Data inversion in MCDM problems

A comprehensive analysis of the procedures for consistent normalization/inversion of benefit and cost attributes for multi-criteria decision making (MCDM) and multivariate classification problems is performed. This study demonstrates that the commonly used 1/a transformation for cost attribute inversion introduces structural inconsistencies in normalized data. Nonlinear data inversion does not have a reasonable interpretation of values and leads to a violation of mutual distances in the original data. The measurement scales of various attributes are not consistent and there is a shift in the domains of normalized values. Elimination of these problems is achieved by using the linear inversion Reverse Sorting algorithm (ReS). The ReS algorithm offers more consistent, linear, and interpretable results for handling cost attributes in MCDM. The ReS algorithm is a linear transformation and preserves the original information about the object: dispositions of attribute values, preserves the relative positions of the domains of different attributes and can be applied to both the original and normalized data sets. The ReS algorithm eliminates all the shortcomings of nonlinear inversion and is recommended for inversion of values when coordinating the optimization goals of a multi-criteria problem, as well as in the weighing methods based on information contained in the decision matrix.