On the conformity of scales of multidimensional normalization: An application for the problems of decision making

The main goal of this paper is to harmonize the scales of normalized values of various attributes for multi-criteria decision-making models (MCDM). A class of models is considered in which the ranking of alternatives is performed based on the performance indicators of alternatives obtained by aggregating private attributes. The displacement of the domains of the normalized values of various attributes relative to each other and the local priorities of the alternatives are the main factors that change the rating when using various normalization methods. Three different linear transformations are proposed, which make it possible to bring the scales of normalized values of various attributes into conformity. The first transformation, the Reverse Sorting (ReS) algorithm, inverts the direction of optimization without displacing the areas of normalized values. The second transformation ‒ IZ-method ‒ allows researchers to align the boundaries of the domains of normalized values of various attributes in each range. The third transformation ‒ MS-method ‒ converts Z-scores into a sub-domain of the interval [0, 1] with the same mean values and the same variance values for all attributes. All transformations preserve the dispositions of the natural values of the attributes of the alternatives and ensure the equality of the contributions of various criteria to the performance indicator of the alternatives. The ReS-algorithm is universal for all normalization methods when converting cost attributes to benefit attributes. IZ and MS transformations expand the range of normalization methods when using nonlinear functions aggregation of attributes.