The chamacteristic polynomiai of a positive reciprecal matrix has sorne notewerthy properties. They are deeply related to the notion of consistency ofa pairwise comparison rnatrix of AHP. Based oll the results, we propose a method ± 'or estimating a rnissing entry of an incomplete pairwise comparison matrix.
The maximization of the third coefficient of the characteristic polynomial has been proved useful. It has a posynomial form as a function of entries of a positive reciprocal matrix.Hence one can transform it to a convex function.We will prove the existence of a solution of the minimization problem of the resulting convex function.
In AHP, a number of consistency indices have been proposed. Saaty's C.I. is a pioneer and generally adopted by users of AHP. We also proposed a new consistency index with the aid of the characteristic polynomial of the pairwise comparison matrix. Surprisingly, 3rd order random matrices make the completely same numerical order of two consistency indices, i.e. Saaty's C.I. and our consistency index. In this short paper, we show this experimental result is theoretically correct.
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