Single-batching-machine scheduling problem with deterioration and release time

We consider a single-batching-machine problem of scheduling deteriorating jobs with release times, where the processing time of a job is dependent on its waiting time, i.e., the time required between the release of the job and the start of the job on the machine. The objective is to minimize the makespan. First, we prove the problem is NP-hard by a reduction from equal-size partition problem. For three special cases, we present polynomial-time algorithms respectively.