The due date assignment in the scheduling problems is concerned with the benefit balance between the assignment cost and the performance criterion. This arises the bicriteria scheduling problems of due date assignment. In single machine bicriteria scheduling of due date assignment with weighted number of tardy jobs, only the linear combination version has been studied in the literature. This paper further studies more models, namely, the constraint version and the Pareto optimization version. The main contribution of this study is the related results on the NP-hardness, polynomially solvable cases, and the polynomial time approximation scheme. By using the proposed manner, the features of a multicriteria optimization problem can be effectively characterized.