A two-agent single-machine scheduling problem with a time-based learning effect

In this paper, we introduce a new scheduling model in which both two-agent and time-based learning effect exist simultaneously. Two agents compete to perform their respective jobs on a common single machine and each agent has his own criterion to optimize. The time-based learning effect of a job is assumed to be a function of the total normal processing time of the jobs scheduled in front of the job. The objective is to minimize the total completion time of the first agent with the restriction that the makespan of the second agent cannot exceed a given upper bound. The optimal properties of the problems are given, and then the optimal polynomial time algorithm is proposed to solve the scheduling problem.