Computational simulation demonstrates non-positive characteristic polynomial values at four in fourth-order comparison matrices, indicating maximum real eigenvalues exceed four under Saaty's scale.
Pairwise comparison matrix (PCM) and its maximum eigenvalue play key roles in analytic hierarchy process (AHP). We shed light on the characteristic polynomial of a PCM of 4th order. By computational simulation, we can confirm that the value of the characteristic polynomial for 4 is non-positive in case which Saaty's discrete scale will be used. Thus we can show the real-number solution of the characteristic equation exists and is greater than 4 in the practical use of AHP.
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Obata et al. (2021) studied this question.
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