On a Maximization Problem Arising From a Positive Reciprocal Matrix in Ahp

Abstract

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.

An application of incomplete pairwise comparison matrices for ranking top tennis players

“…Main results have been discussed by Harker (Harker, 1987), Carmone, Kara and Zanakis (Carmone et al, 1997), Kwiesielewicz and van Uden (Kwiesielewicz, 1996;Kwiesielewicz and van Uden, 2003), Shiraishi, Obata and Daigo (Shiraishi et al, 1998;Shiraishi and Obata, 2002), Takeda and Yu (Takeda and Yu, 1995), Fedrizzi and Giove (Fedrizzi and Giove, 2007).…”

“…, x d are written instead of missing elements as follows: (Saaty, 1980) defined inconsistency index CR as a positive linear transformation of λ max (A) such that CR(A) ≥ 0 and CR(A) = 0 if and only if A is consistent. The idea that larger λ max indicates higher CR inconsistency led Shiraishi, Obata and Daigo (Shiraishi et al, 1998;Shiraishi and Obata, 2002) to consider the eigenvalue optimization problem min x>0 λ max (A(x)).…”

A survey of inconsistency indices for pairwise comparisons

“…Various inconsistency indices have been used as objective functions. Shiraishi and Obata (2002) proposed to maximize the index c 3 , bearing in mind that the interpretation of this index is reversed: the greater its value, the smaller the inconsistency. Peláez and Lamata (2003) proposed something similar with their index.…”

“…Peláez and Lamata (2003) proposed something similar with their index. However, since their index is proportional to c 3 , this approach is equivalent to the one proposed by Shiraishi and Obata (2002). Koczkodaj, Herman, and Orlowski (1999) and Bozóki, Fülöp, and Koczkodaj (2011) followed the same approach for index K. Similar techniques were implemented by Bozóki, Fülöp, and Rónyai (2010) for CI and Khatwani and Kar (2016) for CCI.…”

A note on the proportionality between some consistency indices in the AHP

“…Shiraishi et al [18,19,20] proposed, as a consistency index of a pairwise comparison matrix, the coefficient c 3 of its characteristic polynomial. P A (λ) = λ n + c 1 λ n−1 + · · · + c n−1 λ + c n .…”