Two competing agents with due date assignment scheduling under multitasking environment

We consider multitasking scheduling on a single machine involving two competing agents, in which the due date of each job of the first agent is a decision variable, which is to be determined by the decision maker using unrestricted due date method, whereas the due date of the jobs in the second agent is exogenously given. By multitasking, we mean that the processing of a selected job on the machine may be interrupted by other jobs that are available but unfinished. The first agent's objective value is to minimize an integrated scheduling criterion including the due date assignment cost and the weighted number of its tardy jobs, while the second agent's objective value is to minimize the maximum value of a regular scheduling function of its jobs. The overall objective is to minimize the objective value of the first agent, subject to the objective value of the second agent not exceeding a given threshold. For each of the problems considered, we show that it is NP-hard in the ordinary sense and admits pseudo-polynomial time algorithm.