The ordinary multiple criteria decision making (MCDM) approach requires two types of input, alternative values and criterion weights, and employs two schemes of alternative prioritization, dominance and potential optimality. This paper allows for incomplete information on both types of input and gives rise to the dominance relationships and potential optimality of alternatives. Unlike the earlier studies, we emphasize that incomplete information frequently takes the form of strict inequalities, such as strict orders and strict bounds, rather than weak inequalities. Then the issues of rising importance include: (1) The standard mathematical programming approach to prioritize alternatives cannot be used directly, because the feasible region for the permissible decision parameters becomes an open set. (2) We show that the earlier methods replacing the strict inequalities with weak ones, by employing a small positive number or zeroes, which closes the feasible set, may cause a serious problem and yield unacceptable prioritization results. Therefore, we address these important issues and develop a useful and simple method, without selecting any small value for the strict preference information. Given strict information on both types of decision parameters, we first construct a nonlinear program, transform it into a linear programming equivalent, and finally solve it via a two-stage method. An application is also demonstrated herein.