Equivalent time-dependent scheduling problems

We introduce a general transformation of parallel-machine time-dependent scheduling problems with critical lines. Using the transformation we define the class of equivalent time-dependent scheduling problems. We show that given an initial parallel-machine time-dependent scheduling problem with linear job processing times and the total weighted starting time criterion, the problem can be transformed in a unique way into another problem of this type in such a way that both these problems are mutually dual. We prove that a schedule is optimal for the initial problem if and only if the schedule constructed by this transformation is optimal for the transformed problem. The presented results explain remarkable similarities between different time-dependent scheduling problems and simplify the proofs of properties of such problems.