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,…,JnJ1,…,Jn and the processing time pipi of the i  th job is given by pi=a+bisipi=a+bisi, where sisi is the starting time of the i  th job (i=1,…,n),bi(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>0u>0, then for each ϵ>0ϵ>0 a solution with the value of the goal function that is at most 1+ϵ1+ϵ times greater than the optimal one can be found. We give a FPTAS that finds such a solution in On1+6log1+u21ϵ2log1+u2 time. Consequently, the problem cannot be NP-hard in the strong sense.