Scheduling problems with rejection to minimize the k-th power of the makespan plus the total rejection cost

In this paper, we consider several scheduling problems with rejection on m =1 identical machines. Each job is either accepted and processed on the machines, or it is rejected by paying a certain rejection cost. The objective is to minimize the sum of the k-th power of the makespan of accepted jobs and the total rejection cost of rejected jobs, where k > 0 is a given constant. We also introduce the conception of "job splitting" in our problems. First, we consider the single machine scheduling problem, i.e., m = 1. When job splitting is allowed, we propose an O(n log n)-time optimal algorithm for the problem. When job splitting is not allowed, we show that this problem is polynomially solvable when k E (0, 1] and it becomes binary NP-hard when k > 1. Furthermore, for the NP-hard problem, we propose a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS). Finally, we also extend our problems and some results to m = 2 identical parallel machines.