Minimizing weighted mean absolute deviation of job completion times from their weighted mean

We address a single-machine scheduling problem where the objective is to minimize the weighted mean absolute deviation of job completion times from their weighted mean. This problem and its precursors aim to achieve the maximum admissible level of service equity. It has been shown earlier that the unweighted version of this problem is NP-hard in the ordinary sense. For that version, a pseudo-polynomial time dynamic program and a 2-approximate algorithm are available. However, not much (except for an important solution property) exists for the weighted version. In this paper, we establish the relationship between the optimal solution to the weighted problem and a related one in which the deviations are measured from the weighted median (rather than the mean) of the job completion times; this generalizes the 2-approximation result mentioned above. We proceed to give a pseudo-polynomial time dynamic program, establishing the ordinary NP-hardness of the problem in general. We then present a fully-polynomial time approximation scheme as well. Finally, we report the findings from a limited computational study on the heuristic solution of the general problem. Our results specialize easily to the unweighted case; they also lead to an approximation of the set of schedules that are efficient with respect to both the weighted mean absolute deviation and the weighted mean completion time. Consider a set N of n independent jobs that are available at time zero for non-preempted processing on a continuously available machine which can process at most one job at a time. Let job j, j 2 N, have an integer processing time p j (p j > 0) and an integer weight w j (w j P 0). (For future reference, let p N ¼ P j2N p j and w N ¼ P j2N w j .) Similarly, let S be an ordered set of the job indexes representing a job sequence and [i] be the index of the job in position i in S. Assuming w.l.o.g. a zero start time and no inserted machine idle time, S translates to a schedule in which the completion time of the job in position i in S is given by c ½iŠ ðSÞ ¼ P 16k6i p ½kŠ ðSÞ. We define vðS; dÞ ¼ ð1=w N Þ P 16i6n w ½iŠ ðSÞjc ½iŠ ðSÞ À dj r , for d and r P 0, to be a measure of the variability of the job completion times …