Associated Statistics' Extensions in the p, q-Rung Orthopair Fuzzy Environment: Application in Interactive Multi-Attribute Decision Making

Abstract

In order to systematize statistical analysis in interactive Multi-Attribute Decision Making (MADM) models, the concept of "Monotone Expectation" is considered for the extension of classical statistics in the p, q-Rung Orthopair Fuzzy (p, q-ROF) environment, concretely, the probability mean (expectation), variance, covariance, and correlation coefficient. In the classical view, the operations of subtraction and division are not defined in p, q-ROF values. The operations of pseudo-subtraction and pseudo-division are introduced to define the covariance and correlation coefficient. Based on the Choquet finite integral, extensions of monotone statistics such as F-associated expectation, variance, covariance and correlation coefficient for the p, q-ROF environment are presented. The fuzzy measure Associated Probabilities Class (APC) of a fuzzy measure and the classes of associated statistics: probability averaging class, variances class, covariances class and correlation coefficients class are considered. The relationship between monotone and associated statistics is studied. It is established that monotone statistics can be represented by unique corresponding associated statistics if there is a partial ordering between the arguments of the statistics, as between the p, q-ROF values. That is, the remaining (n!−1) associated statistics of the corresponding associated class, where n is the number of interacting attributes, become useless in the definitions of monotone statistics. In the interactive MADM environment, in decision-making modeling, the aggregations of monotone statistics cannot fully reflect all possible degrees of interaction of attributes. In more detail: the aggregations with monotone statistical parameters under p, q-ROF environment used in interactive MADM models take into account interactions of the focal elements of only one consonant structure from the n! consonant structures of attributes. The extensions of monotone statistics, called F-associated statistics, which take into account the degrees of interaction of elements of all n! consonant structures of attributes in aggregations are defined. The issue of the correctness of the extensions is discussed. The values of monotone and F-associated statistics coincide if there is a partial ordering between the arguments of the statistics and the Choquet second-order extreme capacities are taken in the role of a fuzzy measure. A calculation scheme for monotone and F-associated statistics is constructed for a simple interactive MADM model under p, q-ROF environment. A simple numerical example is presented to illustrate the obtained results.