To describe the values of criteria and to generate a sort of alternatives are two important issues in multi-criteria decision-making (MCDM). A superior tool for the former issue is Pythagorean hesitant fuzzy number (PHFN) and an effective tool for the latter issue is aggregation operator. So far, a number of aggregation operators of PHFNs have been presented within the academia. Each aggregation operator has its own characteristics and can work well for its specific purpose. But there is not yet an aggregation operator of PHFNs that can provide satisfying generality and flexibility in aggregating the values of criteria and capturing the interactions of criteria. The Archimedean t-conorm and t-norm (ATT) are well-known for having the capability to generate versatile and flexible operational rules for fuzzy numbers, while the Muirhead mean (MM) operator is an all-in-one aggregation operator for capturing the interrelationships of the aggregated arguments. To this end, the MM operator and the ATT for PHFNs are combined to present a Pythagorean hesitant fuzzy Archimedean MM (PHFAMM) operator and a weighted PHFAMM operator and a new MCDM method based on the presented operators is proposed in this paper. Firstly, the generalised expressions of the presented operators are provided. The properties of the operators are explored and proved and their specific expressions based on Algebraic, Einstein, Hamacher, and Frank ATTs are then constructed. Based on these specific expressions, a new method for solving the MCDM problems based on PHFNs is developed. Finally, the developed MCDM method is demonstrated via an example, a set of experiments and qualitative and quantitative comparisons.