A heuristic rating estimation algorithm for the pairwise comparisons method

The pairwise comparisons method is a powerful tool used for establishing the relative order between different concepts in situations in which it is difficult (or sometimes even impossible) to provide explicit rating. Appropriate ratings are determined by solving eigenvalue problem for the pairwise comparisons matrix. This study presents a new iterative heuristic rating estimation algorithm that tries to deal with the situation when exact estimations for some concepts (stimulus) CK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{K}$$\end{document} are a priori known and fixed, whilst the estimates for the others (unknown concepts CU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{U}$$\end{document}) need to be computed. The relationship between the local estimation error, understood as the average absolute error E(c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(c)$$\end{document} over all direct estimates for the concept c∈CU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in C_{U}$$\end{document} and the pairwise comparisons matrix inconsistency index is shown. The problem of convergence of subsequent intermediate results is discussed and the convergence conditions are given.