Abstract
The Pythagorean fuzzy number is a new tool for uncertainty and vagueness. It is a generalization of fuzzy numbers and intuitionistic fuzzy numbers. In this paper, we define some Einstein operations on Pythagorean trapezoidal fuzzy set and develop two averaging aggregation operators, which is an induced Pythagorean trapezoidal fuzzy Einstein ordered weighted averaging operator and an induced Pythagorean trapezoidal fuzzy Einstein hybrid averaging (I-PTFEHA) operator. We presented some new methods to deal with the multi-attribute group decision-making problems under the Pythagorean trapezoidal fuzzy environment. Finally, we used some practical examples to illustrate the validity and feasibility of the proposed methods by comparing with existing method. It shows that the proposed I-PTFEHA operator is much better and reliable than the existing one.