Weights from the least squares approximation of pairwise comparison matrices

The method of pairwise comparisons is one of the tools for determining the weights of attributes or evaluating the alternatives in Multi Attribute Decision Making. The decision maker is requested to compare pairwise the importance of the attributes, then the rates are arranged in a matrix. The aim is to find the weight vector which best reflects the values given by the decision maker. There exist many distance minimizing methods, besides the Eigenvector Method (Analytic Hierarchy Process) for determining weights from a pairwise comparison matrix. One of them is the Least Squares Method, the solution of which leads to the optimization of a nonlinear, nonconvex function. In the paper, methods for solving the problem of least squares approximation of pairwise comparison matrices are presented. All methods are suitable for finding all local and global optima.