The calculation of cost efficiency requires complete and accurate information on the input prices at each decision making unit (DMU). In practice, however, exact knowledge of the relevant prices is difficult to come by, and prices may be subject to variation in the short term. To estimate the cost efficiency while taking price uncertainty into account, cone-ratio DEA models incorporating the available price information as weight restrictions can be applied. However, the literature lacks a clear explanation regarding the exact relationships between these two models. In this paper, through a duality study, we establish both the theoretical properties of these relationships and the characteristics of their efficiency solutions between cone-ratio DEA models and CE models, assuming there are imprecise price data. Based on the duality study, we also develop a new approach and design a lexicographic order algorithm to estimate the lower bounds of the cost efficiency measure. Our computational experiments indicate that the proposed models are robust and that the proposed algorithm is computationally simple