A Lower-Bound Heuristic to Minimize Weighted Flowtime WjCj in the Open Shop

Scheduling jobs through machines with no specified order of machine processing poses more sequencing possibilities compared to the jobs which have precedence constraints. This open shop problem with deterministic processing times, and no preemption, with objective of minimizing flowtime can represent the cost of time that the job spends inside the production system. The difference between completion time and the time when the job was released is called the flowtime. For simplicity, it is assumed in this paper that all jobs were available for scheduling at time=0, and hence, flowtime is the completion time Cj of each job. The longer a job is in the system, a certain delay cost over the duration is ascribed by the job's weight. Pinedo (2008) showed that the minimizing weighted completion time sequencing problem is NP-hard, and, therefore, justifies the development of heuristic procedures to possibly shorten the schedule search process. The paper will present a schedule search heuristic that uses the relative values of processing times on the bottleneck and non-bottleneck machines to create an initial set of sequences on the bottleneck machine. A lower-bound operation would be presented that can be used to evaluate the attractiveness of an initial sequence for improvement efforts in a branch-and-bound procedure. The paper concludes by showing certain problem sets where complete enumeration was used to determine optimal sequences. The presented heuristic offers promising results with less computational effort.