Goal Inversion in Multi-Criteria Decision Making Problems

Abstract

Within the framework of multi-criteria decision making (MCDM) problems, a comprehensive analysis of procedures for the consistent normalization of benefit and cost attributes was conducted. Rank-based methods for solving multi criteria decision making (MCDM) problems boil down to criteria convolution, which predetermines the transformation of all criteria to either the maximum or minimum value of the objective function using an inversion procedure. Criteria convolution is performed only for normalized data, so the inversion procedure must be consistent with the normalization procedure. It is shown that thousands of studies in MCDM using nonlinear inversion of cost attributes based on the 1/x transformation should be recognized as inaccurate. The argument is quite simple: distortion of the original data. This study explains what the error is. Nonlinear data inversion 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 areas of normalized values. Also, nonlinear inversion does not have a reasonable interpretation of values. The solution to the above problems is achieved by using the reverse sorting (ReS) algorithm. The ReS algorithm is a linear transformation and preserves the original information about the object: the location of attribute values, preserves the mutual location of the domains of various attributes and can be applied to both original and normalized data sets. The ReS algorithm is recommended for use in the inversion of values when coordinating the optimization goals of a multi-criteria problem.