Pythagorean fuzzy mathematical programming method for multi-attribute group decision making with Pythagorean fuzzy truth degrees

This paper develops a Pythagorean fuzzy (PF) mathematical programming method to solve multi-attribute group decision-making problems under PF environments. The main work is summarized as four aspects: (1) Considering the fuzziness and hesitancy in pairwise comparisons of alternatives, we firstly introduce PF sets to depict the fuzzy truth degrees of alternative comparisons. (2) According to the information entropy, individual subjective attribute weight vectors of decision makers (DMs) are calculated and integrated into a collective one by a cross-entropy optimization model. Then DMs' weights are objectively derived from the collective subjective attribute weight vector. (3) PF group consistency and inconsistency indices are defined based on PF-positive ideal solution (PFPIS) and PF-negative ideal solution (PFNIS), respectively. To determine comprehensive attribute weights, a biobjective PF mathematical programming model is constructed through minimizing two inconsistency indices based on PFPIS and PFNIS simultaneously. A linear programming method is technically developed to solve this model. (4) Using the cross-entropy again, collective relative closeness degrees of alternatives are explicitly derived to rank the alternatives. Finally, an example of green supplier selection is analyzed to verify the effectiveness of the proposed method.