This paper deals with a scheduling problem of independent tasks with common due date where the objective is to minimize the total weighted tardiness. The problem is known to be ordinary NP-hard in the case of a single machine and a dynamic programming algorithm was presented in the seminal work of Lawler and Moore [E.L. Lawler, J.M. Moore, A functional equation and its application to resource allocation and sequencing problems, Management Science 16 (1969) 77–84]. In this paper, this algorithm is described and discussed. Then, a new dynamic programming algorithm is proposed for solving the single machine case. These methods are extended for solving the identical and uniform parallel-machine scheduling problems.