This paper addresses a two-agent scheduling problem with due date assignment under multitasking environment, in which the due dates of the jobs from the first agent are decision variables to be determined using the unrestricted (usually referred to as DIF) due date assignment method. Each agent requests the processing of its own set of jobs on a machine and wishes to minimize a certain scheduling criterion related to the completion times of its jobs only. Under multitasking, when a job (primary job) is processed, it is inevitably interrupted by other jobs (waiting jobs) that are available but unfinished, and the amount of time that each waiting job interrupting the primary job is a linear function of the remaining processing time of the waiting job. The overall objective is to determine the optimal primary job sequence along with the due dates of the jobs from the first agent as to minimize the weighted sum of the due date assignment cost and weighted number of late jobs from the first agent, while maintaining the total completion time of the jobs from the second agent not exceeding a given threshold. We show that the problem is \(\mathcal {NP}\)-hard, devise a pseudo-polynomial time dynamic programming algorithm, establishing that it is \(\mathcal {NP}\)-hard in the ordinary sense, and demonstrate that it admits a fully polynomial-time approximation scheme.

本文解决了在多任务环境下使用到期日分配的两主体调度问题，其中来自第一主体的作业的到期日是要使用无限制（通常称为DIF）确定的决策变量。）到期日分配方法。每个代理都要求在计算机上处​​理自己的一组作业，并希望最小化仅与作业完成时间相关的某些调度标准。在多任务处理中，当处理一个作业（主要作业）时，不可避免地会被其他可用但尚未完成的作业（正在等待的作业）打断，并且每个正在等待的作业中断主要作业的时间是剩余作业的线性函数等待作业的处理时间。总体目标是确定最佳的主要作业顺序以及来自第一代理商的作业的到期日，以最大程度地减少到期日分配成本和来自第一代理商的延迟作业的加权数量的加权总和，同时保持来自第二代理的作业总完成时间不超过给定阈值。我们证明问题是\（\ mathcal {NP} \）- hard，设计一个伪多项式时间动态规划算法，确定它在一般意义上是\（\ mathcal {NP} \）- hard，并证明它接受一个完全多项式时间近似方案。