Priority dispatching for minimizing job tardiness has been the subject of research investigation for several decades. Minimizing weighted tardiness however has considerably more practical relevance, but for this objective only a few dispatching rules have been advanced and scientifically compared. We introduce here a new rule, which we call “Weighted Modified Due Date” (WMDD) and test its effectiveness against other competing rules that have been developed for weighted tardiness. The test is accomplished via a simulation study of a simple queueing system and by static problem analysis. The WMDD rule is found to compare favorably to all the rules tested.