Group Scheduling with Two Competing Agents on a Single Machine

We consider the scheduling problem in which two agents, each with a set of jobs, compete to perform their respective jobs on a single machine under a group technology (GT) environment. The jobs of agents are classified into groups according to their production similarities in advance, all jobs of the same group are required to be processed contiguously on the machine. A sequence-independent setup time precedes the processing of each group. We propose a polynomial time solution for the problem of minimizing the maximum regular cost of one agent, subject to an upper bound on the maximum regular cost of the second agent. We also show that the problem of minimizing the total completion time of the first agent, subject to an upper bound on the maximum lateness of the second agent is strongly $\mathcal {NP}$-hard. The case where all groups of the first agent have the same number of jobs is shown to be polynomially solvable.