Abstract
This paper proposes a makespan minimization procedure for job-shop scheduling problems. The primal objective is to find a minimum makespan schedule by generating and evaluating active schedules implicitly and efficiently under the mechanism of branch and bound. This scheduling method allocates operations from time zero toward the end of the schedule gradually, a characteristic favorable when calculating the lower bound on other criteria. In general, however, it is difficult to calculate a good lower bound on the makespan in earlier branching stages, and thus the branching decision must be made based on considerably under-estimated values. Therefore the approach is time-consuming in general. If this weakness is overcome, the approach can be used to optimize another criterion among minimum makespan schedules. The Lagrangean relaxation technique, which is an effective sub-optimization approach, is adopted in order to improve the branching decisions. Our approach first solves the Lagrangean relaxation problem in order to obtain the average start time of operations. This value is used to determine the priority among sub-problems to be solved. Each sub-problem is generated by placing one candidate operation before all other candidates. The subproblem, of which the front-placed operation has the minimum average start time, is solved first even if it has a greater lower bound on the makespan than others. The effectiveness of the proposed approach is investigated by solving 24 benchmark problems. The results indicate that our approach often shows better performance than an ordinary lower-bound-based branching strategy.
