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We introduce extended component importance measures (Birnbaum importance, RAW, RRW, and Criticality importance) considering aleatory and epistemic uncertainties. The Dempster-Shafer theory, which is considered to be a less restricted extension of probability theory, is proposed as a framework for taking into account both aleatory and epistemic uncertainties. The epistemic uncertainty defined in this paper is the total lack of knowledge of the component state. The objective is to translate this epistemic uncertainty to the epistemic uncertainty of system state, and to the epistemic uncertainty of importance measures of components. Affine arithmetic allows us to provide much tighter bounds in the computing process of interval bounds of importance measures, avoiding the error explosion problem. The efficiency of the proposed measures is demonstrated using a bridge system with different types of reliability data (aleatory uncertainty, epistemic uncertainty, and experts' judgments). The influence of the epistemic uncertainty on the components' rankings is described. Finally, a case study of a fire-detector system located in a production room is provided. A comparison between the proposed measures and the probabilistic importance measures using two-stage Monte Carlo simulations is also made.