Competitive two-agent scheduling with deteriorating jobs on a single parallel-batching machine.

•We study two-agent scheduling on a single batching machine with deteriorating jobs and release dates.•We optimize one objective while ensuring the other objective under a fixed value.•For the unbounded case, optimal algorithms are developed for the problems with identical release dates.•For the bounded case, NP-hardness is proved for different problems.•Several special cases of these NP-hard problems are analyzed, and optimal algorithms are developed for solving them, respectively.