Parallel-machine scheduling with deteriorating jobs, rejection and a fixed non-availability interval

We consider two parallel-machine scheduling problem in which the processing time of a job is a simple linear increasing function of its starting time, jobs can be rejected by paying penalties and the first machine has a fixed non-availability interval. The objective is to minimize the makespan of the accepted jobs plus the total penalty of the rejected jobs. We present a fully polynomial-time approximation scheme (FPTAS) for the case with two identical machines. The time complexity of the FPTAS is O(n6L4/ε3)O(n6L4/ε3) where n is the number of jobs and L   is the input size, εε is the error bound.