The theory of <inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-rung orthopair fuzzy sets (<inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-ROFSs) proposed by Yager effectively describes fuzzy information in the real world. Because <inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-ROFSs contain the parameter <inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula> and can adjust the range of expressed fuzzy information, they are superior to both intuitionistic and Pythagorean fuzzy sets. Archimedean T-norm and T-conorm (ATT) is an important tool used to generate operational rules based on the q -rung orthopair fuzzy numbers (<inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-ROFNs). In comparison, the Bonferroni mean (BM) operator has an advantage because it considers the interrelationships between the different attributes. Therefore, it is an important and meaningful innovation to extend the BM operator to the <inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-ROFNs based upon the ATT. In this paper, we first discuss <inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-rung orthopair fuzzy operational rules by using ATT. Furthermore, we extend BM operator to the <inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-ROFNs and propose the <inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula>-rung orthopair fuzzy Archimedean BM <inline-formula><tex-math notation="LaTeX">(q\hbox{-}{ROFABM})</tex-math></inline-formula> operator and the q -rung orthopair fuzzy weighted Archimedean BM <inline-formula><tex-math notation="LaTeX">(q\hbox{-}{ROFWABM})</tex-math></inline-formula> operator and study their desirable properties. Then, a new multiple-attribute decision-making (MADM) method is developed based on <inline-formula><tex-math notation="LaTeX">q\hbox{-}{ROFWABM}</tex-math></inline-formula> operator. Finally, we use a practical example to verify effectiveness and superiority by comparing to other existing methods.