Due date assignment and two-agent scheduling under multitasking environment

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 $$\mathcal {NP}$$NP-hard, devise a pseudo-polynomial time dynamic programming algorithm, establishing that it is $$\mathcal {NP}$$NP-hard in the ordinary sense, and demonstrate that it admits a fully polynomial-time approximation scheme.