A FPTAS for minimizing total completion time in a single machine time-dependent scheduling problem

In this paper a single machine time-dependent scheduling problem with total completion time criterion is considered. There are given n jobs J1,...,Jn and the processing time pi of the ith job is given by pi=a+bisi, where si is the starting time of the ith job (i=1,...,n),bi is its deterioration rate and a is the common base processing time. If all jobs have deterioration rates different and not smaller than a certain constant u>0, then for each [epsilon]>0 a solution with the value of the goal function that is at most 1+[epsilon] times greater than the optimal one can be found. We give a FPTAS that finds such a solution in time. Consequently, the problem cannot be NP-hard in the strong sense.