Neighborhood relation-based variable precision multigranulation Pythagorean fuzzy rough set approach for multi-attribute group decision making

Nowadays, the complexity of real decision-making problem makes people pay more and more attention to the effective evaluation and selection of decision methods. Considering that the Pythagorean fuzzy set (PFS) owing its better capacity can express the uncertainties of human inherent preferences, multigranulation Pythagorean fuzzy rough set (MGPFRS) model is proposed to handle uncertain multi-attribute group decision making (MAGDM) problem. We firstly build the δ−neighborhood relation over multi-attribute Pythagorean fuzzy decision making (MAPFDM) information system based on the distance measurement. Then, combined with the principle of variable precision rough set and PF information, the rough approximation of crisp and inaccuracy concept are discussed. That is, the variable precision multigranulation rough Pythagorean fuzzy set (VMGRPFS). On this basis, given the special case of fuzzy equivalence relation, we propose the variable precision multigranulation Pythagorean fuzzy rough set (VMGPFRS). Furthermore, some fascinating properties for the VMGRPFS and VMGPFRS model are given. The corresponding models of optimistic and pessimistic are deduced respectively. The interrelationship among of the established VMGRPFS and VMGPFRS with the classical multigranulation rough set are discussed in detail. In addition, a new MAGDM method with Pythagorean fuzzy information is proposed by using VMGPFRS and PROMETHEE methods. This method points out the basic principle and algorithm of decision making. Finally, we perform our established method to select the best photovoltaic cell and implement the comparative analysis which can verify the reliability and superiority of our approach.