We propose asymptotically optimal algorithms for job shop scheduling. We propose a fluid relaxation for the job shop scheduling problem, in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in closed form, obtain a lower bound C/sub max/ to the job shop scheduling problem, and construct a feasible schedule from the fluid relaxation with objective value at most C/sub max/+O(/spl radic/C/sub max/), where the constant in the O(/spl middot/) notation is independent of the number of jobs. However, it depends on the processing times of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase proportionally, then our algorithm produces a schedule with value at most C/sub max/+O(1). In computational experiments our algorithms produce schedules which are within 1% of optimality even for moderately sized problems.
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机译:我们提出用于作业车间调度的渐近最优算法。我们提出了一种针对作业车间调度问题的流体松弛方法,在该问题中,我们用连续的流体流代替了离散的工作。我们以封闭形式计算流体松弛的最佳解决方案,获得作业车间调度问题的下限C / sub max /,并根据目标值最大为C / sub max / + O的流体松弛构造可行的调度(/ spl radic / C / sub max /),其中O(/ spl middot /)表示法中的常数与作业数无关。但是,它取决于作业的处理时间,因此随着作业总数趋于无穷大,从而生成了渐近最优的计划。如果最初存在的作业成比例增加,则我们的算法将生成一个计划,该计划的值最多为C / sub max / + O(1)。在计算实验中,即使对于中等大小的问题,我们的算法所生成的进度表也不在最优值的1%之内。
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