In a recently quoted paper [N. Mahdavi-Amiri and S.H. Nasseri, Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables, Fuzzy Sets and Systems 158 (2007)1961-1978], has been established the dual of linear programming problem with trapezoidal fuzzy variables as a fuzzy number linear programming, using certain general linear ranking functions. In this paper, we define a new dual problem for the linear programming problem with trapezoidal fuzzy variables as a linear programming problem with trapezoidal fuzzy variables and hence deduce the duality results such as weak duality, strong duality and complementary slackness theorems. In throughout of the paper, we apply the trapezoidal fuzzy numbers as a convenient fuzzy numbers which is more practical in comparison of other numbers (and also as a general form of triangular fuzzy numbers) and moreover use the certain linear ranking functions for the sake of illustrating the performance of our approach, but it is no restrict at all to use these linear ranking functions. In particular, we emphasize that the new duality definition and the associated duality results lead us to provide the primal and dual simplex algorithms for solving the linear programming problems with trapezoidal fuzzy numbers.