Selecting an appropriate additive manufacturing (AM) process or machine to fabricate an end-use product is an important issue in design for AM. One of many types of approaches for AM process selection is based on multi-criteria decision making (MCDM). Most of the MCDM based approaches have an advantage in taking into account the relative importance of performance parameter types and a few of them also consider the interrelationships of performance parameter types. Each of these approaches can work well in its specific context. They are however not entirely satisfactory, as they do not have the capabilities to reduce the influence of the deviation of performance parameter values on the decision-making result and to capture the risk attitudes of users in their decision-making models. In this paper, an MCDM approach based on fuzzy Archimedean weighted power Bonferroni aggregation operators with such capabilities is proposed for AM process selection. A fuzzy Archimedean weighted power Bonferroni mean operator and a fuzzy Archimedean weighted power geometric Bonferroni mean operator are firstly constructed. Based on these operators, an MCDM approach for selection of AM processes are then developed. After that, four practical examples are adopted to illustrate the developed approach and a set of sensitivity analysis experiments on the basis of these examples are carried out. Finally, qualitative and quantitative comparisons between the approach and the existing MCDM based approaches are reported to demonstrate its feasibility, effectiveness, and advantages.