To make a scheduling for minimizing the tardiness of delivery, we present an efficient heuristic algorithm for job-shop scheduling problems with due dates. The SDME(Slack Degree Modified Exchange) algorithm, which is the proposed heuristic algorithm, carries out the operation in two phases. In the phase 1, a dispatching rule is employed to generate an active or non-delay initial schedule. In the phase 2, tasks selected from a predetermined set of promising target operations in the initial schedule are tested to ascertain whether, by left-shifting their start times and rearranging some subset of the remaining operations, one can reduce the total tardiness. In the numerical example, the results indicate that the proposed SDME algorithm is capable of yielding notable reductions in the total tardiness for practical size problems.