of Science and Engineering, Otemon Gakuin University |
Abstract
We have demonstrated that we can obtain the maximum eigenvalue of pairwise comparison matrices of general order using Newton's and the secant method. Among classical root-finding methods, the remaining challenge is proving the theoretical convergence of the bisection method. To show the bisection method's convergence, we must show that there is only one eigenvalue greater than n.
In this talk, we provide a theoretical proof that this property holds for n=4. For an inconsistent pairwise comparison matrix of order 4, precisely two real eigenvalues generally exist. It can be shown that the larger one is greater than 4, while the smaller one is less than 4. Therefore, since only the largest eigenvalue exists in the region greater than 4, the bisection method can determine this maximum eigenvalue by choosing an appropriate initial value.
We also demonstrate the convergence of the bisection method for higher-order pairwise comparison matrices through simulations.
Keywords
- Decision Theory
- Decision Support Systems
Status: accepted
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