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The <i>q</i>-rung orthopair fuzzy (<i>q</i>-ROF) set is an efficient tool for dealing with uncertain and inaccurate data in real-world multi-attribute decision-making (MADM). In MADM, aggregation operators play a significant role. The majority of well-known aggregation operators are formed using algebraic, Einstein, Hamacher, Frank, and Yager t-conorms and t-norms. These existing t-conorms and t-norms are some special cases of Archimedean t-conorms (ATCNs) and Archimedean t-norms (ATNs). Therefore, this article aims to extend the ATCN and ATN operations under the <i>q</i>-ROF environment. In this paper, firstly, we present some new operations for <i>q</i>-ROF sets based on ATCN and ATN. After that, we explore a few desirable characteristics of the suggested operational laws. Then, using these operational laws, we develop <i>q</i>-ROF Archimedean weighted averaging (geometric) operators, <i>q</i>-ROF Archimedean order weighted averaging (geometric) operators, and <i>q</i>-ROF Archimedean hybrid averaging (geometric) operators. Next, we develop a model based on the proposed aggregation operators to handle MADM issues. Finally, we elaborate on a numerical problem about site selection for software operating units to highlight the adaptability and dependability of the developed model.