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The circular Pythagorean fuzzy (Cir-PyF) sets (Cir-PyFs) not only contain the membership and non-membership grades, but also have the radius around the circle of each element. Cir-PyFSs can cope with many real-life problems. In this paper, we consider the Hamacher t-norm and t-conorm operational laws for any two Cir-PyF numbers (Cir-PyFNs) and describe their exceptional cases such as algebraic and Einstein operational laws. Furthermore, the Cir-PyF Hamacher averaging (Cir-PyFHA) operator, Cir-PyF Hamacher ordered averaging (Cir-PyFHOA) operator, Cir-PyF Hamacher geometric (Cir-PyFHG) operator, and Cir-PyF Hamacher ordered geometric (Cir-PyFHOG) operator are proposed. Some properties for the above operators are also discussed in detail. Moreover, to select the simplest and best procedure for evaluating the source of gold in mines, we illustrate the application of the multi-attribute decision-making (MADM) technique based on the derived operators. Finally, we demonstrate some examples for comparing the proposed operators with some existing methods to expand the attraction of the proposed way.

Circular Pythagorean Fuzzy Hamacher Aggregation Operators with Application in the Assessment of Goldmines