Matrix balancing problem and binary AHP

A matrix balancing problem and an eigenvalue problem are transformed into two minimum-norm point problems whose difference is only a norm. The matrix balancing problem is solved by scaling algorithms that are as simple as the power method of the eigenvalue problem. This study gives a proof of global convergence for scaling algorithms and applies the algorithm to Analytic Hierarchy process (AHP), which derives priority weights from pairwise comparison values by the eigenvalue method (EM) traditionally. Scaling algorithms provide the minimum χ square estimate from pairwise comparison values. The estimate has properties of priority weights such as right–left symmetry and robust ranking that are not guaranteed by the EM.