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Theoretical and practical issues in single-machine scheduling with two job release and delivery times
Journal of Scheduling ( IF 2 ) Pub Date : 2021-09-09 , DOI: 10.1007/s10951-021-00708-4
Alejandro Reynoso 1 , Nodari Vakhania 1
Affiliation  

We study a single-machine scheduling problem with two allowable job release and delivery times. This special case of a strongly NP-hard single-machine scheduling problem with an arbitrary number of job release and delivery times and with the objective to minimize the maximum job completion time remains NP-hard. Nevertheless, it is more transparent and accessible than the general version. Hence, it permitted us to establish some nice properties that yielded simple and efficient solution methods. On the one hand, the restricted setting is useful by its own right since it fits some real-life applications. On the other hand, the presented case study is helpful in the solution of a more general setting with a constant number of job release and delivery times. In particular, the optimality conditions and the heuristics methods that we propose here for the restricted setting might be generalized to the extended setting. The established optimality criteria are explicit conditions under which the restricted problem can be solved optimally in time \(O(n\log n)\). The extensive computational study showed that at least one of these conditions is satisfied for practically all the randomly generated 50 million problem instances while applying our heuristics to these instances. We report a favorable practical performance of our polynomial-time subroutine that invokes our five heuristics and verifies the proposed optimality conditions consecutively. These conditions were verified for the solutions delivered by the five heuristics individually and in a combined fashion as four pairs of heuristics and as all the five heuristics together. Fifty million problem instances were randomly generated using uniform distribution for each of these ten combinations. For the 50 million instances generated for the last (overall) combination, the schedule created by at least one of the heuristics has satisfied at least one of our optimality conditions (so all these problem instances were solved optimally). We also addressed the (theoretical) possibility that none of our conditions is satisfied, and showed how a known dynamic programming algorithm for SUBSET SUM problem can be used for the solution of the scheduling problem in pseudo-polynomial time. For a considerable part of the tested instances, one of the five scheduling heuristics has solved optimally also SUBSET SUM problem.



中文翻译:

具有两个作业发布和交货时间的单机调度的理论与实践问题

我们研究了具有两个可允许的作业发布和交付时间的单机调度问题。这种具有任意数量的作业发布和交付时间并以最小化最大作业完成时间为目标的强 NP-hard 单机调度问题的特殊情况仍然是 NP-hard。尽管如此,它比普通版本更透明和更易于访问。因此,它允许我们建立一些很好的属性,从而产生简单有效的解决方法。一方面,受限设置本身很有用,因为它适合一些现实生活中的应用程序。另一方面,所呈现的案例研究有助于解决具有恒定数量的工作发布和交付时间的更一般的设置。特别是,我们在此为受限设置提出的最优条件和启发式方法可能会推广到扩展设置。已建立的最优性标准是明确的条件,在这些条件下可以及时最优地解决受限问题\(O(n\log n)\). 广泛的计算研究表明,在将我们的启发式应用于这些实例时,几乎所有随机生成的 5000 万个问题实例都满足这些条件中的至少一个。我们报告了多项式时间子程序的良好实际性能,该程序调用了我们的五个启发式方法并连续验证了所提出的最优条件。这些条件针对由五个试探法单独和以组合方式作为四对试探法和所有五个试探法一起提供的解决方案进行了验证。对于这十个组合中的每一个,使用均匀分布随机生成五千万个问题实例。对于为最后一个(整体)组合生成的 5000 万个实例,由至少一种启发式方法创建的调度至少满足了我们的最优条件之一(因此所有这些问题实例都得到了最佳解决)。我们还解决了我们的条件都不满足的(理论)可能性,并展示了如何使用已知的 SUBSET SUM 问题的动态规划算法来解决伪多项式时间内的调度问题。对于相当一部分的测试实例,五种调度启发式方法中的一种也最优地解决了 SUBSET SUM 问题。并展示了如何使用已知的 SUBSET SUM 问题动态规划算法来解决伪多项式时间内的调度问题。对于相当一部分的测试实例,五种调度启发式方法中的一种也最优地解决了 SUBSET SUM 问题。并展示了如何使用已知的 SUBSET SUM 问题动态规划算法来解决伪多项式时间内的调度问题。对于相当一部分的测试实例,五种调度启发式方法中的一种也最优地解决了 SUBSET SUM 问题。

更新日期:2021-09-10
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