In this paper, an Economic Production Quantity model for deteriorating items with time-dependent demand and shortages including partially back-ordered is developed under a cloudy-fuzzy environment. At first, we develop a crisp model by considering linearly time-dependent demand with constant deterioration rate, constant inflation rate and shortages under partially back-ordered, then we fuzzify the model to archive a decision under the cloudy-fuzzy (extension of fuzziness) demand rate, inflation rate, deterioration rate and the partially back-ordered rate which are followed by their practical applications. In this model, we assume ambiances where cloudy normalized triangular fuzzy number is used to handle the uncertainty in information which is coming from the data. The main purpose of our study is to defuzzify the total inventory cost by applying Ranking Index method of fuzzy numbers as well as cloudy-fuzzy numbers and minimize the total inventory cost of crisp, fuzzy, and cloudy-fuzzy model. Finally, a comparative analysis among crisp, fuzzy and cloudy-fuzzy total cost is carried out in this paper. Numerical example, sensitivity analysis, and managerial insights are elaborated to justify the usefulness of the new approach. A comparative inquiry of the numerical result with a new existing paper is also carried out. This paper ends with a conclusion along with advantages and limitations of our solution approach, and an outlook towards possible future studies.