国家自然科学基金(11001242, 11071220); 浙江省自然科学基金(Y6090554, Y6090175)
讨论了一类两台机流水作业要求最后完工工件完工时间最早的排序问题.问题中每个工件包含两个加工任务:第1 个任务可以在任何一台机器上加工,第2 个任务只能在第1 个任务完成后在第2 台机器上加工.如果要求在加工同一个工件的两个任务时,两个任务之间不能有停顿,则称其为不可等待的模型,记作NSHFS.如果第2 个任务可以在第1 个任务完成后的任意时间加工,则称其为允许等待的模型,记作SHFS.对于SHFS 模型,在魏麒和何勇工作的基础上给出了一种改进的最坏情况界为8/5 的多项式时间近似算法.对于NSHFS 模型,首先证明它是NP-难的,并且给出了一种最坏情况界为5/3 的多项式时间近似算法.
This paper investigates a variant scheduling problem of minimizing makespan in a two-machine flow shop. In this variant, there will be two tasks for each job. The first task can be processed on either machine, and the second task can only be processed on the second machine after the first task has been finished. Furthermore, if the second task should start right after the first task is completed, it is called a no-waited case and is denoted by NSHFS. On the other hand, if the second task is allowed to be processed at any time after the first task is completed, the problem is then denoted as SHFS. In the case of SHFS, based on the result of Wei and He, an improved polynomial time approximation algorithm with worst-case ratio of 8/5 is presented. In the case of NSHFS, this paper shows that it is NP-hard, and presents a polynomial time approximation algorithm with worst-case ratio of 5/3.