This paper presents a new multiattribute decision making method based on intuitionistic fuzzy values. First, we briefly review the concepts of interval-valued intuitionistic fuzzy sets and the Karnik-Mendel algorithms. Then, we propose the interval-valued intuitionistic fuzzy weighted average operator, based on the traditional weighted average method and the Karnik-Mendel algorithms. Then, we propose a fuzzy ranking method for intuitionistic fuzzy values based on likelihood-based comparison relations between intervals. Finally, we present a new method for multiattribute decision making based on the proposed interval-valued intuitionistic fuzzy weighted average operator and the proposed fuzzy ranking method for intuitionistic fuzzy values.