Scheduling problems with rejection and piece-rate maintenance to minimise the total weighted completion time

This paper addresses the single machine scheduling problems with simultaneous consideration of rejection and piece-rate maintenance. Each job is either accepted to be processed on the machine, or rejected in which case a rejection penalty will be incurred. The piece-rate maintenance refers that the machine performs maintenance activity every time it completes a given number of jobs. The objective is to minimise the sum of weighted completion times, rejection costs and maintenance costs. Our contribution is threefold. First, the general case of the considered problem is proved to be NP-hard, and an approximate algorithm is developed to solve the problem. Second, for the case with agreeable condition that jobs with smaller processing times are weighted more, a pseudo-polynomial algorithm is developed to establish that the problem is NP-hard only in the ordinary sense. This pseudo-polynomial algorithm is further converted into a fully polynomial time approximation scheme (FPTAS). In the third, two special cases, in which one with all equal weights and the other one with all equal processing times, are proved to be solved in polynomial time.