Hybrid models for decision-making based on rough Pythagorean fuzzy bipolar soft information

A Pythagorean fuzzy set model is a powerful tool for depicting fuzziness and uncertainty. In this paper, the notions of two novel hybrid models, namely Pythagorean fuzzy bipolar soft set model and rough Pythagorean fuzzy bipolar soft set model are introduced. The Pythagorean fuzzy bipolar soft set is the generalization of different hybrid models including bipolar soft set, fuzzy bipolar soft set and the Pythagorean fuzzy soft set. Our developed concepts are illustrated by examples. Some useful basic properties and operations of the hybrid models are investigated. In particular, multiattribute decision-making problems with numerical examples are solved by our new hybrid models. An algorithm of each approach is developed.