This paper presents a new structure as a simple method at two uncertainties (i.e., aleatory and epistemic) that result from variabilities inherent in nature and a lack of knowledge. Aleatory and epistemic uncertainties use the concept of the entropy and Dempster-Shafer (D-S) theory, respectively. Accordingly, we propose the generalized Shannon entropy in the D-S theory as a measure of uncertainty. This theory has been originated in the work of Dempster on the use of probabilities with upper and lower bounds. We describe the framework of our approach to assess upper and lower uncertainty bounds for each state of a system. In this process, the uncertainty bound is calculated with the generalized Shannon entropy in the D-S theory in different states of these systems. The probabilities of each state are interval values. In the current study, the effect of epistemic uncertainty is considered between events with respect to the non-probabilistic method (e.g., D-S theory) and the aleatory uncertainty is evaluated by using an entropy index over probability distributions through interval-valued bounds. Therefore, identification of total uncertainties shows the efficiency of uncertainty quantification.