A bicriteria approach to scheduling a single machine with job rejection and positional penalties

Single machine scheduling problems have been extensively studied in the literature under the assumption that all jobs have to be processed. However, in many practical cases, one may wish to reject the processing of some jobs in the shop, thus resulting in a rejection cost. In such a framework, the scheduler has to decide …rst which jobs will be rejected and which will be accepted. Then he has to schedule the accepted jobs e¢ ciently. Scheduling with job rejection is essentially a problem with two criteria. The …rst is a scheduling criterion, which is dependent on the completion times of the accepted jobs, and the second is the total rejection cost. Problems of scheduling with rejection have been previously studied, but usually in a narrow framework – focusing on one scheduling criterion at a time. This paper provides a uni…ed bicriteria analysis of a large set of single machine problems sharing a common property: all problems can be represented by or reduced to a scheduling problem with a scheduling criterion which includes positional penalties. Among these scheduling criteria are the minimization of the makespan, the sum of completion times, the sum and variation of completion times, and the total earliness plus tardiness costs where the due dates are assignable. Four di¤erent problem variations for dealing with the two criteria are studied. The variation of minimizing the total integrated cost is shown to be solvable in polynomial time, while all other three variations are shown to be NP-hard. For those hard problems, we provide a pseudo polynomial time algorithm. An FPTAS for obtaining an approximate e¢ cient schedule is provided as well. In addition, we present an interesting special case which is solvable in polynomial time. This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 633/08). Partial support by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev is also gratefully acknowledged. ye-mails: [email protected] and [email protected] ze-mails: [email protected]