带单服务器及装卸操作的并行机调度问题
Parallel Machine Scheduling with a Singler Server and Loading-Unloading Operations
中文总结 AI 辅助
本文研究带单公共服务器装卸操作的并行机调度问题，证明其判定问题强NP完全，分析LS与LPT启发式的最坏情况近似比，得出不同机器数量下的近似界。
中文摘要 AI 辅助
本文研究了一种并行机调度问题，该问题配备一台负责装卸操作的公共服务器。每个作业包含一个单位时间的装卸操作、在\textit{m}台相同机器中某一台上的非抢占式处理，以及由同一服务器执行的单位时间卸载操作。目标是最小化总完成时间（makespan）。与经典的仅含装卸操作的公共服务器模型不同，本设定要求服务器在作业处理阶段结束后立即执行卸载操作。我们证明，当机器数量作为输入的一部分时，对应的判定问题是强NP完全问题。此外，我们分析了列表调度（List Scheduling, LS）和最长处理时间（Longest Processing Time, LPT）启发式算法的最坏情况性能。对于3台机器，我们确定算法LS的近似比至多为5/2；对于任意固定的\textit{m}≥3，我们证明随着作业数量增长，通用LS的界趋近于4-3/\textit{m}，而对于算法LPT，我们证明有限实例比为3-2/\textit{m}+(\textit{m}-1)(\textit{m}-2)/\textit{mn}。因此，对于每个固定的\textit{m}，LPT的界随着作业数量增长趋近于3-2/\textit{m}。
英文摘要
This paper investigates a parallel machine scheduling problem featuring a single common server responsible for both loading and unloading operations. Each job consists of a unit-time loading operation, non-preemptive processing on one of \(m\) identical machines, and a unit-time unloading operation executed by the same server. The objective is to minimize the makespan. Unlike classical loading-only common-server models, our setting requires the server to handle an unloading operation immediately after a job's processing phase concludes. We prove that the corresponding decision problem is strongly NP-complete when the number of machines is given as part of the input. Furthermore, we analyse the worst-case performance of the List Scheduling (LS) and Longest Processing Time (LPT) heuristics. For three machines, we establish that Algorithm LS achieves an approximation ratio of at most \(5/2\). For an arbitrary fixed \(m \ge 3\), we show that the general LS bound approaches \(4-3/m\) as the number of jobs grows, whereas for Algorithm LPT, we prove the finite-instance ratio $3-\frac{2}{m}+\frac{(m-1)(m-2)}{mn}.$ Thus, for each fixed \(m\), the LPT bound approaches \(3-2/m\) as the number of jobs grows.
