Geometric aggregation operators with interval-valued Pythagorean trapezoidal fuzzy numbers based on Einstein operations and their application in group decision making

The aim of this paper is to investigate information aggregation methods under interval-valued Pythagorean trapezoidal fuzzy environment. Some Einstein operational laws on interval-valued Pythagorean trapezoidal fuzzy numbers are defined based on Einstein sum and Einstein product. Based on Einstein operations, we define interval-valued Pythagorean trapezoidal fuzzy aggregation operators, such as interval-valued Pythagorean trapezoidal fuzzy Einstein weighted geometric operator, interval-valued Pythagorean trapezoidal fuzzy Einstein ordered weighted geometric operator and interval-valued Pythagorean trapezoidal fuzzy Einstein hybrid geometric operator. Furthermore, we apply the proposed aggregation operators to deal with multiple attribute group decision making problem. Finally we construct a numerical example for multiple attribute group decision making problem and compare the result with existing methods.