Minimizing Weighted Number of Tardy Jobs and Weighted Completion Time 1

In this paper, we present a novel O(N) algorithm to minimize the weighted number of tardy jobs with unit processing times, integer ready times and deadlines, and M homogeneous parallel machines, where N is the number of jobs to be scheduled. wi is the weight re ecting job i's importance, and Ui is 1 if job i is tardy and 0 otherwise, so P wiUi is the measure of performance. (In standard notation [1], this is the P/ri, pi=1/ P wiUi problem.) This algorithm is extended to minimize weighted completion time P wiCi for some jobs, where completion time Ci is the time job i completes processing, and P wiUi for other jobs, at the same complexity. (The P/ri, pi=1/ P wiUi& P wiCi problem.) Complexity can be reduced to O(N logN) if all ready times are the same (with one additional constraint on weights). (P/pi=1/ P wiUi& P wiCi.) Finally, if one is trying to nd the number of tardy jobs of each weight under optimal scheduling, but an actual schedule is not needed, this (P/ri,pi = 1/ P wiUi) problem can be solved with an O(WN logN) algorithm, where W is the number of di erent weight values. In some cases, this can be done analytically. This material is based upon work supported under Grant NCR-9210626 from the National Science Foundation.