Multiple attribute decision making based on probabilistic generalized orthopair fuzzy sets

The advent of Yager’s generalized orthopair fuzzy sets (also called q-rung orthopair fuzzy sets) has brought more possibilities to accomplish the challenging task of modelling uncertainties. Compared with Pythagorean fuzzy sets and intuitionist fuzzy sets, q-rung orthopair fuzzy sets are more general in theory and more powerful in practice. Nevertheless, these extensions of fuzzy sets all assume that the membership and non-membership grades are equally important for every element. However, membership and non-membership grades may have different degrees of importance in real decision-making applications. Thus it may cause information loss if intuitionistic fuzzy sets or their extensions are used to describe initial decision information in such cases. To fill the gap, we consider adding the probability information which reflects the importance of membership and non-membership grades to further enhance generalized orthopair fuzzy sets in this study. Firstly, we put forth probability generalized orthopair fuzzy numbers, which extend generalized orthopair fuzzy numbers with membership and non-membership probabilities. We define some basic operations of probability generalized orthopair fuzzy numbers and investigate their related properties. We also presented a partial order ≤Lq∗×L∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le _{L_{q}^{*}\times L^{*}}$$\end{document} and a lexicographic order ≤(s,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\le _{(s,h)}$$\end{document} for comparing probability generalized orthopair fuzzy numbers. Also, we define the accuracy and (normalized) score functions of probability generalized orthopair fuzzy numbers, and discuss their basic properties. Secondly, three different probabilistic generalized orthopair fuzzy aggregation operators, namely the probabilistic generalized orthopair fuzzy simple weighted averaging operator, the probabilistic generalized orthopair fuzzy weighted averaging operator and the probabilistic generalized orthopair fuzzy weighted geometric operator are defined, and their fundamental properties are explored in detail. The Minkowski distance and closeness index of probabilistic generalized orthopair fuzzy numbers are proposed, which can be used to develop an attribute weight determination method. Thirdly, we introduce probabilistic generalized orthopair fuzzy sets, and develop two methods for solving multiple attribute decision making problems based on probabilistic generalized orthopair fuzzy sets. As an illustration, the proposed methods are used to solve a green supplier selection problem with unknown attribute weight information. Furthermore, we compare our methods with several existing decision-making approaches to validate the efficacy of the proposed methods.