Abstract
This paper deals with makespan minimization of a job shop based on the active schedule generation procedure of Giffler and Thompson. The makespan is the length of a schedule, and we try to find a minimum makespan schedule under the mechanism of depth-first branch and bound. If two or more operations conflict on a machine when generating active schedules, sub-problems are generated by fixing the start time of each conflicting operation. To select a good sub-problem, which probably leads to a minimum makespan schedule, a lower bound on the makespan is calculated by solving the one-machine sequencing problem for each sub-problem. The obtained preemptive schedules may also give precedence relations between operations to be processed on the same machine. As it is generally difficult, especially in earlier branching stages, to select a good sub-problem based on the calculated lower bounds, we also propose a simple look-ahead method to assist in the selection decision. The look-ahead procedure searches better schedules based on the branch and bound-based procedure within a pre-specified computation time, and our search procedure selects the sub-problem having the best schedule obtained by the look-ahead method. The effectiveness of the proposed approach was examined by solving 44 benchmark instances. For small-scale problems composed of five machines and 10 to 20 jobs, the proposed approach could find minimum makespan schedules within a reasonable computation time even without the aid of the look-ahead method. Middle-scale problems composed of ten or more machines and ten or more jobs were difficult to solve optimally by the proposed approach. However, the look-ahead method improved the solution quality, and the differences between the obtained schedules and optimal schedules were less than 4% on average.
