Minmax weighted earliness-tardiness with identical processing times and two competing agents

A classical single machine scheduling problem is that of minimizing the maximum weighted deviation of the job completion times from a common due-date, assuming identical processing times. We extend this problem to a setting of two competing agents sharing the same machine. We first focus on the case that the objective is of minimizing the maximum weighted deviation of the jobs of the first agent subject to an upper bound on the maximum weighted deviation of the jobs of the second agent. Then we extend this model to a setting of asymmetric cost structure, i.e., the (job- and agent-dependent) earliness and tardiness costs may be different. We also consider a modified model with a minsum measure for the second agent: the objective is of minimizing the maximum weighted deviation of the jobs of the first agent from a common due-date subject to an upper bound on the total weighted deviation of the jobs of the second agent. All these models are also extended to a general setting of job-dependent due-dates. Polynomial time solutions are introduced for all the problems studied in this paper.