Stochastic and fuzzy ordering with the method of minimal transformations

We analyze the methods of stochastic and fuzzy comparison and ordering of random and fuzzy variables. We find simple formulas for computing a number of comparisons and establish the interrelations between various comparisons. We propose and study a new approach to comparing histograms of discrete random (fuzzy) variables based on computing a “directed” minimal transformation that maps one of the compared variables into another. We apply the method of minimal transformations to solving the problem of optimal reduction of discrete random (fuzzy) variables to unimodal form which is considered in the context of ranking the histograms of universities constructed by USE (Unified State Exam) results. We propose a model of “perfect” admission for high school graduates and show that the distribution of admitted graduates to a university in this model will be unimodal under sufficiently general assumptions on the preference function.